University of São Paulo “Luiz de Queiroz” College of ... · “Luiz de Queiroz” College of Agriculture Design and analysis of sugarcane breeding experiments: a case study Alessandra

University of São Paulo“Luiz de Queiroz” College of Agriculture

Design and analysis of sugarcane breeding experiments: a case study

Alessandra dos Santos

Thesis presented to obtain the degree of Doctor in Sci-ence. Area: Statistics and Agricultural Experimenta-tion

Piracicaba2017

Alessandra dos SantosDegree in Mathematics


Advisor:Profa. Dra. CLARICE GARCIA BORGES DEMÉTRIO

Thesis presented to obtain the degree of Doctor in Science.Area: Statistics and Agricultural Experimentation

Piracicaba2017

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Dados Internacionais de Catalogação na PublicaçãoDIVISÃO DE BIBLIOTECA - DIBD/ESALQ/USP

Santos, Alessandra dosDesign and analysis of sugarcane breeding experiments: a case study /

Alessandra dos Santos. – – Piracicaba, 2017 .174 p.

Tese (Doutorado) – – USP / Escola Superior de Agricultura “Luiz deQueiroz”.

1. Modelos mistos 2. Correlação autorregressiva 3. Correlação em banda4. Tombamento 5. Delineamento não replicado 6. Delineamento ótimo 7.Delineamento parcialmente replicado 8.Ganho genético 9. Assertividade naseleção 10. Estudo de simulação I. Título.

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ACKNOWLEDGMENTS

I thank God for all the moments, blessings and opportunities.I thank my family and friends for all their love and care. I could not finish this work without

their efforts to help me and keep my life happy, feeling close every time which it was not possible evenwhen we were far apart. Thank you Leonardo, Aristides, Suely, Andressa, Aristides Neto, Bruna, Lourdes,Domingos, Renata, Roberta, Luís, Alyne, Priscila, Catherine, Pamela and Omar.

I thank my country, Brazil, for investing in education and developing good projects to improvemore. I am proud to be Brazilian and have the opportunity to study in best universities in my countryand work with the best researchers. I thank the Brazilian society and I hope to be able to contribute andreturn all the investments.

I am grateful to Dr. Clarice Garcia Borges Demétrio for accepting to be my supervisor andgiving me incredible opportunities. I thank you very much for trusting me. Also, I am very grateful towork with Dr. Chris Brien. He is a genius and in one year in Australia I can say that my life changed alot and I became better as a professional and a person. Chris, I thank you for all moments, all talking,all new ideas, all effort that you made to help me. You and your wife, Margaret, are incredible peopleand I thank you very much. Dr. Renata Alcarde Sermarini, I also would like to thank you very much,our talking and your points were very important to me.

Teachers and professors from primary school, high school, universities, thank you very much. Ihave only reached this stage because of your efforts. In particular I thank Dr. Vanderli Marino Melem,Londrina State University; because of her I decided to study statistics. Thanks for being an excellentprofessional and person.

For the financial support for this work, I thank CNPq, National Council for Scientific andTechnological Development (38 months) and CAPES, Coordination for the Improvement of Higher Edu-cation Personnel (9 months). Lastly, I thank the University of São Paulo and the Centro de TecnologiaCanavieira for supporting the project through their partnership.

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TABLE OF CONTENTS

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.1 CTC company . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.2 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Individual and joint analysis of sugarcane experiments to select test lines . . . . . . . . . . . . . . 252.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4 Data analysis results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.5 Simulation studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5.1 Single site experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.5.2 Separate versus joint analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.6 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.6.1 Single site experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.6.2 Separate versus joint analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 Negative residual correlation in sugarcane experiments . . . . . . . . . . . . . . . . . . . . . . . 49

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 Motivating example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.3 Material and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Comparison of a systematic design with some spatially optimized designs . . . . . . . . . . . . . 534.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1.1 Optimality criteria for design searches . . . . . . . . . . . . . . . . . . . . . . . . . 544.1.2 Comparing different design types using simulation . . . . . . . . . . . . . . . . . . . 54

4.2 Motivating example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.3 Material and methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5 General conclusions and future works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

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RESUMO

Delineamento e análise de experimentos de melhoramento com cana de açúcar: um estudo de caso

Um dos propósitos dos programas de melhoramento genético é a seleção de novos clones mel-hores (novos materiais). A acurácia de seleção pode ser melhorada usando delineamentos ótimos emodelos bem ajustados. Porém, descobrir isso não é fácil, especialmente, em experimentos grandes quepossuem mais de cem clones sem a possibilidade de repetição devido à limitação de material, área ecustos elevados, dadas as poucas repetições de parcelas com variedades comerciais (testemunhas) e onúmero de parâmetros de complexa variância estrutural que necessitam ser assumidos. Os principaisobjetivos desta tese foram modelar 21 experimentos de cana de açúcar fornecidos pelo Centro de Tec-nologia Canavieira (CTC - empresa brasileira de cana de açúcar) e avaliar o delineamento empregado,o qual usa um número grande de clones não repetidos e testemunhas sistematicamente repetidas. Omodelo linear misto foi usado, identificando três principais componentes de variação espacial nos errosde parcelas e efeitos de competição, em nível genético e residual. Os clones foram assumidos de efeitosaleatórios e as testemunhas de efeitos fixos, pois vieram de processos diferentes. As análises individuais econjuntas foram desenvolvidas neste material pois os experimentos puderam ser agrupados em dois tipos:(i) um delineamento longitudinal (duas colheitas) e (ii) cinco grupos de experimentos (cada grupo umaregião com três locais). Para os estudos de delineamentos, um tamanho fixo de experimento foi assumidopara se avaliar a eficiência do delineamento não replicado (empregado nesses 21 experimentos) com osnão replicados otimizado espacialmente, os parcialmente replicados com testemunhas e os parcialmentereplicados otimizado espacialmente. Quatro estudos de simulação foram feitos para avaliar i) os modelosajustados, sob condições de efeito de competição em nível genético, ii) a acurácia das estimativas vindasdos modelos de análise individual e conjunta; iii) a relação entre tombamento da cana e a correlaçãoresidual negativa, e iv) a eficiência dos delineamentos. Para concluir, as principais informações utilizadasnos estudos de simulação foram: o número de vezes que o algoritmo convergiu; a variância na estimativados parâmetros; a correlação entre os EBLUPs genético direto e os efeitos genéticos reais; a assertividadede seleção ou a semelhança média, sendo semelhança medida como a porcentagem dos 30 clones com osmaiores EBLUPS genético e os 30 melhores verdadeiros clones; e a estimativa da herdabilidade ou doganho genético.

Palavras-chave: Modelos mistos; Correlação autorregressiva; Correlação em banda; Competição; Tomba-mento; Delineamento não replicado; Delineamento ótimo; Delineamento parcialmente replicado; Ganhogenético; Assertividade na seleção; Estudo de simulação

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ABSTRACT


One purpose of breeding programs is the selection of the better test lines. The accuracy ofselection can be improved by using optimal design and using models that fit the data well. Finding this isnot easy, especially in large experiments which assess more than one hundred lines without the possibilityof replication due to the limited material, area and high costs. Thus, the large number of parameters inthe complex variance structure that needs to be fitted relies on the limited number of replicated checkvarieties. The main objectives of this thesis were to model 21 trials of sugarcane provided by “Centro deTecnologia Canavieira” (CTC - Brazilian company of sugarcane) and to evaluate the design employed,which uses a large number of unreplicated test lines (new varieties) and systematic replicated check(commercial) lines. The mixed linear model was used to identify the three major components of spatialvariation in the plot errors and the competition effects at the genetic and residual levels. The test lineswere assumed as random effects and check lines as fixed, because they came from different processes.The single and joint analyses were developed because the trials could be grouped into two types: (i)one longitudinal data set (two cuts) and (ii) five regional groups of experiment (each group was a regionwhich had three sites). In a study of alternative designs, a fixed size trial was assumed to evaluate theefficiency of the type of unreplicated design employed in these 21 trials comparing to spatially optimizedunreplicated and p-rep designs with checks and a spatially optimized p-rep design. To investigate modelsand design there were four simulation studies to assess mainly the i) fitted model, under conditions ofcompetition effects at the genetic level, ii) accuracy of estimation in the separate versus joint analysis; iii)relation between the sugarcane lodging and the negative residual correlation, and iv) design efficiency. Toconclude, the main information obtained from the simulation studies was: the convergence number of thealgorithm model analyzed; the variance parameter estimates; the correlations between the direct geneticEBLUPs and the true direct genetic effects; the assertiveness of selection or the average similarity, wheresimilarity was measured as the percentage of the 30 test lines with the highest direct genetic EBLUPsthat are in the true 30 best test lines (generated); and the heritability estimates or the genetic gain.

Keywords: Mixed models; Autoregressive correlation; Banded correlation; Competition; Lodging; Un-replicated design; Optimal design; p-rep design; Genetic gain; Assertiveness of selection; Simulationstudy

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LIST OF FIGURES

1.1 Map of Brazil with the estimated percentage of sugarcane production in April/2016. . . . 211.2 Map with the six regions where CTC carries out the experiments. On the left is the complete

map of Brazil and right the specific CTC areas. Source: CTC (2013) . . . . . . . . . . . . 22

2.1 Boxplots of the correlations between direct genetic EBLUPs for pairs of models from eachscenario (a, b, c and d). In (b) Model 2 converged only for 90 out of 1000 simulated datasets and hence the density plot for genetic components effects are not displayed (see Table2.6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2 Boxplots of the correlation between the true direct genetic effects and the respective EBLUP’sfrom each model for each scenario (a, b, c and d). In (b) Model 2 converged only for 90 outof 1000 simulated data sets and hence the density plot for genetic components effects arenot displayed (see Table 2.6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3 Densities of the selection gain from each model for each scenario (a, b, c and d). In (b)Model 2 converged only for 90 out of 1000 simulated data sets and hence the density plotfor genetic components effects are not displayed (see Table 2.6). . . . . . . . . . . . . . . . . 37

2.4 Percentage of average similarity of the 30 test lines selected as best compared to the truebest 30 lines in each model and scenario (a, b, c and d). In scenario b, Model 2 convergedonly for 90 out of 1000 simulated data sets and hence the density plot for genetic componentseffects are not displayed (see Table 2.6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5 Densities of the direct genetic (first column of plots), neighbour genetic (third) compo-nents along with their covariance (second column) for each scenario and model. Each rowrepresents a different scenario and the vertical lines represent the assumed values of theparameters. Remember that the convergence was different in each one (see Table 2.6) andonly Models 2 and 4 assumed neighbour genetic effect. In scenario b Model 2 converged onlyfor 90 out of 1000 simulated data sets and hence the density plot for genetic componentseffects are not displayed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.6 Densities of the residual component(first column of plots) and the column (second column)and row (third column) correlations for each scenario and model. Each row represents adifferent scenario and the vertical lines represent the assumed values of the parameters.Remember that the convergence was different in each one (see Table 2.6) and only Model3 and 4 assumed residual correlation. In scenario b Model 2 converged only for 90 out of1000 simulated data sets and hence the density plot for genetic components effects are notdisplayed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.7 Densities of the global components, column (in the left) and row (in the right), for eachscenario and model. Each row represents a different scenario and the vertical lines representthe assumed values of the parameters. Remember that the convergence was different in eachone (see Table 2.6). In scenario b Model 2 converged only for 90 out of 1000 simulated datasets and hence the density plot for genetic components effects are not displayed. . . . . . . 41

2.8 Average EBLUP correlations in the individual (light symbols) and joint (dark symbols)analysis for autoregressive models in each scenario. The panels refer to comparison between(a) pairs of Locals; (b) individual and joint analysis at the same Local. . . . . . . . . . . . 42

2.9 Average similarity between the true and estimated 30 best test lines (around top 7%). Thedark symbols represent the result from joint analysis and the light are the individuals forautoregressive models in each scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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2.10 Densities of the heritability for each site from the joint analysis for autoregressive modelsin each scenario. The panels refer to the scenarios. . . . . . . . . . . . . . . . . . . . . . . . 43

2.11 Densities of the direct genetic components from autoregressive and banded models for eachLocal from scenarios (a, b, c and d). The vertical lines represent the assumed values of theparameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.12 Densities of the residual variance components from autoregressive and banded models foreach Local from scenario a. The vertical lines represent the assumed values of the parameters. 45

2.13 Densities of the residual variance components from autoregressive and banded models foreach Local from scenario b. The vertical lines represent the assumed values of the parameters. 45

2.14 Densities of the residual variance components from autoregressive and banded models foreach Local from scenario c. The vertical lines represent the assumed values of the parameters. 46

2.15 Densities of the residual variance components from autoregressive and banded models foreach Local from scenario d. The vertical lines represent the assumed values of the parameters. 46

3.1 Densities of the residual component and the column and row correlation for each scenario.The curves within a panel are for the different percentage of plots lodged. The plot of theresidual components have been truncated to exclude a few estimated values that were inexcess of 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1 Layout of the Design 0, where there are four commercial varieties, named “A”, “B”, “C”and “T” checks, and 414 test lines in the unnamed plots. . . . . . . . . . . . . . . . . . . . 56

4.2 Layouts from the optimal designs 1 (panels (a) and (b)), 2 (panels (c) and (d)), 3 (panels(e) and (f))). Panels (a), (c) and (e) used ϕr = 0.4 and the other panels used ϕr = −0.25.“A”, “B”, “C” and “T” are the checks and the numbers are the duplicated test lines as wellthe unnamed plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3 Layouts from the optimal designs 4 (panels (a) and (b)) and 5 (panels (c) and (d)). Panels(a) and (c) used ϕr = 0.4 and the other panels used ϕr = −0.25. “A”, “B”, “C” and “T”are the checks and the numbers are the duplicated test lines as well the unnamed plots. . . 59

4.4 Densities of the correlation between the true and estimated genetic effects (first column ofplots), relative genetic gain (second column) and prediction error variance - PEV (third)for each scenario and design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.5 Similarity percentage between true and estimated top 7% of test lines. The horizontal grayline indicates 50%. The panels represent scenarios the different scenarios. . . . . . . . . . . 62

B.1 Boxplot of TCH for each group of lines for Paraná cuts. Panel (a) refers to first cut and (b)second. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

B.2 Heat map for TCH in the first cut of the Paraná. . . . . . . . . . . . . . . . . . . . . . . . . 76B.3 Heat map for TCH in the second cut of the Paraná. . . . . . . . . . . . . . . . . . . . . . . 76B.4 Plots of the row and column faces of the empirical variograms for the residuals referring to

Models 4 (panels (a) and (d)); 6 (panels (b) and (e)) and 18 (panels (c) and (f)). Panels(a), (b) and (c) refer to the column direction and panels (d), (e) and (f) refer to the rowdirection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

B.5 Dispersion plot of the genetic EBLUPs values of the Model 18 and the traditional non-spatial analysis model for this design (without spatial dependence) for 1st cut of Paraná.The cut-offs for the 30 best test lines (7 % upper) in each Model are indicated by the dottedline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

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B.6 Panels (a) and (b) refer respectively to the plots of the column and row faces of the empiricalvariograms for the residuals referring to Model 3. Panel (c) is the semi-variogram of theModel 14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

B.7 Dispersion plot of the genetic EBLUPs values of the Model 14 and the traditional non-spatial analysis model for this design (without spatial dependence) for 2nd cut of Paraná.The cut-offs for the 30 best test lines (7 % upper) in each Model are indicated by the dottedline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

B.8 Dispersion plot of the predicted values from selected model in each cut of Paraná. Thecut-offs for the 30 best test lines (7% upper) in each Model are indicated by the dotted line. 85

B.9 Predicted values from individual (indiv) and joint analysis of Paraná cuts. The left panelis for the first cut and the second is to the right. . . . . . . . . . . . . . . . . . . . . . . . . 88

C.1 Boxplot of TCH for each group of lines for Locals of Paraná. . . . . . . . . . . . . . . . . . 90C.2 Heat map relating to TCH for Local 651 of Paraná. . . . . . . . . . . . . . . . . . . . . . . 90C.3 Heat map relating to TCH for Local 851 of Paraná. . . . . . . . . . . . . . . . . . . . . . . 91C.4 Heat map relating to TCH for Local 852 of Paraná. . . . . . . . . . . . . . . . . . . . . . . 91C.5 Plots of the row and column faces of the empirical variogram for the residuals for the Local

651 of the Paraná experiment referring to models 19 (panels (a) and (c)) and 20 (panels (b)and (d)). The panels (a) and (b) are for the column direction and the others are for therow direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

C.6 Dispersion plot of the genetic EBLUPs values of the Model 20 and the traditional non-spatial analysis model for this design (without spatial dependence) for Local 651 of theParaná. The cut-offs for the 14 best test lines (7 % upper) in each Model are indicated bythe dotted line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

C.7 Plot of the empirical semi-variogram for the residuals for the Local 851 of the Paranáexperiment referring to Model 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98


C.9 Plot of the empirical semi-variogram for the residuals for the Local 852 of the Paranáexperiment referring to Model 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101


C.11 Dispersion plots of the genetic EBLUPs from model selected for each Paraná Local. Panelsthe relationship between pairs of Locals (a) 651 and 851; (b) 651 and 852; and (c) 851 and852. The cut-offs for the 14 best test lines in each Local are indicated by the dotted line. . 103

D.1 Boxplot of TCH for each group of lines for Locals of Goiás. . . . . . . . . . . . . . . . . . . 110D.2 Heat map relating to TCH for Local 3 of Goiás. . . . . . . . . . . . . . . . . . . . . . . . . . 110D.3 Heat map relating to TCH for Local 521 of Goiás. . . . . . . . . . . . . . . . . . . . . . . . 111D.4 Heat map relating to TCH for Local 533 of Goiás. . . . . . . . . . . . . . . . . . . . . . . . 111D.5 Heat map relating to fall down for Local 3 of Goiás. . . . . . . . . . . . . . . . . . . . . . . 112D.6 Plots of the empirical semi-variogram for the residuals for the Local 3 of the Goiás experi-

ment referring to models 19 and 21. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

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D.7 Dispersion plot of the genetic EBLUPs values of the Model 19 and the traditional non-spatial analysis model for this design (without spatial dependence) for Goiás Local 3. Thecut-offs for the 15 best test lines (7 % upper) in each model are indicated by the dotted line. 115

D.8 Plots of the empirical semi-variogram for the residuals for Local 521 of Goiás experimentreferring to Models (a) 20 and (b) 23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

D.9 Dispersion plot of the genetic EBLUPs values of the Model 20 and the traditional non-spatial analysis model for this design (without spatial dependence) for Goiás Local 521.The cut-offs for the 15 best test lines (7 % upper) in each model are indicated by the dottedline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

D.10 Plots of the empirical semi-variogram for residuals from Model 15 for Local 533 of the Goiásexperiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

D.11 Dispersion plot of the genetic EBLUPs values of the Model 20 and the traditional non-spatial analysis model for this design (without spatial dependence) for Goiás Local 533.The cut-offs for the 15 best test lines (7 % upper) in each model are indicated by the dottedline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

D.12 Dispersion plots for the genetic EBLUPs values from model selected for each Local in Goiás.Panels the relationship between pairs of Locals (a) 3 and 521; (b) 3 and 533; and (c) 521and 533. The cut-offs for the 15 best test lines in each Local are indicated by the dotted line. 123

E.1 Boxplot of TCH for each group of lines for Locals of Ribeirão. . . . . . . . . . . . . . . . . . 128E.2 Heat map relating to TCH for Local 20 of Ribeirão. . . . . . . . . . . . . . . . . . . . . . . 128E.3 Heat map relating to TCH for Local 72 of Ribeirão. . . . . . . . . . . . . . . . . . . . . . . 129E.4 Heat map relating to TCH for Local 140 of Ribeirão. . . . . . . . . . . . . . . . . . . . . . . 129E.5 Plots of the column and row faces of the empirical variogram for the residuals for the Local

20 of the Ribeirão experiment for models 2 (panels (a) and (c)), and 14 (panels (b) and(d)). The panels (a) and (b) are for the column direction and the others for row. . . . . . . 132

E.6 Dispersion plot of the genetic EBLUPs values of the Model 14 and the traditional non-spatial analysis model for this design (without spatial dependence) for Ribeirão Local 20.The cut-offs for the 30 best test lines (7 % upper) in each model are indicated by the dottedline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

E.7 Plot of the empirical semi-variogram for the residuals for Local 140 of Ribeirão experimentreferring to Model 14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

E.8 Dispersion plot of the genetic EBLUPs values of the Model 14 and the traditional non-spatial analysis model for this design (without spatial dependence) for Ribeirão Local 140.The cut-offs for the 30 best test lines (7 % upper) in each model are indicated by the dottedline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

E.9 Dispersion plot of the genetic EBLUPs from model selected for Ribeirão Locals 20 and 140.The cut-offs for the 30 best test lines in each Local are indicated by the dotted line. . . . . 138

F.1 Boxplot of TCH for each group of lines for Locals of Piracicaba. . . . . . . . . . . . . . . . 142F.2 Heat map relating to TCH for Local 54 of Piracicaba. . . . . . . . . . . . . . . . . . . . . . 142F.3 Heat map relating to TCH for Local 58 of Piracicaba. . . . . . . . . . . . . . . . . . . . . . 143F.4 Heat map relating to TCH for Local 76 of Piracicaba. . . . . . . . . . . . . . . . . . . . . . 143F.5 Plots of the row and column faces of the empirical variogram for the residuals for the Local

54 of Piracicaba experiment for models 8 (panels (a) and (c)) and 18 (panels (b) and (d)).The panels (a) and (b) are column direction and the others are row direction. . . . . . . . . 145

11

F.6 Dispersion plot of the genetic EBLUPs values of the Model 18 and the traditional non-spatial analysis model for this design (without spatial dependence) for Piracicaba Local 54.The cut-offs for the 30 best test lines (7 % upper) in each model are indicated by the dottedline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

F.7 Plot of the empirical semi-variogram for the residuals of the Local 58 of Piracicaba. Thisrefers to Model 19. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149


F.9 Plots of the column and row faces of the empirical variogram for the residuals for Local 54of Piracicaba experiment for models 18 (panels (a) and (c)) and 19 (panels (b) and (d)).The panels (a) and (b) are column direction and the others are row direction. . . . . . . . . 153


F.11 Dispersion plots for the genetic EBLUPs values from selected model for each PiracicabaLocal. Panels refer the relation between Local (a) 54 and 58; (b) 54 and 76; (c) 58 and 76.The cut-offs for the 30 best test lines in each Local are indicated by the dotted line. . . . . 155

G.1 Boxplot of TCH in each area for Local 101 of Araçatuba. . . . . . . . . . . . . . . . . . . . 160G.2 Heat maps relating to TCH in Local 101 of Araçatuba. The panel refers to (a) area 1; (b)

area 3 and (c) area 2. The names PAD1, PAD2, PAD3 and INTERC are the check plotsand the test lines are labeled alpha-numerically (AR plus number). . . . . . . . . . . . . . . 160

G.3 Plots of the row and column faces of the empirical variogram for the residuals for Local 101area 1 of Araçatuba experiment for models 4 (panels (a) and (c)) and 15 (panels (b) and(d)). The panels (a) and (b) are column direction and the others are row direction. . . . . . 163

G.4 Boxplot of TCH in each area for Local 130 of Araçatuba. . . . . . . . . . . . . . . . . . . . 166G.5 Heat maps relating to TCH in Local 130 of Araçatuba. The panel refers to (a) area 1 and

(b) area 2. The names PAD1, PAD2, PAD3 and INTERC are the check plots and the testlines are labeled alpha-numerically (AR plus number). “x” represents the empty plots. . . . 167

G.6 Plots of the column and row faces of the empirical variogram for the residuals for Local 130area 2 of Araçatuba experiment for Model 8. The panels (a) is column direction and (b) isrow direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

G.7 Boxplot of TCH in each area for Local 551 of Araçatuba. . . . . . . . . . . . . . . . . . . . 172G.8 Heat maps relating to TCH in the Local 551 of Araçatuba. The panel (a) refers to area 1

and (b) area 2. The names PAD1, PAD2, PAD3 and INTERC are the check plots and thetest lines are labeled alpha-numerically (AR plus number). “x” represents the empty plots. 172

G.9 Plots of the column and row faces of the empirical variogram for the residuals for Local 551area 1 of Araçatuba experiment for Model 3. The panels (a) is column direction and (b) isrow direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

12

LIST OF TABLES

2.1 Summary of the analyzed experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2 Some possible covariance structures for direct genetic effects over cut or sites (i). . . . . . . 302.3 Summary of the selected models for all experiments. All the models have a fixed check

varieties effects and random effects for test lines denominated as the direct genetic effects(G). The other effects are represented with the symbols: Cext (co-variate, external effectin the 3 east columns); H (harvest); C (column); R (row); N (neighbour genetic); spl(.)(spline in some direction indicated in parentheses) and lin(.) (linear trend in some directionindicated in parentheses). The structures are US(.) - unstructured; AR(1) - first-orderautoregressive, Band(.) - banded correlation, the number indicates the order of correlation,SAR(2) - constrained autoregressive and Id - identity. . . . . . . . . . . . . . . . . . . . . . 31

2.4 REML estimates of variance parameters from each fitted model of the individual analysisin relation to the residual component (σ2). γ. = σ2

. /σ2; given that each letter represent: c

-column; r - row; g- direct genetic; n- neighbour genetic; gn- genetic covariance; u - nugget.ϕ. are the local residual parameters related to the correlations. Observe Table 2.3 to checkif ϕ. is the direct correlation (banded models) or not (autoregressive models). . . . . . . . . 32

2.5 Correlations (a) between the direct genetic EBLUPs from each fitted model of the individualanalysis; (b) between the direct genetic effects from the joint analysis. . . . . . . . . . . . . 33

2.6 Number of convergence cases in each model and simulation scenario for single site experi-ments, given 1000 data sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.7 Number of convergence cases in each model and scenario for separate versus joint analyses,given 1000 data sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1 Convergence number of the lodging models algorithms in each scenario and percentage ofplots lodged, given 1000 simulated data sets. . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.1 Numbers of check lines and duplicated test lines for five spatially optimized designs. Design1 has the same numbers of check lines as Design 0. All designs were on grid of 500 plotsand had 414 test lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2 Relative efficiencies of spatially optimized designs to unreplicated designs with systematicchecks. In parenthesis are the PEV values for Design 0. . . . . . . . . . . . . . . . . . . . . 60

4.3 Number of estimation algorithm converged for each design-scenario combination, given 1000data sets for each. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

B.1 Descriptive analysis of the groups of sugarcane carried out in Paraná in the first and secondcut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

B.2 Summary of the fitted models to the Paraná first cut data with restricted log likelihood(Log-lik.) and the p-value of the REML ratio test. All models include a random directgenetic effects for the test lines (G), fixed check varieties effects and co-variate. The othereffects are represented with the symbols: H (harvest); Co(cone); S (sowing); C (column);R (row); N (neighbour genetic); spl(.) (spline in some direction indicated in parentheses)and lin(.) (linear trend in some direction indicated in parentheses). The structures canbe: US(.)- unstructured; RR(.)- reduced rank; AR(1) - first-order autoregressive; Band(.) -banded correlation, the number indicates the order of correlation; and Id - identity. . . . . . 78

B.3 REML variance parameters estimates from fitted Model 18 to the experiment of Paraná,first cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

13

B.4 Wald F tests for fixed terms for Model 18. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . . . . 80

B.5 Estimates of the fixed effects for Model 18 and their respective standard errors. . . . . . . . 80B.6 The 30 best test lines with predicted values (pred. value) and standard errors (stand. error)

from Model 18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80B.7 Summary of the fitted models to the Paraná second cut data with restricted log likelihood

(Log-lik.) and the p-value of the REML ratio test. All models include a random directgenetic effects for the test lines (G), fixed check varieties effects and co-variate. The othereffects are represented with the symbols: H (harvest); Co (cone); S (sowing); C (column);R (row); N (neighbour genetic); spl(.) (spline in some direction indicated in parentheses)and lin(.) (linear trend in some direction indicated in parentheses). The structures can be:US(.) - unstructured; RR(.) - reduced rank; sar(2) - constrained autoregressive; AR(.) -autoregressive; Band(.) - banded correlation; and Id - identity. The number indicates theorder of correlation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

B.8 REML estimates of variance parameters from fitted Model 14 for the experiment of Paraná,second cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

B.9 Wald F tests for fixed terms for Model 14. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . . . . 84

B.10 Estimates of the fixed effects for Model 14 and their respective standard errors. . . . . . . . 84B.11 The 30 best test lines with predicted values (pred. value) and respective standard errors

(stand. error) for Model 14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84B.12 Summary of the fitted models to joint analysis of the Paraná experiments in the first (1) and

second (2) cut, with restricted log likelihood (Log-lik.); p-value of the REML ratio test andAIC. All the models have the same fixed; local and global effects modeled in the individualanalysis plus the cut fixed effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

B.13 REML estimates of variance parameters from fitted Model 4 to the joint analysis of thefirst-cut (1) and second-cut (2) of the experiment carried out in the Paraná. . . . . . . . . . 86

B.14 Wald F tests for fixed terms from Model 4. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . . . . 87

B.15 Predicted values and standard errors of the 30 best test lines of the Model 4 for each cut. . 87

C.1 Descriptive analysis of the groups of clones carried out in Paraná. . . . . . . . . . . . . . . 89C.2 Summary of the fitted models to Local 651 with restricted log-likelihood (log-lik.) and the

p-value of the REML ratio test. All models include a random direct genetic effects for thetest lines (G) and fixed check varieties effects. The other effects are represented with thesymbols: H (harvest); Co (cone); S (sowing); C (column); R (row); N (neighbour genetic);spl(.) (spline in some direction indicated in parentheses) and lin(.) (linear trend in somedirection indicated in parentheses). The structures can be: US(.) - unstructured; AR(1) -first-order autoregressive; Band(.) - banded correlation, the number indicates the order ofcorrelation; and Id - identity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

C.3 REML estimates of variance parameters for fitted Model 20 for the experiment in the Local651 of the Paraná. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

C.4 Wald F tests for fixed terms for Model 20. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . . . . 94

C.5 Estimates of the fixed effects for Model 20 and their respective standard errors. . . . . . . . 95C.6 The 14 best test lines with predicted values and respective standard errors for Model 20. . . 95

14

C.7 Summary of the fitted models to Local 851 with restricted log-likelihood (log-lik.) and thep-value of the REML ratio test. All models include a random direct genetic effects for thetest lines (G) and fixed check varieties effects. The other effects are represented with thesymbols: H (harvest); Co (cone); S (sowing); C (column); R (row); N (neighbour genetic);spl(.) (spline in some direction indicated in parentheses) and lin(.) (linear trend in somedirection indicated in parentheses). The structures can be: RR(.) - reduced rank; AR(1) -first-order autoregressive; Band(.) - banded correlation, the number indicates the order ofcorrelation; and Id - identity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

C.8 REML estimates of variance parameters for fitted Model 15 for the experiment in the Local851 of the Paraná. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

C.9 Wald F tests for fixed terms from Model 15. Df means degrees of freedom and DenDF isthe approximate denominator degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . . 98

C.10 Estimates of the fixed effects for Model 15 and their respective standard errors. . . . . . . . 98C.11 The 14 best test lines with predicted values and respective standard errors for Model 15. . 99C.12 Summary of the fitted models to Local 852 with restricted log-likelihood (log-lik.) and the

p-value of the REML ratio test. All models include a random direct genetic effects for thetest lines (G) and fixed check varieties effects. The other effects are represented with thesymbols: H (harvest); Co (cone); S (sowing); C (column); R (row); N (neighbour genetic);spl(.) (spline in some direction indicated in parentheses) and lin(.) (linear trend in somedirection indicated in parentheses). The structures can be: US(.) - unstructured; RR(.) -reduced rank; AR(.) - autoregressive structure; Band(.) - banded correlation, the numberindicates the order of correlation; sar(2) constrained autoregressive; and Id - identity. . . . . 100

C.13 REML estimates of variance parameters for Model 15 to the experiment in the Local 852of the Paraná. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

C.14 Wald F tests for fixed terms for Model 15. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . . . . 101

C.15 Estimates of the fixed effects for Model 15 and their respective standard errors. . . . . . . . 102C.16 The 14 best test lines with predicted values and respective standard errors for Model 15. . 102C.17 Summary of the fitted models to joint analysis of the Paraná experiments carried out in the

Locals 851 (2) and 852 (3), with REML log (log-lik.) and the p-value of the REML ratiotest. All the models have the same effects as modeled in the individual analysis plus thefixed Local effect. Here σ2

gi is the genetic variance at the ith Local and ρgii′ is the geneticcorrelation between the ith and i′th Locals. . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

C.18 Summary of the models fitted in joint analysis of the Paraná experiments carried out in theLocals 561 (1), 851 (2) and 852 (3), with REML ratio tests (log-lik.) and the p-values ofthe REML ratio tests. All the models have the same effects as modeled in the individualanalysis plus the fixed Local effect. Here σ2

gi is the genetic variance at the ith Local, ρii′is the genetic correlation between the ith and i′th Locals and σg∗ is the genetic covariancebetween two locals which have been hypothesis to be equal for two or more pairs of Locals. 104

C.19 REML estimates of the variance parameters from fitted Model 3 in the joint analysis of theLocals in Paraná. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

C.20 Wald F test for fixed terms of the Model 3. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . . . . 105

C.21 Estimates of the fixed effects from Model 3 and their respective standard errors. . . . . . . 106C.22 Predicted values (pred.) and standard error (error) of the 14 best test lines from Model 3

for each Local. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

15

D.1 Descriptive analysis of TCH for the groups of clones carried out in Goiás. . . . . . . . . . . 109

D.2 Summary of the fitted models to Local 3 of the Goiás experiment with restricted log-likelihood (log-lik.) and the p-value of the REML ratio test. All models include a randomdirect genetic effect for the test lines (G) and fixed check varieties effect. The other effectsare represented with the letters: H (harvest); Co (cone); S (sowing); C (column); R (row);N (neighbour genetic); spl(.) (spline in some direction indicated in parentheses) and lin(.)(linear trend in some direction indicated in parentheses). The structures can be: US(.) -unstructured; AR(.) - autoregressive; sar(2) - constrained autoregressive; Band(.) - bandedcorrelation, the number indicates the order of correlation; and Id - identity. . . . . . . . . . 113

D.3 REML estimates of variance parameters from fitted Model 19 to the experiment in the Local3 of the Goiás. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

D.4 Wald F tests for fixed terms for Model 19. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . . . . 114

D.5 Estimates of the fixed effects for Model 19 and their respective standard errors. . . . . . . . 115

D.6 The 15 best test lines with predicted values and respective standard errors from Model 19. 115

D.7 Summary of the fitted models to Local 521 of the Goiás experiment with restricted log-likelihood (log-lik.) and the p-value of the REML ratio test. All the models include a randomdirect genetic effects for the test lines (G) and fixed check varieties effects. The other effectsare represented with the letters: H (harvest); Co(cone); S (sowing); C (column); R (row);N (neighbour genetic); spl(.) (spline in some direction indicated in parentheses) and lin(.)(linear trend in some direction indicated in parentheses). The structures can be: US(.)-unstructured; RR(.)- reduced rank; AR(.) - autoregressive; Band(.) - banded correlation,the number indicates the order of correlation; sar(2) - constrained autoregressive; and Id -identity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

D.8 REML estimates of variance parameters for Model 20 fitted to the data from Local 521from the Goiás experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

D.9 Wald F tests for fixed terms from Model 20. Df means degrees of freedom and DenDF isthe approximate denominator degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . . 118

D.10 Estimates of the fixed effects for Model 20 and their respective standard errors. . . . . . . . 118

D.11 The 15 best test lines with predicted values and respective standard errors from Model 20. 119

D.12 Summary of the fitted models to Local 533 of the Goiás experiment with restricted log-likelihood (log-lik.) and the p-value of the REML ratio test. All the models include arandom direct genetic effects for the test lines (G) and fixed check varieties effects. The othereffects are represented with the letters: H (harvest); Co (cone); S (sowing); C (column);R (row); N (neighbour genetic); spl(.) (spline in some direction indicated in parentheses)and lin(.) (linear trend in some direction indicated in parentheses). The structures can be:RR(.) - reduced rank; AR(.) - first-order autoregressive; sar(2) - constrained autoregressive;Band(.) - banded correlation, the number indicates the order of correlation; and Id - identity. 120

D.13 REML estimates of variance parameters from fitted Model 15 for the experiment in theLocal 533 of the Goiás. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

D.14 Wald F tests for fixed terms from Model 15. Df means degrees of freedom and DenDF isthe approximate denominator degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . . 121

D.15 Estimates of the fixed effects from Model 15 and their respective standard errors. . . . . . . 122

D.16 The 15 best test lines with predicted values and respective standard errors for Model 15. . . 122

16

D.17 Summary of the fitted models to joint analysis of the Goiás experiments carried out in theLocals 3 (1) and 533 (3), with REML test (log-lik.) and the p-value of the REML ratio test.All the models have the same effects as modeled in the individual analysis plus the fixedLocal effects. Here σ2

gi is the genetic variance at the ith Local, σgii′ is the genetic covariancebetween the ith and i′th Locals which have been hypothesized to be equal for two or morepairs of Locals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

D.18 REML estimates of variance parameters for fitted Model 3 to the joint analysis of the Local3 (1) and 533 (3) of the experiment carried out in Goiás. . . . . . . . . . . . . . . . . . . . . 124

D.19 Wald F tests for the fixed terms in Model 3. Df means degrees of freedom and DenDF isthe approximate denominator degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . . 124

D.20 The 15 best test lines with predicted values and respective standard error from Model 3 foreach Local. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

E.1 Descriptive analysis of the groups of clones carried out in Ribeirão. . . . . . . . . . . . . . . 127E.2 Summary of the fitted models for Local 20 with restricted log-likelihood (log-lik.) and the

p-value for the REML ratio test. All the models include a random direct genetic effectsfor the test lines (G) and fixed checks varieties effects. The other effects are representedwith the symbols: H (harvest); Co (cone); S (sowing); C (column); R (row); N (neighbourgenetic); spl(.) (spline in some direction indicated in parentheses) and lin(.) (linear trendin some direction indicated in parentheses). The structures can be: US(.) - unstructured;AR(1) - autoregressive; Band(.) - banded correlation, the number indicates the order ofcorrelation; and Id - identity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

E.3 REML estimates of variance parameters for fitted Model 14 for the experiment in Local 20from Ribeirão. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

E.4 Wald F test for fixed terms for Model 14. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . . . . 132

E.5 Estimates of the fixed effects for Model 14 and their respective standard errors. . . . . . . . 133E.6 The 30 best test lines with predicted values and respective standard errors for Model 14. . . 133E.7 Estimates of the variance parameters and standard error for the model with first-order

autoregressive structure in row and column direction, and direct genetic, row and columnrandom effects. The log-likelihood of the model is -1851.92. . . . . . . . . . . . . . . . . . . 134

E.8 Summary of the models fitted for Local 140 with restricted log-likelihood (log-lik.) and thep-value for the REML ratio test. All the models include a random direct genetic effectsfor the test lines (G) and fixed checks varieties effects. The other effects are representedwith the symbols: H (harvest); Co (cone); S (sowing); C (column); R (row); N (neighbourgenetic); spl(.) (spline in some direction indicated in parentheses) and lin(.) (linear trendin some direction indicated in parentheses). The structures can be: US(.) - unstructured;AR(1) - first-order autoregressive; Band(.) - banded correlation, the number indicates theorder of correlation; and Id - identity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

E.9 REML estimates of variance parameters for fitted Model 14 to the experiment in the Local140 of the Ribeirão. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

E.10 Wald F test of fixed terms from Model 14. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . . . . 136

E.11 Estimates of the fixed effects for Model 14 and their respective standard errors. . . . . . . . 137E.12 The 30 best test lines with predicted values (pred. value) and respective standard errors

(stand. error) for Model 14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

17

E.13 Summary of the models fitted to joint analysis for the Ribeirão experiments carried out forthe Local 20 (1) and 140 (3), with REML log (log-lik.) and the p-value of the REML ratiotest. All the models have the same fixed; local and global effects modeled in the individualanalysis plus the local fixed effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

E.14 REML estimates of variance parameters for fitted Model 3 for the joint analysis of Local 20(1) and 140 (3) of the experiment carried out in Ribeirão. . . . . . . . . . . . . . . . . . . . 138

E.15 Wald F test for fixed terms of the Model 3. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . . . . 139

E.16 Predicted values and standard errors of the 30 best test lines of the Model 3 for each local. 139

F.1 Descriptive analysis of the groups of clones carried out in Piracicaba. . . . . . . . . . . . . . 141

F.2 Summary of the fitted models to Local 54 of Piracicaba with restricted log-likelihood (log-lik.) and the p-value of the REML ratio test. All models include a random direct geneticeffects for the test lines (G) and fixed checks varieties effects. The other effects are rep-resented with the symbols: H (harvest); Co (cone); S (sowing); C (column); R (row); N(neighbour genetic); spl(.) (spline in some direction indicated in parentheses) and lin(.)(linear trend in some direction indicated in parentheses). The structures can be: US(.) -unstructured; RR(.) - reduced rank; AR(1) - first-order autoregressive; Band(.) - bandedcorrelation, the number indicates the order of correlation; and Id - identity. . . . . . . . . . 144

F.3 REML estimates of variance parameters for fitted Model 18 for the experiment of Piracicaba,Local 54. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

F.4 Wald F test for fixed factors for Model 18. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . . . . 146

F.5 Estimates of the fixed effects for Model 18 and their respective standard errors. . . . . . . . 146

F.6 The 30 best test lines with predicted values (pred. value) and respective standard errors(stand. error) for Model 18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

F.7 Summary of the fitted models to Local 58 of Piracicaba with restricted log-likelihood (log-lik.) and the p-value of the REML ratio test. All models have a random direct genetic effectsfor the test lines (G) and fixed check varieties effects. The other effects are represented withthe symbols: H (harvest); Co(cone); S (sowing); C (column); R (row); N (neighbour genetic);spl(.) (spline in some direction indicated in parentheses) and lin(.) (linear trend in somedirection indicated in parentheses). The structures can be: US(.) - unstructured; RR(.) -reduced rank. AR(.) - autoregressive; Band(.) - banded correlation, the number indicatesthe order of correlation; sar(2) - constrained autoregressive 3; and Id - identity. . . . . . . . 148


F.9 Wald F test for fixed terms from Model 19. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . . . . 150

F.10 Estimates of the fixed effects for Model 19 and their respective standard errors. . . . . . . . 150

F.11 The 30 best test lines with predicted values (pred. value) and respective standard errors(stand. error) for Model 19. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

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F.12 Summary of the fitted models to Local 76 of Piracicaba with restricted log-likelihood (log-lik.) and the p-value of the REML ratio test. All models include a random direct geneticeffects for the test lines (G) and fixed checks varieties effects. The other effects are rep-resented with the symbols: H (harvest); Co (cone); S (sowing); C (column); R (row); N(neighbour genetic); spl(.) (spline in some direction indicated in parentheses) and lin(.)(linear trend in some direction indicated in parentheses). The structures can be: US(.) -unstructured; RR(.) - reduced rank; AR(.) - autoregressive; Band(.) - banded correlation,the number indicates the order of correlation; sar(2) - constrained autoregressive; and Id -identity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152


F.14 Wald F test for fixed terms from Model 19. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . . . . 154

F.15 Estimates of the fixed effects from Model 19 and their respective standard errors. . . . . . . 154F.16 The 30 best test lines with predicted values (pred. value) and respective standard errors

(stand. error) for Model 19. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154F.17 Summary of the fitted models to joint analysis of the Piracicaba experiments carried out in

the Local 54 (1), 58 (2) and 76 (3), with REML log (log-lik.) and the p-value of the REMLratio test. All the models have the same effects modeled in the individual analysis plus thefixed Local effects. Here σ2

gi is the genetic variance at the ith Local, σgij is the geneticcorrelation between the ith and jth Locals and σg∗ is the genetic correlation between twolocals which has been hypothesis to be equal for two or more pairs of Locals. . . . . . . . . 156

F.18 REML estimates of the variance parameters for fitted Model 5 in the joint analysis of theLocals in Piracicaba. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

F.19 Predicted values (pred.) and standard errors (error) of the 30 best test lines from Model 5for each Local. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

G.1 Descriptive analysis of the groups of clones carried out in Local 101 of Araçatuba. . . . . . 159G.2 Summary of the fitted models to Local 101 area 1 with restricted log-likelihood (log-lik.) and

the p-value of the REML ratio test. All the models include a random direct genetic effectsfor the test lines (G) and fixed checks varieties effects. The other effects are representedwith the symbols: H (harvest); Co (cone); S (sowing); C (column); R (row); N (neighbourgenetic); spl(.) (spline in some direction indicated in parentheses) and lin(.) (linear trendin some direction indicated in parentheses). The structures can be: US(.) - unstructured;RR(.) - reduced rank; AR(1) - autoregressive; Band(.) - banded correlation, the numberindicates the order of correlation; and Id - identity. . . . . . . . . . . . . . . . . . . . . . . . 162

G.3 Wald F test for fixed effects for Model 15. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . . . . 163

G.4 Estimates of the fixed effects and their respective standard errors for Model 15. . . . . . . . 164G.5 The 15 best test lines with predicted values (pred. values) and respective standard errors

(stand. errors) for Model 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164G.6 REML estimates of variance parameters from models 1 and 2 for the experiment in Local

101 of Araçatuba in area 2. The values in brackets are the standard errors of the estimates. 165G.7 REML estimates of variance parameters from models 1 and 2 for the experiment in Local

101 of Araçatuba in area 3. The values in brackets are the standard errors of the estimates. 165G.8 Descriptive analysis of the groups of clones carried out in Local 130 of Araçatuba. . . . . . 166

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G.9 REML estimates of variance parameters for models 1 and 2 for the experiment in Local 130of Araçatuba in area 1. The values in brackets are the standard errors of the estimates. . . 168

G.10 Summary of the models fitted to area 2 of the Local 130 with restricted log-likelihood (log-lik.) and the p-value of the REML ratio test. All the models include a random directgenetic effects for the test lines (G) and fixed checks varieties effects. The other effects arerepresented with the symbols: H (harvest); Co (cone); S (sowing); C (column); R (row);N (neighbour genetic); spl(.) (spline in some direction indicated in parentheses) and lin(.)(linear trend in some direction indicated in parentheses). The structures can be: US(.) -unstructured; AR(1) - first-order autoregressive; Band(.) - banded correlation, the numberindicates the order of correlation; and Id - identity. . . . . . . . . . . . . . . . . . . . . . . . 169

G.11 REML estimates of variance parameters for fitted Model 8 to the experiment in the Local130 of the Araçatuba. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

G.12 Wald F tests for the fixed effects for Model 8. Df means degrees of freedom and DenDF isthe approximate denominator degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . . 170

G.13 Estimated fixed parameters and their respective standard errors for Model 8. . . . . . . . . 170G.14 Predicted values and standard errors of the test lines for Model 8. . . . . . . . . . . . . . . 171G.15 Descriptive analysis of the groups of clones carried out in Local 551 of Araçatuba. . . . . . 171G.16 Summary of the models fitted to Local 551 area 1 of the Araçatuba experiment with re-

stricted log (log-lik.). All the models include a random direct genetic effect for the testlines, fixed effects for checks and there is one model with linear trend in column direction(lin(Col)). AR(1) is the first-order autoregressive structure and Id is the identity structure. 173

G.17 REML estimates of variance parameters for models 1, 2 and 3 for the experiment in Local551 of Araçatuba in area 1. The values in brackets are the standard errors (std.error) ofthe estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

G.18 Summary of the models fitted to Local 551 area 2 of the Araçatuba experiment with re-stricted log-likelihood (log-lik.). All the models include a random direct genetic effectsfor the test lines, fixed effects for checks and there is one model with linear trend in col-umn direction (lin(Col)). AR(1) is the first-order autoregressive structure; Band(3) is thethird-order banded correlation and Id is the identity structure. . . . . . . . . . . . . . . . . 174

G.19 REML estimates of variance parameters from models 1 and 3 for the experiment in Local551 of Araçatuba in area 2. The values in brackets are the standard errors (std.error) ofthe estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

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21

1 INTRODUCTION

Important food and bioenergy source, the sugarcane can be a significant component of theeconomy of many countries in the tropics and subtropics, mainly because of three attributes: highproductivity; inputs efficient agriculture; and local processing into value-added products (such as sugar,molasses, ethanol and energy) (Moore et al., 2013).

Brazil is the largest producer of sugarcane, being responsible for more than 45% of worldwideproduction (FAOSTAT, 2013). This is one of Brazil’s most important trade commodities, although itis used mainly to get ethanol fuel, considered to be a sustainable biofuels economy. For 2016/17 theCompanhia Nacional de Abastecimento (Conab, 2016) estimates a yield of 690.98 million tonnes, anincrease of 3.8% relative to the previous years, with a cultivated area of 9,073.7 thousand hectares.

The outlook for agribusiness is favorable due to the petroleum price and its attractiveness as arenewable energy source, which is better for environmental protection. Moreover, the world consumptionof sugar and ethanol grows at significant rates. In Brazil, the cultivation of sugarcane continues to growin areas of traditional plants and dozens of new projects are being carried out in culture expansion regions(Dinardo-Miranda et al., 2008).

The total area by percentage of sugarcane in Brazil is higher in the southeast and central-westwhere around 65.8% of the crop is produced. As shown in Figure 1.1, by Federation Unit, São Paulo(SP) is the biggest producer with 55.2%; followed by Goiás (GO) with 10.7%; Minas Gerais (MG) with10.1%; Mato Grosso do Sul (MS) with 7.0%; Paraná (PR) with 6.7%; Alagoas (AL) with 2.6%; MatoGrosso (MG) with 2.1% and Pernambuco (PE) with 2.0%. These eight states are responsible for 96.4%of the national production (Conab, 2016).

Figure 1.1. Map of Brazil with the estimated percentage of sugarcane production in April/2016.

To increase annual yield, breeding companies work every day pursuing lower costs for productionand better crop yield. The release and recommendation of new commercial varieties requires severalbreeding trials with potential genotypes obtained from crossings of known varieties with highly performingestablished ones. At the initial stage of the breeding program there are many genotypes and little materialfor planting in experiments, which are usually in large fields. The experiments are often highly unbalancedand involve substantial sources of non-genetic variation and correlation. This is similar to the situationdescribed by Stringer (2006), who reports experiments with 100 to 1232 clones.

In Brazil, the main sugarcane breeding programs belong to: RIDESA - Rede Interuniversitáriapara o Desenvolvimento do Setor Sucroenergético; CTC - Centro de Tecnologia Canavieira; and the IAC- Instituto Agronômico de Campinas. With these breeding programs the sugar industry is provided with

22

the best varieties to cover a wide environmental range, adapted to specific conditions, either more adverseor favourable.

1.1 CTC company

The CTC is one of the biggest research centres of sugarcane in the world, and holds a verylarge germplasm bank. Their efforts cover all processes in the sugarcane chain, investing in geneticimprovement, biotechnology and new technologies for the sugar energy sector (CTC, 2013).

The development of a new variety of sugar cane is divided into the following steps: I) crossings;II) phase 1, where the best seedlings are planted in the field and evaluated as individual plants, and thebest 5 to 10% are selected; III) these clones are planted in phase 2 in 2-row plots, and evaluated foragronomic performance. The best 10% of clones are selected to plant in phase 3. IV) phase 3 clones areplanted in 3 locations, but without replication in each location. V) The best 7% of clones from phase 3are planted in multilocation trials in phase 4 where they are evaluated for potential release as commercialvarieties.

The research takes around eight years and requires large areas to accommodate the number ofgenotypes tested. The results come from trials set up in more than one environment and year. The mainproduction area of Brazil can be divided into six parts due to edaphoclimatic conditions, as Figure 1.2illustrates.

Figure 1.2. Map with the six regions where CTC carries out the experiments. On the left is thecomplete map of Brazil and right the specific CTC areas. Source: CTC (2013)

In phase 1 the test lines are compared with commercial varieties (checks), and a percentage ofthe best new materials are selected. The clones are evaluated as experiments by families. Within andbetween plots of the same family are different individuals.

For phases 2 and 3 the unreplicated designs with systematic checks (Kempton, 1984; Clarkeand Stefanova, 2011) are used. These designs have been used commonly for plant breeding trials andthey include a large number of test lines and commercial varieties. The new test lines normally are notreplicated, because of the available area and small amount of clonal material or seeds, but checks arereplicated what allows to capture the residual effect. The advantage of evaluating the treatments withonly a small number of replications is a significant economic gain because they need a smaller area andlower production resources. However, decreasing the number of experimental units implies a less accurateanalysis, with a smaller number of residual degrees of freedom, when compared with designs in which alllines are replicated.

In these early stages of selection trials (phases 1 to 3), it has been an usual practice to have avisual analysis to be made by a sugarcane expert technician, who observes a series of plant development

23

characteristics, giving scores for height, colour, number of tillering, presence of disease, lodging, etc. Toimprove the visual accuracy, a “calibration” process is carried out using checks. For this, in phase 3,three conventional checks (equally replicated and placed normally in neighbouring row plots) are usedplus one special check allocated systematically on a diagonal grid throughout the trial, so that in thesame column there are always eight or fewer test lines between checks. For this special check is allocatedan area of around 11% of the experimental area.

The idea behind this systematic arrangement of the check varieties was that it would contributeto capture better the spatial effects and facilitate the visual evaluation on the selection of the better lines.However, more recently CTC decided to move to a selection of lines based on results given by mixed modelanalysis.

1.2 Outline of thesis

The purpose of this thesis is to (I) develop the quantitative evaluation of phase 3 experimentsbased on the fitting of appropriate mixed models, and (II) investigate how the design of the experimentsfor this phase can be improved.

The thesis is organized as follows. Chapter 2 describes and analyzes several CTC experimentsusing mixed models that account for the local, global and extraneous variation. When an experimentinvolves multiple sites or cuts, both individual and joint analyses are conducted. The experimental resultswere used to develop a simulation study to assess the capacity of the design being used by CTC to fitthe models required to describe the genetic and non-genetic variation in their breeding experiments. Theaccuracy of individual and joint analyses are also compared.

After, the lodging effect is evaluated in Chapter 3 using simulated data and Chapter 4 presentssix different designs and their prediction error variance (PEV) optimality criteria measure. At the end,Chapter 5 has general conclusions and recommendations for future research.

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25

2 INDIVIDUAL AND JOINT ANALYSIS OF SUGARCANE EXPERIMENTS TOSELECT TEST LINES

One purpose of the breeding programs is the selection of the best test lines. With the ex-perimental results, the accuracy of selection can be improved by using the best fitted model. Findingthe best model is not easy, especially in large experiments which assess more than one hundred lineswithout replication, mainly because of few replicated check plots and the number of parameters in thecomplex variance structure that needs to be assumed. The main objective of this chapter is to discussthe models fitted to the data from 21 trials of sugarcane which can be grouped into two types: (i) onelongitudinal data set (two cuts) and (ii) five regional groups of experiment (each group is a region withthree sites). The single and joint analyses were carried out using mixed models that allowed for geneticand/or residual competition and spatial correlation. In the single-site analyses, evidence was found ofcompetition effects at the residual or genetic level. The correlations between the genetic EBLUPs fromthe individual analysis were not very strong (less than 0.55), resulting in a different group of selectedtest lines in each cut or site. Another important observation is that the banded correlations were amore realistic structure than the autoregressive process to model the local effects. The joint analysis,conducted based on the mixed models selected for the individual cuts or sites, showed significant directgenetic correlation between cuts or sites within the same region; the direct correlations were more thandouble the measure of the individual analysis from EBLUPs. Two simulation studies were conducted.The first verified that, using the type of design employed in the experiments, it is possible to estimatewith accuracy the competition effect from individual analysis. The second assessed the individual andjoint analysis from three sites with different local effects. In this second study, both the banded andautoregressive structures in the row direction were used in the residual model for the same data set. Theresults suggested that the competition genetic effects from individual analysis were captured, but theconvergence rates were between 47.0% and 68.7%; the difference in estimated parameters and selectionbetween the true simulated model (with competition effect) and the simplest (only check and test lineseffects) was not large; the joint analysis achieved better precision in the estimation of the genetic meritof the test lines; and there was no obvious difference between the models fitted with autoregressive orbanded structure. In particular, the simulation study identified that the group of selected true best testlines of this experiment can be 55% or less.Keywords: Assertiveness of selection; Autoregressive correlation; Banded correlation; Competition ef-fect; Mixed models; Simulation study.

2.1 Introduction

Test line selection from breeding programs can depend on the model chosen and it is necessaryto identify the effects present in the experiment to ensure the maximum precision in selecting genotypes,especially in large experiments, because of the residual variability. Indeed, as well as the model, theselection precision will be better.

Using mixed linear models, Gilmour et al. (1997) identify three major components of spatialvariation in plot errors from field experiments: local, global and extraneous variation. Local variationreflects short distance spatial effects. Global variation ranges across the entire field. Extraneous variationis associated with trial management. They used a separable first-order autoregressive process (AR(1) ×AR(1)) in the row and column directions to model the local variation. They affirmed that a more accurateassessment of the presence of global and extraneous variation or outliers can be obtained when using thistwo-dimensional spatial procedure rather than assuming independent plot errors.

Global variation is modelled using polynomial functions of the spatial coordinates in row and/or

26

column direction and/or smoothing splines. Extraneous variation can arise from the practice of serpentineharvesting, i.e., harvesting rows of plots in alternating directions, inaccurate plot trimming methods,leading to plots of unequal length, or the use of multiplot seeders (Gilmour et al., 1997; Stefanovaet al., 2009).

Then, the autoregressive models for residual variation have become common for researchers(Stringer and Cullis, 2002; Atkin et al., 2009; Hunt et al., 2013; Liu et al., 2015). The variationmay occur because of management practices impacting on the experiment, non-stationary spatial trendoccurring across the field, and neighbouring plots being more similar than those further apart related tosoil fertility or moisture levels (De Faveri et al., 2015). Smith et al. (2005) presented an overview ofmixed model approaches to analyze the crop cultivar breeding and evaluation trials, merging the spatialand randomization models.

For sugarcane experiments, the plots are generally long and it is to be expected that thecorrelation between plots sharing the longest boundary (here different rows) will be greater than thosesharing the shorter boundary (here different columns). Stringer et al. (2011) found that the moststriking feature was the occurrence of a negative residual correlation between neighbouring plots indifferent rows. This is evidence of competition, as would be a correlation value in row direction that issmall, compared to the value in column direction.

Several researchers (Besag and Kempton, 1986; Stringer and Cullis, 2002; Stringeret al., 2011; Silva and Kerr, 2013; Hunt et al., 2013) argued that the interplot competition betweenplants of different varieties or treatments is one effect that can seriously bias the assessment of varietalperformance and thus reduce the accuracy of the genetic prediction.

Silva and Kerr (2013), with a simulated study on genetic forestry trials, found out thatranking genotypes on the basis of predicted breeding values using models that ignore genetic competitiondoes not correlate well with a ranking based on predicted total breeding values, which are a function ofboth direct and competitive additive effects. This happens because the variances estimated for geneticand residual effects tended to be biased under models that ignored genetic competition.

Stringer et al. (2011) extends Gilmour et al. (1997) to include the Random TreatmentInterference Model (R-TIM), which incorporates direct and neighbour genetic effects. They concluded,using autoregressive models, that the competition at the residual level can be modelled by second-order(AR(2)) and if there are both effects, trend and competition effects, the third-order, or the constrainedversion of this model (SAR(2), notation in Butler et al. (2009)) can be adequate. Stringer et al.(2011) showed a disagreement between classical approaches and competition models in terms of selection,given that a correlation between predicted values in one trial was 0.76 and in another (presented insupplementary material) was 0.95.

As some agricultural experiments are conducted in more than one harvest and/or site, thejoint analysis can be used. Cullis et al. (2000), assessing several experiments from Australia cropvariety evaluation programs in different years and locations, advised that the joint analysis is betterthan individual. However, they did not discuss competition and there are few papers which do this withreal data from groups of experiments and/or longitudinal data. Another relevant issue not found in theworks quoted is to compare and show how much the joint analysis improves in heritability estimationand assertiveness selection, with this kind of experiment.

Hence, the aim in this chapter is to find which was the best model to select the most productivetest lines of sugarcane from five regions of Brazil, where there are two harvests or three sites each, ina systematic design with up to 21% of check plots (commercial varieties). With the chosen individualmodels, the joint analysis is investigated, evaluating the genetic correlation among sites or year of harvest.

Additionally, simulations are performed to establish that the parameters of the complex modelsbeing proposed for sugarcane experiments can be estimated using designs such as those typically used

27

by “Centro de Tecnologia Canavieira” (CTC) in their experiments, in Brazil. For this simulation studyit was used the gain in precision of estimated variance components, heritability or selection gain, andassertiveness of selection (based on 30 best test lines, around the top 7%), of the models from individualand joint analysis.

The chapter is arranged as follows. First, the experiments are described in Section 2.2. Section2.3 presents the general model for these data, including an extension of competition effects of Stringeret al. (2011) and correlation structures for the residual variation. Section 2.4 gives the results of theindividual and joint analysis of all experiments. Section 2.5 and Section 2.6 describe and show the resultsof the simulation studies, respectively. Some concluding remarks are made in Section 2.7.

2.2 Material

The experimental data were provided by the CTC, Piracicaba - Brazil, in a partnership withthe University of São Paulo and are summarized in Table 2.1. The experiments were carried out in fiveregions in the southeastern and central-west regions of Brazil (see Figure 1.2). These areas were chosenby the company based on climatic and soil conditions. Each region had three sites, called Locals whichwere numbered, with the same sugarcane lines to test. In the Paraná region there was an exception withone site only that was assessed in two consecutive years. Because of space problems, Araçatuba hadmultiple experimental areas at each Local.

All experiments had 79% or more of the area planted with new sugarcane lines, denominatedtest lines. These lines had only one plot in each site due to insufficient material for replication and thelarge area that would be necessary to accommodate more plots.

The aim of the company project was to select the best seven percent of the test lines in eacharea. It does this by comparing them with four commercial varieties, which were called checks and werereplicated such that they occupied at most 21% of the plots. One special check, nominated interspersedplot, was planted in almost 11% of the plots and it was allocated systematically on a diagonal gridthroughout the trial. The other three checks were equally replicated and each replicate was spread outin three neighbouring row plots. The idea behind this systematic arrangement of the check varietieswas that it would contribute to capture better the spatial effects and facilitate the visual evaluation andselection of the better lines, because the same column always had eight or fewer test lines between checks.

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Table 2.1. Summary of the analyzed experiments.

Region Site Number of Number of plotsharvests columns × rows test lines interspersed total

Paraná — 2 26 × 19 406 55 488

ParanáLocal 651 1 10 × 24 200 26 238Local 851 1 10 × 24 200 28 240Local 852 1 10 × 24 200 28 240

RibeirãoLocal 20 1 19 × 28 429 57 516Local 72 1 21 × 25 429 58 517Local 140 1 14 × 38 429 59 519

PiracicabaLocal 54 1 14 × 37 422 57 506Local 58 1 14 × 37 422 57 506Local 76 1 15 × 35 422 57 506

GoiásLocal 3 1 13 × 20 212 30 260Local 521 1 13 × 20 212 30 260Local 533 1 13 × 20 212 30 260

Araçatuba

Local 1011 3 × 18 42 6 511 15 × 21 252 35 3021 3 × 15 31 5 39

Local 130 1 26 × 12 259 35 3091 5 × 15 62 10 75

Local 551 1 9 × 21 109 19 1371 11 × 24 209 29 250

The plots were 12m long, double-furrows with 0.9m between furrows within the plot and 1.5mspacing between different plots. There was 1m of space between columns. Therefore, the plots whichwere row-neighbours (i.e., within the same column) shared the longest plot boundary.

The unnamed Paraná site was the first experiment studied. It was planted in 2013, there weretwo cuts, with the first harvest being obtained in 2014 and the second in 2015. The other trials wereplanted in 2014 and harvested in 2015.

The yields in tonnes of cane per hectare (TCH) from each cut or Local were analyzed individ-ually. Afterwards, the results were used to perform the joint analysis for the experiments from a region.Other variables were measured by CTC, and one problem, related by the company’s breeder, was thatsome plots exhibited lodging. However, they did not have the information about the direction in whichplants fell down, only the score of each plot that rated the amount of lodging. The impact of plantlodging will be simulated and reported in Chapter 3.

2.3 Methodology

To provide reliable predictions of test line performance across harvest or environments in an-alyzing Brazilian sugarcane experiments, we propose to follow Gilmour et al. (1997) and to use mixedmodels that incorporate terms for local, global and extraneous variation. In addition, as Stringer et al.(2011) and Hunt et al. (2013), terms for genetic and residual competition will be included. Then, thegeneral form of the model for n× 1 vector of yields, y (assumed ordered as rows within columns), wheren is the total number of plots, can be written as:

Y∼

= Xτ∼+ Zgug

∼+ Zouo

∼+ ε

∼; (2.1)

where X is the incidence matrix associated with vector τ∼

of fixed effects, Zg is the incidence matrixassociated with the vector ug

∼of random genetic effects of the test lines, Zo is the incidence matrix

associated with the vector uo∼

of non-genetic random effects, and ε∼

is the vector of residual effects.

29

We assume that the joint distribution of (ug∼

, uo∼

, ε∼

) is Gaussian with zero mean and variancematrix

σ2

Gg(γg) 0 00 Go(γo) 00 0 R(ϕ)

.For a single experiment, the effects include the global mean, commercial varieties (checks) and

linear trend, if any, that is, the effects of the checks, which are generally standard released varieties, areconsidered fixed, as in Santos et al. (2002); Piepho et al. (2008); Peternelli et al. (2009); Pastinaet al. (2012); Ali et al. (2013). For the genetic effects of m test lines from a single experiment, weconsider (i) random genetic effects only for which Gg(γg) = γgIm; and (ii) the Random-effects TreatmentInterference Model (R-TIM) (Stringer et al, 2011) in which ug

∼

t = (ugd∼

t,ugn∼

t) contains the direct and

neighbour genetic effects and

Gg(γg) =

[γgd γgdn

γgdn γgn

]⊗ Im;

where σ2gd = γgdσ

2, σ2gn = γgnσ

2 and σgdn = γgdnσ2 are the variance components for the direct and

neighbour genetic effects, and covariance component between these effects, respectively.For this model the incidence matrix for the genetic effects is Zg = [Zgd NgZgd], where Zgd is

the incidence matrix associated with the direct genetic effect and Ng is a first-order neighbour incidencematrix. For between-row genetic competition, Ng = Ic ⊗ Nr, where Nr is the within-row neighbourincidence matrix.

A model for the local spatial variation can be written as ε∼

= η∼+ ζ

∼, where η

∼is a vector of

technical or measurement error (nugget effect) and ζ∼

is a vector that represents a spatially dependentprocess, which is usually modeled using a separable process in the row and column directions such asR = (

∑c ⊗

∑r) + γψIn, with

∑c and

∑r representing the spatial correlation matrix in the column (c)

and row (r) direction (Gilmour et al., 1997; Stefanova et al., 2009; Stringer et al., 2011).In terms of the starting model for the model fitting process, the approach of Smith et al. (2005)

is followed and a model that includes all the terms that experience tell us are likely to occur in sugarcaneexperiments. It includes an R-TIM model, an AR(1) × AR(1) spatial model and random row and columneffects. Autoregressive models are used initially because often close neignbours are more highly correlatedthan far neighbours. Here, they will also be compared with banded correlation structures as a check thatthe pattern in the bands of observed correlations conforms to the more restrictive patterns assumed foralternative correlation structures. The genetic and non-genetic effects are assessed, in order, as follows:

(i) Neighbour genetic effects, by replacing the R-TIM with independent genetic effects only;(ii) Local effects in the column and row directions, by replacing the AR(1) × AR(1) spatial

model with AR(1)×Band(3) in the model selected in (i);(iii) Management practice effects, such as harvesting direction; sowing direction; cone seeder,

by including fixed effects for these in the selected model in (ii);(iv) Global variation, by splitting the column and row global effects in the model selected in

(iii) into cubic smoothing spline and non-smooth deviations;(v) Retest the neighbour genetic effect in the context of model (iv).The models were compared formally, using a REML ratio test, for selecting between nested

random models, or Wald F tests, for the fixed effects, assuming a 5% significance level. Each selectedmodel is assessed using diagnostic plots, including plots of faces of the empirical sample variogram and/orthe empirical semi-variogram (Stefanova et al., 2009).

The models from the individual analysis are used to formulate the model for the joint analysis,but with the inclusion of cut or site main effects and the interaction of cut or site with the genetic effects.

30

Beginning with the model for independent direct genetic effects (id), the covariance structures shown inTable 2.2 are tested.

Table 2.2. Some possible covariance structures for direct genetic effects over cut or sites (i).

Structures parametersidh() heterogeneous identity γgicorg() general correlation γgi and ρgicor() uniform correlation γg and ρgid() identity γg

All computations were done in the R software (R Core Team, 2014) using the ASReml-R(Butler et al., 2009) and asremlPlus (Brien, 2016) packages.

2.4 Data analysis results

Each individual analysis showed the peculiarities of each site within the region. In Table 2.3,the selected individual model is shown and Table 2.4 displays the REML estimates of variance parametersfor these models. In most trials there were no significant global or local effects in the column direction.However, in Piracicaba Local 58 and Goiás Local 3 there were negative correlations in this directionwhich Stringer (2006) also found in sugarcane experiments. However, competition does not seema plausible biological explanation for such correlation in the column direction, in contrast to the rowdirection. We believe that, in our case, it perhaps was because material from different plots becomemixed up as a result of lodging so that separation of the material from different plots at harvest becamedifficult. Also, the correlation in row direction, when it was significant, normally did not conform toan autoregressive process as suggested by some authors (Gilmour et al., 1997; Stringer et al., 2011).We compared autoregressive with banded models and out of 11 experiments, eight were modeled morerealistically using a banded correlation structure. In all of these 11 experiments there were significantrow correlations and because, except for Paraná Local 651, the first neighbour correlation was negativeor smaller than the second neighbour correlation, it was concluded that there was competition at theresidual level. For Paraná Local 651 the magnitude of the first neighbour correlation was similar to thecolumn correlation so that it is concluded that there is not competition at the residual level at this site.

Only in Goiás Local 521 there was evidence of significant genetic competition, with γgd =

3.44. This value was more than three times the ratios found in the other two sites at the same region.Consequently, we believe that this is most likely a spurious result.

In the Araçatuba experiments, the direct genetic effects were close to zero or the standard errors(s.e) for the estimated components were high when compared with the magnitude of the parameters (Local101-1: s.e genetic= 111.61 and s.e residual= 72.66. Local 130-2: s.e genetic= 123.19 and s.e residual=105.53). Therefore, in this case, two explanations were proposed. First, in every Local there was atleast one experiment with less than 10 rows, which makes it difficult to estimate spatial dependence,and/or less than 100 plots which limits the ability to estimate variances. Second, because even the largerexperiments performed poorly, it could be that the small areas are inherently more variable or there weregreater difficulties encountered in carrying out the operations for the experiment. The improvement ofthis type of trials will require more uniform areas and/or more careful performance of the operationsduring the experiment.

One problem found using ASReml-R is that the structure Band()×Band() is not availableand, for this reason, in one direction we still used the autoregressive process. Another is that, for themodels using spl(), the response variance matrix cannot be written with this effect to obtain the plotof the empirical variogram using asremlPlus. Hence, in this case, the empirical semi-variogram, fromASReml-R, was used.

31

Table 2.3. Summary of the selected models for all experiments. All the models have a fixed checkvarieties effects and random effects for test lines denominated as the direct genetic effects (G). Theother effects are represented with the symbols: Cext (co-variate, external effect in the 3 east columns);H (harvest); C (column); R (row); N (neighbour genetic); spl(.) (spline in some direction indicated inparentheses) and lin(.) (linear trend in some direction indicated in parentheses). The structures are US(.)- unstructured; AR(1) - first-order autoregressive, Band(.) - banded correlation, the number indicatesthe order of correlation, SAR(2) - constrained autoregressive and Id - identity.

Experiments Fixed effects Random effects Residual variationGlobal/extraneous Genetic Local

column × row

Paraná

1st. cut Cext Row G Id × Band(4)2nd. cut Cext+H+lin(C)+lin(R) spl(C)+spl(R) G Id × Band(3)651 nugget G AR(1) × AR(1)851 lin(C) spl(C)+R G Id × Id852 lin(C)+lin(R) spl(R) G Id × Band(2)

Ribeirão20 lin(C)+lin(R) R G Id × Band(1)72 direct genetic effects (σ̃2

g=14.27) less than its standard error (79.92).140 lin(C)+lin(R) spl(R) G AR(1) × Band(1)

Piracicaba54 lin(C) R G Id × Band(1)58 lin(R) spl(R)+R G Band(2) × SAR(2)76 R G Id × Band(2)

Goiás3 H+lin(R) C+spl(R) G AR(1) × Band(2)521 lin(C)+lin(R) C+spl(R) US(G:N) Id × Id533 lin(R) spl(R) G Id × AR(1)

Araçatuba

101 -1 lin(C) G Id × Id101 -2 direct genetic effects close to zero101 -3 direct genetic effects close to zero130 -1 direct genetic effects close to zero130 -2 G Id × Id551 -1 direct genetic effects (σ̃2

g=95.40) less than its standard error (141.74).551 -2 direct genetic effects close to zero

32Ta

ble

2.4.

REM

Les

timat

esof

varia

nce

para

met

ers

from

each

fitte

dm

odel

ofth

ein

divi

dual

anal

ysis

inre

latio

nto

the

resid

ualc

ompo

nent

(σ2).γ.=σ2 ./σ2;

give

nth

atea

chle

tter

repr

esen

t:c

-col

umn;r

-row

;g-d

irect

gene

tic;n

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etic

cova

rianc

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-nug

get.ϕ.

are

the

loca

lres

idua

lpar

amet

ers

rela

ted

toth

eco

rrel

atio

ns.

Obs

erve

Tabl

e2.

3to

chec

kifϕ.

isth

edi

rect

corr

elat

ion

(ban

ded

mod

els)

orno

t(a

utor

egre

ssiv

em

odel

s).

Reg

ion

Para

náR

ibei

rão

Pira

cica

baG

oiás

Ara

çatu

baLo

cal

1st

cut

2nd

cut

651

851

852

2014

054

5876

352

153

310

1-1

130

-2

parameters

γ̃c

——

——

——

——

——

0.12

0.18

——

—γ̃r

0.09

——

0.48

—0.

07—

0.25

0.07

0.46

——

——

—sp

l(C)

—0.

67—

0.04

——

——

——

——

——

—sp

l(R)

—0.

15—

—0.

03—

0.11

—0.

04—

0.02

0.41

0.10

——

γ̃gd

0.52

1.31

0.78

1.10

1.30

0.85

0.88

0.77

0.30

0.66

0.84

3.44

1.35

2.48

0.57

γ̃gdn

——

——

——

——

——

—-0

.97

——

—γ̃gn

——

——

——

——

——

—0.

67—

——

γ̃u

——

1.15

——

——

——

——

——

——

σ̃2

238.

8718

7.27

127.

9223

8.74

195.

1217

8.00

205.

5223

5.89

320.

2431

7.95

320.

0989

.35

262.

2410

5.71

223.

87ϕ̃c1

——

0.67

——

—0.

23—

-0.1

4—

-0.2

8—

——

—ϕ̃c2

——

——

——

——

0.14

——

——

——

ϕ̃r1

0.02

-0.1

90.

56—

-0.4

0-0

.35

-0.4

0-0

.27

0.39

-0.2

4-0

.61

—-0

.53

——

ϕ̃r2

0.17

0.03

——

0.29

——

—-0

.22

0.26

0.21

——

——

ϕ̃r3

0.18

0.19

——

——

——

——

——

——

—ϕ̃r4

0.19

——

——

——

——

——

——

——

33

At the same region, the joint analyses presented a significant estimated correlation betweendirect genetic effects in all cases. Table 2.5 compares the correlations between direct genetic effects fromthe joint analyses with the calculated correlations of the direct genetic EBLUPs from the individualanalyses. In all cases the correlations from the joint analyses are around twice, or more, than those fromthe corresponding individual analysis. For Paraná first and second cuts, the 7% best test lines selectedusing the direct genetic effects from the joint analysis were the same in both cuts, because the correlationbetween the direct genetic effects was approximately one. For the experiments at different sites withina region, the selected group of test lines using the genetic effects from the joint analysis were rathermore similar than when the genetic effects from the individual analyses were used. Cullis et al. (2000),in examining the efficiency of Australian crop variety evaluation programs, concluded that the effect ofincreasing the number of locations is much less than that of increasing the number of years.

Table 2.5. Correlations (a) between the direct genetic EBLUPs from each fitted model of the individualanalysis; (b) between the direct genetic effects from the joint analysis.

Region (a) (b)

Cuts 1st,2nd 1st,2ndParaná 0.54 — — 1.00 — —

Sites 1st,2nd 1st,3rd 2nd,3rd 1st,2nd 1st,3rd 2nd,3rdParaná 0.37 0.33 0.49 0.84 0.84 0.91Ribeirão — 0.40 — — 0.82 —Piracicaba 0.31 0.30 0.28 0.84 0.84 0.84Goiás 0.17 -0.02 0.03 — 0.39 —

2.5 Simulation studies

Two simulation studies were conducted. The first involved a single-site experiment that em-ployed an unreplicated design with systematic checks and involved a model for simulation that alwaysincluded genetic competition. It sought to investigate the effect on the selection of test lines using in-dividual analyses of (i) the magnitude of the direct genetic variation, (ii) the presence or absence ofnon-genetic competition, and (iii) the model fitted. The second simulation study involved a three-siteexperiment that used an unreplicated design with systematic checks at each site, but did not includecompetition at the genetic level. In this case, the study investigated the effect on the selection of testlines of (i) the magnitude of between-sites genetic correlation, (ii) unequal versus equal site variances,(iii) fitting autoregressive versus banded correlation structures, and (iv) individual versus joint analysesof the sites. It was also of interest in these studies to see how well the variance components are estimatedwith the type of unreplicated design used by CTC.

In all cases, 1000 data sets were simulated for a design grid of 25 columns by 20 rows, totaling500 plots. An unreplicated design design of the type employed by CTC was used to assign 418 linesto this grid. The 418 lines consisted of 414 unreplicated test lines and four check lines; three of thecheck lines were replicated 10 times each and the fourth had 56 replicates laid out diagonally every eightcolumns and eight rows; the test lines were randomized to the plots remaining after the check lines hadbeen assigned.

2.5.1 Single site experiments

The model for simulation had: (i) global column (γc) and row (γr) variation; (ii) direct (γgd) andneighbour (γgn) genetic variation, as well as covariance (γgdn) between these two sources of variation; and(iii) residual correlation with AR(1) in the column (ϕc) direction and Band(1) in the row (ϕr) direction.

34

Four different scenarios for simulations were conducted. For all scenarios γc = γr = 0.3, γgdn = 0.3,γgn = 0.1, ϕc = 0.5 and the scale parameter, σ2, was set at a value of 1.0. The values of γgd and ϕr werevaried to yield the different simulation scenarios:

a. γgd = 1 and ϕr = −0.3;

b. γgd = 1 and ϕr = 0.3;

c. γgd = 1.5 and ϕr = −0.3;

d. γgd = 1.5 and ϕr = 0.3.

Thus the simulations differed in the magnitude of the direct genetic variation and in whether or not therewas competition at the residual level.

In generating the data for each scenario a single set of direct and neighbour genetic effectswas generated for that using the values of the genetic variance parameters for the simulation. This setof effects constituted the true genetic effects for the simulation. They were added to each of the 1000simulated data sets obtained for a model with all the non-genetic parameters included.

For each simulation, the data were analyzed using four models:

Model 1 only random column, row and direct genetic terms;

Model 2 Model 1 plus neighbour genetic and direct-neighbour genetic covariance terms;

Model 3 Model 1 plus residual correlation;

Model 4 Model 2 plus residual correlation.

Clearly, Model 1 is the simplest model, while Model 4 is the same as the simulation model.For each simulation and each analysis model, the following information was collected:

• the number of converged analyses;

• the correlation between the direct genetic EBLUPs and the true (generated) direct genetic effects,at the same model and between pairs of models;

• the selection gain measured as the average of the true direct genetic effects for the 30 test lines(around 7%) with the largest estimated direct genetic EBLUPs;

• the assertiveness of selection or average similarity where similarity is measured as the percentage ofthe 30 test lines with the largest direct genetic EBLUPs that are in the true 30 best test lines; thetrue 30 best test lines are the 30 test lines with the highest true (generated) direct genetic effects;

• the estimates of the variance parameters.

2.5.2 Separate versus joint analyses

This study was based on an experiment with three sites, as in a CTC experiment in oneregion. Each site employed the same unreplicated design with systematic checks, although test lines werererandomized at each site. In all simulations, all sites assumed the same direct genetic variance (γgd = 0.8,which is approximately the average for the three-site CTC experiments) and an AR(1) residual correlationin the column direction (ϕc = 0.2). Further, the assumed structure for the residual row correlation (

∑ri,

i = 1 . . . 3) differed between sites, then R = diag(σ21

∑c⊗

∑r1, σ2

2

∑c⊗

∑r2, σ2

3

∑c⊗

∑r3). At Local 1

it was assumed to be SAR(2) (ϕr11 = 0.6 and ϕr12 = −0.2), at Local 2 to be AR(1) (ϕr21 = 0.4), and

35

at Local 3 to be AR(2) (ϕr31 = −0.2 and ϕr32 = 0.1). Hence, Locals 1 and 3 were assumed to havenon-genetic competition of different types and Local 2 to not have residual competition.

Four simulation scenarios were conducted in which the sites were assumed to have either equalor unequal residual variance (σ2

i ) and one of two values for the genetic correlation between sites (ρgs).The values of the parameters used in the simulations were:

a. ρgs = 0.70, σ21 = 1.15, σ2

2 = 1.30, and σ23 = 1.00;

b. ρgs = 0.70 and σ21 = σ2

2 = σ23 = 1.00;

c. ρgs = 0.90, σ21 = 1.15, σ2

2 = 1.30, and σ23 = 1.00;

d. ρgs = 0.90 and σ21 = σ2

2 = σ23 = 1.00.

In generating the data for each scenario a set of direct genetic effects was generated for thethree sites for that simulation scenario using the values of the genetic variance parameters. This set ofeffects constituted the true genetic effects for the simulation scenario. Then, 1000 simulated data sets forthe whole experiment were obtained for a model that included all the non-genetic parameters. Finally,the set of true genetic effects was added to each of the 1000 simulated data sets.

Two joint analyses was performed on each of the 1000 data sets for each scenario. The twomixed models used included all the terms in the model for simulation, but differed in their assumedresidual row correlation structures: (i) as in the simulated data, the autoregressive structures SAR(2),AR(1), and AR(2) used for Locals 1, 2 and 3, respectively; and (ii) the banded structures Band(3),Band(2) and Band(1) used for Locals 1, 2 and 3, respectively.

For each scenario and each analysis model, the following information was collected:

• the number of converged analyses;

• the correlations between the direct genetic EBLUPs and the true direct genetic effects for each siteobtained from the individual and joint analyses;

• the assertiveness of selection or the average similarity for each site obtained from the individual andjoint analyses, where similarity is measured as the percentage of the 30 test lines with the largestdirect genetic EBLUPs that are in the true 30 best test lines; the true 30 best test lines are the 30test lines with the highest true (generated) direct genetic effects ;

• the heritability estimates (as proposed by Cullis et al. (2006));

• the estimates of the direct genetic variance parameter.

2.6 Simulation results

2.6.1 Single site experiments

The estimation process for the simplest model converged without problems for all simulateddata sets. However, when the number of parameters increased convergence was not obtained in some ofthe cases, see Table 2.6. Note that for the fitted of the true model, Model 4, the algorithm convergedbetween 470 to 687 times in each scenario and also Model 2 had a smaller convergence rate in some cases.Therefore, with this data, fitting the models with competition genetic effects resulted in more problemsthan the simplest model or the model with residual correlation (Model 3). Observe that scenario b Model2 converged only 90 times. This number is very small and the results were not used to compare with theother models.

36

Table 2.6. Number of convergence cases in each model and simulation scenario for single site experi-ments, given 1000 data sets.

Simulation scenarioa b c d

mod

els 1 1000 1000 1000 1000

2 789 90 749 3433 997 682 997 7714 470 578 530 687

Assessing the correlation of the direct genetic EBLUPs between pairs of models in each scenario,the average correlations were superior to 0.80 (see Figure 2.1) although the correlations were better whenthere was negative row correlation rather than positive row correlation. This means that on average therewere small differences between the models.

Figure 2.2 illustrates the correlations between true simulated direct genetic effects and the directgenetic EBLUPs. The results were around 0.65 to 0.80 which means that the estimates are not that closeto the true values. Also, there was little difference in the EBLUPs among the models. Although thecorrelation for Model 4 was a little better in all scenarios. Assessing the selection gain, related to theaverage EBLUPs of the 30th best test lines in each model and scenario, the results from Model 4 werebetter, see Figure 2.3. Also, the best selections came from Model 4, which selected in average around 44to 51% of the true 30 best test lines (Figure 2.4). In general, the greater the genetic variance componentthe greater will be the selection gain and assertiveness. Note that there are some differences betweenthe results from scenarios a and b with c and d, because of the direct genetic effects. Moreover, theseresults do not represent a big difference among the models from the genetic criteria (selection gain andsimilatity), because of the magnitude of the parameters and maybe the convergence problems when fittingthe models.

0.6

0.7

0.8

0.9

1.0

corr

elat

ion

1:2 1:3 1:4 2:4 3:4

(a)

0.6

0.7

0.8

0.9

1.0

1:2 1:3 1:4 2:4 3:4

(b)

0.6

0.7

0.8

0.9

1.0

Models

corr

elat

ion

1:2 1:3 1:4 2:4 3:4

(c)

0.6

0.7

0.8

0.9

1.0

Models

1:2 1:3 1:4 2:4 3:4

(d)

Figure 2.1. Boxplots of the correlations between direct genetic EBLUPs for pairs of models from eachscenario (a, b, c and d). In (b) Model 2 converged only for 90 out of 1000 simulated data sets and hencethe density plot for genetic components effects are not displayed (see Table 2.6).

37

0.6

0.7

0.8

0.9

1.0

corr

elat

ion

1 2 3 4

(a)

0.6

0.7

0.8

0.9

1.0

1 2 3 4

(b)

0.6

0.7

0.8

0.9

1.0

Models

corr

elat

ion

1 2 3 4

(c)

0.6

0.7

0.8

0.9

1.0

Models

1 2 3 4

(d)

Figure 2.2. Boxplots of the correlation between the true direct genetic effects and the respectiveEBLUP’s from each model for each scenario (a, b, c and d). In (b) Model 2 converged only for 90 outof 1000 simulated data sets and hence the density plot for genetic components effects are not displayed(see Table 2.6).

0.0 0.5 1.0 1.5 2.0 2.5

0.0

0.5

1.0

1.5

2.0

Den

sity

(a)

0.0 0.5 1.0 1.5 2.0 2.5

0.0

0.5

1.0

1.5

2.0

selection gain

Den

sity

(c)

0.0 0.5 1.0 1.5 2.0 2.5

0.0

0.5

1.0

1.5

2.0

(b)

0.0 0.5 1.0 1.5 2.0 2.5

0.0

0.5

1.0

1.5

2.0

selection gain

(d)

Model 1Model 2Model 3Model 4

Figure 2.3. Densities of the selection gain from each model for each scenario (a, b, c and d). In (b)Model 2 converged only for 90 out of 1000 simulated data sets and hence the density plot for geneticcomponents effects are not displayed (see Table 2.6).

38

3540

4550

scenario

aver

age

sim

ilarit

y (%

)

a b c d


Figure 2.4. Percentage of average similarity of the 30 test lines selected as best compared to the truebest 30 lines in each model and scenario (a, b, c and d). In scenario b, Model 2 converged only for 90 outof 1000 simulated data sets and hence the density plot for genetic components effects are not displayed(see Table 2.6).

Cullis et al. (1998), with barley and wheat experiments, affirmed that models with complexvariance structure can affect estimation of the genetic merit of breeding lines. However, for the simulationstudy, Model 4 was not strongly affected. Figure 2.5 displays the densities of the genetic estimates of thegenetic parameters for all scenarios and models (exception scenario b Model 2). Each plot represents theresults for a combination a scenario and a component. Note that only Models 2 and 4 assumed geneticcovariance and neighbour components. It can be confirmed that with these simulated models, it is possibleto obtain the competition genetic effect with this design. Model 4 had some deviation in relation to thetrue parameters, but it still displayed the best results, when the model algorithm converged, comparedwith the other models. Investigating the residual and global row and column variance parameters, aswell as the row and column correlation (Figures 2.6 and 2.7), this true model presents a better fit thanModels 2 and 3. However, overall Model 1 tends to give the best estimates of the row and column variancecomponents (Figure 2.7), whereas Model 4 tends to underestimate them.

39

0.0 1.0 2.0 3.0

0.0

1.0

2.0

Den

sity

γgd

scen

ario

a

0.0 1.0 2.0 3.0

0.0

1.0

2.0

genetic component

Den

sity

γgd

scen

ario

b

0.0 1.0 2.0 3.0

0.0

1.0

2.0

Den

sity

γgd

scen

ario

c

0.0 1.0 2.0 3.0

0.0

1.0

2.0

direct genetic component

Den

sity

γgd

scen

ario

d

0.0 0.4 0.8 1.2

02

46 γgdn

0.0 0.4 0.8 1.2

02

46 γgdn

0.0 0.4 0.8 1.2

02

46 γgdn

0.0 0.4 0.8 1.2

02

46

genetic covariance

γgdn

0.0 0.2 0.4 0.6

02

46

8 γgn

0.0 0.2 0.4 0.6

02

46

8 γgn

0.0 0.2 0.4 0.6

02

46

8 γgn

0.0 0.2 0.4 0.6

02

46

8

neighbour genetic component

γgn


Figure 2.5. Densities of the direct genetic (first column of plots), neighbour genetic (third) componentsalong with their covariance (second column) for each scenario and model. Each row represents a differentscenario and the vertical lines represent the assumed values of the parameters. Remember that theconvergence was different in each one (see Table 2.6) and only Models 2 and 4 assumed neighbour geneticeffect. In scenario b Model 2 converged only for 90 out of 1000 simulated data sets and hence the densityplot for genetic components effects are not displayed.

40

0.5 1.0 1.5 2.0 2.5

0.0

1.0

2.0

3.0

Den

sity

σ2

scen

ario

a

0.5 1.0 1.5 2.0 2.5

0.0

1.0

2.0

3.0

Den

sity

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scen

ario

b

0.5 1.0 1.5 2.0 2.5

0.0

1.0

2.0

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ario

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0.2 0.4 0.6 0.8

01

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45

6 φc

0.2 0.4 0.6 0.8

01

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6 φc

0.2 0.4 0.6 0.8

01

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6 φc

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column correlation

φc

−0.6 −0.2 0.2

02

46

812

φr

−0.1 0.1 0.3 0.5

02

46

812

φr

−0.6 −0.2 0.2

02

46

812

φr

−0.1 0.1 0.3 0.5

02

46

812

row correlation

φr


Figure 2.6. Densities of the residual component(first column of plots) and the column (second column)and row (third column) correlations for each scenario and model. Each row represents a different scenarioand the vertical lines represent the assumed values of the parameters. Remember that the convergencewas different in each one (see Table 2.6) and only Model 3 and 4 assumed residual correlation. In scenariob Model 2 converged only for 90 out of 1000 simulated data sets and hence the density plot for geneticcomponents effects are not displayed.

41

0.0 0.2 0.4 0.6 0.8

01

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45 γr

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0.0 0.2 0.4 0.6 0.8

01

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45 γr

0.0 0.2 0.4 0.6 0.8

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row component

γr


Figure 2.7. Densities of the global components, column (in the left) and row (in the right), for eachscenario and model. Each row represents a different scenario and the vertical lines represent the assumedvalues of the parameters. Remember that the convergence was different in each one (see Table 2.6). Inscenario b Model 2 converged only for 90 out of 1000 simulated data sets and hence the density plot forgenetic components effects are not displayed.

2.6.2 Separate versus joint analyses

There was no big difference between the number of simulated data set and the convergencemodels (Table 2.7). However, the convergence rates were slightly smaller for the banded models, mainlyin the joint analysis. This was not good because better predictions were expected from the joint analysis,but the convergence can be obtained with the update of the model or by increasing the number ofiterations.

Table 2.7. Number of convergence cases in each model and scenario for separate versus joint analyses,given 1000 data sets.

Scenarios ModelsLocal 1 Local 2 Local 3 Joint

auto

reg. a 1000 1000 1000 999

b 997 1000 1000 998c 998 1000 1000 999d 998 1000 1000 999

band

ed

a 999 999 1000 962b 999 998 999 931c 999 1000 1000 951d 999 997 1000 914

42

As the results from autoregressive and banded models were similar, for the following criteriawe present only the results for the autoregressive models.

Examining the average genetic EBLUP correlations between pairs of Local in the individual andjoint analysis (Figure 2.8(a)), those for the joint analyses are twice those for the individual analyses. Sincethe true values of this parameter is either 0.7 or 0.9, it is clear that the individual analyses underestimatethis correlation and that the joint analyses tend to overestimate it a little. An unexpected result wasthe strong correlation for average genetic EBLUPs between individual and joint analysis at the sameLocal (Figure 2.8(b)). These were a little bit more for scenario with small genetic correlation component(0.70, scenarios a and b) than the others (0.90, scenarios c and d), and means that there were nostrong differences among genetic EBLUPs from individual and joint analysis. On the other hand, theassertiveness of selection, reported in Figure 2.9, showed differences of approximately 6 to 12% amongthe models from individual and joint analysis at the same Local. Regarding the number of test lines, thisrepresented a small difference, however all improvements in the selection are welcome.

0.2

0.4

0.6

0.8

1.0

Locals

corr

elat

ion

1:2 1:3 2:3

(a)

scenario ascenario bscenario cscenario d

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

Local

corr

elat

ion

1 2 3

(b)

Figure 2.8. Average EBLUP correlations in the individual (light symbols) and joint (dark symbols)analysis for autoregressive models in each scenario. The panels refer to comparison between (a) pairs ofLocals; (b) individual and joint analysis at the same Local.

43

3540

4550

5560

Local

aver

age

sim

ilarit

y (%

)

1 2 3

scenario ascenario bscenario cscenario d

Figure 2.9. Average similarity between the true and estimated 30 best test lines (around top 7%). Thedark symbols represent the result from joint analysis and the light are the individuals for autoregressivemodels in each scenario.

For the joint analysis from autoregressive models, the densities of the heritability for eachscenario are presented in Figure 2.10. There are differences among Locals when the scenario assumeddifferent residual variances between sites (scenarios a and c) and a little bit more heritability fromsimulated scenarios which set out the larger genetic correlation (γgs = 0.9 - scenarios c and d). Forinstance, the average heritability estimates were around 0.6 and 0.7.

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

02

46

810

Den

sity

(a)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

02

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810

heritability

Den

sity

(c)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

02

46

810

(b)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

02

46

810

heritability

Den

sity

(d)

Local 1Local 2Local 3

Figure 2.10. Densities of the heritability for each site from the joint analysis for autoregressive modelsin each scenario. The panels refer to the scenarios.

The densities of the direct genetic components for all scenario did not differ between Localsand/or correlation structures from fitting autoregressive and banded correlation structures, but between

44

the joint and individual analysis there were differences, (see Figure 2.11). The estimated values weremore accurate for the joint analysis, independently of residual variance or genetic correlation parameters.

0.2 0.4 0.6 0.8 1.0 1.2 1.4

01

23

45

direct genetic component− autoregressive models

Den

sity

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scen

ario

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autoregressive models

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ario

d

0.2 0.4 0.6 0.8 1.0 1.2 1.40

12

34

5 γgd

banded models

0.2 0.4 0.6 0.8 1.0 1.2 1.4

01

23

45 γgd

0.2 0.4 0.6 0.8 1.0 1.2 1.4

01

23

45 γgd

0.2 0.4 0.6 0.8 1.0 1.2 1.4

01

23

45


γgd

Local 1

Local 2

Local 3

joint

Figure 2.11. Densities of the direct genetic components from autoregressive and banded models for eachLocal from scenarios (a, b, c and d). The vertical lines represent the assumed values of the parameters.

When investigating the residual variance components (Figures 2.12, 2.13, 2.14 and 2.15) it canbe noted that there was no strong difference between the estimated variance components from fittingautoregressive and banded correlation structures. However, the estimates of the residual variance com-ponents showed greater spread, and so are less accurate, for the individual analyses than for the jointanalyses. The difference seemed greater when the genetic correlation between sites is higher (scenarios cand d).

45

0.6 1.0 1.4

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regr

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ded

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σ2

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σ2

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01

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45

Local 3

residual component

σ2

0.6 1.0 1.4

01

23

45

residual component

Den

sity

σ2

indivjoint

Figure 2.12. Densities of the residual variance components from autoregressive and banded models foreach Local from scenario a. The vertical lines represent the assumed values of the parameters.

0.6 1.0 1.4

01

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Local 1

Den

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ve

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σ2

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σ2

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01

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Local 3

residual

σ2

0.6 1.0 1.4

01

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residual component

Den

sity

σ2

indivjoint

Figure 2.13. Densities of the residual variance components from autoregressive and banded models foreach Local from scenario b. The vertical lines represent the assumed values of the parameters.

46

0.6 1.0 1.4

01

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Local 1D

ensi

ty

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Local 3

residual

σ2

0.6 1.0 1.40

12

34

5residual component

Den

sity

σ2

indivjoint

Figure 2.14. Densities of the residual variance components from autoregressive and banded models foreach Local from scenario c. The vertical lines represent the assumed values of the parameters.

0.6 1.0 1.4

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Local 1

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residual component

Den

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indivjoint

Figure 2.15. Densities of the residual variance components from autoregressive and banded models foreach Local from scenario d. The vertical lines represent the assumed values of the parameters.

2.7 Conclusions

Like Smith et al. (2005), the strategy employed for selecting a mixed model for the data froma single site was to start with a model that includes all the terms that experience tells us are likely tooccur in the experiment. Then diagnostic plots were used to assess model adequacy and the need tovary the model. Also, we advocate the fitting of banded correlation structures with about three bandsas a diagnostic technique for checking that the correlation structure conforms to the assumed patternfor autoregressive models such as AR(1), AR(2) and SAR(2). If it was found that the autoregressive

47

models do not describe the pattern then banded correlation models were used. In addition to modellingthe correlation structure, the R-TIM model for genetic competition was considered and models for globaland extraneous variation were tested. A joint analysis of the sites within a region was conducted, themodel for the spatial variation being a combination of those selected for the individual analyses.

Of the 21 experiments investigated, seven had no significant direct genetic effects (componentclose to zero and/or standard error larger than the estimate). For the others, in 11 there was significantcompetition, but at the residual level rather than at the genetic level. Each individual trial had its ownpeculiarity as far as the residual variance structure is concerned and so it was impossible to determine onespecific model for the residual variation that applied to all the experiments. Only one experiment hadthe competition at the genetic level significant. It would appear that there is more general competitionin Brazilian experiments rather than it being genotype specific.

The test lines selected within a region differed between sites and the genetic EBLUP correlationswere less than 0.54. However, the genetic correlations were significant in the joint analyses and thesecorrelations were stronger than in the individual analyses. Hence, using the joint analyses, better accuracyin the selection of the test lines was obtained. Higher predicted values were found with lower standarderrors because the observation of genotypes at multiple sites provides replication of them. Thus a jointanalysis is preferable to individual analyses whenever it is possible, whether it is based on different sitesor more than one harvest from the same trial. As Cullis et al. (2000) have already shown, using modelssuggested by Gilmour et al. (1997), the effect of increasing the number of locations is much less efficientthan increasing the number of years for crops such as barley and wheat. However, in our case, the Paranátrial with two cuts resulted in an estimated correlation between the direct genetic effects (ρgd), obtainedfrom the joint analysis, that was close to 1.00 (see Table 2.5). It remains to establish whether this resultsin more efficient selection of lines.

The simulation study for single sites showed that, when there is genetic competition, it isdifficult to fit a model that incorporates a term for it. When there is also residual competition, suchthat the row autocorrelation is negative, the true model will fit in only about 50% of cases, but a modelwithout genetic competition will fit almost surely. When there is no residual competition, a model withgenetic competition will fit around 60% of the time, but the only model that is guaranteed to fit is onewith no genetic competition term and no residual correlation. The simulation study demonstrated clearlythat a strategy of fitting a single model to the data from single sites or cuts, even if it is the true model,is not a viable strategy and that detecting genetic competition in an experiment using the CTC typeof design is fraught. Hence, it cannot be stated conclusively that genetic competition does not occur inBrazilian sugarcane experiments.

Fortunately, it would appear that the estimation of the direct genetic effects is little affectedby the model selected and the presence or not of residual competition, although it is affected by themagnitude of the genetic variance. While, in general, Model 4 gives the best correlation between trueand estimated genetic effects, selection gain and average similarity, the differences between it and theother models are small.

The results of the simulation study for separate versus joint analyses shows there were noproblems in fitting either single-site or joint models when there is no genetic competition and that betterestimates of the genetic effects come from joint analyses. There was little effect of heterogeneous variances,but there were differences in the results that depended on the magnitude of the genetic variance. Forthe models simulated, there was little difference between fitting autocorrelation and banded correlationstructures.

Given the difficulties in fitting models that include both genetic and residual competition andthat only around 45 to 55% of the true best test lines were selected even for the joint analyses, moreresearch into improved design and management practices is necessary, to see if model fitting can be

48

improved and to increase the direct genetic variance so as to achieve better selection results.

49

3 NEGATIVE RESIDUAL CORRELATION IN SUGARCANE EXPERIMENTS

The negative residual correlation in the row direction in sugarcane experiments is associatedwith competition effects between plots in different rows and is taken to result from the narrowness ofthe rows. On the other hand, negative correlation between columns is generally not observed. However,when such correlation occurs in an experiment then the question arises as to what might have causedit. It was conjectured that one possibility is that lodging, which occurs in dry weather, could lead tonegative correlation in the column direction. Our aim in this chapter is to show, with a simulated study,that this is the case. Then, careful harvest processing is necessary to minimize the effect of lodging onthe recorded results and avoid confusing it with competition effects.

Keywords: Competition, Harvest process, Lodging, Simulation study

3.1 Introduction

Sugarcane is generally large, perennial, tropical or subtropical grasses that evolved under con-ditions of high sunlight, high temperatures, and large quantities of water (Moore et al., 2013). Duringthe production year, the plant can be around 2 meters high. It is not surprising, that under climaticeffects (mainly strong winds and/or dry weather), the sugarcane can lose its erectness (Singh et al.,2002; Van Heerden et al., 2015).

These effects were investigated by some authors (Singh et al., 2002; Van Heerden et al.,2011, 2015), which have shown a decrease in yield under some conditions of lodging. As a result, Singhet al. (2002) affirmed that the impact of lodging on cane yield tends to be masked given that the bettergrown crops of sugarcane are more likely to lodge than poorly grown crops. Van Heerden et al. (2015)declared that the lodging close to harvest had no seriously deleterious effects on yields.

Stringer et al. (2011), working with models assuming competition at the residual level, showedthat this biological effect results in negative or smaller correlation for the first neighbour than the secondin one direction, because the plots are long and thin. Hence the idea of competition in both directions isnot expected. Moreover, Stringer and Cullis (2002) and Stringer (2006) found significant negativecorrelation for both directions working with autoregressive models in some sugarcane experiments. Themagnitude of the correlation was smaller for the direction that shared a smaller boundary and Stringerand Cullis (2002) affirmed that a special plant-breeding program, which results in a higher competitioneffect, was the dry one and a particular trial there did not receive irrigation. Then, both competitionand lodging effects can be related to dry weather, which causes plant stress.

We noted, in some breeding experiments conducted by the “Centro de Tecnologia Canavieira”(CTC - Brazil), that sometimes the lodged sugarcane fell down over adjacent plots, causing seriousproblems in separating the material from different plots at harvest. Given this and that the process ismechanical, it is possible that part of lodged plot yield was ascribed to its neighbouring plot. It was alsonoted that in the experiments there was significant negative residual correlation in the column direction.We suspect that, even in the absence of competition, lodging could lead to negative correlation in thecolumn direction. Thus, our aim in this chapter is to prove that the negative residual correlation incolumn direction can be associated with inaccurate mechanical harvest that might occur when plants arelodged in the row direction.

The chapter is arranged as follows. First, we describe the motivating example (Section 3.2).Section 3.3 presents the material and methods. Section 3.4 gives the results and section 3.5 some con-cluding remarks.

50

3.2 Motivating example

Our motivation came from a partnership between the CTC and the University of São Paulo(Brazil) where a series of sugarcane breeding experiments were provided to facilitate the development ofnew models for the company and to understand the “real” experimental effects. One particular exper-iment, on which the lodging score was recorded, had 5.4% of plots with score 9 and 10.7% with score8. These represent high levels of lodging and it means that in these plots most of the plant were eitherprostrate or nearly so.

The selected model for this experiment included harvest, row spline and random column effects,with negative residual correlation in both directions. This was not expected given that the plot dimensionswere 12m long, double-furrows with 0.9m between furrows within the plot and 1.5m between differentplots, and 1m of space between columns.

Then, if a plot with 2m high cane became extensively lodged, it can be expected that 1m ofsome stalks came into the neighbouring plot. It has been noted that this can result in the plot bordersbecoming obscured and making it difficult to determine during the harvesting the plot to which some ofthe cane belongs. This is particularly a problem for the column borders.

Other experiments, with the same design structure, also resulted in negative correlation in bothdirections, however lodging scores were not available.

3.3 Material and methods

We begin by assuming that, at the harvest, lodging results in a loss of tonnes of cane per hectare(TCH) are lost to a neighbouring plot in the same row. Two cases were assumed, one with a 5% loss ofTCH and the second with 15%. In addition, four percentages of lodged plots for the experiments werestudied: 0%, 1%, 5% and 10%.

The design array has 25 columns by 20 rows, totaling 500 plots, with 414 unreplicated test linesand four checks. Three checks were replicated 10 times each and one 56 times in a diagonal grid, eacheight columns or rows, respecting the similar characteristics of the original design from CTC. In practice,the checks (commercial varieties) are stable and erect, so the lodging only occurs with test line plots.

The model for the data vector, Y∼

, is:

Y∼

= Xτ∼+ Zgug

∼+ ε

∼; (3.1)

given that: X is the incidence matrix associated with vector τ∼

of fixed effects; Zg is the incidence matrixassociated with the vector ug

∼of random genetic effects of the test lines; ε

∼is the vector of residual effects.


, ε∼


σ2

[Gg(γg) 0

0 R(ϕ)

]=

[Gg 00 R

].

where, Gg(γg) = γgIm and R =∑c⊗

∑r = AR(1) ⊗ Band(1), assuming Y

∼are in field order with rows

within columns. γg = σ2g/σ

2, Im is the identity matrix of order m, such as m is the number of testlines, AR(1) indicates a first-order autoregressive correlation matrix and Band(1) the first-order bandedcorrelation.

For all simulations γg = 1, ρr = −0.3 and the scale parameter, σ2, was set at a value of 1.0.Two types of data sets are provided assuming: (a) ρc = 0.5 and (b) ρc = 0. For each, 1000 data setswere generated. The data set combined with the percentage of lodged plants result in the following fourscenarios.

1) ρc = 0.5 and 5% of TCH lost;

51

2) ρc = 0.5 and 15% of TCH lost;3) ρc = 0 and 5% of TCH lost;4) ρc = 0 and 15% of TCH lost;The density plots of the residual variance parameters were used to show how the lodging affects

the estimation. All computations were done in the R software (R Core Team, 2014) using the ASReml-Rpackage (Butler et al., 2009).

3.4 Results

In most of the cases the algorithm converged (Table 3.1). An exception was for 1% lodging, aloss of 15% and ρc = 0.5, when nearly half of the plots failed to converge.

However, observing the densities of the residual components (Figure 3.1), the residual varianceincreases as the percentage of plots lodged increases, but only for a 5% loss of TCH. In general, the columncorrelation becomes increasingly more negative as the percentage of plots that are lodged increases. Notethat the row correlation is also affected by the lodging of plots.

Table 3.1. Convergence number of the lodging models algorithms in each scenario and percentage ofplots lodged, given 1000 simulated data sets.

Percentage of plots lodged0% 1% 5% 10%

scen

ario 1 ρc = 0.5 and 5% of TCH lost 986 943 1000 1000

2 ρc = 0.5 and 15% of TCH lost 552 984 9733 ρc = 0 and 5% of TCH lost 984 982 1000 9974 ρc = 0 and 15% of TCH lost 923 989 997

52

1 2 3 4

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ρc

−0.6 −0.2 0.2 0.60

24

6

ρc

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46

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column correlation

ρc

−0.6 −0.4 −0.2 0.0

04

8 ρr

−0.6 −0.4 −0.2 0.0

04

8 ρr

−0.6 −0.4 −0.2 0.0

04

8 ρr

−0.6 −0.4 −0.2 0.0

04

8

row correlation

ρr

0%1%5%10%

Figure 3.1. Densities of the residual component and the column and row correlation for each scenario.The curves within a panel are for the different percentage of plots lodged. The plot of the residualcomponents have been truncated to exclude a few estimated values that were in excess of 4.

3.5 Conclusions

In summary, the negative correlation can be due to the lodging effect in the experiment ifthis results in material from on plot being associated with a plot from a neighbouring column at theharvest. As the dry weather can also cause more competition between plots, the negative correlation canbe related to either one or both of these effects. Hence, when there is lodging care in separating materialfrom different plots will lead to improved results.

53

4 COMPARISON OF A SYSTEMATIC DESIGN WITH SOME SPATIALLY OPTIMIZEDDESIGNS

This chapter considers a trial of fixed size, similar to those used by the “Centro de TecnologiaCanavieira” (CTC), a Brazilian sugarcane company. The efficiency of the type of unreplicated design thatthey typically employ is compared to spatially optimized unreplicated and p-rep designs with checks anda spatially optimized p-rep design. A simulation study is conducted to compare the designs as far as thegenetic gain from selection is concerned. The comparison is done by using the prediction error variances(PEVs), computed from the simulated data, to compute relative efficiencies of the designs to a CTCdesign. Other criteria calculated to evaluate performance are the relative genetic gain, the correlationbetween the true and estimated genetic effects and the similarity percentage that gives the percentageof the test lines that are in the top 7% of both the true and estimated genetic effects. The results showthat differences in the number of check varieties have little effect on the performance of unreplicateddesigns with checks. On the other hand, experiments that employ a p-rep designs are better than usingunreplicated designs.Keywords: Unreplicated design, Optimal design, p-rep design, Prediction error variances, Genetic gain,Assertiveness of selection, Mixed models, Simulation study

4.1 Introduction

Breeding companies continuously pursue lower production costs and better crop yield. Toachieve this, new varieties are recommended from several breeding trials with potential genotypes; theserecommended varieties become the test lines in subsequent experiments. However, at the initial phasesof the breeding program there are many test lines and limited material for planting in the experiments.Smith et al. (2006) pointed out that, in such circumstance, fully replicated designs are prohibitivelyexpensive and are unnecessary from a statistical perspective.

One solution, discussed by Kempton (1984), is to employ unreplicated designs in which thereare unreplicated test lines and replicated commercial varieties, called checks. Cullis et al. (2006) callthese designs grid-plot designs. As Clarke and Stefanova (2011) describe, there are two types ofunreplicated designs: (i) designs in which the replicates of the check are systematically allocated and (ii)augmented designs (Federer, 1956) in which a standard randomized design (e.g. randomized complete-block design, Latin square design) is used to allocate check lines and the test lines are added to thisdesign.

A problem is, given there may be global, local and/or extraneous variation (Gilmour et al.,1997; Stefanova et al., 2009), the precision with which parameters can be estimated may depend on thespatial distribution of the check lines plots (Müller et al., 2010). It may be that neither unreplicateddesigns with systematic check nor complete randomization of checks are optimal for parameter estimation.In short, the statistical methods used for design and analysis must be as accurate, efficient and informativeas possible (Smith et al., 2005). To improve the precision of parameter estimation, various arrangementsof the checks have been suggested (Lin and Poushinsky, 1983, 1985; Martin et al., 2006; Mülleret al., 2010; Clarke and Stefanova, 2011). Of these, Martin et al. (2006) is unique in consideringthe use of optimality criteria to search for an optimal unreplicated design for a single site.

A more recent approach, pioneered by Cullis et al. (2006), is to partially replicate the lines.They introduced p-rep designs in which p% of the lines have two replicates and the remaining lines areunreplicated; no checks are included. A search is made for an optimal design for assigning the lines toplots. Clarke and Stefanova (2011) have compared unreplicated designs with p-rep designs and haveconcluded that the latter are superior.

54

4.1.1 Optimality criteria for design searches

The optimal design depends on the criterion that it is sought to optimize. For linear models, theinformation concerning the precision of the parameter estimates is contained in the variance-covariancematrix of parameter estimates. In general, optimal designs are those experimental layouts that optimizesome function of this matrix (Hooks et al., 2009).

The most commonly used criterion for comparative experiments is the A-optimality criterion.It is the average variance of pairwise differences of effects (AVPD) and the optimal design is the onethat minimizes it. It is certainly appropriate when the researcher wishes to estimate differences betweenspecific lines and, as Smith et al. (2005) outline, in this case the lines effects should be assumed to befixed. Federer and Raghavarao (1975) introduced three additional A-criteria that might be moreappropriate in the context of breeding experiments: (i) the AVPD between test line effects, Att; (ii) theAVPD between checks, Acc (not useful in breeding experiments); and (iii) the AVPD between test linesand checks, Atc. Which of these is relevant depends on the comparisons that are of most importance tothe researcher.

On the other hand, Smith et al. (2006) argue that the lines effects should be assumed asrandom if the objective of the experiment is to select the best lines so that the accuracy of the linerankings is paramount. Piepho et al. (2008) agree, but note that this is not practical when the geneticvariance components cannot be accurately estimated. It is generally accepted that an optimal designfor random genetic effects is one that maximizes the genetic gain due to selection from field trials (forexample, Kempton, 1984; Sarker and Singh, 2015; Cullis et al., 2006; Clarke and Stefanova,2011). Cullis et al. (2006) and Bueno Filho and Gilmour (2007) demonstrate that maximizing theexpected genetic gain (EGG) is equivalent to minimizing the average pairwise prediction error variance(AVPD, albeit applied to random effects). On the other hand, Smith et al. (2015) conjecture, in thecontext of multiphase experiments that display autocorrelation, that minimizing the AVPD does not leadto a design that gives maximum genetic gain.

Another approach to assessing the accuracy of the parameter estimates is based on consideringthe correlation between true and observed genetic effects. Fouilloux and Laloë (2001) propose thatthe accuracy of the estimates be assessed by the prediction error variance (PEV), or some function ofit. Butler (2013) notes that PEV is proportional to the squared correlation coefficient trace of thePEV matrix for test line effects. This leads to suggesting the sum of the PEV of each genetic effect, or,equivalently, the average of these variances (APEV), as potential criteria for assessing competing designs.

Hooks et al. (2009) points out that, while the AVPD is equivalent to the APEV when the effectsfor each random term in a model and those for different random terms can be assumed independent; theyare not when one or more random terms involve autocorrelation.

Piepho and Williams (2006) declared that finding a universally optimal design that capitalizeson genetic correlation structure is difficult, because of the genetic variance components are typically notaccurately known, and they differ among traits. However, BLUE and BLUP are valid methods of analysis,when lines can be regarded as random.

4.1.2 Comparing different design types using simulation

A topic, related to criteria for searching for designs, is that of criteria for comparing designsof different types that have been selected as possible contenders for use in an experiment. In breedingexperiments, the aim is to evaluate the genetic gain resulting from selection and so this guides the choiceof criteria. Here we suppose that the selected design is optimal, according to either the AVPD or thePEV, for a particular mixed model with nominated values for its variance parameters. Then, simulationis to be used to obtain values for one or more criteria for comparing designs.

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Obvious criteria for comparing designs are one or both of the criteria used in searching for anoptimal design: the AVPD or the PEV. However, as Clarke and Stefanova (2011) noted, a number ofauthors have suggested that the AVPD may not be appropriate for breeding experiments. They compareddesigns using the SE ratio, or the relative error, which is the standard error of the comparison betweentest lines and control for a design as a proportion of that standard error in a completely randomizeddesign. Cullis et al. (2006) used the relative genetic gain which is the realized genetic gain (RGG),calculated as the mean of the top s% of test line EBLUPs, to the true genetic gain, calculated as themean of the top s% of the true test line effects. They show that the RGG is well correlated with theexpected genetic gain (EGG) used by Kempton (1984). An alternative measure of realized genetic gain,described by Piepho and Möhring (2007) and Moehring et al. (2014), is the response to selection,calculated as the mean of the true genetic effects for the test lines with the top s% of EBLUPs. Piephoand Möhring (2007) also propose another measure of genetic gain, namely the square of the correlationbetween the true and predicted test line effects; this is the coefficient of determination discussed byFouilloux and Laloë (2001) and others. Smith et al. (2011) and Moehring et al. (2014) use the(unsquared) correlation. A related measure used by Moehring et al. (2014) is the mean square error ofa difference between simulated and estimated test lines effects (MSED), i.e. the root mean square error(RMSE). Finally, the similarity percentage might also be used, it being the percentage of test lines thatare in the top s% of both the true genetic effects and the EBLUPs (Peternelli et al., 2009).

Note that many of these measures require the true genetic effects and so, in conducting asimulation for a design, the same set of genetic effects must be used for every simulated data set.

The main goal of the study reported below is to evaluate the performance of an unreplicateddesign with systematic checks, as used by a Brazilian sugarcane company, in comparison to spatiallyoptimized unreplicated and p-rep designs with checks and spatially optimized p-rep designs. The optimaldesigns are obtained using a mixed model, with the values of its variance parameters set to values likethose found for the experiments analysed in Chapter 2. A simulation study was conducted to comparethe genetic gain from selection for each of the designs.

4.2 Motivating example

The motivation for this work came from a partnership between CTC and University of SãoPaulo (Brazil). CTC provided the data from a series of sugarcane breeding experiments to form the basisof a scientific study about the design and analysis of such experiments.

There were 21 trials available, where all had 79% or more of the area planted with new sugarcanelines, denominated test lines. These lines had only one plot in each site due to insufficient material forreplication and the large area that is required. The best seven percent of the test lines were chosen bycomparing them with four commercial varieties, which were called checks and were replicated such thatthey occupied at most 21% of the plots. One special check, nominated as the interspersed check (Check“T”), was planted in almost 11% of the plots and it was allocated systematically on diagonals throughoutthe trial. The other three checks (Check “A”, Check “B”, Check “C”) were equally replicated and a singlereplicate, made up a plot of each of these three checks, was placed in three neighbouring row plots; thereplicates are spread throughout to minimize, as much as possible, the distance of the test lines from acheck.

CTC believed that this distribution of check lines would better capture the environmental effectsin the statistical models, and that it facilitated visual selection using them, because the same columnalways had eight or fewer test lines between checks. However, no objective study to confirm this has beendone.

Typically, a CTC experiment would occupy a rectangular array of 20 rows (r) by 25 columns

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(c). In general, individual plots in these experiments are rectangular, being narrow in the row direction.There would be, say, 56 replicates of the interspersed check and 10 replicates of each of the other checks.This would leave space for 414 test lines. The Check “T” was allocated to diagonal plots in every eighthrow and eighth column; a replicate of the other three checks was placed in every second or third column.The test lines were randomized to plots not occupied by the checks. Figure 4.1 gives this design, whichis referred to as Design 0.

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C

Figure 4.1. Layout of the Design 0, where there are four commercial varieties, named “A”, “B”, “C”and “T” checks, and 414 test lines in the unnamed plots.

From the results presented in Chapter 2, it can be expected that, in such experiments, therewill be residual competition in the row direction and occasionally either autocorrelation or residualcompetition in the column direction. Global variation can also be expected in the row direction and lessfrequently in the column direction.

4.3 Material and methodology

In addition to Design 0, described in Section 4.2, five spatially optimized designs were identifiedfor investigation:

1. a spatially optimized, unreplicated design with four check lines;

2. a spatially optimized, unreplicated design with a single check line;

3. a spatially optimized, unreplicated design with three check lines;

4. a spatially optimized, p-rep design with with three check lines;

5. a spatially optimized, p-rep design.

All of the designs assumed a rectangular array of 20 rows (r) by 25 columns (c) to which 414 testlines (m) were assigned. The differences between the design were in the numbers of the checks and theduplicated test lines, as presented in Table 4.1. Given these numbers the error degrees of freedom willbe approximately equal for all designs.

57

Table 4.1. Numbers of check lines and duplicated test lines for five spatially optimized designs. Design1 has the same numbers of check lines as Design 0. All designs were on grid of 500 plots and had 414test lines.

Number of plots withCheck “A” Check “B” Check “C” Check “T” Duplicated test lines

Design 1 10 10 10 56 —Design 2 — — — 86 —Design 3 29 29 28 — —Design 4 10 10 10 — 56Design 5 — — — — 86

The layouts for the spatially optimized were obtained using the od package (Butler, 2014)and the PEV optimality criterion. The number of random exchanges in searching for a layout was set at10000. A completely randomized design was supplied as an initial design. In order to obtain a layout, amixed model with nominated values for its variance parameters must be supplied and this model is calledthe design model.

The design model for the data vector, Y∼

, is:

Y∼

= Xτ∼+ Zgug

∼+ Zouo

∼+ ε

∼; (4.1)

given that: X is the incidence matrix associated with vector τ∼

of fixed effects; Zg is the incidence matrixassociated with the vector ug

∼of random genetic effects of the test lines; Zo is the incidence matrix

associated with the vector uo∼

of column and row random effects; ε∼

is the vector of residual effects.


, uo∼

, ε∼


σ2

Gg(γg) 0 00 Go(γo) 00 0 R(ϕ)

=

Gg 0 00 Go 00 0 R

.

where, assuming Y∼

are in field order with rows within columns, Gg(γg) = γgIm, Go(γo) = γc(Ic ⊗Jr)+

γr(Jc ⊗ Ir), R =∑c⊗

∑r = AR(1) ⊗ AR(1), γ. = σ2

. /σ2, I∗ is an identity matrix of order ∗, J∗ is an

unitary matrix of order ∗, and AR(1) indicates a first-order autoregressive correlation matrix. It is notedthat the results presented in Chapter 2 indicate that a banded correlation structure is more likely in therow direction than a first-order autoregressive correlation structure. However, the latter is used becausethe specification of banded correlation structures is not implemented in od. For small values of ϕr, thedifferences between the two correlation structures are small.

The values used for the variance parameters for these models are roughly consistent with thevalues obtained in the analyses conducted in Chapter 2. For all models σ2 was set to 1, γg to 1.0, γc andγr to 0.3, and ϕc to 0.5.

For the design generation, two versions of each of Designs 1–5 were obtained, one in which ϕr

was set to 0.40 (positive residual row correlation) and the other in which ϕr was set to −0.25 (negativeresidual row correlation, representing competition at the residual level). Thus 10 layouts were obtainedin all and those for the spatially optimized designs are presented in Figure 4.2.

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(a)

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A

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C

(b)

1

2

3

4

5

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11

12

13

14

15

16

17

18

19

20

Row

Column

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

T

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(c)

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T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

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T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

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T

T

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T

T

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T

T

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T

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T

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T

T

T

T

T

T

T

T

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T

T

T

T

T

T

T

T

T

T

T

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T

T

T

T

T

T

T

T

T

T

T

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T

T

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T

T

T

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T

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T

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T

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T

T

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T

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T

T

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T

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T

T

T

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T

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T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

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T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

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T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

(d)

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Row

Column

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

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B

C

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C

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C

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C

C

C

C

C

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C

C

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C

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C

C

C

C

C

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C

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C

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C

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C

C

C

C

C

C

C

C

C

C

C

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C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

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C

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C

C

C

C

C

C

C

C

C

C

C

C

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C

C

C

C

C

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C

C

C

C

C

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C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

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C

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C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

(e)

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Row

Column

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B B

B

B

B

B

B

B

B

B

B

B

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B

B

B

B

B

B

B

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B

B B

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B B

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B B

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B B

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B

B

B

B

B

B

B

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(f)

Figure 4.2. Layouts from the optimal designs 1 (panels (a) and (b)), 2 (panels (c) and (d)), 3 (panels(e) and (f))). Panels (a), (c) and (e) used ϕr = 0.4 and the other panels used ϕr = −0.25. “A”, “B”, “C”and “T” are the checks and the numbers are the duplicated test lines as well the unnamed plots.

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(d)

Figure 4.3. Layouts from the optimal designs 4 (panels (a) and (b)) and 5 (panels (c) and (d)). Panels(a) and (c) used ϕr = 0.4 and the other panels used ϕr = −0.25. “A”, “B”, “C” and “T” are the checksand the numbers are the duplicated test lines as well the unnamed plots.

A simulation study was conducted to compare the designs in terms of the genetic gain fromselection achievable with them. It used a mixed model, called the simulation model. There were fourscenarios for each of Designs 1–5 for which simulated data sets were obtained:

++: design and simulation models with positive row correlation;

+−: design model with positive row correlation, but simulation model with negative row correlation;

−+: design model with negative row correlation, but simulation model with positive row correlation;

−−: design model and simulation model with negative row correlation.

There were two scenarios for Design 0 that differed in their row correlation values.The simulation model is the same as the design model in Equation 4.1, except that R is of

the form∑c⊗

∑r = AR(1)⊗Band(1), where Band(1) indicates a first-order banded correlation matrix.

The two values used for the row correlation in the first off-diagonal banded were ϕr = 0.3 or ϕr = −0.3.The values for the other parameters are the same as those assumed for the design model.

To construct simulated data sets, the simulation model is separated into the genetic effects, ug∼

,

and the non-genetic simulation model:

Y∼

∗ = Xτ∼+ Zouo

∼+ ε

∼;

60

Only one set of genetic effects is generated for use in all simulations. These become the true predictedgenetic effects, ug

∼. Then two simulations are performed, in each of which 1000 non-genetic data sets

are generated; one using negative and the other positive row correlation. For each design and scenariocombination, the ith simulated data set, yi

∼, is obtained by identifying (i) the Zg matrix for its layout

and (ii) the ith non-genetic data set, say y∗i

∼, for its simulation model; then yi

∼= Zgug

∼+ y∗

i∼

.

For every simulated data set, the simulation model is fitted and, if the estimation algorithmconverged, the following criteria are calculated from the results and stored: PEV; the relative geneticgain; the correlation between the true and estimated genetic effects; and the similarity percentage for thetop 7% of test lines. For this example, the top 7% of test lines is equivalent to the top 30 test lines, thenumber that CTC selects in its breeding experiments.

PEV is calculated as:

trace (Zgt(Go + R)−Zg + Gg

− − Zgt(Go + R)−X(Xt(Go + R)−X)−Xt(Go + R)−Zg

t)−1.

The other measures are obtained as described in Section 4.1.2.For each design and scenario combination there are 1000 simulated data sets and the results of

the converged algorithm are summarized as follows: the density of each criterion, except the similaritypercentage is produced; for the similarity percentage, boxplots are produced. In addition, since the aimis to obtain a spatially optimized design that improves on Design 0, the relative efficiency of the ithsimulated data set for a target design is calculated as the ratio of two PEVs. The numerator is the PEVfor the ith data set of Design 0 for the same simulation model as the target design; the denominator isthe PEV for the target design. A mean of the relative efficiencies for each design-scenario combinationis calculated. Relative efficiencies greater than one indicate that the spatially optimized design is moreefficient than Design 0.

4.4 Results

Table 4.2 gives the mean relative efficiencies of the systematically optimized designs whencompared to Design 0. Designs 1 and 0 are equally efficient, the only difference in the designs beingthat Design 1 is spatially optimized. Designs 2 and 3 are slightly less efficient than Design 0. They havefewer check lines (one and three, respectively), but the same number of plots with checks. In comparisondesigns 4 and 5, that involve p-rep designs, are more efficient than Design 0 by around 2% and 5%,respectively.

Note that the PEVs from designs assuming ϕr = 0.4 are smaller values than the same designwith ϕr = −0.25.

Table 4.2. Relative efficiencies of spatially optimized designs to unreplicated designs with systematicchecks. In parenthesis are the PEV values for Design 0.

Row residual correlationDesign ϕr = 0.4 ϕr = −0.25

0 1.0000 (0.4086) 1.0000 (0.4422)1 1.0079 1.00322 0.9813 0.98553 0.9791 0.98354 1.0197 1.02415 1.0477 1.0516

Table 4.3 presents the number of estimation algorithm converged for each design-scenario com-bination. While the convergence numbers differ, the differences are not large.

61

Table 4.3. Number of estimation algorithm converged for each design-scenario combination, given 1000data sets for each.

Design Scenario++ +− −+ −−

0 993 977 993 9771 987 973 987 9702 990 970 987 9813 990 977 992 9764 985 999 982 9745 990 986 992 982

Figure 4.4 presents the densities of the correlation between the true and estimated geneticeffects, the relative genetic gain and the PEV values for each scenario and design. Figure 4.5 givesboxplots of the similarity percentage. Overall, there is little difference between the densities and boxplotsof the four scenarios. Further, there is little difference between the Designs in relative genetic gain.However, Designs 4 and 5 are superior in terms of the correlation between true and predicted geneticeffects (larger values), PEV (smaller values) and similarity percentage (higher medians). For Designs 4and 5, in half of the data-set analyzed, the similarity percentage is between 50 to 80%.

0.65 0.70 0.75 0.80 0.85

05

15

Den

sity

scen

ario

++

0.65 0.70 0.75 0.80 0.85

05

15

Den

sity

scen

ario

+−

0.65 0.70 0.75 0.80 0.85

05

15

Den

sity

scen

ario

−+

0.65 0.70 0.75 0.80 0.85

05

15

correlation

Den

sity

scen

ario

−−

0.8 1.2 1.6 2.0

0.0

1.0

2.0

0.8 1.2 1.6 2.0

0.0

1.0

2.0

0.8 1.2 1.6 2.0

0.0

1.0

2.0

0.8 1.2 1.6 2.0

0.0

1.0

2.0

relative genetic gain

0.30 0.40 0.50 0.60

04

812

0.30 0.40 0.50 0.60

04

812

0.30 0.40 0.50 0.60

04

812

0.30 0.40 0.50 0.60

04

812

PEV

des.0des.1des.2des.3des.4des.5

Figure 4.4. Densities of the correlation between the true and estimated genetic effects (first column ofplots), relative genetic gain (second column) and prediction error variance - PEV (third) for each scenarioand design.

62

020

4060

80

sim

ilarit

y (%

)

des.0 des.1 des.2 des.3 des.4 des.5

(a)

020

4060

80


(b)

020

4060

80

sim

ilarit

y (%

)


(c)

020

4060

80


(d)

Figure 4.5. Similarity percentage between true and estimated top 7% of test lines. The horizontal grayline indicates 50%. The panels represent scenarios the different scenarios.

4.5 Conclusions

For the experimental situation considered here, an experimental design that included partialreplication of the test lines was superior to the others. It resulted in better correlation between the trueand predicted genetic effects, better relative genetic gain, smaller PEV values and higher median valuesof similarity percentages. This is in accordance with Clarke and Stefanova (2011).

However, if a researcher wants to be able to compare test lines with commercial checks toensure that the test lines are better than commercial varieties then a design with both checks and partialreplication of test lines is to be preferred to design with unreplicated test lines. Greater genetic gain willbe achieved. This agrees with the conclusion by Moehring et al. (2014) that an augmented p-rep designis better than an augmented design for individual environments.

Of interest is that, independent of the numbers of check lines, there is no advantage in usingone of the spatially optimized, unreplicated designs over the unreplicated design with systematic checks.It may be that the latter is preferred because it is practically or economically more efficient. This too isin accordance with the conclusion by Clarke and Stefanova (2011) that the distribution of the checksis not important. Further, it would appear that the number of check lines has only a slight effect onperformance of the unreplicated designs which agrees with the conclusions of Müller et al. (2010) foraugmented designs. The effect of the number of check varieties in the case of the p-rep design with checks(Design 4) could be investigated. However, Kempton (1984) and Müller et al. (2010) suggest that aminimum of two genotypes, of similar genetic background to the test material, be used to guard againstdifferential response of lines to soil fertility variations.

63

It is noted that no attempt was made in this study to find improved designs for the situationin which establishing that test lines are superior to check lines is considered to be more important thanselecting the best test lines. In this case, Atc-optimal designs would be important.

In this simulated study, the models to generated the optimal designs and the data sets weredifferent. However, the results from these combinations were not different given residual row correlationpositive or negative, which means that in this case does not matter if the model used to generate thedesign is the same to the data.

64

65

5 GENERAL CONCLUSIONS AND FUTURE WORKS

In this work, we have fitted models for 21 sugarcane trials, where some of them had similarlines but they were planted in different sites or harvest year at the same region. With this, we found thateach trial had its own peculiarity as far as the residual variance structure was concerned and so it wasimpossible to determine one specific model for the residual variation that applied to all the experiments.The strategy to select a mixed model was to include all the terms that the experience tells us are likelyto occur in the experiment. However, it was found that some data set did not have significant directgenetic effects (component close to zero and/or standard error larger than the estimate) and the bandedcorrelation structure models were more realistically for row direction than the autoregressive models. Forthe competition effect, it would appear that there was more general competition in Brazilian experimentsrather than it being genotype specific, because there was only one experiment with significant neighbourgenetic effect while ten others had significant residual competition. One response not outcome wasthe high genetic correlation value obtained from the joint analyses. This was twice or more than thecorrelation between genetic EBLUPs from individual analyses. Hence, using the joint analyses, betteraccuracy in the selection of the test lines was obtained. Higher predicted values were found with lowerstandard errors, so that the observation of genotypes at multiple sites provides replication of them. Thusa joint analysis was preferable to individual analyses whenever it is possible, whether it is based ondifferent sites or more than one harvest from the same trial.

The simulation study for single sites or cuts demonstrates clearly that a strategy of fitting amodel, even if it is the true model, was not a viable strategy and that detecting genetic competition inan experiment using the CTC type of design was fraught. Hence, it cannot be stated conclusively thatgenetic competition does not occur in Brazilian sugarcane experiments. Fortunately, it would appearthat the prediction of the direct genetic effects was little affected by the model selected and the presenceor not of residual competition, although it was affected by the magnitude of the genetic variance. While,in general, the simulated model (Model 4) gave the best correlation between true and predicted geneticeffects, selection gain and average similarity, the differences between it and the models without neighbourgenetic effects and/or residual correlation are small.

The results of the simulation study for separate versus joint analyses shows there were no prob-lems in fitting either single-site or joint models when there was no genetic competition and that betterpredicted values of the genetic effects come from joint analyses. There was little effect of heterogeneousvariances, but there were differences in the results that depended on the magnitude of the genetic vari-ance. For the models simulated, there was little difference between fitting autocorrelation and bandedcorrelation structures.

In the chapter 3, it was showed that the negative correlation can be due to the lodging effectif this results in material from plot being associated with a plot from a neighbouring column at theharvest. As the dry weather can also cause more competition between plots, the negative correlation canbe related to either one or both of these effects. Hence, when there is lodging care in separating materialfrom different plots will lead to improved results.

In the end, when designs were studied, it was presented, under specific conditions, that thepartial replication of the test lines was superior to the others. It resulted in better correlation betweenthe true and predicted genetic effects, better relative genetic gain, smaller PEV values and higher medianvalues of similarity percentages. However, the design does not allow to compare test lines with commercialchecks to ensure that the test lines are better than commercial varieties. Then a design with both checksand partial replication of test lines is to be preferred to design with unreplicated test lines. Otherimportant result was that, independent of the numbers of check lines, there is no advantage in using oneof the spatially optimized unreplicated designs over the unreplicated design with systematic checks. It

66

may be that the latter is preferred because it is practically or economically more efficient. Further, itwould appear that the number of check lines has only a slight effect on performance of the unreplicateddesigns.

Future works could be the development of new experiments using different designs and space inthe plots. It is conjectured that double furrow used in the Brazilian sugarcane experiments plots decreasesthe competition effect, as this indicates an extra space in each plot. Then, we suggest new experimentswith the same lines but different design and single furrow to see if there is significant neighbour geneticeffects and the direct genetic variance increase to achieve better selection results. Given that the truebest simulated test lines are selected with average around 45 to 55%, even for the individual and jointanalyses or optimal design. Another study could be with the test power to identify the significance ofthe variance and covariance components.

In the software ASReml-R the structure Band()×Band() is not available, and the responsevariance matrix cannot be written with spline effects to obtain the plot of the empirical variogram usingasremlPlus. Then new functions could be developed.

67

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70

APPENDIX

71

A CORRELATION MODELS

Assuming the general matrix of dimension n× n as:

ρ0 ρ1 ρ2 ρn−1

ρ1 ρ0 ρ1 ρn−2

ρ2 ρ1 ρ0 ρn−3

. . .ρn−1 ρn−2 ρn−3 ρ0

Given i = 0, 1, 2, ..., n− 1. The structures are:

Identity (id):

1 0 0 0

0 1 0 0

0 0 1 0

. . .0 0 0 1

;

ρ0 = 1 and ρi,i>0 = 0

First order autoregressive (AR(1)):

1 ϕ1 ϕ21 ϕn−11

ϕ1 1 ϕ1 ϕn−21

ϕ21 ϕ1 1 ϕn−31

. . .ϕn−11 ϕn−2

1 ϕn−31 1

;

ρi = ϕi1; |ϕ1| < 1

Second order autoregressive (AR(2)):

1 ϕ1

(1−ϕ2)ϕ21

(1−ϕ2)+ ϕ2 ρn−1

ϕ1

(1−ϕ2)1 ϕ1

(1−ϕ2)ρn−2

ϕ21

(1−ϕ2)+ ϕ2

ϕ1

(1−ϕ2)1 ρn−3

. . .ρn−1 ρn−2 ρn−3 1

;

ρ0 = 1, ρ1 = ϕ1

(1−ϕ2)and ρi,i>1 = ϕ1ρi−1 + ϕ2ρi−2, |ϕ1 ± ϕ2| < 1, |ϕ1| < 1, |ϕ2| < 1

72

Third order autoregressive (AR(3)):

ρ0 = 1

ρ1 = ϕ1+ϕ2ϕ3

ω

ρ2 = ϕ1(ϕ1+ϕ3)+ϕ2(1−ϕ2

ω

ρi,i>2 = ϕ1ρi−1 + ϕ2ρi−2 + ϕ3ρi−3

ω = 1− ϕ2 − ϕ3(ϕ1 + ϕ3)

|ϕ1| < (1− ϕ2), |ϕ2| < 1, |ϕ3| < 1

Constrained autoregressive 3 (sar(2)):

as for AR(3) where:

ϕ1 = γ1 + 2γ2

ϕ2 = −γ2(2γ1 + γ2)

ϕ3 = γ1γ22

first-order banded (Band(1)):

1 ρ1 0 0

ρ1 1 ρ1 0

0 ρ1 1 0

. . .0 0 0 1

;

ρ0 = 1, ρi,i>1 = 0.

Second-order banded (Band(2)):

1 ρ1 ρ2 0

ρ1 1 ρ1 0

ρ2 ρ1 1 0

. . .0 0 0 1

;

ρ0 = 1, ρi,i>2 = 0.

Unstructured general covariance (us(n)):

ϕ11 ϕ12 ϕ13 ϕ1n

ϕ12 ϕ22 ϕ23 ϕ2n

ϕ13 ϕ23 ϕ33 ϕ3n. . .

ϕ1n ϕ2n ϕ3n ϕnn

;

73

Factor analytic order k (fa(k)):

γ1

γ2...γk

[γ1 γ2 · · · γk

]+

ψ1 0 0

0 ψ2 0

. . .0 0 ψk

= ΓΓ′ +Ψ;

Γ contains covariance factors and Ψ contains specific variance.

Reduced rank (RR(2)):

σ

[1 ρ

ρ ρ2

];

74

75

B PARANÁ - LONGITUDINAL DATA

This experiment assessed 406 test lines with 4 commercial varieties nominated “PAD1”, “PAD2”,“PAD3” and “Inter”. It was planted in 2013 in a rectangular array of 26 columns by 19 rows and theyield in TCH was obtained in 2014 as first cut and 2015 in a second cut. The results were organized insubsections where B.1 presents the descriptive analysis of the cuts, B.2 shows the individual analysis foreach one and B.3 the joint analysis. Finally, we make concluding remarks in subsection B.4.

B.1 Descriptive analysis

A summary about TCH in the first and second cut is presented in Table B.1. The average ofTCH decrease in the first to second cut and this may be due to lower environmental effect. The variancesin the sugarcane check plots is smaller for the second cut when compared to the first and the largestvariance (1588.89) was found in “PAD2” variety. For the plots with test lines there are lower average ofyield and the variance increases when comparing the cuts. Looking at Figure B.1, the boxplots of TCHof the groups of clones in each cut reveal some possible outliers. Note that one of the “PAD2” varietyreplicates presented one plot with a super yield for the first cut.

Table B.1. Descriptive analysis of the groups of sugarcane carried out in Paraná in the first and secondcut.

Groups of clones Minimum Maximum Mean Variance

First cut

PAD1 59.50 131.00 96.57 547.98PAD2 59.50 192.00 89.29 1588.89PAD3 68.50 110.10 85.50 261.89Interspersed 53.60 138.40 92.57 330.91Test lines (Reg) 20.80 142.90 74.88 414.69

Second cut

PAD1 55.10 104.20 82.34 279.19PAD2 40.20 89.30 68.29 250.90PAD3 56.50 83.30 69.43 89.03Interspersed 20.80 105.70 80.96 237.13Test lines (Reg) 8.90 141.40 65.80 448.62

TC

H

50

100

150

200

INTER PAD1 PAD2 PAD3 Reg

(a)

TC

H

50

100

150

200

INTER PAD1 PAD2 PAD3 Reg

(b)

Figure B.1. Boxplot of TCH for each group of lines for Paraná cuts. Panel (a) refers to first cut and(b) second.

Figures B.2 and B.3 display the yield for each plot in the first and second cut, respectively.The names “PAD1”, “PAD2”, “PAD3” and “INTER” are the checks, the alpha-numerical labels (PRplus number) are the test lines, “x” are empty plots. There is at 3 last column yielded less than the

76

others because an interference effect extern. Then, a co-variate for the difference between these and othercolumns was included (namely in the analysis, Cext). Also, it is possible to observe a high yield in the“PAD2” plot (first column and second row). This same plot presented a regular yield in the second cut.When discussing this with the company, they informed that this measure for first cut was incorrect, andhence was removed from the analysis.

Column

Row

INTER

PAD2

PAD3

PAD1

PR38

PR199

PR56

PR143

PR31

INTER

PR197

PR142

PR215

PR217

PR164

PR218

PR144

PR198

INTER

PR222

INTER

PR172

PR247

PR220

PR203

PR224

PR204

PR202

PR171

INTER

PR196

PR179

PR200

PR178

PR166

PR29

PR195

PR34

PR234

PR233

INTER

PR201

PR141

PR225

PR165

PR223

PR221

PR177

PR15

INTER

PR16

PR174

PR2

PR168

PR32

PR310

PR14

PR173

PR170

PR176

INTER

PR169

PR9

PR323

PR342

PR270

PR343

PR281

PR30

INTER

PR24

PR334

PR260

PAD3

PAD1

PAD2

PR313

PR39

PR302

PR291

INTER

PR156

PR72

PR55

PR50

PR238

PR61

PR186

PR145

INTER

PR25

PR167

PR35

PR1

PR175

PR373

PR312

PR388

PR378

PR357

INTER

PR368

PR311

PR381

PR308

PR367

PR344

PR356

PR375

INTER

PR123

PR91

PR28

PR318

PR341

PR374

PAD1

PAD2

PAD3

PR112

INTER

PR354

PR102

PR325

PR326

PR249

PR309

PR22

PR26

INTER

PR33

PR307

PR206

PR37

PR226

PR397

PR17

PR350

PR124

PR352

INTER

PR353

PR347

PR387

PR5

PR348

PR376

PR27

PR349

INTER

PR351

PR44

PR365

PR333

PR355

PR43

PR114

PR42

PR54

PR335

INTER

PR58

PR49

PR86

PR48

PR336

PR45

PR46

PR52

INTER

PR62

INTER

PR47

PR41

PR59

PR337

PR51

PR60

PR70

PR65

INTER

PR74

PR66

PR68

PR63

PAD2

PAD3

PAD1

PR81

INTER

PR99

INTER

PR82

PR40

PR95

PR80

PR6

PR78

PR96

PR93

INTER

PR53

PR73

PR97

PR76

PR64

PR88

PR113

PR94

PR75

PR104

INTER

PR101

PR67

PR106

PR87

PR105

PR98

PR111

PR100

INTER

PR79

PR126

PR132

PR92

PR125

PR85

PR117

PR71

PAD3

PAD1

INTER

PAD2

PR129

PR116

PR83

PR380

PR405

PR122

PR89

INTER

PR135

PR396

PR118

PR383

PR77

PR395

PR379

PR384

PR110

PR382

INTER

PR7

PR136

PR403

PR398

PR400

PR369

PR21

PR103

INTER

PR370

PR401

PR90

PR372

PR402

PR119

PR371

PR386

PR115

PR399

INTER

PR314

PR389

PR385

PR108

PR10

PR8

PR390

PR216

INTER

PR227

PR109

PR84

PR331

PR130

PR134

PR187

PR107

PR139

PR140

INTER

PR137

PR332

PR128

PR133

PR394

PR392

PR391

PR359

INTER

PAD1

PAD2

PAD3

x

PR360

PR393

PR363

PR377

PR362

PR366

INTER

PR404

PR345

PR18

PR364

PR361

PR131

PR358

PR121

INTER

PR153

PR316

PAD2

PR158

PR292

PR305

PR147

PR296

PR148

PR306

INTER

PR304

PR13

PR317

PR301

PR303

PR283

PR279

PR300

INTER

PR288

INTER

PAD3

PAD1

PR284

PR298

PR297

PR290

PR271

PR294

INTER

PR286

PR277

PR293

PR280

PR273

PR278

PR268

PR276

INTER

PR239

INTER

PR315

PR275

PR243

PR263

PR287

PR258

PR241

PR267

INTER

PR272

PR261

PR245

PR264

PR248

PR259

PR244

PR274

PR269

PR246

INTER

PR252

PR329

PR185

PR12

PR265

PR295

PR253

PR262

INTER

PR256

PR322

PR266

PR327

PR254

PR338

PAD3

PR240

PR321

PR324

INTER

PR255

PR251

PR330

PR242

PR320

PR328

PR319

PR138

INTER

PR236

PR299

PR230

PR229

PAD2

PAD1

PR211

PR214

PR235

PR339

INTER

PR213

PR237

PR212

PR232

PR207

PR340

PR231

PR209

INTER

PR191

PR188

PR228

PR205

PR181

PR192

PR190

PR194

PR189

PR210

INTER

PR182

PR157

PR163

PR208

PR155

PR193

PR180

PR184

INTER

PR11

PR160

PR154

PR162

PR183

PAD2

PAD3

PAD1

PR19

PR161

INTER

PR159

PR150

PR151

PR152

PR149

PR219

PR36

PR4

INTER

PR57

PR69

PR285

x

x

x

x

x

x

PR250

INTER

PR257

PR289

PR282

PR23

PR346

PR120

PR127

PR3

INTER

PR146

PR2020

40

60

80

100

120

140

160

180

200

Figure B.2. Heat map for TCH in the first cut of the Paraná.

Column

Row

INTER

PAD2

PAD3

PAD1

PR38

PR199

PR56

PR143

PR31

INTER

PR197

PR142

PR215

PR217

PR164

PR218

PR144

PR198

INTER

PR222

INTER

PR172

PR247

PR220

PR203

PR224

PR204

PR202

PR171

INTER

PR196

PR179

PR200

PR178

PR166

PR29

PR195

PR34

PR234

PR233

INTER

PR201

PR141

PR225

PR165

PR223

PR221

PR177

PR15

INTER

PR16

PR174

PR2

PR168

PR32

PR310

PR14

PR173

PR170

PR176

INTER

PR169

PR9

PR323

PR342

PR270

PR343

PR281

PR30

INTER

PR24

PR334

PR260

PAD3

PAD1

PAD2

PR313

PR39

PR302

PR291

INTER

PR156

PR72

PR55

PR50

PR238

PR61

PR186

PR145

INTER

PR25

PR167

PR35

PR1

PR175

PR373

PR312

PR388

PR378

PR357

INTER

PR368

PR311

PR381

PR308

PR367

PR344

PR356

PR375

INTER

PR123

PR91

PR28

PR318

PR341

PR374

PAD1

PAD2

PAD3

PR112

INTER

PR354

PR102

PR325

PR326

PR249

PR309

PR22

PR26

INTER

PR33

PR307

PR206

PR37

PR226

PR397

PR17

PR350

PR124

PR352

INTER

PR353

PR347

PR387

PR5

PR348

PR376

PR27

PR349

INTER

PR351

PR44

PR365

PR333

PR355

PR43

PR114

PR42

PR54

PR335

INTER

PR58

PR49

PR86

PR48

PR336

PR45

PR46

PR52

INTER

PR62

INTER

PR47

PR41

PR59

PR337

PR51

PR60

PR70

PR65

INTER

PR74

PR66

PR68

PR63

PAD2

PAD3

PAD1

PR81

INTER

PR99

INTER

PR82

PR40

PR95

PR80

PR6

PR78

PR96

PR93

INTER

PR53

PR73

PR97

PR76

PR64

PR88

PR113

PR94

PR75

PR104

INTER

PR101

PR67

PR106

PR87

PR105

PR98

PR111

PR100

INTER

PR79

PR126

PR132

PR92

PR125

PR85

PR117

PR71

PAD3

PAD1

INTER

PAD2

PR129

PR116

PR83

PR380

PR405

PR122

PR89

INTER

PR135

PR396

PR118

PR383

PR77

PR395

PR379

PR384

PR110

PR382

INTER

PR7

PR136

PR403

PR398

PR400

PR369

PR21

PR103

INTER

PR370

PR401

PR90

PR372

PR402

PR119

PR371

PR386

PR115

PR399

INTER

PR314

PR389

PR385

PR108

PR10

PR8

PR390

PR216

INTER

PR227

PR109

PR84

PR331

PR130

PR134

PR187

PR107

PR139

PR140

INTER

PR137

PR332

PR128

PR133

PR394

PR392

PR391

PR359

INTER

PAD1

PAD2

PAD3

x

PR360

PR393

PR363

PR377

PR362

PR366

INTER

PR404

PR345

PR18

PR364

PR361

PR131

PR358

PR121

INTER

PR153

PR316

PAD2

PR158

PR292

PR305

PR147

PR296

PR148

PR306

INTER

PR304

PR13

PR317

PR301

PR303

PR283

PR279

PR300

INTER

PR288

INTER

PAD3

PAD1

PR284

PR298

PR297

PR290

PR271

PR294

INTER

PR286

PR277

PR293

PR280

PR273

PR278

PR268

PR276

INTER

PR239

INTER

PR315

PR275

PR243

PR263

PR287

PR258

PR241

PR267

INTER

PR272

PR261

PR245

PR264

PR248

PR259

PR244

PR274

PR269

PR246

INTER

PR252

PR329

PR185

PR12

PR265

PR295

PR253

PR262

INTER

PR256

PR322

PR266

PR327

PR254

PR338

PAD3

PR240

PR321

PR324

INTER

PR255

PR251

PR330

PR242

PR320

PR328

PR319

PR138

INTER

PR236

PR299

PR230

PR229

PAD2

PAD1

PR211

PR214

PR235

PR339

INTER

PR213

PR237

PR212

PR232

PR207

PR340

PR231

PR209

INTER

PR191

PR188

PR228

PR205

PR181

PR192

PR190

PR194

PR189

PR210

INTER

PR182

PR157

PR163

PR208

PR155

PR193

PR180

PR184

INTER

PR11

PR160

PR154

PR162

PR183

PAD2

PAD3

PAD1

PR19

PR161

INTER

PR159

PR150

PR151

PR152

PR149

PR219

PR36

PR4

INTER

PR57

PR69

PR285

x

x

x

x

x

x

PR250

INTER

PR257

PR289

PR282

PR23

PR346

PR120

PR127

PR3

INTER

PR146

PR200

20

40

60

80

100

120

140

Figure B.3. Heat map for TCH in the second cut of the Paraná.

B.2 Individual analysis

For the first cut, it was observed one wrong observation of the standard check (with TCH was192) which was omitted. Then, the experiment was analyzed with 486 and 487 plots in the first andsecond cut, respectively.

77

A summary of the sequence of models fitted to Paraná first cut data is presented in Table B.2.Our analysis began with the model denominated random effects Treatment Interference Model (R-TIM),used by Stringer et al. (2011). This model includes random column and row effects, and unconstraineddirect and neighbour genetic effects with an AR(1) correlation structure in row and column direction.However, the genetic effects can be written as a special case, denominated reduced rank. From the REMLratio test, we conclude that the neighbour genetic effect was not significant. Then, the local and globaleffects were assessed, respectively. Looking at the empirical variograms of the residuals, Figure B.4, weobserved that for Model 4, the semivariances of the residuals in both the column and row direction werewithin the expected values, although, for some models in the row direction, some semivariances wereclose to the envelope limits. Finally, the selected model was Model 18. This model contains random rowand direct genetic effects, fixed checks and co-variate effects and strange local correlation in row directionmodelled with a fourth-order banded structure. The REML estimates of variance parameters from theselected model are shown in Table B.3.

78Ta

ble

B.2

.Su

mm

ary

ofth

efit

ted

mod

elst

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raná

first

cutd

ata

with

rest

ricte

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glik

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ood

(Log

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incl

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vest

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ne);

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(col

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spl(.

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indi

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79

0.0

0.5

1.0

1.5

2.0

0 5 10 15 20 25Col differences

Variogram face of Standardized conditional residuals for Col

(a)

0.0

0.5

1.0

1.5

2.0



(b)

0.0

0.5

1.0

1.5

2.0



(c)

0.0

0.5

1.0

1.5

0 5 10 15Row differences

Variogram face of Standardized conditional residuals for Row

(d)

0.0

0.5

1.0

1.5



(e)

0.0

0.5

1.0

1.5



(f)

Figure B.4. Plots of the row and column faces of the empirical variograms for the residuals referringto Models 4 (panels (a) and (d)); 6 (panels (b) and (e)) and 18 (panels (c) and (f)). Panels (a), (b) and(c) refer to the column direction and panels (d), (e) and (f) refer to the row direction.

Observe in Table B.3 that the direct genetic component is approximately half of the residualcomponent and even through the random row component is very small when compared with the residualcomponent (ratio of 0.09), it is significant. The residual correlation in row direction between the second,third and fourth neighbours are similar, approximately 0.18. These were not expected and look like theyreflect a machinery or procedural effect, but we do not have information from the company about this.There is a competition effect at the residual level because the correlation between the first neighbours inrow direction is close to zero (0.02) and the second is approximately 8 times more.

Table B.3. REML variance parameters estimates from fitted Model 18 to the experiment of Paraná,first cut.

Variance parameters Ratios (γ) Estimates Std.errorsRow (σ̃2

r) 0.09 20.50 11.23Direct genetic (σ̃2

g) 0.52 123.16 43.67Residual (σ̃2) 1.00 238.87 37.28

Spatial (ρ̃r1) — 0.02 0.08Spatial (ρ̃r2) — 0.17 0.08Spatial (ρ̃r3) — 0.18 0.07Spatial (ρ̃r4) — 0.19 0.07

Table B.4 shows the Wald F tests for fixed terms from the selected model (Model 18). Theseshowed no evidence to affirm that there was a significant difference between the group of test lines andchecks (Control) or among the checks (Control:Check). The estimated fixed effects from this model arein Table B.5.

80

Table B.4. Wald F tests for fixed terms for Model 18. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom.

Fixed terms Df denDF F test p-value(Intercept) 1.00 473.00 2315.00 0.00Control 1.00 473.00 2.02 0.16Cext 1.00 473.00 36.33 0.00Control:Check 3.00 473.00 1.56 0.20

Table B.5. Estimates of the fixed effects for Model 18 and their respective standard errors.

Effects Estimates Standard errorINTERC 0.00 —PAD1 4.76 5.41PAD2 -10.08 5.60PAD3 -1.82 5.28co-variate (Cext)-1 level 0.00 —co-variate (Cext)-2 level -20.65 3.45overall Checks 0.00 —overall Test lines -16.73 11.30overall mean (µ) 93.60 11.34

The 30 best test lines from Model 18 and their respective predicted values and standard errorsare in Table B.6.

Table B.6. The 30 best test lines with predicted values (pred. value) and standard errors (stand. error)from Model 18.

Test line Pred. value Stand. error Test line Pred. value Stand. errorPR171 91.08 9.07 PR377 78.18 9.08PR279 84.20 9.10 PR327 78.07 9.07PR295 83.78 9.04 PR90 78.00 9.13PR217 83.60 9.05 PR344 77.55 9.04PR14 82.77 9.18 PR5 77.34 9.05PR164 80.32 9.06 PR91 77.31 9.14PR304 80.20 9.07 PR82 77.29 9.15PR331 79.83 9.16 PR361 77.15 9.05PR383 79.63 9.13 PR190 77.11 9.06PR247 79.25 9.09 PR113 77.06 9.18PR157 79.21 8.98 PR7 77.06 9.08PR96 79.16 9.06 PR143 77.05 9.05PR71 78.99 9.13 PR118 76.92 9.09PR246 78.65 9.18 PR133 76.91 9.05PR250 78.27 9.08 PR72 76.65 9.06

The correlation of EBLUPs when comparing the selected model (Model 18) and the traditionalmodel with only global effects (row, column and direct genetic) is high, approximately 0.967. Although,the group of test lines selected by each model is different (see Figure B.5).

81

−10 0 10 20

−10

010

20

EBLUPs − traditional

EB

LUP

s −

sel

ecte

d m

odel cor = 0.967

Figure B.5. Dispersion plot of the genetic EBLUPs values of the Model 18 and the traditional non-spatial analysis model for this design (without spatial dependence) for 1st cut of Paraná. The cut-offsfor the 30 best test lines (7 % upper) in each Model are indicated by the dotted line.

For the second cut, the same procedure in first cut was done. However, the genetic structureeffect can not be written as reduced rank. The neighbour genetic effect is close to zero and using theREML test it was verified that it is not significant. Figure B.6 presents the empirical variograms. Wheninvestigating alternative models, harvest and splines effects were significant. In the end, the best modelfound was Model 14 with: fixed harvest, check and linear trend effects in row and column; random directgenetic and splines effects in row and column; and third-order banded structure in row direction. Theempirical semi-variogram for this model shows an improvement over that for Model 3 in Figure B.6 andindicates that the model is adequate.

82Ta

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26

83

0.0

0.5

1.0

1.5

2.0



(a)

0.0

0.5

1.0

1.5



(b)

05

1015

2025

0

5

10

15

0.0

0.5

1.0

1.5

2.0

2.5

Col (lag)

Row (lag)

(c)

Figure B.6. Panels (a) and (b) refer respectively to the plots of the column and row faces of theempirical variograms for the residuals referring to Model 3. Panel (c) is the semi-variogram of the Model14.

In Table B.8 can be seen the REML estimates of variance parameters of the Model 14. Thedirect genetic component for the second cut is almost twice those for the first cut. With Model 14, thedirect genetic component is 1.31 times the residual component, this means better test line selection forthe second cut than first. Note that the correlations in row direction do not represent autoregressiveprocess. The correlation for the second band is close to zero.

Table B.8. REML estimates of variance parameters from fitted Model 14 for the experiment of Paraná,second cut.

Variance parameters Ratios (γ) Estimates Std. errorsspl(Column) 0.67 125.77 102.91spl(Row) 0.15 28.21 25.69Direct genetic (σ̃2

g) 1.31 245.05 42.42Residual (σ̃2) 1.00 187.27 30.51

Spatial (ρ̃r1) — -0.19 0.09Spatial (ρ̃r2) — 0.03 0.09Spatial (ρ̃r3) — 0.19 0.09

84

Table B.9 shows the Wald F tests for fixed terms. Also, there was no evidence of significa-tive difference between the group of test lines and checks (Control), but within checks this is not true(Control:Check). The estimated fixed terms from this model are in Table B.10.

Table B.9. Wald F tests for fixed terms for Model 14. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom.

Fixed terms Df denDF F test p-value(Intercept) 1.00 471.00 4013.00 0.00Control 1.00 471.00 0.70 0.40Cext 1.00 471.00 8.84 0.00Harvest 1.00 471.00 6.82 0.01lin(Col) 1.00 471.00 2.66 0.10lin(Row) 1.00 471.00 3.47 0.06Control:Check 3.00 471.00 2.84 0.04

Table B.10. Estimates of the fixed effects for Model 14 and their respective standard errors.

Effects Estimates Standard errorsINTERC 0.00 —PAD1 1.67 4.87PAD2 -9.76 4.84PAD3 -10.49 4.81lin(Row) 0.26 0.15lin(Col) -0.37 0.21Harvest - 1 level 0.00 —Harvest - 2 level -4.52 1.78co-variate (Cext)-1 level 0.00 —co-variate (Cext)-2 level -1.20 7.19overall Checks 0.00 —overall Test lines -15.32 15.79overall mean (µ) 85.29 15.98

In Table B.11 are the 30 best test lines from Model 14 and their respective predicted valuesand standard errors.

Table B.11. The 30 best test lines with predicted values (pred. value) and respective standard errors(stand. error) for Model 14.

Test line Pred. value Stand. error Test line Pred. value Stand. errorPR96 108.59 10.97 PR204 90.13 11.07PR56 103.58 11.09 PR150 90.03 10.95PR358 98.64 11.17 PR1 89.77 11.30PR331 95.03 11.20 PR400 89.75 10.80PR52 94.65 11.25 PR207 89.50 11.53PR279 94.40 11.19 PR6 89.41 11.10PR232 94.15 11.56 PR318 89.21 11.36PR90 93.44 10.98 PR233 89.10 11.30PR386 93.26 11.06 PR122 88.99 10.83PR98 92.88 10.84 PR342 88.77 11.12PR133 92.77 10.97 PR170 88.46 11.31PR9 92.53 11.21 PR14 88.26 11.36PR327 91.92 11.19 PR143 88.20 11.20PR171 90.85 10.98 PR189 87.71 10.85PR309 90.38 11.10 PR109 87.40 11.10

The correlation between the EBLUPs from selected model and traditional also is high, 0.975.However the group of test lines selected is different in each model. See this relation in Figure B.7.

85

−20 0 20 40

−40

−20

020

40


EB

LUP

s −

sel

ecte

d m

odel cor = 0.975

Figure B.7. Dispersion plot of the genetic EBLUPs values of the Model 14 and the traditional non-spatial analysis model for this design (without spatial dependence) for 2nd cut of Paraná. The cut-offsfor the 30 best test lines (7 % upper) in each Model are indicated by the dotted line.

B.3 Joint analysis

Comparing the predicted values from selected model, for first and second cut, the correlation is0.542. Then, in the joint analysis is expected a significant correlation between the direct genetic effects.Figure B.8 displays the predicted values from the selected models. Note that there are less than half testlines selected in common between the cuts.

50 60 70 80 90

4060

8010

0

predicted value − 1º cut

pred

icte

d va

lue

− 2

º cu

t

cor = 0.542

Figure B.8. Dispersion plot of the predicted values from selected model in each cut of Paraná. Thecut-offs for the 30 best test lines (7% upper) in each Model are indicated by the dotted line.

The joint analysis started with the same effects as the individual analysis to which was added thefixed cut effect. Then, as summarized in Table B.12, models that assumed different covariance matricesfor the genetic effects for the different cuts were compared. However, the models using unstructuredgeneral covariance matrix (us) and covariance matrices with heterogeneous variances (corh) had someconvergence problems, because they estimate the correlation between genetic effects to be one. Then, thefactor analytic model (fa(2)) was used to get a convergenced model and to verify the estimated value forthe correlation. See in Table B.12 the models tested with REML log, the p-value of the REML ratio testand the AIC.

86

Table B.12. Summary of the fitted models to joint analysis of the Paraná experiments in the first (1)and second (2) cut, with restricted log likelihood (Log-lik.); p-value of the REML ratio test and AIC. Allthe models have the same fixed; local and global effects modeled in the individual analysis plus the cutfixed effect.

Model Genetic effects Constraint Log-lik. Test p-value AIC

1[σ2g1 00 σ2

g2

]-3333.46 6694.92

2[σ2g 00 σ2

g

]σ2g = σ2

g1 = σ2g2 -3335.41 M1 vs M2 0.02 6696.82

3[

σ2g1 ρgσg1σg2

ρgσg1σg2 σ2g2

]-3264.52 6559.04

4[

σ2g1 σg1σg2

σg1σg2 σ2g2

]ρg=1 -3464.51 M2 vs M3 1.00 6557.03

Using the AIC, Model 4 is chosen as the best to explain the effects in these experiments. Thisindicates that the correlation between the first and second cut is 1, then the test lines effects are thesame, indicating that the best genotypes are the same for both cuts. Table B.13 displays the REMLestimates of the variance parameters and their respective standard errors from fitted Model 4. As, thefactor analytic model has been assumed the estimated genetic covariance matrix is:

[fa1fa2

].[

fa1 fa2]+

[var1 0

0 var2

]=

[154.15 204.45

204.45 271.17

].

Table B.13. REML estimates of variance parameters from fitted Model 4 to the joint analysis of thefirst-cut (1) and second-cut (2) of the experiment carried out in the Paraná.

Variance parameters Estimate Standard errorsCut-1:Row (σ̃2

r1) 20.58 10.34Cut-2:spl(Column) 38.05 42.59Cut-2:spl(Row) 25.31 22.79fa(Cut-1):Genetic var1 0.00 —fa(Cut-2):Genetic var2 0.00 —fa(Cut-1):Genetic fa1 12.42 1.06fa(Cut-2):Genetic fa2 16.47 1.12Cut-1:variance (σ̃2

1) 204.42 20.34Spatial:row direction (ρ̃r11) 0.05 0.07Spatial:row direction (ρ̃r12) 0.14 0.06Spatial:row direction (ρ̃r13) 0.16 0.06Spatial:row direction (ρ̃r14) 0.12 0.06

Cut-2:variance (σ̃22) 172.10 24.20

Spatial:row direction (ρ̃r21) -0.16 0.08Spatial:row direction (ρ̃r22) 0.02 0.08Spatial:row direction (ρ̃r23) 0.22 0.08

Table B.14 contains the Wald-F test for fixed terms of the Model 4. Note there are evidences ofsignificant difference between the cuts, checks and test lines (Control) and within checks (Control:Check).The predicted values and standard errors of the 30 best test lines from Model 4 for each cut are in TableB.15. It can be observed that in the joint analysis the precision in the selection of the test lines isbetter, given that the predicted values are highest and standard errors are smaller, when compared withindividual analysis (Tables B.6 and B.11). However, the correlation between the genetic predicted valuesfrom individual and joint analysis is high, 0.787 and 0.934 for first and second cut,respectively. Thisrelationship can be observed in Figure B.9.

87

Table B.14. Wald F tests for fixed terms from Model 4. Df means degrees of freedom and DenDF isthe approximate denominator degrees of freedom.

Fixed terms Df denDF F test p-value(Intercept) 1.00 944.00 4766.00 0.00Cut 1.00 944.00 46.71 0.00Control 1.00 944.00 10.27 0.00Cext 1.00 944.00 42.65 0.00Control:Check 3.00 944.00 5.66 0.00Cut:Control 1.00 944.00 0.34 0.56Cut:Cext 1.00 944.00 3.48 0.06Cut-2:harvest 1.00 944.00 5.04 0.03Cut-2:lin(Col) 1.00 944.00 0.37 0.54Cut-2:lin(Row) 1.00 944.00 2.74 0.10Cut:Control:Check 3.00 944.00 0.62 0.60

Table B.15. Predicted values and standard errors of the 30 best test lines of the Model 4 for each cut.First cut Second cut

Test line Predicted value Standard error Test line Predicted value Standard errorPR96 100.55 7.01 PR96 109.82 9.57PR171 96.25 7.02 PR171 104.11 9.62PR279 92.73 7.02 PR279 99.44 9.77PR331 91.65 7.11 PR331 98.02 9.79PR295 91.15 6.96 PR295 97.36 9.64PR14 90.06 7.12 PR14 95.91 9.94PR90 89.78 6.99 PR90 95.53 9.62PR56 89.63 6.91 PR56 95.34 9.66PR358 89.18 7.01 PR358 94.74 9.74PR327 88.76 6.97 PR327 94.18 9.82PR217 88.67 7.06 PR217 94.06 9.87PR133 88.03 7.01 PR133 93.21 9.54PR204 87.77 6.97 PR204 92.87 9.67PR232 87.49 7.07 PR232 92.49 10.05PR143 87.27 7.05 PR143 92.20 9.77PR386 86.68 6.99 PR386 91.42 9.66PR52 86.45 7.04 PR52 91.12 9.84PR98 86.23 6.95 PR98 90.82 9.47PR6 85.30 7.02 PR6 89.60 9.70PR150 85.22 6.94 PR150 89.48 9.55PR207 84.47 7.06 PR207 88.50 10.01PR197 84.28 7.07 PR197 88.25 9.71PR298 83.94 6.94 PR298 87.79 9.70PR9 83.86 7.01 PR9 87.68 9.81PR71 83.68 7.04 PR71 87.44 9.72PR109 83.53 7.04 PR109 87.24 9.71PR208 83.49 6.97 PR208 87.19 9.33PR309 83.43 6.95 PR309 87.11 9.68PR304 83.20 7.01 PR304 86.81 9.54PR113 83.03 7.07 PR113 86.58 9.80

88

Figure B.9. Predicted values from individual (indiv) and joint analysis of Paraná cuts. The left panelis for the first cut and the second is to the right.

B.4 Conclusions

For the first cut a model with fourth-order banded correlation structure in row direction waschosen. It is unusual for there to be as many as four bands and unexpected that because the correlationsin the second to fourth bands are the similar magnitude, at around 0.18. This looks like an external effectof some machine or procedure. However the CTC company had no idea about what it could be. There issome evidence of competition at the residual level, because the row correlation for the first band is closeto zero (0.02) and the second (0.17) is significant. The genetic variance is small, being approximatelyhalf of the residual variance.

For the second cut, the genetic variance is higher than the residual variance and so test-lineselection will be more successful. The ratio between genetic and residual components is 1.31, morethan the double of the first-cut. The residual correlation in row direction was modeled with third-orderbanded correlation structure and it shows more competition than in the first cut, because the first-ordercorrelation is negative and significant. The correlations were -0.19, 0.03 and 0.19 for first to third order,respectively. The chosen model assumed spline effects in row and column directions and also the harvesteffect was significant, differing from the first-cut that assumed a global row effect.

Comparing the best test lines selected in each cut from the individual analyses, there are onlynine of the selected in common between the two cuts and the correlation between all the genetic predictedvalues was 0.542. This relation is stronger in the joint analysis, which found that genetic variance differedbetween the two cuts and the correlation between them was one in each cut and correlation one betweenthese. Hence, using the joint analysis, it can be concluded that while the test-line effects differed betweenthe two cuts, but that the ranking of the test lines is similar for the two cuts.

89

C PARANÁ - GROUP OF EXPERIMENTS

There were three sites, nominated: Local 651, 851 and 852. It one assessed the same 200 testlines (N) with 4 commercial varieties nominated “PAD2”, “PAD7”, “PAD8” and “Interc”. Planted in2014 in a rectangular array of 10 columns by 24 rows and the yield in TCH was obtained in 2015, in aunique cut. The results were organized in subsections where C.1 presents the descriptive analysis of thesites, C.2 shows the individual analyses for each site and C.3 gives the joint analysis. The concludingremarks are in subsections C.4.

C.1 Descriptive analysis

The summary of the TCH for the groups of clones by each site is presented in Table C.1. Itis possible to observe that the average yield is smaller in Local 651 than the others. The Local 851resulted in higher average yield. Also the “PAD2” variance is higher, which is unexpected because it is acommercial variety and it should be stable. When the boxplot, Figure C.1, is observed, the outlier valuesand the distribution of the data can be verified. Only one plot was considered outlier within the plots ofeach commercial varieties and this was from Local 651.

Table C.1. Descriptive analysis of the groups of clones carried out in Paraná.


Loca

l651

Interspersed 26.80 95.20 72.23 287.98PAD2 43.20 89.30 76.27 489.70PAD7 67.00 86.30 77.37 71.91PAD8 40.20 75.90 51.32 272.71Test lines (N) 13.40 122.00 61.59 386.62

Loca

l851


Loca

l852


90T

CH

50

100

150

200

INTERC N PAD2 PAD7 PAD8

651


851


852

Figure C.1. Boxplot of TCH for each group of lines for Locals of Paraná.

The Figures C.2, C.3 and C.4 show the heat maps relating to TCH in each site, and sometrends can be observed in Local 651 (Figure C.2) and 851 (Figure C.3). It seems that there is one specificand unusual environmental effect in these sites, not only test treatment effect (test lines and checks).However the company were unaware of problems that could affect the results. The names “PAD2”,“PAD7”, “PAD8” and “INTERC” are the check plots, the test lines are labeled alpha-numerically (PRplus number) and “x” represents the empty plots.

Column

Row

INTERC

PR104

PR102

PR148

PR128

PR115

PR112

PR107

PR114

INTERC

PR126

PR100

PR118

PR103

PR105

PR125

PR8

PR143

INTERC

PR149

PR131

PR153

PR124

X

PR121

INTERC

PR146

PR162

PR151

PR123

PR106

PR122

PR6

PR150

INTERC

PR145

PR156

PR196

PR141

PR154

PR130

PR144

PR120

INTERC

PR33

PR155

PR129

PR152

PR136

PR200

INTERC

PR137

PR197

PAD7

PAD2

PAD8

PR199

PR132

PR193

INTERC

PR9

PR135

PR133

PR194

PAD2

PAD8

PAD7

PR127

INTERC

PR159

PR195

PR164

PR7

PR163

PR139

INTERC

PR158

PR31

PR183

PR45

PR198

PR17

PR30

PR12

INTERC

PR140

PR165

PR168

PR157

PR40

PR138

PR134

PR184

INTERC

PR147

PR1

PR166

PR181

PR169

PR167

INTERC

PR34

PR14

PR36

PR38

PR13

PR35

PR171

PR170

INTERC

PR41

PR15

PR19

PR39

PR178

PR161

PR174

PR176

INTERC

PR173

PR22

PR20

PR160

PR25

PR18

INTERC

PR177

PR172

PR189

PR37

PR23

PR32

PR21

PR186

INTERC

PR175

PR191

PR28

PR10

PR190

PR51

PR179

PR182

INTERC

PR16

PR58

PR29

PR180

PR26

PR187

INTERC

PR27

PR46

PR43

PR24

PR42

PR47

PR56

PR49

INTERC

PR53

PR52

PR57

PR50

PR185

PR192

PR55

PR48

PR11

PR60

PR54

PAD2

PAD8

PAD7

PR188

INTERC

PR44

PR59

PR89

PR72

PR61

PR77

PR74

PR79

INTERC

PR2

PAD8

PAD7

PAD2

PR78

PR85

PR80

PR70

PR65

PR88

PR63

PR62

PR68

PR4

PR82

INTERC

PR76

PR113

PR86

PR81

PR119

PR90

PR73

PR67

INTERC

PR3

PR87

PR83

PR71

PR64

PR84

X

PR5

PR75

PR110

PR66

PR69

PR91

PR111

PR116

INTERC

PR93

PR109

PR142

PR117

PR92

PR98

PR96

PR97

INTERC

PR95

PR108

PR101

PR99

PR94

20

40

60

80

100

120

Figure C.2. Heat map relating to TCH for Local 651 of Paraná.

91

Column

Row

INTERC

PR177

PR25

PR24

PR18

PR178

PR51

PR10

PR17

INTERC

PR48

PR54

PR52

PR16

PR21

PR182

PR176

PR47

INTERC

PR192

PR53

PR180

PR23

INTERC

PR179

INTERC

PR55

PR181

PR22

PR20

PR173

PR14

PR13

PR15

INTERC

PR171

PR189

PR191

PR170

PR190

PR172

PR174

PR175

INTERC

PR188

PR50

PR59

PR11

PR58

PR30

INTERC

PR33

PR45

PAD8

PAD7

PAD2

PR49

PR185

PR44

INTERC

PR46

PR1

PR9

PR43

PAD8

PAD7

PAD2

PR40

INTERC

PR56

PR57

PR60

PR42

PR41

PR35

INTERC

PR39

PR184

PR19

PR26

PR31

PR187

PR186

PR34

INTERC

PR28

PR29

PR162

PR38

PR151

PR156

PR152

PR150

INTERC

PR144

PR141

PR167

PR140

PR6

PR153

INTERC

PR154

PR158

PR155

PR159

PR37

PR166

PR163

PR165

INTERC

PR160

PR27

PR183

PR169

PR32

PR157

PR164

PR161

INTERC

PR36

PR168

PR149

PR148

PR120

PR121

INTERC

PR123

PR125

PR147

PR145

PR143

PR146

PR142

PR130

INTERC

PR137

PR7

PR135

PR134

PR8

PR132

PR128

PR131

INTERC

PR103

PR105

PR110

PR124

PR199

PR115

INTERC

PR113

PR126

PR118

PR112

PR195

PR136

PR12

PR129

INTERC

PR119

PR108

PR116

PR127

PR138

PR122

PR109

PR133

PR193

PR200

PR114

PAD8

PAD7

PAD2

PR106

INTERC

PR197

PR102

PR107

PR100

PR104

PR101

PR98

PR92

INTERC

PR139

PAD8

PAD7

PAD2

PR117

PR79

PR82

PR72

PR80

PR91

PR95

PR88

PR196

PR86

PR84

INTERC

PR99

PR87

PR111

PR96

PR97

PR90

PR194

PR94

INTERC

PR3

PR198

PR73

PR83

PR89

PR4

INTERC

PR69

PR74

PR75

PR71

PR2

PR76

PR81

PR77

INTERC

PR93

PR85

PR5

PR78

PR68

PR70

PR66

PR67

INTERC

PR61

PR62

PR63

PR64

PR65

80

100

120

140

160

180

200

220


Column

Row

INTERC

PR62

PR82

PR124

PR74

PR63

PR2

PR78

PR84

INTERC

PR80

PR67

PR73

PR64

PR79

PR75

PR72

PR70

INTERC

PR65

PR69

PR68

PR166

INTERC

PR85

INTERC

PR66

PR86

PR61

PR77

PR6

PR71

PR76

PR87

INTERC

PR100

PR116

PR81

PR91

PR104

PR126

PR128

PR119

INTERC

PR101

PR3

PR117

PR125

PR118

PR130

INTERC

PR99

PR114

PAD8

PAD7

PAD2

PR133

PR120

PR94

INTERC

PR129

PR90

PR95

PR122

PAD8

PAD7

PAD2

PR121

INTERC

PR93

PR89

PR127

PR105

PR115

PR136

INTERC

PR134

PR110

PR123

PR83

PR4

PR92

PR146

PR145

INTERC

PR5

PR96

PR97

PR107

PR112

PR102

PR88

PR158

INTERC

PR98

PR135

PR180

PR142

PR131

PR154

INTERC

PR26

PR7

PR108

PR164

PR163

PR139

PR162

PR200

INTERC

PR141

PR148

PR106

PR157

PR113

PR160

PR140

PR168

INTERC

PR132

PR109

PR111

PR152

PR151

PR196

INTERC

PR171

PR181

PR169

PR137

PR175

PR176

PR28

PR143

INTERC

PR33

PR138

PR41

PR167

PR103

PR173

PR38

PR177

INTERC

PR37

PR45

PR29

PR159

PR179

PR42

INTERC

PR34

PR165

PR198

PR170

PR40

PR174

PR9

PR52

INTERC

PR172

PR149

PR43

PR12

PR161

PR144

PR183

PR32

PR44

PR27

PR35

PAD8

PAD7

PAD2

PR60

INTERC

PR55

PR1

PR30

PR178

PR150

PR39

PR155

PR195

INTERC

PR186

PAD8

PAD7

PAD2

PR185

PR182

PR31

PR17

PR193

PR49

PR190

PR191

PR192

PR184

PR188

INTERC

PR8

PR48

PR23

PR189

PR156

PR199

PR14

PR46

INTERC

PR187

PR147

PR53

PR54

PR25

PR36

INTERC

PR153

PR197

PR10

PR59

PR22

PR58

PR11

PR51

INTERC

PR50

PR57

PR56

PR47

PR21

PR15

PR19

PR24

INTERC

PR16

PR20

PR18

PR13

PR194

60

80

100

120

140

160


C.2 Individual analysis

For Local 651 a summary of the sequence of fitted models is presented in Table C.2. Analysiscommenced with the model denominated random effects Treatment Interference Model (R-TIM), usedby Stringer et al. (2011). This contains random column, row and direct and neighbour genetic effectsand first-order autoregressive variance model for the residuals in the row and column direction. UsingREML tests and residual graphics new models were developed and in the end Model 20 was chosen toexplain the data. Hence, in the chosen model, there are random direct and nugget genetic effects andfirst-order autoregressive structure for the residuals. The log-likelihood did not show strong differenceamong the models. However, in Figure C.5, it is clear that the spatial dependence, related to Model 20,fitted better.

92

Here, previous studies with banded structure were done, but it was observed no competitionat the residual level. In this Local, the AR(1) structure appropriately described the spatial dependence.The magnitude of the correlation in the column and row directions is similar. See the REML estimatesin Table C.3.

93

Tabl

eC

.2.

Sum

mar

yof

the

fitte

dm

odel

sto

Loca

l651

with

rest

ricte

dlo

g-lik

elih

ood

(log-

lik.)

and

thep-v

alue

ofth

eR

EML

ratio

test

.A

llm

odel

sin

clud

ea

rand

omdi

rect

gene

ticeff

ects

for

the

test

lines

(G)

and

fixed

chec

kva

rietie

seff

ects

.T

heot

her

effec

tsar

ere

pres

ente

dw

ithth

esy

mbo

ls:H

(har

vest

);C

o(c

one)

;S

(sow

ing)

;C

(col

umn)

;R

(row

);N

(nei

ghbo

urge

netic

);sp

l(.)

(spl

ine

inso

me

dire

ctio

nin

dica

ted

inpa

rent

hese

s)an

dlin

(.)(li

near

tren

din

som

edi

rect

ion

indi

cate

din

pare

nthe

ses)

.T

hest

ruct

ures

can

be:

US(

.)-u

nstr

uctu

red;

AR

(1)

-firs

t-or

der

auto

regr

essiv

e;Ba

nd(.)

-ban

ded

corr

elat

ion,

the

num

ber

indi

cate

sth

eor

der

ofco

rrel

atio

n;an

dId

-ide

ntity

.

Mod

elEff

ects

Log-

lik.

Test

p-v

alue

Glo

bal/

extr

aneo

usG

enet

icLo

cal

Fixe

dR

ando

m(c

olum

n×

row

)1

C+

RU

S(G

:N)

AR

(1)×

AR

(1)

-796

.70

2U

S(G

:N)

AR

(1)×

AR

(1)

-796

.70

M1

vsM

21.

003

GA

R(1

)×

AR

(1)

-797

.89

M2

vsM

30.

214

C+

RG

AR

(1)×

Band

(3)

-797

.49

5G

AR

(1)×

Band

(3)

-797

.49

M4

vsM

51.

006

GA

R(1

)×

Band

(2)

-797

.67

M5

vsM

60.

557

GA

R(1

)×

Band

(1)

-798

.50

M6

vsM

70.

198

GA

R(1

)×

Id-8

01.9

1M

7vs

M8

0.01

9G

Id×

Band

(1)

-806

.18

M7

vsM

80.

0010

GA

R(1

)×

AR

(1)

-797

.89

11H

GA

R(1

)×

AR

(1)

-795

.63

not

signi

fican

t12

Co

GA

R(1

)×

AR

(1)

-795

.24

not

signi

fican

t13

SG

AR

(1)×

AR

(1)

-796

.82

not

signi

fican

t14

lin(C

)+lin

(R)

spl(C

)+sp

l(R)

GA

R(1

)×

AR

(1)

-796

.77

15lin

(C)+

lin(R

)sp

l(R)

GA

R(1

)×

AR

(1)

-796

.78

M14

vsM

151.

0016

lin(C

)+lin

(R)

GA

R(1

)×

AR

(1)

-797

.99

M15

vsM

160.

0617

lin(C

)G

AR

(1)×

AR

(1)

-798

.30

not

signi

fican

t18

lin(R

)G

AR

(1)×

AR

(1)

-797

.57

not

signi

fican

t19

GA

R(1

)×

AR

(1)

-797

.89

20un

itsG

AR

(1)×

AR

(1)

-795

.36

M20

vsM

190.

0121

units

US(

G:N

)A

R(1

)×

AR

(1)

-795

.27

M21

vsM

200.

80

94

0.0

0.5

1.0

1.5

0.0 2.5 5.0 7.5Col differences


(a)

0.0

0.5

1.0

0.0 2.5 5.0 7.5Col differences


(b)

0.0

0.5

1.0

1.5

2.0

0 5 10 15 20Row differences


(c)

0.0

0.5

1.0

1.5



(d)

Figure C.5. Plots of the row and column faces of the empirical variogram for the residuals for the Local651 of the Paraná experiment referring to models 19 (panels (a) and (c)) and 20 (panels (b) and (d)).The panels (a) and (b) are for the column direction and the others are for the row direction.

Table C.3. REML estimates of variance parameters for fitted Model 20 for the experiment in the Local651 of the Paraná.

Variance parameters Ratios (γ) Estimates Standard errorsDirect genetic (σ̃2

g) 0.78 100.31 62.71Nugget (σ̃2

u) 1.15 147.68 61.37Residual (σ̃2) 1.00 127.92 45.92

Spatial (ρ̃c) — 0.67 0.16Spatial (ρ̃r) — 0.56 0.18

Note that the direct genetic component is smaller than the residual (ratio of 0.78) and the nuggetcomponent is high (ratio of 1.15). Table C.4 presents the Wald F-tests for fixed terms. These shownthere were no evidence of significative difference between the group of checks and test lines (Control) oramong checks (Control:Check). The coefficients and standard errors of the factors levels from Model 20are in Table C.5.

Table C.4. Wald F tests for fixed terms for Model 20. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom.

Fixed terms Df denDF F test p-value(Intercept) 1 6.80 500.80 0.00Control 1 6.10 0.76 0.42Control:Check 3 30.40 2.49 0.08

95

Table C.5. Estimates of the fixed effects for Model 20 and their respective standard errors.

Effects Estimates Standard errorsINTERC 0.00 —PAD2 6.37 8.03PAD7 4.04 7.94PAD8 -17.65 7.98overall Test lines 0.00 —overall Checks 9.61 10.47overall mean (µ) 61.68 2.77

The top 7% of the test lines for Local 651 of the Paraná experiment, along with their predictedvalues and standard errors, are presented in Table C.6. In Figure C.6 can be observed the relationbetween the genetic EBLUPs from Model 20 and the traditional non-spatial analysis, the latter havingonly fixed checks varieties and random direct genetic, column and row effects. The correlation betweenthem is high, 0.911, but the group of selected test lines was not the same.

Table C.6. The 14 best test lines with predicted values and respective standard errors for Model 20.

Test line Predicted value Standard errorPR58 78.25 8.71PR65 75.98 8.71PR156 75.17 8.70PR61 75.02 8.71PR136 73.94 8.71PR46 73.67 8.70PR151 72.81 8.70PR123 72.65 8.70PR143 72.49 8.72PR135 71.83 8.70PR169 71.75 8.70PR174 71.65 8.70PR7 71.39 8.71PR116 70.93 8.70

−10 −5 0 5 10 15

−10

−5

05

1015

Genetic EBLUPs − Traditional

Gen

etic

EB

LUP

s −

sel

ecte

d m

odel

cor = 0.911

Figure C.6. Dispersion plot of the genetic EBLUPs values of the Model 20 and the traditional non-spatial analysis model for this design (without spatial dependence) for Local 651 of the Paraná. Thecut-offs for the 14 best test lines (7 % upper) in each Model are indicated by the dotted line.

For Local 851, we started with the same procedure. However, the neighbour genetic effectscan be written as a special case, reduced rank, but using REML test (Table C.7) it was not significant.

96

Then testing local and global effects and using residual graphics, the better fit was found for Model 15,which has random spline effects in column direction and row and direct genetic effects. The model doesnot show significant competition in the residual or genetic level and when the empirical semi-variogramin Figure C.7 is observed there is any strong effect to be modeled, only some extraneous variability notdetected by the selected model.

97

Tabl

eC

.7.

Sum

mar

yof

the

fitte

dm

odel

sto

Loca

l851

with

rest

ricte

dlo

g-lik

elih

ood

(log-

lik.)

and

thep-v

alue

ofth

eR

EML

ratio

test

.A

llm

odel

sin

clud

ea

rand

omdi

rect

gene

ticeff

ects

for

the

test

lines

(G)

and

fixed

chec

kva

rietie

seff

ects

.T

heot

her

effec

tsar

ere

pres

ente

dw

ithth

esy

mbo

ls:H

(har

vest

);C

o(c

one)

;S

(sow

ing)

;C

(col

umn)

;R

(row

);N

(nei

ghbo

urge

netic

);sp

l(.)

(spl

ine

inso

me

dire

ctio

nin

dica

ted

inpa

rent

hese

s)an

dlin

(.)(li

near

tren

din

som

edi

rect

ion

indi

cate

din

pare

nthe

ses)

.T

hest

ruct

ures

can

be:

RR

(.)-r

educ

edra

nk;A

R(1

)-fi

rst-

orde

rau

tore

gres

sive;

Band

(.)-b

ande

dco

rrel

atio

n,th

enu

mbe

rin

dica

tes

the

orde

rof

corr

elat

ion;

and

Id-i

dent

ity.

Mod

elEff

ects

Log-

lik.

Test

p-v

alue

Glo

bal/

extr

aneo

usG

enet

icLo

cal

Fixe

dR

ando

m(c

olum

n×

row

)1

C+

RR

R(G

:N)

AR

(1)×

AR

(1)

-858

.11

2C

+R

GA

R(1

)×

AR

(1)

-859

.16

M1

vsM

20.

153

C+

RG

AR

(1)×

Band

(3)

-858

.08

4C

+R

GA

R(1

)×

Band

(2)

-859

.16

M3

vsM

40.

145

C+

RG

AR

(1)×

Band

(1)

-859

.20

M4

vsM

50.

786

C+

RG

AR

(1)×

Id-8

59.5

7M

5vs

M6

0.39

7C

+R

GId

×Id

-859

.57

M6

vsM

70.

958

HG

Id×

Id-8

57.9

1no

tsig

nific

ant

9C

oG

Id×

Id-8

57.8

9no

tsig

nific

ant

10S

GId

×Id

-856

.20

not

signi

fican

t11

lin(C

)+lin

(R)

spl(C

)+C

+sp

l(R)+

RG

Id×

Id-8

58.7

412

lin(C

)+lin

(R)

spl(C

)+R

GId

×Id

-858

.74

M11

vsM

121.

0013

lin(C

)+lin

(R)

RG

Id×

Id-8

61.0

4M

12vs

M13

0.01

14lin

(C)+

lin(R

)sp

l(C)

GId

×Id

-871

.73

M13

vsM

140.

0015

lin(C

)sp

l(C)+

RG

Id×

Id-8

57.8

616

lin(C

)sp

l(C)+

RR

R(G

:N)

Id×

Id-8

57.8

2M

16vs

M15

0.95

98

0

2

46

8

0

5

10

15

20

0.0

0.5

1.0

1.5

2.0

Col (lag)

Row (lag)

Figure C.7. Plot of the empirical semi-variogram for the residuals for the Local 851 of the Paranáexperiment referring to Model 15.

Table C.8 presents the REML estimates of variance parameters from Model 15, where it canbe observed that for this model the direct genetic component is 1.1 times the residual component. Thespline column component is very small, with the standard error being greater than the estimated value;however this term was significant using REML ratio test.

Table C.8. REML estimates of variance parameters for fitted Model 15 for the experiment in the Local851 of the Paraná.

Variance parameters Ratios (γ) Estimates Standard errorsspl(Column) 0.04 8.43 11.09Row (σ̃2


g) 1.10 261.76 85.36Residual (σ̃2) 1.00 238.74 64.88

Table C.9 presents the Wald F-tests for fixed terms of the Model 15 and Table C.10 shows theestimated fixed effects. As it can be seen, there is no evidence of a significant difference between thecheck and test lines groups (Control) or among the checks (Control:Check).

Table C.9. Wald F tests for fixed terms from Model 15. Df means degrees of freedom and DenDF isthe approximate denominator degrees of freedom.

Fixed terms Df denDF F test p-value(Intercept) 1.00 28.10 2441.00 0.00Control 1.00 20.60 0.71 0.41lin(Col) 1.00 179.60 1.10 0.29Control:Check 3.00 32.70 3.95 0.02


Effects Estimates Standard errorsINTERC 0.00 —PAD2 20.91 8.74PAD7 19.29 8.77PAD8 18.05 8.74lin(Col) -0.54 0.47overall test lines 0.00 —overall checks -19.28 16.52overall mean (µ) 136.51 3.74

99

In Table C.11 there are the 14 best test lines from Model 15 with their respective predictedvalues and standard errors. Figure C.8 compares the genetic EBLUPs from Model 15 and the traditionalnon-spatial analysis, the latter including fixed check varieties and random direct genetic, column and roweffects. The correlation between them is 0.931 which means there is a high similarity between the results.


Test line Predicted values Standard.errorsPR135 163.33 11.81PR139 154.52 11.97PR53 154.20 12.14PR180 152.44 12.17PR156 152.05 11.85PR49 151.95 11.98PR65 151.76 12.15PR23 151.70 12.18PR173 151.05 12.04PR186 149.95 11.90PR184 149.95 11.88PR25 148.76 12.16PR174 148.60 12.03PR12 146.65 11.92

−20 −10 0 10 20 30 40

−20

010

2030


Gen

etic

EB

LUP

s −

sel

ecte

d m

odel

cor = 0.931


For Local 852, Table C.12 shows some fitted models starting with the model R-TIM, withrandom column, row and genetic (direct, neighbour and covariance) effects and first-order autoregressivestructure in the row and column direction. The random row, column, genetic and neighbour covarianceeffects are very small components. They do not represent significant effects as can be observed in TableC.12 and it is not a special case of reduced rank. However, a negative correlation was found in the rowdirection which indicates competition among the plots. The model with third-order banded structurehas problems of convergence. Hence, it is difficult to analyze the correlation levels among the plots andto decide which is the better structure to explain the data. Using the empirical semi-variogram for theresiduals, it was possible to compare the models and to choose Model 15 as the best to explain the data.Figure C.9 shows the empirical semi-variogram for the residuals from the model 15.

100Ta

ble

C.1

2.Su

mm

ary

ofth

efit

ted

mod

els

toLo

cal8

52w

ithre

stric

ted

log-

likel

ihoo

d(lo

g-lik

.)an

dth

ep-v

alue

ofth

eR

EML

ratio

test

.A

llm

odel

sin

clud

ea

rand

omdi

rect

gene

ticeff

ects

for

the

test

lines

(G)

and

fixed

chec

kva

rietie

seff

ects

.T

heot

her

effec

tsar

ere

pres

ente

dw

ithth

esy

mbo

ls:H

(har

vest

);C

o(c

one)

;S

(sow

ing)

;C

(col

umn)

;R

(row

);N

(nei

ghbo

urge

netic

);sp

l(.)

(spl

ine

inso

me

dire

ctio

nin

dica

ted

inpa

rent

hese

s)an

dlin

(.)(li

near

tren

din

som

edi

rect

ion

indi

cate

din

pare

nthe

ses)

.T

hest

ruct

ures

can

be:

US(

.)-u

nstr

uctu

red;

RR

(.)-r

educ

edra

nk;A

R(.)

-aut

oreg

ress

ive

stru

ctur

e;Ba

nd(.)

-ban

ded

corr

elat

ion,

the

num

ber

indi

cate

sth

eor

der

ofco

rrel

atio

n;sa

r(2)

cons

trai

ned

auto

regr

essiv

e;an

dId

-ide

ntity

.

Mod

elEff

ects

Log-

lik.

Test

p-v

alue

Glo

bal/

extr

aneo

usG

enet

icLo

cal

Fixe

dR

ando

m(c

olum

n×

row

)1

C+

RU

S(G

:N)

AR

(1)×

AR

(1)

-825

.74

2C

+R

GA

R(1

)×

AR

(1)

-826

.31

M1

vsM

20.

423

C+

RG

AR

(1)×

Band

(3)

not

conv

erge

d-sin

gula

rity

4C

+R

GA

R(1

)×

Band

(2)

-824

.45

5C

GA

R(1

)×

Band

(2)

-824

.45

M4

vsM

51.

006

CG

AR

(1)×

Band

(1)

-828

.14

M5

vsM

60.

007

CG

AR

(1)×

sar(

2)-8

24.2

48

CG

AR

(1)×

AR

(2)

-825

.53

9C

GId

×Ba

nd(1

)-8

28.1

4M

5vs

M9

0.14

10H

CG

Id×

Band

(2)

-824

.05

not

signi

fican

t11

Co

CG

Id×

Band

(2)

-824

.59

not

signi

fican

t12

SC

GId

×Ba

nd(2

)-8

24.2

5no

tsig

nific

ant

13lin

(C)+

lin(R

)sp

l(C)+

C+

spl(R

)G

Id×

Band

(2)

-822

.67

14lin

(C)+

lin(R

)C

+sp

l(R)

GId

×Ba

nd(2

)-8

22.6

7M

13vs

M14

1.00

15lin

(C)+

lin(R

)sp

l(R)

GId

×Ba

nd(2

)-8

22.7

6M

14vs

M15

0.33

16lin

(C)+

lin(R

)G

Id×

Band

(2)

-824

.87

M15

vsM

160.

0217

lin(C

)+lin

(R)

spl(R

)R

R(G

:N)

Id×

Band

(2)

-821

.51

M17

vsM

150.

11

101

0

2

46

8

0

5

10

15

20

0.0

0.5

1.0

1.5

Col (lag)

Row (lag)

Figure C.9. Plot of the empirical semi-variogram for the residuals for the Local 852 of the Paranáexperiment referring to Model 15.

Table C.13 presents the REML estimates of variance parameters from selected model. Thedirect genetic component is 1.3 times the residual component and there is a negative correlation in rowdirection shown that have the competition at the residual level. The spline row component is very small,with standard error grater than the component, although this term was significant using REML test.

Table C.13. REML estimates of variance parameters for Model 15 to the experiment in the Local 852of the Paraná.

Variance parameters Ratios (γ) Estimates Standard errorsspl(Row) 0.03 6.03 8.17Direct genetic (σ̃2

g) 1.30 252.75 57.22Residual (σ̃2) 1.00 195.12 43.15

Spatial (ρ̃r1) — -0.40 0.13Spatial (ρ̃r2) — 0.29 0.12

The Wald F tests for fixed terms are present in Table C.14. They show that there were noevidence to assume difference between the group of test lines and checks (Control), but that there wasamong the checks (Control:Check). The estimated fixed effects from this model are in Table C.15.

Table C.14. Wald F tests for fixed terms for Model 15. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom.

Fixed terms Df denDF F test p-value(Intercept) 1.00 228.00 5169.00 0.00Control 1.00 228.00 0.03 0.87lin(Col) 1.00 228.00 9.98 0.00lin(Row) 1.00 228.00 0.03 0.87Control:Check 3.00 228.00 4.95 0.00

102


Effects Estimates Standard errorsINTERC 0.00 —PAD2 -0.79 6.72PAD7 19.60 6.93PAD8 12.52 6.82lin(Col) 1.21 0.45overall test lines 0.00 —overall checks -1.69 16.65overall mean (µ) 95.04 2.93

In Table C.16 are the 14 best test lines selecting using Model 15, along with their respectivepredicted values and standard errors. Comparing, in Figure C.10, the genetic EBLUPs from Model 15and the traditional non-spatial analysis, that contain fixed checks and random direct genetic, row andcolumn effects, there is a linear trend and the correlation is 0.938. However, note that the group ofselected test lines is not the same.


Test line Predicted values Standard errorsPR50 139.26 9.83PR66 135.03 10.15PR173 131.05 10.14PR86 126.50 10.02PR136 123.85 10.06PR64 122.68 10.02PR151 122.24 10.01PR122 122.12 9.79PR49 121.21 10.27PR37 119.69 10.64PR177 118.14 10.29PR119 118.11 9.98PR35 117.57 10.02PR100 116.97 9.75

−20 −10 0 10 20 30

−20

020

40


Gen

etic

EB

LUP

s −

sel

ecte

d m

odel

cor = 0.938


103

C.3 Joint analysis

Based on the separate analysis for each site, it was observed that the genetic EBLUPs are verydifferent. Figure C.11 presents the relation of these values between pairs of Locals; the correlation ismedium, among 0.330 to 0.493. This means that there is a substantial Local effect on the test lines.

−10 −5 0 5 10 15

−20

010

2030

Genetic EBLUPs − Local 651

Gen

etic

EB

LUP

s −

Loc

al 8

51 cor = 0.376

(a)

−10 −5 0 5 10 15

−20

020

40


Gen

etic

EB

LUP

s −

Loc

al 8

52 cor = 0.330

(b)

−20 −10 0 10 20 30

−20

020

40


Gen

etic

EB

LUP

s −

Loc

al 8

52 cor = 0.493

(c)

Figure C.11. Dispersion plots of the genetic EBLUPs from model selected for each Paraná Local.Panels the relationship between pairs of Locals (a) 651 and 851; (b) 651 and 852; and (c) 851 and 852.The cut-offs for the 14 best test lines in each Local are indicated by the dotted line.

Then, the joint analysis of all three sites was performed in order to investigate and to comparethe magnitudes of the genetic variances at each Local and to assess the genetic covariance betweenLocals. However, there were some convergence problems. Hence, firstly only Local 851 and 852 werejointly analyzed because their genetic components were closer in the individual analyses. Table C.17presents the models and the REML test for the joint analysis between Local 851 and 852. Using theREML test, it is possible to conclude that the genetic components are similar and there is a correlationof 0.9 between Locals 851 and 852 (Model 3). These results were used in the joint analysis with all sites.

104

Table C.17. Summary of the fitted models to joint analysis of the Paraná experiments carried out inthe Locals 851 (2) and 852 (3), with REML log (log-lik.) and the p-value of the REML ratio test. Allthe models have the same effects as modeled in the individual analysis plus the fixed Local effect. Hereσ2gi is the genetic variance at the ith Local and ρgii′ is the genetic correlation between the ith and i′th

Locals.Model Genetic effects Constraint log-lik. Test p-value

1[σ2g2 00 σ2

g3

]-1690.63

2[σ2g 00 σ2

g

]σ2g = σ2

g2 = σ2g3 -1680.64 M1 vs M2 0.46

3[σ2g σgg

σgg σ2g

]σgg = ρgσ

2g -1651.92 M3 vs M2 0.00

For the joint analysis, there are convergence problems with some models, mainly when unstruc-tured variance model is used for genetic effects. Therefore, the results from the models in Table C.17were used as initial parameters in fitting some models. The tested models are presented in Table C.18.Within the possible models, Model 3 was considered as the best to explain the data. It has the sameenvironmental parameters of the individual analysis, plus the Local fixed effect and covariance betweengenetic and Local effect. The genetic variances for Local 851 and 852 are considered equal, as are thecovariances between the genetic effect for all Locals. Table C.19 presents the REML estimates of variancecomponents of the Model 3. Note that the direct genetic component for Local 651 was less than halfwhen compared to the ones for Local 851 and 852. However site 651 had the smaller residual component.

Table C.18. Summary of the models fitted in joint analysis of the Paraná experiments carried out inthe Locals 561 (1), 851 (2) and 852 (3), with REML ratio tests (log-lik.) and the p-values of the REMLratio tests. All the models have the same effects as modeled in the individual analysis plus the fixedLocal effect. Here σ2

gi is the genetic variance at the ith Local, ρii′ is the genetic correlation between theith and i′th Locals and σg∗ is the genetic covariance between two locals which have been hypothesis tobe equal for two or more pairs of Locals.

Model Genetic effects Constrained Log-lik. Test p-value

1

σ2g1 0 00 σ2

g2 00 0 σ2

g3

-2476.00

2

σ2g1 σg12 σg13

σg12 σ2g2 σg23

σg13 σg23 σ2g3

σgii′ = ρgσgiσgi′ -2427.85 M2 vs M1 0.00

3

σ2g1 σg1∗ σg1∗

σg1∗ σ2g σgg

σg1∗ σgg σ2g

σg1∗ = ρg1σg1σgσgg = ρg23σ

2gg

-2428.59 M2 vs M3 0.35

105

Table C.19. REML estimates of the variance parameters from fitted Model 3 in the joint analysis ofthe Locals in Paraná.

Variance parameters Estimates Standard errorsLocal651:Genetic (σ̃2

g1) 91.35 60.34Local851-852:Genetic (σ̃2

g) 261.58 41.38cov(Local 651-...):New (σ̃g1∗) 130.79 20.69cov(Local 851-852):New (σ̃gg) 239.13 37.62Local 851:spl(Col) 5.63 7.60Local 851:Row (σ̃2

r2) 108.75 41.82Local 852:spl(Row) 5.54 7.10Local 561: Nugget (σ̃2

u1) 154.53 58.62Local 561: Residual (σ̃2

1) 120.37 40.23Spatial:Column (ρ̃1c) 0.72 0.14Spatial:Row (ρ̃1r) 0.52 0.17

Local 851:Residual (σ̃22) 230.43 39.21

Local 852: Residual (σ̃23) 187.52 32.39

Spatial:Row (ρ̃3r1) -0.32 0.10Spatial:Row (ρ̃3r2) 0.11 0.10

Table C.20 presents the Wald F-tests for fixed terms. Observe that there is the significantdifference between the Locals and between check and test lines group. The coefficients and standarderrors of these fixed effects from Model 3 are in Table C.21.

Table C.20. Wald F test for fixed terms of the Model 3. Df means degrees of freedom and DenDF isthe approximate denominator degrees of freedom.

Fixed terms Df denDF F test p-value(Intercept) 1.00 683.00 5972.00 0.00Site 2.00 683.00 192.20 0.00Control 1.00 683.00 2.37 0.12Control:Check 3.00 683.00 5.44 0.00Site:Control 2.00 683.00 3.19 0.04at(Site, 851):lin(Col) 1.00 683.00 0.00 0.98at(Site, 852):lin(Col) 1.00 683.00 12.90 0.00at(Site, 852):lin(Row) 1.00 683.00 0.59 0.44Site:Control:Check 6.00 683.00 2.93 0.01

106

Table C.21. Estimates of the fixed effects from Model 3 and their respective standard errors.

Effects Estimates Standard errorsoverall mean (µ) 61.89 2.71

INTERC 0.00 —PAD2 5.11 7.96PAD7 4.08 7.89PAD8 -18.28 7.91Checks 9.89 10.03

Local 651 0.00 —INTERC 0.00 —PAD2 0.00 —PAD7 0.00 —PAD8 0.00 —Checks 0.00 —

Local 851 74.29 4.33INTERC 0.00 —PAD2 17.25 11.65PAD7 16.08 11.61PAD8 36.51 11.61lin(Col) -0.49 0.42Checks -29.15 10.50

Local 852 34.11 3.78INTERC 0.00 —PAD2 -7.19 10.67PAD7 16.50 10.71PAD8 28.04 10.63lin(Row) -0.09 0.13lin(Col) 1.23 0.34Checks -10.69 10.39

Comparing the predicted values and their standard errors from Model 3 for the 14 best test linesfrom the different Locals, given in Table C.22, it is observed that (i) these results have small standarderrors when compared to the separate analysis of the sites, and (II) a large number of clones in commonamong the Locals within the top 14 test lines.

Table C.22. Predicted values (pred.) and standard error (error) of the 14 best test lines from Model 3for each Local.

Local 651 Local 851 Local 852Test lines Pred. Error Test lines Pred. Error Test lines Pred. ErrorPR135 80.04 6.57 PR173 164.68 9.44 PR173 133.91 8.81PR173 79.00 6.52 PR135 164.22 9.38 PR50 132.54 8.65PR156 78.89 6.53 PR50 160.71 9.43 PR66 128.74 8.87PR151 78.71 6.53 PR156 158.36 9.35 PR135 127.77 8.99PR136 77.63 6.51 PR151 157.88 9.34 PR151 127.24 8.79PR50 77.47 6.53 PR66 156.72 9.51 PR136 124.01 8.78PR61 75.73 6.54 PR53 155.28 9.49 PR156 123.99 8.68PR37 75.62 6.58 PR136 153.88 9.33 PR37 121.72 9.02PR174 75.36 6.54 PR139 153.70 9.40 PR49 121.23 8.88PR53 75.32 6.54 PR49 153.63 9.43 PR23 120.62 8.68PR7 74.89 6.52 PR23 153.47 9.49 PR184 119.82 8.78PR123 74.82 6.53 PR174 152.95 9.43 PR122 119.68 8.69PR66 74.77 6.54 PR184 152.84 9.35 PR53 119.54 8.84PR65 74.41 6.53 PR37 152.78 9.39 PR86 119.50 8.83

107

C.4 Conclusions

The individual analyses resulted in different fitted models for each site. The model in Local651 shows that there is no residual competition, because the correlations, modeled with AR(1) in bothdirections, are positive and similar in value. The neighbour genetic effects were not significant, neitherrandom column or row effect, but there is a nugget effect. For Local 851 there is not significant spatialdependence neither competition at the genetic or residual level, the selected model had a significantspline term in the column direction, random row and direct genetic effects. Local 852 exhibited residualcompetition effects, with first-order correlation in row direction equal to -0.40 while the second-order is0.29. The selected model had linear trend effect in column direction, spline term in the row direction anddirect genetic effects. For local effect the second-order banded structure in row direction was adequate.The ratios between genetic and residual components were approximately 0.78, 1.10 and 1.30 for Locals561, 851 and 852, respectively. This results in better selection of genotypes for Local 851 and 852 becausethey have larger genetic variance relative to the residual variance. When comparing the 14 best-predictedvalues for the test lines from each Local there were few similarities among the lines selected; there werearound 2 to 5 common test lines and the correlation between the sites for all values was among 0.33 and0.49.

However, in the joint analysis the selected model indicates the same genetic component forLocal 851 and 852 (261.58) and smaller component for Local 651 (91.35). There is genetic covariancebetween Locals, which for 651 with 851 or 852 is 239.13, while between 851 and 852 is estimated to be130.79. The number of common test lines in the best 14 lines is greater; more than a half is the same inall three Locals. In addition, better accuracy in the selection of the test lines is obtained, with predictionshaving lower standard errors.

108

109

D GOIÁS - GROUP OF EXPERIMENTS

There were three sites, nominated: Local 3, 521 and 533. The same 212 test lines (N) with 4commercial varieties nominated “PAD1”, “PAD4”, “PAD9” and “Interc” were assessed at all sites. Theywere planted in 2014 in a rectangular array of 13 columns by 20 rows and the yield in TCH was obtainedin 2015, in a unique cut. The results were organized in subsections where D.1 presents the descriptiveanalysis of the sites, D.2 shows the individual analyses for each Local, D.3 gives the joint analysis andD.4 present some conclusions.

D.1 Descriptive analysis

A summary of TCH for the groups of clones by each Local is shown in Table D.1. Consideringaverages yield, there were no great differences among the sites and the commercial varieties (checks). ForLocal 3, “PAD4” yields between 97.6 and 176.2; Local 521 and 533 also exhibit similar minimum andmaximum yield values, but is for “PAD1”. When the boxplot is observed, Figure D.1, some outlier valuesare found for each group of clones given each site. These plots will be carefully monitored in the analysesto check whether they result in problems such as being an influential plot or an outlier. Note that thevariance and average yield of the commercial varieties (checks) were different among the Locals.

Table D.1. Descriptive analysis of TCH for the groups of clones carried out in Goiás.


Loca

l3

Inerspersed 97.60 202.40 145.91 436.31PAD1 119.00 181.00 142.65 451.29PAD4 97.60 176.20 137.70 659.41PAD9 116.70 160.70 138.30 352.20Test lines (N) 67.90 264.30 125.37 637.07

Loca

l521


Loca

l533


110T

CH

50

100

150

200

250


3


521


533

Figure D.1. Boxplot of TCH for each group of lines for Locals of Goiás.

The Figures D.2, D.3 and D.4 show the heat maps relating to TCH in each site. The names“PAD1”, “PAD4”, “PAD9” and “INTERC” are the check plots and the test lines are labeled alpha-numerically (GS plus number). It seems there is in Local 3 a false measure in the “GS51” plot, becausethis had a much higher yield than expected, 264.30 TCH. Observe that its neighbour, “GS33”, had smallyield, but it also was small in Local 521 and 533. Then, it was supposed that there were problems in theharvest. Maybe, “GS51” gained yield from neighbours and its value is flawed. However, this suppositionwas not confirmed by the CTC. For the other two sites there were no strong visual problems or discrepantmeasures.

Column

Row

INTERC

GS2

GS201

GS186

GS189

GS7

GS202

GS122

GS144

INTERC

GS119

PAD1

GS118

GS147

GS120

GS157

GS145

GS124

INTERC

GS151

GS156

INTERC

GS159

PAD9

GS13

GS158

GS153

GS123

GS116

GS142

INTERC

GS148

GS154

GS113

GS140

GS141

GS110

PAD4

GS114

INTERC

GS111

GS132

INTERC

GS134

GS112

GS130

GS117

GS138

GS135

GS129

GS12

INTERC

GS136

GS99

GS139

GS108

GS143

GS146

GS109

GS115

PAD4

GS133

GS121

INTERC

GS92

GS105

GS93

GS107

GS131

PAD1

GS10

GS89

INTERC

GS101

GS98

GS91

GS94

GS104

GS103

PAD9

GS95

GS102

GS106

GS11

INTERC

GS96

GS86

GS97

GS5

GS21

GS27

GS90

GS30

INTERC

GS16

PAD4

GS20

GS24

GS28

GS126

GS19

PAD1

GS25

GS125

GS18

INTERC

GS15

GS17

PAD9

GS87

GS23

GS88

GS22

GS203

INTERC

GS1

GS194

GS200

GS209

GS193

GS214

GS206

GS192

GS213

GS210

GS32

INTERC

GS190

GS6

GS41

GS29

GS8

GS26

GS38

GS31

INTERC

GS61

GS56

GS65

GS55

GS57

GS76

GS52

PAD4

GS44

GS48

GS43

INTERC

GS33

GS51

GS46

PAD1

GS47

GS49

GS35

GS40

INTERC

GS77

PAD9

GS80

GS34

GS54

GS58

GS59

GS50

GS62

GS39

GS37

INTERC

GS199

GS162

GS69

GS74

GS161

GS160

GS152

GS75

INTERC

GS60

GS42

INTERC

GS64

GS70

GS68

GS71

GS155

GS82

PAD4

GS45

INTERC

GS53

GS79

GS63

GS85

PAD4

GS67

GS164

GS78

INTERC

GS66

GS100

INTERC

GS163

GS81

GS73

GS84

GS72

GS150

GS83

GS175

INTERC

GS165

GS128

GS180

GS127

GS166

GS174

GS173

GS172

INTERC

GS170

GS184

INTERC

GS188

GS171

GS198

GS3

GS176

PAD1

GS177

GS178

INTERC

GS179

GS197

GS191

GS149

GS187

GS204

PAD9

GS181

PAD9

GS167

GS182

INTERC

GS211

GS168

GS196

GS169

GS205

GS183

PAD1

GS14

INTERC

GS215

GS185

GS212

GS195

GS207

GS4

GS208

100

150

200

250

Figure D.2. Heat map relating to TCH for Local 3 of Goiás.

111

Column

Row

INTERC

GS131

GS112

GS193

GS195

GS186

GS214

GS200

GS169

INTERC

GS198

PAD9

GS201

GS4

GS183

GS212

GS191

GS134

INTERC

GS159

GS164

INTERC

GS205

PAD4

GS162

GS3

GS151

GS211

GS202

GS139

INTERC

GS150

GS130

GS197

GS181

GS184

GS196

PAD1

GS204

INTERC

GS182

GS209

INTERC

GS127

GS140

GS128

GS132

GS188

GS172

GS133

GS46

INTERC

GS50

GS167

GS14

GS175

GS178

GS179

GS52

GS125

PAD9

GS96

GS38

INTERC

GS48

GS83

GS43

GS25

GS63

PAD1

GS80

GS37

INTERC

GS57

GS59

GS73

GS215

GS67

GS1

PAD4

GS213

GS192

GS34

GS207

INTERC

GS19

GS208

GS65

GS189

GS21

GS69

GS47

GS18

INTERC

GS174

PAD4

GS194

GS187

GS203

GS170

GS210

PAD4

GS168

GS199

GS206

INTERC

GS185

GS180

PAD1

GS16

GS39

GS123

GS22

GS113

INTERC

GS41

GS62

GS35

GS28

GS32

GS45

GS29

GS78

GS68

GS56

GS42

INTERC

GS8

GS33

GS7

GS61

GS44

GS49

GS31

GS84

INTERC

GS71

GS64

GS74

GS54

GS76

GS85

GS53

PAD9

GS60

GS77

GS58

INTERC

GS55

GS82

GS70

PAD9

GS81

GS75

GS51

GS40

INTERC

GS79

PAD1

GS115

GS98

GS122

GS107

GS94

GS95

GS88

GS12

GS17

INTERC

GS11

GS27

GS108

GS100

GS99

GS89

GS97

GS86

INTERC

GS110

GS104

INTERC

GS6

GS124

GS15

GS26

GS103

GS90

PAD9

GS13

INTERC

GS111

GS109

GS23

GS20

PAD1

GS114

GS87

GS24

INTERC

GS91

GS101

INTERC

GS116

GS10

GS190

GS30

GS126

GS171

GS135

GS177

INTERC

GS156

GS155

GS143

GS154

GS149

GS163

GS158

GS176

INTERC

GS5

GS72

INTERC

GS66

GS173

GS2

GS152

GS141

PAD1

GS129

GS147

INTERC

GS148

GS142

GS161

GS166

GS165

GS153

PAD4

GS144

PAD4

GS121

GS106

INTERC

GS119

GS120

GS102

GS118

GS93

GS105

PAD9

GS117

INTERC

GS92

GS157

GS145

GS136

GS138

GS146

GS160

60

80

100

120

140

160

180


Column

Row

INTERC

GS20

GS61

GS55

GS19

GS9

GS59

GS45

GS21

INTERC

GS27

PAD1

GS41

GS51

GS5

GS22

GS30

GS34

INTERC

GS6

GS46

INTERC

GS57

PAD9

GS42

GS39

GS63

GS47

GS26

GS31

INTERC

GS49

GS38

GS8

GS56

GS24

GS48

PAD4

GS43

INTERC

GS37

GS23

INTERC

GS25

GS16

GS40

GS52

GS17

GS33

GS28

GS15

INTERC

GS18

GS32

GS58

GS29

GS89

GS100

GS1

GS67

PAD1

GS62

GS198

INTERC

GS91

GS98

GS87

GS75

GS81

PAD4

GS73

GS90

INTERC

GS66

GS65

GS99

GS86

GS88

GS69

PAD9

GS82

GS71

GS77

GS145

INTERC

GS78

GS84

GS85

GS79

GS80

GS83

GS72

GS74

INTERC

GS64

PAD1

GS68

GS76

GS70

GS60

GS44

PAD1

GS50

GS54

GS35

INTERC

GS95

GS92

PAD9

GS166

GS142

GS159

GS123

GS116

INTERC

GS152

GS148

GS10

GS214

GS190

GS94

GS149

GS114

GS121

GS156

GS120

INTERC

GS153

GS154

GS151

GS181

GS13

GS117

GS115

GS118

INTERC

GS93

GS112

GS126

GS119

GS125

GS113

GS155

PAD4

GS140

GS197

GS209

INTERC

GS164

GS136

GS206

PAD4

GS127

GS134

GS3

GS211

INTERC

GS189

PAD9

GS208

GS157

GS104

GS12

GS135

GS110

GS187

GS193

GS192

INTERC

GS108

GS194

GS133

GS111

GS2

GS160

GS141

GS150

INTERC

GS163

GS137

INTERC

GS143

GS138

GS128

GS144

GS146

GS165

PAD9

GS131

INTERC

GS162

GS107

GS195

GS196

PAD4

GS213

GS4

GS205

INTERC

GS122

GS215

INTERC

GS188

GS109

GS161

GS207

GS147

GS186

GS183

GS185

INTERC

GS130

GS158

GS132

GS53

GS124

GS170

GS169

GS171

INTERC

GS177

GS178

INTERC

GS173

GS176

GS14

GS168

GS172

PAD4

GS175

GS167

INTERC

GS174

GS179

GS180

GS96

GS11

GS182

PAD1

GS184

PAD1

GS105

GS102

INTERC

GS106

GS103

GS97

GS101

GS201

GS199

PAD9

GS200

INTERC

GS202

GS204

GS210

GS203

GS212

GS36

GS19140

60

80

100

120

140

160

180

200

220


D.2 Individual analysis

For the Local 3, the plot “GS51” was analyzed with care, because there was indication ofmeasurement problems. The CTC send to us the lodging score of this experiment and the plot did nothave the high score (classified as medium) but their neighbours “GS41” and “GS46” had. However thereis no information about how much the lodging effect could affect the yield and if it may have causedserious problems in separating the material from different plots at harvest. As “GS51” had an uniqueinformation in this site and the residuals of the models did not have any problems, no plot was removed.

112

But, observing Figure D.5, that there the lodging score, it is possible to note that are more than 15% ofplots with strong lodging (score among 8 and 9).

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Row

Column

1 2 3 4 5 6 7 8 9 10 11 12 13

9

8

7

6

5

4

3

Figure D.5. Heat map relating to fall down for Local 3 of Goiás.

In Table D.2 a sequence of fitted models is presented. We started with the R-TIM model withrandom genetic, column and row effects and AR(1) in both directions. The final model was selected usingthe REML ratio test and the residual plots. At the beginning there is not significant competition effectat the genetic level, but a negative correlation in row direction represents the presence of competition atthe residual level. In sequence, different local and global effects were tested and the chosen model wasModel 19, with: fixed check varieties and harvest effects; spline and linear trend in row direction; randomcolumn and direct genetic effects; and residual structure modeled with AR(1) and Band(2) in column androw direction, respectively. Figure D.6 has the plots of empirical semi-variogram from models 19 and 21.Note that Figure D.6(a) shows a competition effect which is accounted for in Model 21, Figure D.6(b).Model 21 had small residual and direct genetic components when compared with Model 19. Hence, giventhat the neighbour genetic component was not significant using a REML ratio test, Model 21 was notchosen.

113

Tabl

eD

.2.

Sum

mar

yof

the

fitte

dm

odel

sto

Loca

l3of

the

Goi

ásex

perim

entw

ithre

stric

ted

log-

likel

ihoo

d(lo

g-lik

.)an

dth

ep-v

alue

ofth

eR

EML

ratio

test

.A

llm

odel

sinc

lude

ara

ndom

dire

ctge

netic

effec

tfor

the

test

lines

(G)a

ndfix

edch

eck

varie

tiese

ffect

.T

heot

here

ffect

sare

repr

esen

ted

with

the

lett

ers:

H(h

arve

st);

Co

(con

e);S

(sow

ing)

;C(c

olum

n);R

(row

);N

(nei

ghbo

urge

netic

);sp

l(.)

(spl

ine

inso

me

dire

ctio

nin

dica

ted

inpa

rent

hese

s)an

dlin

(.)(li

near

tren

din

som

edi

rect

ion

indi

cate

din

pare

nthe

ses)

.T

hest

ruct

ures

can

be:

US(

.)-u

nstr

uctu

red;

AR

(.)-a

utor

egre

ssiv

e;sa

r(2)

-con

stra

ined

auto

regr

essiv

e;Ba

nd(.)

-ban

ded

corr

elat

ion,

the

num

ber

indi

cate

sth

eor

der

ofco

rrel

atio

n;an

dId

-ide

ntity

.

Mod

elEff

ects

Log-

lik.

Test

p-v

alue

Glo

bal/

extr

aneo

usG

enet

icLo

cal

Fixe

dR

ando

m(c

olum

n×

row

)Ex

tran

eous

(col

umn×

row

)1

R+

CU

S(G

:N)

AR

(1)×

AR

(1)

-929

.19

2R

+C

GA

R(1

)×

AR

(1)

-930

.31

M2

vsM

10.

233

R+

CG

AR

(1)×

Band

(3)

not

conv

erge

d-s

ingu

larit

y4

R+

CG

AR

(1)×

Band

(2)

-928

.80

5R

+C

GA

R(1

)×

Band

(1)

not

conv

erge

d-s

ingu

larit

y6

R+

CG

AR

(1)×

AR

(2)

-929

.70

M6

vsM

20.

277

R+

CG

AR

(1)×

AR

(3)

-929

.01

M7

vsM

60.

248

R+

CG

AR

(1)×

sar(

2)no

tco

nver

ged

-sin

gula

rity

9R

+C

GA

R(2

)×

Band

(2)

-927

.76

M9

vsM

40.

1510

R+

CG

Id×

Band

(2)

-930

.77

M4

vsM

100.

0411

Co

R+

CG

AR

(1)×

Band

(2)

-927

.32

not

signi

fican

t12

HR

+C

GA

R(1

)×

Band

(2)

-925

.16

signi

fican

t13

SR

+C

GA

R(1

)×

Band

(2)

-926

.17

not

signi

fican

t14

H+

lin(C

)+lin

(R)

spl(C

)+C

+sp

l(R)+

RG

AR

(1)×

Band

(2)

-923

.29

15H

+lin

(C)+

lin(R

)sp

l(C)+

C+

spl(R

)G

AR

(1)×

Band

(2)

-923

.39

M14

vsM

150.

3316

H+

lin(C

)+lin

(R)

spl(C

)+sp

l(R)

GA

R(1

)×

Band

(2)

-925

.88

M15

vsM

160.

0117

H+

lin(C

)+lin

(R)

spl(C

)+C

GA

R(1

)×

Band

(2)

-929

.77

M15

vsM

170.

0018

H+

lin(C

)+lin

(R)

C+

spl(R

)G

AR

(1)×

Band

(2)

-923

.58

M15

vsM

180.

2719

H+

lin(R

)C

+sp

l(R)

GA

R(1

)×

Band

(2)

-923

.12

20H

+lin

(R)

C+

spl(R

)U

S(G

:N)

AR

(1)×

Band

(2)

not

conv

erge

d-s

ingu

larit

y21

H+

lin(R

)C

+sp

l(R)

US(

G:N

)A

R(1

)×

AR

(1)

-923

.89

114

02

46

810

12

0

5

10

15

0.0

0.5

1.0

1.5

2.0

Col (lag)

Row (lag)

(a)

02

46

810

12

0

5

10

15

0.0

0.5

1.0

1.5

2.0

2.5

Col (lag)

Row (lag)

(b)

Figure D.6. Plots of the empirical semi-variogram for the residuals for the Local 3 of the Goiásexperiment referring to models 19 and 21.

The REML estimates of the variance parameters for Model 19 are presented in Table D.3. Notethat there is negative correlation in the column direction. However, in this direction there is 1 meter ofspace between plots and the boundary is small (2.4 meters). Then, this negative correlation in the columndirection is unlikely to arise from competition between plots, as it does in the row direction. The directgenetic component is small than the residual component (ratio of 0.84) and the first-order correlation inrow direction is negative and the bigger found in all experiments. The standard error for the spline isbiggest than its value, although the REML ratio test is significant for this term.

Table D.3. REML estimates of variance parameters from fitted Model 19 to the experiment in the Local3 of the Goiás.

Variance parameters Ratios (γ) Estimates Standard errorsColumn (σ̃2

c ) 0.12 39.22 21.62spl(Row) 0.02 7.62 8.80Direct genetic (σ̃2

g) 0.84 269.54 61.19Residual (σ̃2) 1.00 320.09 57.42

Spatial (ρ̃c1) — -0.28 0.11Spatial (ρ̃r1) — -0.61 0.10Spatial (ρ̃r2) — 0.21 0.08

The Wald F tests for the fixed terms are in Table D.4. Note that there are no evidence ofsignificant difference between the group of checks and test lines (Control) or among the checks (Con-trol:Checks). Table D.5 presents the estimated fixed effects for Model 19.

Table D.4. Wald F tests for fixed terms for Model 19. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom.

Fixed terms Df denDF F test p-value(Intercept) 1.00 246.00 3403.00 0.00Control 1.00 246.00 0.92 0.34colheita 1.00 246.00 5.86 0.02lin(Row) 1.00 246.00 0.11 0.74Control:Check 3.00 246.00 1.95 0.12

115

Table D.5. Estimates of the fixed effects for Model 19 and their respective standard errors.

Effects Estimates Standard errorsPAD1 -7.75 6.46PAD4 -9.47 6.48PAD9 8.60 6.63lin(Row) -0.09 0.16harvest-level 1 0.00 —harvest-level 2 8.51 3.16overall test lines 0.00 —overall check 17.04 16.75overall mean (µ) 122.58 3.11

The 15 best test lines with predicted values and respective standard errors from Model 19are presented in Table D.6. When comparing the genetic EBLUPs values from this model with the onesfrom the traditional non-spatial analysis, which contains fixed checks varieties and random direct genetic,column and row effects, there are some trends and the correlation between them is 0.854. See Figure D.7.

Table D.6. The 15 best test lines with predicted values and respective standard errors from Model 19.

Test lines Predicted values Standard errorsGS51 186.41 10.93GS204 167.82 11.01GS87 160.93 10.66GS53 160.15 10.39GS132 157.63 11.08GS99 156.24 11.12GS151 155.64 11.61GS141 155.21 11.16GS210 154.84 11.17GS2 154.50 11.09GS79 154.15 10.98GS43 153.41 10.57GS102 152.55 11.37GS165 150.69 10.53GS201 149.94 11.33

−20 −10 0 10 20 30 40

−20

020

40


Gen

etic

EB

LUP

s −

sel

ecte

d m

odel

cor = 0.854

Figure D.7. Dispersion plot of the genetic EBLUPs values of the Model 19 and the traditional non-spatial analysis model for this design (without spatial dependence) for Goiás Local 3. The cut-offs forthe 15 best test lines (7 % upper) in each model are indicated by the dotted line.

116

For Local 521 some models were tested and some of them are presented in Table D.7. Withthis data set several models had convergence problems. Then, it was difficult to decide about what effectshould be true and which was confounded effect. To assess the local effect, the genetic competition andcorrelation genetic were removed in some models to obtain convergence, but these were not efficient for allstructures tested. Using REML ratio test and residuals plots, the Model 20 was chosen as the best to fitto the data. It assumes: fixed check varieties; linear trend in the column and row directions; row spline;random direct, neighbour and direct-neighbour covariance genetic effects. See in Figure D.8 the plots ofempirical semi-variogram from fitted model 20 and 23, where we did not assume the competition at thegenetic level. Table D.8 presents the REML estimates of variance parameters from this fitted models.

117

Tabl

eD

.7.

Sum

mar

yof

the

fitte

dm

odel

sto

Loca

l521

ofth

eG

oiás

expe

rimen

tw

ithre

stric

ted

log-

likel

ihoo

d(lo

g-lik

.)an

dth

ep-v

alue

ofth

eR

EML

ratio

test

.A

llth

em

odel

sin

clud

ea

rand

omdi

rect

gene

ticeff

ects

for

the

test

lines

(G)

and

fixed

chec

kva

rietie

seff

ects

.T

heot

her

effec

tsar

ere

pres

ente

dw

ithth

ele

tter

s:H

(har

vest

);C

o(co

ne);

S(s

owin

g);C

(col

umn)

;R(r

ow);

N(n

eigh

bour

gene

tic);

spl(.

)(s

plin

ein

som

edi

rect

ion

indi

cate

din

pare

nthe

ses)

and

lin(.)

(line

artr

end

inso

me

dire

ctio

nin

dica

ted

inpa

rent

hese

s).

The

stru

ctur

esca

nbe

:US(

.)-un

stru

ctur

ed;R

R(.)

-red

uced

rank

;AR

(.)-a

utor

egre

ssiv

e;Ba

nd(.)

-ban

ded

corr

elat

ion,

the

num

ber

indi

cate

sth

eor

der

ofco

rrel

atio

n;sa

r(2)

-con

stra

ined

auto

regr

essiv

e;an

dId

-ide

ntity

.

Mod

elEff

ects

Log-

lik.

Test

p-v

alue

Glo

bal/

extr

aneo

usG

enet

icLo

cal

fixed

rand

om(c

olum

n×

row

)1

R+

CU

S(G

:N)

AR

(1)×

AR

(1)

-901

.42

2R

+C

RR

(G:N

)A

R(1

)×

AR

(1)

-903

.54

M2

vsM

10.

023

R+

CG

AR

(1)×

AR

(1)

-904

.18

M3

vsM

10.

044

R+

CU

S(G

:N)

AR

(1)×

Band

(3)

not

conv

erge

d-s

ingu

larit

y5

R+

CU

S(G

:N)

AR

(1)×

Band

(2)

not

conv

erge

d-s

ingu

larit

y6

R+

CU

S(G

:N)

AR

(1)×

Band

(1)

not

conv

erge

d-s

ingu

larit

y7

R+

CU

S(G

:N)

AR

(1)×

ar(2

)no

tco

nver

ged

-sin

gula

rity

8R

+C

US(

G:N

)A

R(1

)×

sar(

2)no

tco

nver

ged

-sin

gula

rity

9R

+C

GA

R(1

)×

Band

(3)

-903

.70

10R

+C

GA

R(1

)×

Band

(2)

not

conv

erge

d-s

ingu

larit

y11

R+

CG

AR

(1)×

Band

(1)

not

conv

erge

d-s

ingu

larit

y12

R+

CG

AR

(1)×

AR

(2)

-904

.02

M12

vsM

30.

5713

R+

CG

AR

(1)×

sar(

2)pr

oble

ms

ofco

nver

genc

e14

R+

CU

S(G

:N)

AR

(1)×

Id-9

01.6

2M

1vs

M14

0.52

15R

+C

US(

G:N

)Id

×Id

-902

.40

M14

vsM

150.

2116

Co

R+

CU

S(G

:N)

Id×

Id-9

01.0

1no

tsig

nific

ant

17H

R+

CU

S(G

:N)

Id×

Id-9

00.8

8no

tsig

nific

ant

18S

R+

CU

S(G

:N)

Id×

Id-9

00.9

8no

tsig

nific

ant

19lin

(C)+

lin(R

)sp

l(C)+

C+

spl(R

)+R

US(

G:N

)Id

×Id

-900

.40

20lin

(C)+

lin(R

)C

+sp

l(R)

US(

G:N

)Id

×Id

-900

.40

M19

vsM

201.

0021

lin(C

)+lin

(R)

CU

S(G

:N)

Id×

Id-9

02.1

5M

20vs

M21

0.03

22lin

(C)+

lin(R

)sp

l(R)

US(

G:N

)Id

×Id

-902

.28

M20

vsM

220.

0223

lin(C

)+lin

(R)

C+

spl(R

)G

Id×

Id-9

14.4

7M

20vs

M23

0.00

24lin

(C)+

lin(R

)C

+sp

l(R)

RR

(G:N

)Id

×Id

-914

.47

M20

vsM

240.

04

118

02

46

810

12

0

5

10

15

0.0

0.5

1.0

Col (lag)

Row (lag)

(a)

02

46

810

12

0

5

10

15

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Col (lag)

Row (lag)

(b)

Figure D.8. Plots of the empirical semi-variogram for the residuals for Local 521 of Goiás experimentreferring to Models (a) 20 and (b) 23.

Table D.8. REML estimates of variance parameters for Model 20 fitted to the data from Local 521 fromthe Goiás experiment.

Variance parameters Ratios (γ) Estimates Standard errorsColumn (σ̃2

c ) 0.18 15.91 12.21spl(Row) 0.41 36.33 34.31Direct genetic (σ̃2

g) 3.44 307.26 60.98Covariance genetic (σ̃gn) -0.97 -86.31 20.46Neighbour genetic (σ̃2

n) 0.67 59.76 25.39Residual (σ̃2) 1.00 89.35 47.76

Table D.9 shows the Wald F tests for the fixed terms and Table D.10 the estimated fixed effectsfrom the chosen model. Note that there are no evidences of significant difference between the group ofchecks with test lines (Control) or among the checks (Control:Checks).

Table D.9. Wald F tests for fixed terms from Model 20. Df means degrees of freedom and DenDF isthe approximate denominator degrees of freedom.

Fixed terms Df denDF F test p-value(Intercept) 1.00 15.60 4147.00 0.00Control 1.00 53.30 2.23 0.14lin(Col) 1.00 10.70 12.70 0.00lin(Row) 1.00 57.00 0.04 0.85Control:Check 3.00 34.20 1.72 0.18

Table D.10. Estimates of the fixed effects for Model 20 and their respective standard errors.

Effects Estimates Standard errorsINTERC 0.00 —PAD1 -2.78 5.82PAD4 -12.59 5.65PAD9 -3.93 5.67lin(Row) -0.03 0.16lin(Col) 1.40 0.39overall test lines 0.00 —overall checks 28.70 17.74overall mean (µ) 93.94 3.61

119

The 15 best test lines from Model 20 with predicted values and respective standard errors arepresented in Table D.11. Figure D.9 compares the genetic EBLUPs from Model 20 and the ones from thetraditional non-spatial analysis, that contain fixed checks varieties and random direct genetic, row andcolumn effects; there is a linear trend and the correlation between them is 0.884. However, note that thegroup of selected test lines is not the same.

Table D.11. The 15 best test lines with predicted values and respective standard errors from Model 20.


−20 0 20 40

−20

020

40


Gen

etic

EB

LUP

s −

sel

ecte

d m

odel

cor = 0.971


In Local 533 several models were tested and they are presented in Table D.12. In the begin-ning, it was possible observe the genetic effects as a special case nominated reduced rank. However theneighbour genetic component is not significant using REML ratio test. Hence, some new models weredeveloped with different local and global effects. Using the REML ratio test and residual plots the Model15 is chosen that best explains the data.

120Ta

ble

D.1

2.Su

mm

ary

ofth

efit

ted

mod

elst

oLo

cal5

33of

the

Goi

ásex

perim

entw

ithre

stric

ted

log-

likel

ihoo

d(lo

g-lik

.)an

dth

ep-v

alue

ofth

eR

EML

ratio

test

.A

llth

em

odel

sin

clud

ea

rand

omdi

rect

gene

ticeff

ects

for

the

test

lines

(G)

and

fixed

chec

kva

rietie

seff

ects

.T

heot

her

effec

tsar

ere

pres

ente

dw

ithth

ele

tter

s:H

(har

vest

);C

o(c

one)

;S(s

owin

g);C

(col

umn)

;R(r

ow);

N(n

eigh

bour

gene

tic);

spl(.

)(s

plin

ein

som

edi

rect

ion

indi

cate

din

pare

nthe

ses)

and

lin(.)

(line

artr

end

inso

me

dire

ctio

nin

dica

ted

inpa

rent

hese

s).

The

stru

ctur

esca

nbe

:R

R(.)

-red

uced

rank

;AR

(.)-fi

rst-

orde

rau

tore

gres

sive;

sar(

2)-c

onst

rain

edau

tore

gres

sive;

Band

(.)-b

ande

dco

rrel

atio

n,th

enu

mbe

rin

dica

tes

the

orde

rof

corr

elat

ion;

and

Id-i

dent

ity.

Mod

elEff

ects

Log-

lik.

Test

p-v

alue

Glo

bal/

extr

aneo

usG

enet

icLo

cal

Fixe

dR

ando

m(c

olum

n×

row

)1

RR

R(G

:N)

AR

(1)×

AR

(1)

-931

.95

2R

GA

R(1

)×

AR

(1)

-932

.15

M2

vsM

10.

523

RG

AR

(1)×

Band

(3)

not

conv

erge

d-s

ingu

larit

y4

RG

AR

(1)×

Band

(2)

-933

.52

5R

GA

R(1

)×

AR

(2)

-932

.00

M5

vsM

20.

586

RG

AR

(1)×

sar(

2)-9

31.9

87

RG

Id×

AR

(1)

-932

.37

M2

vsM

70.

508

Co

RG

Id×

AR

(1)

-930

.79

not

signi

fican

t9

HR

GId

×A

R(1

)-9

30.7

5no

tsig

nific

ant

10S

RG

Id×

AR

(1)

-930

.99

not

signi

fican

t11

lin(C

)+lin

(R)

spl(C

)+sp

l(R)+

RG

Id×

AR

(1)

-929

.94

12lin

(C)+

lin(R

)sp

l(C)+

spl(R

)G

Id×

AR

(1)

-929

.94

M11

vsM

121.

0013

lin(C

)+lin

(R)

spl(R

)G

Id×

AR

(1)

-930

.13

M12

vsM

130.

2714

lin(C

)+lin

(R)

GId

×A

R(1

)-9

32.4

1M

13vs

M14

0.01

15lin

(R)

spl(R

)G

Id×

AR

(1)

-929

.34

16lin

(R)

spl(R

)R

R(G

:N)

Id×

AR

(1)

-929

.23

M16

vsM

150.

64

121

The plots of the empirical semi-variogram for the Model 15 are presented in Figure D.10. Itis observed that there is no strong trend given that the first neighbours show the flat residual effectsin the graphics. The REML estimates of the variance parameters are presented in Table D.13. Hereit was found more direct genetic component than residual (1.35 ratio) and also negative correlation inrow direction. The standard error of row splines is a bit bigger than its component, but this term wassignificant when assessed using the REML ratio test.

02

46

810

12

0

5

10

15

0.0

0.5

1.0

Col (lag)

Row (lag)

Figure D.10. Plots of the empirical semi-variogram for residuals from Model 15 for Local 533 of theGoiás experiment.

Table D.13. REML estimates of variance parameters from fitted Model 15 for the experiment in theLocal 533 of the Goiás.


g) 1.35 353.97 69.20Residual (σ̃2) 1.00 262.24 52.31

Spatial (ρ̃r) — -0.53 0.10

Table D.14 shows the Wald F tests for fixed terms and Table D.15 the estimated fixed effectsfrom this model. Note that there is no evidence of significant difference between the group of checks andtest lines (Control), but there is among the checks (Control:Checks).

Table D.14. Wald F tests for fixed terms from Model 15. Df means degrees of freedom and DenDF isthe approximate denominator degrees of freedom.

Fixed terms Df denDF F test p-value(Intercept) 1.00 141.80 8337.00 0.00Control 1.00 55.30 0.94 0.34lin(Row) 1.00 196.50 10.74 0.00Control:Check 3.00 49.00 18.25 0.00

122

Table D.15. Estimates of the fixed effects from Model 15 and their respective standard errors.

Effects Estimates Standard errorsINTERC 0.00 —PAD1 -36.85 6.60PAD4 -33.11 6.39PAD9 3.77 6.40lin(Row) -0.65 0.19overall test lines 0.00 —overall checks 26.46 19.10overall mean (µ) 142.18 2.52

The 15 best test lines from Model 15 with predicted values and respective standard errors arepresented in Table D.16. Comparing the genetic EBLUPs values from Model 15 and the traditional non-spatial analysis, that contain fixed checks varieties and random direct genetic, column and row effects,in Figure D.11, there is a linear trend and the correlation between them is 0.941. However, note that thegroup of selected test lines is not the same.

Table D.16. The 15 best test lines with predicted values and respective standard errors for Model 15.


−40 −20 0 20 40

−40

−20

020

4060


Gen

etic

EB

LUP

s −

sel

ecte

d m

odel

cor = 0.933


123

D.3 Joint analysis

Based on the separate analysis for each Local, it was observed that the genetic EBLUPs valuesare very different. Figure D.12 presents the relation of these values between pairs of Locals; the correlationis close to zero. This means that there is a Local effect on the test lines and the correlation among geneticeffects should be very small.

−20 0 20 40

−20

020

40


Gen

etic

EB

LUP

s −

Loc

al 5

21 cor = 0.171

(a)

−20 0 20 40

−40

−20

020

4060


Gen

etic

EB

LUP

s −

Loc

al 5

33 cor = −0.019

(b)

−20 0 20 40

−40

−20

020

4060


Gen

etic

EB

LUP

s −

Loc

al 5

33 cor = 0.032

(c)

Figure D.12. Dispersion plots for the genetic EBLUPs values from model selected for each Local inGoiás. Panels the relationship between pairs of Locals (a) 3 and 521; (b) 3 and 533; and (c) 521 and 533.The cut-offs for the 15 best test lines in each Local are indicated by the dotted line.

Hence, based on the results of the individual analyses, Local 521 is the ones that exhibits moregenetic correlation with the other Locals. However, we believe that the chosen model can be biased toestimate the neighbour genetic effects. The residual component is much smaller at Local 521, almost 3times greater than that found for the other Locals. We considered that some components as residual andneighbour genetic can be wrong estimate (underestimate and overestimate) and because of this Local 521will not be include in the joint analysis.

Locals 3 and 533 were joint analyzed in order to investigate and to compare the magnitudes ofgenetic variance, assessing if there was genetic covariance between Locals. To do this the models shownin Table D.17 were tested and the results of the REML tests are also given in this Table.

124

Table D.17. Summary of the fitted models to joint analysis of the Goiás experiments carried out inthe Locals 3 (1) and 533 (3), with REML test (log-lik.) and the p-value of the REML ratio test. All themodels have the same effects as modeled in the individual analysis plus the fixed Local effects. Here σ2

gi

is the genetic variance at the ith Local, σgii′ is the genetic covariance between the ith and i′th Localswhich have been hypothesized to be equal for two or more pairs of Locals.

Model Genetic effects Log-lik. Test p-value

1[σ2g1 00 σ2

g3

]-1852.47

2[σ2g1 σg13

σg13 σ2g3

]-1847.32 M2 vs M1 0.00

3[σ2g ρσ2

g

ρσ2g σ2

g

]-1847.86 M2 vs M3 0.15

Model 3 was chosen as the best to explain the data. It has the same environmental parameters,plus Local fixed effect and correlation between the direct genetic effects in Locals 3 and 533. Table D.18showed the REML estimates of the variance parameters and their respective standard error from fittedModel 3.

Table D.18. REML estimates of variance parameters for fitted Model 3 to the joint analysis of theLocal 3 (1) and 533 (3) of the experiment carried out in Goiás.

Variance parameters Estimates Standard errorsCorrelation between genetic effects (ρ̃g) 0.39 0.12direct genetic (σ̃2

g) 299.81 48.73Local 3:Col (σ̃2

c1) 35.86 20.39Local 3:Spline(Row) — 13.50 15.41Local 533:Spline(Row) — 19.59 22.82Local 3:Residual (σ̃2

1) 292.25 50.32Spatial:column direction (ρ̃c1) -0.29 0.11Spatial:row direction (ρ̃r11) -0.57 0.10Spatial:row direction (ρ̃r12) 0.18 0.09

Local 533:Residual (σ̃23) 284.01 51.23

Spatial:row direction (ρ̃r31) -0.44 0.09

The Wald F tests for fixed terms in Model 3 are presented in Table D.19. Note that there is nota significant difference between the Locals. The predicted values with their standard error of the 15 besttest lines of the Model 3 for each local are in Table D.20. It can be observed that in the joint analysis theprecision in the selection of the test lines is similar than the individual analysis. For some test lines, thepredicted values are a bit higher and standard error are smaller, when compared with individual analysis(Tables D.6 and D.16).

Table D.19. Wald F tests for the fixed terms in Model 3. Df means degrees of freedom and DenDF isthe approximate denominator degrees of freedom.

Fixed terms Df denDF F test p-value(Intercept) 1.00 23.80 10680.00 0.00Local 1.00 1.10 16.51 0.14Control 1.00 0.00 1.45 0.14Control:Check 3.00 47.80 12.67 0.00Local:Control 1.00 0.00 0.01 0.00Local 3:harvest 1.00 0.00 8.75 0.00Local:lin(Row) 2.00 — 6.15 —Local:Control:Check 3.00 76.60 4.54 0.01

125

Table D.20. The 15 best test lines with predicted values and respective standard error from Model 3for each Local.

Local 3 Local 533Test lines Pred.value Stand.error Test lines Pred.value Stand.errorGS51 192.03 10.96 GS194 186.42 11.18GS204 168.00 11.09 GS189 178.99 11.18GS87 163.80 10.73 GS211 169.92 11.24GS132 161.88 11.15 GS40 167.16 11.39GS2 160.26 11.17 GS210 166.21 11.45GS99 159.52 11.19 GS213 165.57 11.32GS210 159.43 11.23 GS173 165.04 11.25GS151 158.11 11.61 GS64 162.94 11.06GS53 157.65 10.50 GS13 158.93 11.23GS79 156.90 11.00 GS132 158.61 11.37GS162 155.42 10.99 GS198 158.16 11.24GS198 154.37 11.15 GS176 156.29 11.41GS141 153.66 11.23 GS51 155.84 11.38GS102 153.07 11.39 GS87 155.49 11.36GS48 153.00 11.13 GS157 154.31 11.87

D.4 Conclusions

The individual analyses resulted in different models for Locals. The main difference was theselected models: for Local 3 and 533, there were residual competition effects in the row direction, whilefor Local 521 there was no significant residual correlation, but competition at the genetic level. Thesame test lines were assessed in different sites although the ratios between direct genetic and residualcomponents were different: they were 0.84, 3.44 and 1.35 for Local 3, 521 and 533, respectively. It wasexpected that the models with genetic competition effect would have lower residual and high geneticcomponents. However, this was the only experiment for which there was a significant competition effectand it is extraordinary that the direct genetic ratio is so much larger for this Local when compared toother Locals, given that the same test lines were used as all Locals. Consequently a simulation study willbe conducted for this case.

The selected model for Local 3 included fixed effect for harvest and random effects for a rowspline; column effects and direct genetic effects; residual correlation was modeled with AR(1) and Band(2)in column and row direction, respectively. For Local 521, it was included term for a column linear trend,a row spline, random column effects and unstructured genetics effects (direct and neighbour variancesand their covariance). For Local 533, the model included a row spline term and direct genetic effectswith residual correlation only in row direction modeled with AR(1) structure.

In the joint analysis, for Locals 3 and 533, the selected model indicates a common geneticcomponent for Locals (σ2

g = 299.81) and a genetic correlation term (ρg = 0.39). The number of commontest lines considering the 15 best lines is a little bigger; 5 of the 15 best predicted values for the test lineswere common for the Locals. That is, the joint analysis did not result in greater accuracy of selection,as is to be expected given the low genetic correlation between Locals.

126

127

E RIBEIRÃO - GROUP OF EXPERIMENTS

There were three sites, nominated: Local 20, 72 and 140. Each site had different rectangulararray and number of filler plot. For Local 20, there were 19 columns by 28 rows; Local 72, 21 columns by25 rows, and, Local 140, 14 columns by 38 rows. All assessed Locals had the same 429 test lines (R) with4 commercial varieties nominated “PAD1”, “PAD2”, “PAD3” and “TI” or “INTER”. The experimentswas planted in 2014 with the yield in TCH obtained in 2015, in a unique cut. The results were organizedin subsections where E.1 presents the descriptive analysis of the sites, E.2 shows the individual analysesfor each site and E.3 gives the joint analysis. The concluding remarks are in subsection E.4.

E.1 Descriptive analysis

In the descriptive analysis presented in Table E.1, it is possible to observe that the commercialvariety “PAD2” is not so stable, the average yield was close to the other groups, but it has the highestvariance in all sites. For Local 72 the interspersed group presents also greater TCH variance, endingup with the highest TCH value for that Local. Considering the mean values, it seems there is no greatdifference among sites, however Local 140 is the one with had smaller yield average. Figure E.1 showsthat some plots can be considered outliers. Then, it can expected that some environmental effects haveinfluenced the productivity and the residuals from the selected models should be examined. On the otherhand, the test lines outliers are expected, mainly because our interest is selection of the best sugarcanelines.

Table E.1. Descriptive analysis of the groups of clones carried out in Ribeirão.

Group of clones Minimum Maximum Mean Variance

Loca

l20 Interspersed (TI or INTER) 100.00 158.30 128.63 155.32

PAD1 90.50 127.40 112.38 172.07PAD2 66.70 133.30 99.76 527.39PAD3 107.10 145.20 124.28 170.20Test lines (R) 66.70 173.80 111.96 351.49

Loca

l72 Interspersed (TI or INTER) 71.40 210.70 121.28 577.45

PAD1 103.60 152.40 127.60 218.82PAD2 81.00 175.00 110.95 657.55PAD3 95.20 147.60 131.29 187.89Test lines (R) 58.30 246.40 125.72 508.83

Loca

l140

Interspersed (TI or INTER) 78.60 138.10 109.62 129.95PAD1 72.60 121.40 102.86 316.88PAD2 63.10 136.90 85.48 466.35PAD3 95.20 135.70 108.56 183.43Test lines (R) 57.10 167.90 106.67 431.66

128T

CH

50

100

150

200

250

INTER PAD1 PAD2 PAD3 R

20


72


140

Figure E.1. Boxplot of TCH for each group of lines for Locals of Ribeirão.

Figures E.2, E.3 and E.4 show the heat maps relating to TCH in each Local. The names“PAD1”, “PAD2”, “PAD3” and “TI” are the check plots; the numbers refer to the test lines and ´´filler”indicate plots that were ignored in the experiments. Figure E.3 reports that line 193 has a higher yieldthan the rest and it is not clear that there is some trend in this area.

Column

Row

TI

PAD2

PAD1

PAD3

127

128

138

112

110

TI

199

192

194

121

43

56

41

38

TI

288

39

53

37

34

54

11

49

TI

213

TI

142

144

214

212

124

202

201

132

TI

126

131

125

57

28

35

45

33

TI

40

294

30

14

13

23

31

50

134

133

TI

211

310

140

139

205

218

149

120

TI

143

154

18

22

46

20

21

48

TI

PAD2

PAD1

PAD3

2

32

55

309

151

148

147

TI

123

136

217

185

135

311

159

209

TI

210

307

279

36

278

265

304

47

TI

17

302

283

293

19

52

160

164

220

158

TI

145

141

137

155

44

152

146

74

TI

247

308

246

PAD1

PAD2

PAD3

12

275

TI

303

277

287

10

286

163

90

150

77

92

TI

60

89

61

96

165

312

161

219

TI

26

29

25

51

301

300

237

7

TI

253

299

225

9

315

PAD2

PAD1

PAD3

162

222

TI

203

198

153

93

97

166

167

24

TI

262

281

252

306

280

290

8

276

TI

284

270

254

98

80

100

73

157

99

206

TI

197

156

75

175

169

314

228

257

TI

241

251

239

274

295

256

297

285

TI

272

305

82

313

168

174

177

215

101

91

TI

3

94

PAD1

PAD2

PAD3

259

249

245

TI

269

296

238

298

16

273

291

231

TI

232

TI

85

84

83

81

72

223

207

178

TI

173

172

176

95

235

266

6

267

TI

264

271

243

244

248

255

292

250

TI

216

TI

170

171

181

68

204

195

208

78

TI

86

88

87

226

236

258

268

233

TI

242

229

224

289

282

261

15

227

76

79

TI

190

70

63

196

71

64

180

184

TI

182

179

108

104

107

106

102

230

TI

260

234

PAD3

PAD2

PAD1

240

263

1

183

69

TI

58

42

66

62

65

221

67

59

TI

27

113

116

105

188

103

186

189

TI

115

109

114

5

117

4

372

370

379

373

TI

382

380

381

375

376

378

383

384

TI

193

191

119

PAD2

PAD1

PAD3

118

200

TI

122

130

129

111

187

374

369

PAD2

PAD3

PAD1

TI

371

349

340

347

368

348

345

344

TI

395

414

416

396

397

391

390

385

TI

393

389

386

387

335

399

317

341

339

350

TI

336

338

337

342

420

330

320

394

TI

392

417

367

415

413

346

412

411

TI

409

408

364

388

377

366

343

355

319

325

TI

334

323

324

332

333

PAD3

PAD1

PAD2

TI

404

362

360

361

402

410

365

406

TI

405

407

424

PAD2

PAD3

PAD1

426

425

328

429

TI

427

331

329

321

322

359

363

401

TI

358

356

403

400

354

398

352

353

TI

351

filler

filler

filler

filler

428

423

327

421

422

TI

316

318

419

418

326

357

filler

filler

filler

filler

filler

filler

filler

filler

filler

filler

filler

filler 60

80

100

120

140

160

180

Figure E.2. Heat map relating to TCH for Local 20 of Ribeirão.

129

Column

Row

TI

331

422

320

423

424

414

426

425

TI

413

417

332

PAD2

PAD1

PAD3

333

334

TI

316

328

329

327

323

322

352

TI

390

363

408

391

359

411

325

321

TI

318

410

392

362

393

360

361

404

TI

402

324

412

403

401

389

353

TI

379

351

342

348

340

350

373

372

TI

319

336

370

420

388

400

398

364

TI

365

387

383

367

386

384

378

TI

385

349

PAD3

PAD2

PAD1

376

371

347

TI

343

377

338

317

341

369

330

421

TI

429

326

416

418

419

427

428

TI

415

407

346

406

356

358

405

357

TI

409

354

397

396

395

394

381

344

TI

345

374

243

238

235

237

233

TI

229

244

240

228

355

366

399

335

TI

368

337

380

375

382

PAD3

PAD2

PAD1

TI

339

6

236

232

226

241

224

TI

239

227

272

260

234

230

280

242

TI

225

270

231

258

269

278

257

14

TI

279

26

7

251

250

12

286

TI

13

273

PAD2

PAD1

PAD3

246

276

274

TI

255

267

281

254

245

249

287

256

268

110

248

264

105

261

108

176

TI

263

252

313

106

61

114

221

186

TI

113

111

72

193

201

192

112

TI

98

122

PAD3

PAD2

PAD1

132

188

74

TI

115

128

202

130

97

145

81

119

TI

199

194

60

123

66

116

117

TI

103

118

247

190

253

200

104

131

TI

121

107

102

191

109

133

73

129

TI

4

187

83

126

209

210

100

TI

138

216

92

141

136

96

89

134

TI

212

77

142

101

99

135

91

139

TI

5

127

124

88

218

152

90

TI

137

94

150

147

215

151

217

160

TI

93

143

95

211

PAD3

PAD2

PAD1

213

TI

125

310

223

158

166

222

87

TI

311

161

149

205

140

153

204

207

TI

148

312

208

120

154

220

86

219

TI

146

156

184

214

144

206

181

TI

PAD3

PAD2

PAD1

159

155

84

163

315

TI

182

197

157

162

78

170

167

175

TI

65

185

85

3

179

28

62

TI

34

173

76

171

172

183

79

164

TI

75

165

169

314

180

174

203

168

TI

198

82

64

32

63

67

80

TI

40

52

36

38

37

259

35

178

TI

31

71

177

33

189

PAD3

PAD2

PAD1

45

15

2

48

30

50

27

51

TI

22

57

55

56

29

59

69

25

TI

68

58

23

70

54

24

53

TI

PAD3

PAD2

PAD1

39

47

49

288

42

TI

44

46

308

290

43

304

282

8

TI

41

296

303

11

275

299

283

TI

284

17

262

19

292

265

266

291

TI

302

294

293

271

295

297

9

277

TI

300

289

10

301

298

20

21

TI

18

285

16

309

305

306

307

195

TI

196

filler

PAD3

PAD2

PAD1

filler

filler

filler

filler

filler

1

filler

filler 50

100

150

200

250


Column

Row

TI6066939091929689TI7497

2216873778367TI

255243244232238257245251TI

252268239229

PAD3PAD1PAD2

6TI

filler

101TI8272863

87886180TI99595895949862

260TI

237233226242236235234256TI

230248228227241224250266TI

10057TI5664848544817875TI76

PAD2PAD3PAD1

7951

249175TI

267177169312179182178181TI

180184170149185183253225

541840TI3941424337492

50TI6569717063

261162176TI

172152171

PAD1PAD2PAD3315146TI

154222167168165158153

55345253TI4538484716201921TI12

30415

29946

157159126TI

142160173124143217213211TI

310219313164166314

27729717

302280TI

3032982963632352922TI2826

156PAD1PAD2PAD3174163TI

144151161311220150214145TI

134140141155147

93331111330TI

3082527

30730528627523TI14

131188190129148130205TI

192191138136200202127199TI

137139212128

282285283284288309300TI24

306301278292

8291295TI10

122108104105125123201TI

133210209135

PAD1PAD2PAD3194TI

118132109

281276294


273271270269

7287290289TI

1865

119116206

4115193TI

121113196103189107187106TI

102195

TI265264262258259279240263TI

246231254


204197203117120198216114TI

223112208110111207215218TI1

423TI

326418412419421422318408TI

332329328334316327417388TI

369341339


324325322346353352410411TI

359361TI

360362424364365367407409TI

414415416331333413401321TI

323426345381383379398342TI

335336317376374351375354

357397363TI

396395389392391350368338TI

355320358356

PAD1PAD3PAD2387TI

386385400429380382347348TI

402403427404390330420

399340372370TI

PAD2PAD1PAD3371377349319343TI

366337393394406344428425TI

384405fillerfillerfillerfillerfillerfillerfillerfillerfillerfillerfillerfillerfiller

60

80

100

120

140

160


E.2 Individual analysis

For Local 20, the algorithms for the models with neighbour genetic effect did not converge,there were singularity problems with the Average Information Matrix in fitting the R-TIM model. TableE.2 presents the sequence of models fitted to find one that describes the global and local effects for Local20. Ultimately, Model 14 was chosen and had: linear trend in row and column factor; random row anddirect genetic effects; and residual with first-order banded correlation in the row direction. The row and

130

column faces of the empirical variogram, presented in Figure E.5, shows the residuals for the Models 2and 14.

131

Tabl

eE

.2.

Sum

mar

yof

the

fitte

dm

odel

sfor

Loca

l20

with

rest

ricte

dlo

g-lik

elih

ood

(log-

lik.)

and

thep-v

alue

fort

heR

EML

ratio

test

.A

llth

em

odel

sinc

lude

ara

ndom

dire

ctge

netic

effec

tsfo

rth

ete

stlin

es(G

)an

dfix

edch

ecks

varie

ties

effec

ts.

The

othe

reff

ects

are

repr

esen

ted

with

the

sym

bols:

H(h

arve

st);

Co

(con

e);

S(s

owin

g);

C(c

olum

n);

R(r

ow);

N(n

eigh

bour

gene

tic);

spl(.

)(s

plin

ein

som

edi

rect

ion

indi

cate

din

pare

nthe

ses)

and

lin(.)

(line

artr

end

inso

me

dire

ctio

nin

dica

ted

inpa

rent

hese

s).

The

stru

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132

0.0

0.5

1.0

1.5

0 5 10 15COL differences

Variogram face of Standardized conditional residuals for COL

(a)

0.0

0.5

1.0

1.5

2.0

0 5 10 15COL differences

Variogram face of Standardized conditional residuals for COL

(b)

0.0

0.5

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Variogram face of Standardized conditional residuals for ROW

(c)

0.0

0.5

1.0

1.5

0 10 20ROW differences

Variogram face of Standardized conditional residuals for ROW

(d)

Figure E.5. Plots of the column and row faces of the empirical variogram for the residuals for the Local20 of the Ribeirão experiment for models 2 (panels (a) and (c)), and 14 (panels (b) and (d)). The panels(a) and (b) are for the column direction and the others for row.

The REML estimates of variance parameters from fitted Model 14 are presented in Table E.3.Note that the genetic component is smaller than the residual (this ratio is 0.85) and the row correlationis negative, it indicates competition at the residual level.

Table E.3. REML estimates of variance parameters for fitted Model 14 for the experiment in Local 20from Ribeirão.

Variance parameters Ratios (γ) Estimates Standard errorsRow (σ̃2


g) 0.85 151.18 32.02Residual (σ̃2) 1.00 178.00 26.34

Spatial (ρ̃r) — -0.35 0.06

Table E.4 gives the Wald F tests for the fixed terms in Model 14 and the coefficients of thefixed effects with their respective standard errors are in Table F.5. With the test, there is no evidence ofsignificant difference between the group of test lines and checks (Control), but there is within the checks(Control:Check).

Table E.4. Wald F test for fixed terms for Model 14. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom.

Fixed terms Df denDF F test p-value(Intercept) 1.00 505.00 12940.00 0.00Control 1.00 505.00 0.38 0.54lin(COL) 1.00 505.00 7.53 0.01lin(ROW) 1.00 505.00 6.58 0.01Control:Check 3.00 505.00 21.76 0.00

133

Table E.5. Estimates of the fixed effects for Model 14 and their respective standard errors.

Effects Estimates Standard errorsTI 18.74 4.46PAD1 0.00 —PAD2 -10.14 6.56PAD3 14.25 6.50lin(ROW) -0.31 0.11lin(COL) 0.30 0.10overall Checks 0.00 —overall Test lines 2.16 12.99overall mean (µ) 111.39 13.11

The 30 best test lines with predicted values and respective standard errors, using Model 14, arepresented in Table E.6. The correlation between the EBLUPs from selected model 14 and the traditionalmodel which has fixed checks effects and random direct genetic, row and column effects, is high, 0.953,but as it can be possible to observe in Figure E.8 that the group of select test lines is different in eachmodel.

Table E.6. The 30 best test lines with predicted values and respective standard errors for Model 14.

Test lines Pred. values Stand. errors Test lines Pred. values Stand. errors350 138.85 8.72 8 127.58 8.8449 137.65 8.77 314 127.54 8.84

358 133.67 8.72 43 127.14 8.8761 133.49 8.85 97 126.84 8.84

282 133.36 8.85 151 126.62 9.02375 131.99 8.85 69 126.41 8.70215 131.89 8.84 291 126.28 8.84323 130.92 8.85 381 125.60 8.85220 130.10 8.84 166 125.60 8.84364 129.89 9.02 356 125.55 8.86259 129.15 8.70 380 125.43 8.84384 128.91 8.70 293 125.32 8.86411 128.90 8.72 130 124.88 8.8539 128.17 8.87 412 124.87 8.85

429 128.03 8.73 117 124.77 8.86

−20 −10 0 10 20

−20

−10

010

20


EB

LUP

s −

sel

ecte

d m

odel cor = 0.953

Figure E.6. Dispersion plot of the genetic EBLUPs values of the Model 14 and the traditional non-spatial analysis model for this design (without spatial dependence) for Ribeirão Local 20. The cut-offsfor the 30 best test lines (7 % upper) in each model are indicated by the dotted line.

For Local 72, the initial model fitted included the variance parameters shown in Table E.7:random row, column and genetic effects; residual effects modelled using AR(1) in both the row and

134

column directions. It is clear that the estimated value of the residual component is much larger thanthose for the other components and, in particular, that there is very little genetic variation. This was thecase irrespective of whether or not observations considered to be outliers were removed. Hence, selectionof test lines is not viable for this Local and it will not be included in the joint analysis.

Table E.7. Estimates of the variance parameters and standard error for the model with first-orderautoregressive structure in row and column direction, and direct genetic, row and column random effects.The log-likelihood of the model is -1851.92.

Variance parameters Estimates Standard errorsRow σ2

r 19.92 11.47Column σ2

c 9.81 7.97Direct genetic σ2

g 14.27 79.92Residual σ2 467.41 72.13

Spatial:column ρ2c -0.11 0.05Spatial:row ρ2r -0.13 0.05

For Local 140, as in Local 20, the same problem with the Average Information Matrix happenedwith the models R-TIM and third-order banded correlation. Table E.8 shows a sequence of fitted models.Using REML ratio test and graphics of residuals diagnostic, Model 14 was considered the best fitted withcolumn linear trend; row spline; direct genetic effect and residual AR(1) and Band(1) in column and rowdirection, respectively. See in Figure E.7, the empirical semi-variogram for this model.

135

Tabl

eE

.8.

Sum

mar

yof

the

mod

els

fitte

dfo

rLo

cal1

40w

ithre

stric

ted

log-

likel

ihoo

d(lo

g-lik

.)an

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ep-v

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for

the

REM

Lra

tiote

st.

All

the

mod

els

incl

ude

ara

ndom

dire

ctge

netic

effec

tsfo

rthe

test

lines

(G)a

ndfix

edch

ecks

varie

tiese

ffect

s.T

heot

here

ffect

sare

repr

esen

ted

with

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sym

bols:

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arve

st);

Co

(con

e);

S(s

owin

g);

C(c

olum

n);

R(r

ow);

N(n

eigh

bour

gene

tic);

spl(.

)(s

plin

ein

som

edi

rect

ion

indi

cate

din

pare

nthe

ses)

and

lin(.)

(line

artr

end

inso

me

dire

ctio

nin

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ted

inpa

rent

hese

s).

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02

46

810

12

0

10

20

30

0.0

0.2

0.4

0.6

0.8

1.0

1.2

COL (lag)

ROW (lag)

Figure E.7. Plot of the empirical semi-variogram for the residuals for Local 140 of Ribeirão experimentreferring to Model 14.

REML estimates of the variance parameters with γ and standard errors from fitted Model 14are presented in the Table E.9. The direct genetic component also is smaller than the residual, with ratioequal 0.88. There is competition at the residual level, because the correlation in row direction is negative,-0.40.

Table E.9. REML estimates of variance parameters for fitted Model 14 to the experiment in the Local140 of the Ribeirão.


g) 0.88 180.48 32.90Residual (σ̃2) 1.00 205.52 28.66

Spatial (ρ̃c) — 0.23 0.08Spatial (ρ̃r) — -0.40 0.06

The Wald F test for fixed terms of the Model 14 are presented in Table E.10. There is noevidence of significant difference between the group of checks and test lines (Control), but there is withinthe checks (Control:Checks). The estimates of these fixed effects are in Table E.11 with their respectivestandard error.

Table E.10. Wald F test of fixed terms from Model 14. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom.

Fixed terms Df denDF F test p-value(Intercept) 1.00 506.00 18960.00 0.00Control 1.00 506.00 0.08 0.77lin(COL) 1.00 506.00 12.86 0.00lin(ROW) 1.00 506.00 2.99 0.08Control:Check 3.00 506.00 12.59 0.00

137

Table E.11. Estimates of the fixed effects for Model 14 and their respective standard errors.

Effects Estimates Standard errorsTI 12.33 4.50PAD1 0.00 —PAD2 -9.96 6.78PAD3 7.56 6.44lin(ROW) -0.09 0.06lin(COL) 0.55 0.15overall Checks 0.00 —overall Test lines 9.21 14.09overall mean (µ) 95.16 14.17

The 30 best test lines with their respective predicted values and standard errors are presentedin Table E.12. The correlation between the EBLUPs from Model 14 and the traditional is high, 0.932,but also the group of the best test lines is different for each model.

Table E.12. The 30 best test lines with predicted values (pred. value) and respective standard errors(stand. error) for Model 14.

Test lines Pred. values Stand. errors Test lines Pred. values Stand. errors152 137.26 9.32 49 131.26 9.4043 136.13 9.40 266 131.07 9.2522 135.80 9.05 315 130.87 9.19

223 134.82 9.11 298 130.50 9.39255 134.44 9.03 426 129.76 9.31

8 134.38 9.35 337 129.38 9.39188 133.81 9.21 4 129.21 9.3339 133.58 9.12 219 129.09 9.39

416 132.90 9.34 325 129.01 9.40348 132.76 9.21 279 128.68 9.40245 132.40 9.45 274 128.37 9.18310 132.15 9.11 24 128.30 9.11314 131.95 9.67 186 127.82 8.94410 131.82 9.43 15 127.75 9.29352 131.55 9.42 36 127.18 9.39

−30 −20 −10 0 10 20 30

−20

−10

010

20


EB

LUP

s −

sel

ecte

d m

odel cor = 0.932

Figure E.8. Dispersion plot of the genetic EBLUPs values of the Model 14 and the traditional non-spatial analysis model for this design (without spatial dependence) for Ribeirão Local 140. The cut-offsfor the 30 best test lines (7 % upper) in each model are indicated by the dotted line.

138

E.3 Joint analysis

Based on the separate analysis for each Local, it was observed that the genetic EBLUPs arevery different. Figure E.9 presents the relationship between the values for Locals 20 and 140 in the formof a scatterplot; the correlation is 0.401. This means that there is a substantial Local effect on the testlines and only limited correlation between the genetic effects is expected.

90 100 110 120 130 140

9010

011

012

013

0

predicted value − Local 20

pred

icte

d va

lue

− L

ocal

140 cor = 0.401

Figure E.9. Dispersion plot of the genetic EBLUPs from model selected for Ribeirão Locals 20 and 140.The cut-offs for the 30 best test lines in each Local are indicated by the dotted line.

We started the joint analysis with the models chosen in the individual analysis, adding onlyfixed effects for Local. Then, three models for genetic components and covariance (see Table E.13) werefitted and the REML ratio tests used to select the model to use for the data.

Table E.13. Summary of the models fitted to joint analysis for the Ribeirão experiments carried outfor the Local 20 (1) and 140 (3), with REML log (log-lik.) and the p-value of the REML ratio test. Allthe models have the same fixed; local and global effects modeled in the individual analysis plus the localfixed effects.

Model Genetic Effects Constraint Log-lik. Test p-value

1[σ2g1 00 σ2

g3

]-3473.15

2[σ2g 00 σ2

g

]-3473.33 M2 vs M1 0.26

3[

σ2g σg13

σg12 σ2g

]σg13 = ρgσ

2g -3434.87 M2 vs M3 0.00

Using the REML ratio test, the Model 3 was chosen as the best to explain the effects thathappened in these experiments. Table E.14 shows the REML estimates of the variance parameters andtheir respective standard errors from fitted Model 3.

Table E.14. REML estimates of variance parameters for fitted Model 3 for the joint analysis of Local20 (1) and 140 (3) of the experiment carried out in Ribeirão.

Variance parameters Estimates Standard errorsCorrelation between genetic effects (ρ̃g) 0.82 0.12Direct genetic (σ̃2

g) 162.09 24.43Local 20:Row (σ̃2

r1) 16.21 8.21Local 140:Spline(Row) 9.48 10.36Local 20:Residual (σ̃2

1) 166.23 21.43Spatial:row direction (ρ̃r1) -0.31 0.06

Local 140:Residual (σ̃23) 213.31 25.23

Spatial:column direction (ρ̃c3) 0.21 0.06Spatial:row direction (ρ̃r3) -0.32 0.05

139

The Wald F test for fixed terms of the Model 3 are presented in Table E.15. Note that there isa significant difference between the Locals, and the 30 best test lines predicted values with their standarderrors from Model 3 are in Table E.16 for each Local. It can be observed that in the joint analysisthe precision in the selection of the test lines is better, given that the standard errors are smaller thancompared with individual analysis (Tables E.6 and E.12).

Table E.15. Wald F test for fixed terms of the Model 3. Df means degrees of freedom and DenDF isthe approximate denominator degrees of freedom.

Fixed terms Df denDF F test p-value(Intercept) 1.00 1011.00 21230.00 0.00Local 1.00 1011.00 25.94 0.00Control 1.00 1011.00 0.04 0.84Control:Check 3.00 1011.00 32.38 0.00Local:Control 1.00 1011.00 2.09 0.15Local:lin(COL) 2.00 1011.00 9.61 0.00Local:lin(ROW) 2.00 1011.00 6.22 0.00Local:Control:Check 3.00 1011.00 0.75 0.52

Table E.16. Predicted values and standard errors of the 30 best test lines of the Model 3 for each local.Local 20 Local 140

Test lines Predicted values Standard errors Test lines Predicted values Standard errors49 140.56 7.99 49 138.18 8.20350 138.28 7.99 152 136.82 8.21282 135.17 8.07 8 136.55 8.178 134.82 8.03 43 136.28 8.2439 134.06 7.99 314 135.48 8.36314 134.01 8.08 39 135.40 8.15152 133.43 8.04 350 134.65 8.1843 133.21 8.05 282 133.18 8.3661 131.23 7.99 245 132.68 8.26245 130.79 7.98 266 131.91 8.18215 130.69 8.05 352 130.27 8.28384 130.22 7.97 384 130.11 8.16291 129.85 8.02 325 130.10 8.21220 129.80 8.04 348 130.09 8.20411 129.54 7.96 416 129.95 8.16259 129.38 7.96 291 129.57 8.14151 129.27 8.16 410 129.43 8.23325 129.05 8.00 426 129.41 8.17266 128.39 8.00 223 129.18 8.13130 127.81 8.04 151 128.98 8.2697 127.68 8.05 315 128.93 8.22364 127.54 8.15 310 128.70 8.17410 127.16 8.05 22 128.69 8.07426 127.16 8.01 411 128.46 8.15382 126.93 8.06 217 128.39 8.23429 126.81 8.02 4 128.30 8.24352 126.58 8.10 215 128.04 8.23358 126.41 8.05 382 127.89 8.30375 125.85 8.06 219 127.81 8.21380 125.84 8.12 220 127.78 8.22

140

E.4 Conclusions

The individual analyzes reflect the peculiarities of each local. For this region, convergenceproblems were experienced when fitting models with competition at the genetic level. For Local 72, itwas impossible to select test lines, because all the models indicated a high residual component comparedwith the genetic component which was not significant. Both Locals 20 and 140 were modeled with trendeffect in row and column, and similar correlation in row direction using first-order banded structure (-0.35 and -0.40), indicating competition at the residual level. The difference between the Locals was thatLocal 20 had a significant random row effect, while Local 140, there is a significant row spline term andAR(1) in column direction. The direct genetic component for both is similar when compared to the ratiobetween this parameter and the residual, around 0.86, and the correlation between the predicted valuesfrom selected models is 0.401. Hence, better accuracy in the selection of the test lines is achieved whenit is based on the joint analysis. Lower standard errors values are found. Model 3 was chosen to jointlymodel Locals 20 and 140, indicates a common genetic component (162.09) for both locals and a highcorrelation (0.82) between the genetic effects.

141

F PIRACICABA - GROUP OF EXPERIMENTS

There were three sites, nominated: Local 54, 58 and 76. In the first two sites the experimentswere carried out in a rectangular array of 14 columns by 37 rows while in the last there were 15 columnby 35 rows. All assessed the same 422 test lines (R) with 4 commercial varieties nominated “PAD1”,“PAD5”, “PAD6” and “Interc”. Planted in 2014, the yield in TCH was obtained in 2015, in a unique cut.The results were organized in subsections where F.1 presents the descriptive analysis of the Locals. F.2shows the individual analysis for each Local and F.3 the joint analysis. The concluding remarks are insubsection F.4.

F.1 Descriptive analysis

Table F.1 shows a summary of the TCH of the groups of clones by each Local. It is possible toobserve that the greatest yield for each group is found in Local 76, but this also had the greatest varianceamong sugarcane groups, except for “PAD5”. Local 54 has the smallest average value for all groups andthe biggest variance value referring to “PAD5”.

Table F.1. Descriptive analysis of the groups of clones carried out in Piracicaba.


Loca

l54 Interspersed 80.40 145.80 110.78 266.25

PAD1 86.40 136.90 112.42 280.11PAD5 59.50 144.30 99.03 788.48PAD6 78.80 139.90 110.45 347.36Test Lines (R) 44.60 194.90 104.08 475.51

Loca

l58 Interspersed 92.90 192.90 156.01 304.22


Loca

l76 Interspersed 110.70 225.00 171.61 568.03


The boxplot in Figure F.1 illustrates the outliers values for each group of lines given each Local.

142

TC

H

50

100

150

200

INTERC PAD1 PAD5 PAD6 R

54


58


76

Figure F.1. Boxplot of TCH for each group of lines for Locals of Piracicaba.

For all the Locals there are test lines considered outliers. However, as these did not haverepetition and external information, the values were accepted. Also some check plots were consideredoutliers, in Local 58 and 76. The yield of the interspersed plot in Local 58 was found in Local 54. Then,this value must be real. The same is thought for “PAD5” plots Local 58 and “PAD1” Local 76. The“PAD5” plot in Local 76 did not occur in other Locals, hence it will be observed if it represents a possibleproblem in the analysis, verifying the residuals of the models.

See in Figures F.2, F.3 and F.4, the heat maps relating to TCH in each Local. The names“PAD1”, “PAD5”, “PAD6” and “INTERC” are the check plots, the test lines are labeled alpha-numerically(PI plus number) and “x” represents the empty plots.

Column

Row

INTERCPI250PI47PAD5PI220PI252PI256PI257PI44

INTERCPI341PI290PI378PAD1PI377PI336PI340PI307

INTERCPI330PI321PI283PI324PI49PI48

PI279PI339

INTERCPI197

PI3PI4

PI201PAD1PI203

PI1PI18

INTERC

PI229INTERCPI223PI394PI393PI276PI396PI218PI284PI245

INTERCPI247PI379PI338PI381PI61

PI344PI337PI343


PI225PI278

PI6INTERC

PI25PI30PI27PI22PI24PI23PAD5PI26

PI255PI274


PI395PI34

INTERCPI33

PI392PI246PI391PI42

PI227PI36


INTERCPI16

PI195PI202PI19

PI204PI8

PI21

PI280PI228PI219

INTERCPI281PI224PI221PI249PI248PI241PI264PI243

INTERCPI240PI244PI119PAD5PI128PI242PI32

PI131INTERCPI112PI277PI40

PI111PI132PI367PI369PI356

INTERCPI358PI359PI196PAD6PI31

PI191

PI267PAD6PI287PI260

INTERCPI266PI222PI265PI262PAD1PI120PI235PI384

INTERCPI41

PI385PI383PAD6PI390PI238PI39PI37

INTERCPI386PI38PI35


INTERCPI193PI15

PI194PI198PI215

PI263PI269PI51

PI272PAD1


PI7PI20



PI361PI354PI360

INTERCPI13

PI368PI362PI192

PI55PI304PI282PI285PI289PI288


INTERCPI5

PI234PAD1PI410PI409PI407PI136PI412


PI342PI114PI365

INTERCPI366PI113PI84

PAD5PI270PI292PI53

PI306PI302PI303



INTERCPI117PI363PI95PI99

PI370PI380PI348PI346

INTERCPI160PI345

PAD6PI333PI332PI317PI331PI329PI56


INTERCPI416PAD6PI149PI199PAD5PI200PAD6PI405

INTERCPI376PI88PI94

PI351PI350PAD5PI352PAD1

INTERCPI371





INTERCPI148PI171

PI2PI150PAD5PI175PI212PI174


PI382PI65

PI158INTERC

PI327INTERC

PAD6PI326PI313PI312PI50

PI322PI323PI137



INTERCPI208PAD1PI10

PI106PI375PAD6PI165PI155

PI52PI300

INTERCPI296PI314PI318PI334PI335PI319PAD5PI207


PI9PI170PI141PI406

INTERCPI399PAD6PI110PI103PAD5PI422PI78PI75

INTERCPI71PI80PI69PI70PI63PI72PI73

PI325PI309PI308


PI154PI162PI12PAD1PI169


INTERCPI104PI96


PI349INTERC


XXXXXXXXXXXX

PI100INTERCPI163PI161PAD1PI156PI153PI93

PI373PI92

INTERCPI91

PI408PI107PI109PI372PI374PI101PI102

INTERCPI89PI90

PI353PI85PI98

40

60

80

100

120

140

160

180

200

Figure F.2. Heat map relating to TCH for Local 54 of Piracicaba.

143

Column

Row


PI275PI263PI272

INTERCPI44


PI271INTERCPI269PI268PAD5PI261PI258PI260PI243PI259

INTERCPI267PI247PI254PI249PI244PI252PI253PAD6

INTERC





PI385PI36

PI386INTERC

PI35PI118PI32


PI340PI45

INTERCPI339PI321PI330PI266PI245PAD6PI255PI248


PI396PI257

INTERCPI221PAD6PI112PI219PI394PI395PI228PI227


PI380

PI250PI387PI392


INTERCPI111PI41

PI393PI108PI38

PI379PI103PI42



INTERCPAD1PI234PI40

PI390PI119PI33


INTERCPAD1PI135PI251PI136PI331PI235PI376PI240

INTERCPI329PI217PI332PI83PAD1PI95

PI224PI317

INTERCPI97

PI223PI351PI270PAD5PI389PI328PI88


PI353PAD5PI313PI290PI312

INTERCPI56

PI391PAD6PI299PI298PI115PI292PI133


PI326PI295



INTERCPI91PI50

PI110PI281

PI303PI310PI54PI55

PI304PI286


INTERCPI308PI52

PI309PI284PAD5PI301PI102PI53

INTERCPI106PI89

PI302PAD5PI92PI94PI93PI64

INTERCPI374PAD1PI80

PI160PI161PI349PI99PAD1PI79

PI121INTERC

PI90PI73

PI370PI48PI86PAD1PI288PI350



PI346PI67PI63PI66

INTERCPI69PI65

PI82PI305PI105PI289PI101PI285PI85PI72

INTERCPI11PI75

PI152PI207PI76

PI159PI57PI78

INTERCPI77PI70



INTERCPI149



INTERCPI137PAD5PI418PI148

PI9PI146PI347PI145


PI154PAD1PI158

INTERC

PI421INTERCPI403PI420PI405PAD1PI419PI414PAD6PI400


INTERCPI179

PI1PI213PAD1PI214PI209PI186PI15

INTERCPI30

PI366PI21

PI185PAD6PI193PI198PI13

PI206PI200

INTERCPAD5PI362



PI201PI114

INTERCPI113PI199PAD6PI204PI203PI169PI23

PI175INTERC


PI10PI171PI397



PI363PI174

INTERCPI12


PI5INTERC

PI3PI184PI191PI20

PI187PI2

XXXXXXXXXXXX

PI358INTERC

PI25PI24PI4

PI359PI177PI356PI29

PI357INTERCPI360PI355PAD5PI354PI196PI26PI18

PI194INTERC

PI31PI202

PI8PI27PI22

80

100

120

140

160

180

200


Column

Row



INTERCPAD1PI374PI100PI90PI85

PI232PI258PI372


PI254PI77

PI271PI264

PI333INTERCPI113PI46

PI266PI44




PI248PI92

PI105INTERC

PAD6PI281PI99

PI161PI49

PI418

PI334PI367

INTERCPI320PI368PI313PI364PI316PAD6PI231PI314



INTERCPI236PI400PI302PAD5PI404PI306PI18

PI180INTERCPI405PI286PI410PI299PI412

PI47PI74PAD5

INTERCPI136PI62

PI335PI134PI284PI116PI327PAD1


INTERCPI237PAD6PI253PI279PI398PI288PI280PI417


PI200

PI263PI66

PI383PI34




PI394PI56

PI305PI293PI319

INTERCPI292PAD1PI296

PI1PI51PI3

PI291PI403


PAD6PI340PI87PI35

PI135INTERC

PI83PI117PAD5PI88

PI329PI55PI32PI84


PI311PI53

PI125PI141PI304

INTERCPI206PI54


INTERCPI80

PI159


INTERCPI353PI40PI42

PI245PI337PI343PI67

PI342INTERCPI336PI61PI63PI81PAD6PI68PI69

PI407INTERC

PAD5PI131PI129PI145PI420PI127PI86PI79

INTERCPI395

PI219PI352PI348PI76

PI246PI371PI64

INTERCPI118PAD1PI338PI380PI373PI382PI29

PI381INTERCPI370PI347PAD5PI406PI70

PI205PAD1PI144

INTERCPI413PI139PI146PAD6PI57PI78

PI151PI199

INTERC

PI75PI396PI350PI98

PI109PI65PI71PI93

INTERCPI91PI95PI96

PI269PI225PI102

PI9PI97


PI158PI162PI10

PI138PI156

INTERCPI399PI137PI140PI148PAD1PI207PI150PI149




INTERCPI397

PI7PI411PI201PI160PAD5PI163





INTERCPI197PI19PI2

PI184PI402PI153

PI285PI233


INTERCPI238PI39PAD5PI387PI388PI247PI349PAD1


INTERCPI164PI177PAD6PI188PI214

PI41PAD5PI298

INTERCPI295PI38PAD6PI345PI379PI217PAD1PI59

INTERCPI216PAD6PI60


INTERCPI17




INTERCPI52

PI339PI94

PI378PI50

PI377PI278PI393


PI215PI167PI196

INTERCPI22PI30PI16

PI191PI5

PI189PI193

PI6INTERC

PI15PAD1PI31

XXXXXXXXXXXXXXXXXXX

PI8PI14PI43PI24

INTERCPI26PI23

PI194PI25

PI190PI21PI13

PI192INTERC

PI27PI28 60

80

100

120

140

160

180

200

220


F.2 Individual analysis

For Local 54, it is shown, in Table F.2, a summary of the sequence of fitted models. This beganwith a model that included random effect for direct, neighbour and covariance (G:N) genetic, rows andcolumns; fixed effect for the check varieties; and spatial dependence, fitted with first-order autoregressivestructure in the row and column direction. However, model with competition at the genetic level werenot significant. Hence, the local and global effects were assessed without genetic competition.

144Ta

ble

F.2

.Su

mm

ary

ofth

efit

ted

mod

els

toLo

cal5

4of

Pira

cica

baw

ithre

stric

ted

log-

likel

ihoo

d(lo

g-lik

.)an

dth

ep-v

alue

ofth

eR

EML

ratio

test

.A

llm

odel

sin

clud

ea

rand

omdi

rect

gene

ticeff

ects

for

the

test

lines

(G)

and

fixed

chec

ksva

rietie

seff

ects

.T

heot

her

effec

tsar

ere

pres

ente

dw

ithth

esy

mbo

ls:H

(har

vest

);C

o(c

one)

;S(s

owin

g);C

(col

umn)

;R(r

ow);

N(n

eigh

bour

gene

tic);

spl(.

)(s

plin

ein

som

edi

rect

ion

indi

cate

din

pare

nthe

ses)

and

lin(.)

(line

artr

end

inso

me

dire

ctio

nin

dica

ted

inpa

rent

hese

s).

The

stru

ctur

esca

nbe

:U

S(.)

-un

stru

ctur

ed;R

R(.)

-re

duce

dra

nk;A

R(1

)-

first

-ord

erau

tore

gres

sive;

Band

(.)-

band

edco

rrel

atio

n,th

enu

mbe

rin

dica

tes

the

orde

rof

corr

elat

ion;

and

Id-i

dent

ity.

Mod

elEff

ects

Log-

lik.

Test

p-v

alue

Glo

bal/

extr

aneo

usG

enet

icLo

cal

Fixe

dR

ando

m(c

olum

n×

row

)1

C+

RU

S(G

:N)

AR

(1)×

AR

(1)

-176

2.97

2C

+R

RR

(G:N

)A

R(1

)×

AR

(1)

-176

2.97

3C

+R

GA

R(1

)×

AR

(1)

-176

3.39

M2

vsM

30.

364

C+

RG

AR

(1)×

Band

(3)

-176

2.80

5C

+R

GA

R(1

)×

Band

(2)

-176

2.83

M4

vsM

50.

816

C+

RG

AR

(1)×

Band

(1)

-176

2.83

M5

vsM

60.

977

C+

RG

AR

(1)×

Id-1

770.

29M

6vs

M7

0.00

8C

+R

GId

×Ba

nd(1

)-1

763.

49M

6vs

M8

0.25

9C

oC

+R

GId

×Ba

nd(1

)-1

762.

88no

tsig

nific

ant

10S

C+

RG

Id×

Band

(1)

-176

2.93

not

signi

fican

t11

HC

+R

GId

×Ba

nd(1

)-1

762.

18no

tsig

nific

ant

12lin

(C)+

lin(R

)sp

l(C)+

C+

spl(R

)+R

GId

×Ba

nd(1

)-1

764.

1013

lin(C

)+lin

(R)

spl(C

)+C

+R

GId

×Ba

nd(1

)-1

764.

10M

12vs

M13

1.00

14lin

(C)+

lin(R

)sp

l(C)+

RG

Id×

Band

(1)

-176

4.38

M13

vsM

140.

2315

lin(C

)+lin

(R)

RG

Id×

Band

(1)

-176

4.63

M14

vsM

150.

2416

lin(C

)+lin

(R)

GId

×Ba

nd(1

)-1

778.

09M

15vs

M16

0.00

17lin

(R)

RG

Id×

Band

(1)

-176

6.58

not

signi

fican

t18

lin(C

)R

GId

×Ba

nd(1

)-1

762.

93sig

nific

ant

19lin

(C)

RU

S(G

:N)

Id×

Band

(1)

-176

2.62

M19

vsM

180.

57

145

Note in Table F.2 that after testing for global and lodal environmental effects, the neighbourgenetic and covariance between the genetics effects were tested again. However, they were not significant.

Using the REML ratio test, the chosen model was Model 18, with column linear trend, fixedcheck varieties effects, random rows and direct genetic effects, and residual competition modeled with afirst-order banded structure in the row direction. Figure F.5 has the plots of the row and column faces ofthe empirical variogram for the Models 8 and 18; it shows the effect of the fitted model on the residuals.

0.0

0.5

1.0

1.5

0 5 10Col differences


(a)

0.0

0.5

1.0

1.5



(b)

0.0

0.5

1.0

1.5



(c)

0.0

0.5

1.0

1.5



(d)

Figure F.5. Plots of the row and column faces of the empirical variogram for the residuals for the Local54 of Piracicaba experiment for models 8 (panels (a) and (c)) and 18 (panels (b) and (d)). The panels(a) and (b) are column direction and the others are row direction.

In table F.3 can be seen the REML estimates of variance parameters of the Model 18. Notethat the direct genetic component is smaller than the residual (the ratio is 0.77) and the row correlationis negative 0.27, this indicates residual competition.

Table F.3. REML estimates of variance parameters for fitted Model 18 for the experiment of Piracicaba,Local 54.

Variance parameters Ratios (γ) Estimates Standard errorsRow (σ̃2


g) 0.77 182.15 47.25Residual (σ̃2) 1.00 235.89 38.76

Spatial:Row (ρ̃r) — -0.27 0.07

The Wald F tests for fixed terms and the estimated fixed effects from selected Model 18 arepresented in Table F.4 and Table F.5, respectively. In this model for this data, there are no evidences

146

of significant difference between the group of test lines and checks (Control) or within the checks (Con-trol:Check).

Table F.4. Wald F test for fixed factors for Model 18. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom.

Fixed terms Df denDF F test p-value(Intercept) 1 496 4568.00 0.00Control 1 496 0.28 0.60lin(Col) 1 496 8.03 0.00Control:Check 3 496 1.71 0.16

Table F.5. Estimates of the fixed effects for Model 18 and their respective standard errors.

Effects Estimates Standard errorsINTERC 0.00 —PAD1 -0.03 5.45PAD5 -11.11 5.43PAD6 3.70 5.51lin(Col) 0.49 0.18overall Checks 0.00 —overall Test lines -7.88 13.69overall mean (µ) 108.11 13.76

The 30 best test lines with predicted values and respective standard errors from Model 18 arepresented in Table F.6. Note in Figure F.6 the correlation between genetic EBLUPs from selected modeland the traditional model (with random row, column and direct genetic effects) which are very strong,0.966. However the group of test lines selected in each model are different.

Table F.6. The 30 best test lines with predicted values (pred. value) and respective standard errors(stand. error) for Model 18.

Test line Pred. values Stand. errors Test lines Pred. values Stand. errorsPI318 136.13 10.23 PI254 119.43 10.19PI356 129.61 10.14 PI308 119.17 10.15PI342 124.94 10.19 PI37 119.14 10.13PI203 123.11 10.19 PI391 119.01 10.22PI296 123.06 10.14 PI301 118.84 10.22PI100 122.96 10.26 PI388 118.76 10.21PI320 122.46 10.15 PI235 118.56 10.21PI22 121.71 10.21 PI260 118.19 10.13PI136 121.60 10.19 PI160 118.19 10.13PI412 121.28 10.13 PI395 118.09 10.20PI408 121.12 10.22 PI1 117.90 10.21PI340 120.36 10.22 PI128 117.87 10.16PI256 119.87 10.24 PI357 117.85 10.20PI115 119.74 10.20 PI294 117.80 10.15PI248 119.64 10.22 PI259 117.80 10.23

147

−20 −10 0 10 20 30

−20

010

2030

EBLUPs − traditionalE

BLU

Ps

− s

elec

ted

mod

el cor = 0.966

Figure F.6. Dispersion plot of the genetic EBLUPs values of the Model 18 and the traditional non-spatial analysis model for this design (without spatial dependence) for Piracicaba Local 54. The cut-offsfor the 30 best test lines (7 % upper) in each model are indicated by the dotted line.

For Local 58 the same process than Local 54 was conducted, but the random column componentis boundary. Then, the models were analyzed without this effect and the sequence of the models ispresented in Table F.7.

148Ta

ble

F.7

.Su

mm

ary

ofth

efit

ted

mod

els

toLo

cal5

8of

Pira

cica

baw

ithre

stric

ted

log-

likel

ihoo

d(lo

g-lik

.)an

dth

ep-v

alue

ofth

eR

EML

ratio

test

.A

llm

odel

sha

vea

rand

omdi

rect

gene

ticeff

ects

for

the

test

lines

(G)

and

fixed

chec

kva

rietie

seff

ects

.T

heot

her

effec

tsar

ere

pres

ente

dw

ithth

esy

mbo

ls:H

(har

vest

);C

o(co

ne);

S(s

owin

g);C

(col

umn)

;R(r

ow);

N(n

eigh

bour

gene

tic);

spl(.

)(s

plin

ein

som

edi

rect

ion

indi

cate

din

pare

nthe

ses)

and

lin(.)

(line

artr

end

inso

me

dire

ctio

nin

dica

ted

inpa

rent

hese

s).

The

stru

ctur

esca

nbe

:U

S(.)

-uns

truc

ture

d;R

R(.)

-red

uced

rank

.A

R(.)

-aut

oreg

ress

ive;

Band

(.)-b

ande

dco

rrel

atio

n,th

enu

mbe

rin

dica

tes

the

orde

rof

corr

elat

ion;

sar(

2)-c

onst

rain

edau

tore

gres

sive

3;an

dId

-ide

ntity

.

Mod

elEff

ects

Log-

lik.

Test

p-v

alue

Glo

bal/

extr

aneo

usG

enet

icLo

cal

Fixe

dR

ando

m(c

olum

n×

row

)1

RU

S(G

:N)

AR

(1)×

AR

(1)

-177

0.82

2R

GA

R(1

)×

AR

(1)

-177

1.98

M1

vsM

20.

223

RG

AR

(1)×

Band

(3)

-176

9.47

4R

GA

R(1

)×

Band

(2)

-176

9.63

M3

vsM

40.

575

RG

AR

(1)×

Band

(1)

-177

2.05

M4

vsM

50.

026

RG

AR

(1)×

AR

(2)

-176

9.53

7R

GA

R(1

)×

sar(

2)-1

769.

418

RG

AR

(2)×

sar(

2)-1

768.

37M

8vs

M7

0.15

9R

GBa

nd(3

)×

sar(

2)-1

767.

5210

RG

Band

(2)×

sar(

2)-1

767.

87M

9vs

M10

0.40

11R

GBa

nd(1

)×

sar(

2)-1

769.

92M

10vs

M11

0.04

12C

oR

GBa

nd(2

)×

sar(

2)-1

767.

36no

tsig

nific

ant

13S

RG

Band

(2)×

sar(

2)-1

767.

19no

tsig

nific

ant

14H

RG

Band

(2)×

sar(

2)-1

766.

99no

tsig

nific

ant

15lin

(C)+

lin(R

)sp

l(C)+

spl(R

)+R

GBa

nd(2

)×

sar(

2)-1

766.

5816

lin(C

)+lin

(R)

spl(R

)+R

GBa

nd(2

)×

sar(

2)-1

766.

58M

15vs

M16

1.00

17lin

(C)+

lin(R

)sp

l(R)

GBa

nd(2

)×

sar(

2)-1

768.

77M

16vs

M17

0.02

18lin

(C)+

lin(R

)R

GBa

nd(2

)×

sar(

2)-1

769.

95M

16vs

M18

0.00

19lin

(R)

spl(R

)+R

GBa

nd(2

)×

sar(

2)-1

765.

4920

lin(R

)sp

l(R)+

RU

S(G

:N)

Band

(2)×

sar(

2)-1

765.

4421

lin(R

)sp

l(R)+

RR

R(G

:N)

Band

(2)×

sar(

2)-1

765.

44M

21vs

M19

0.44

149

Using the REML test and the plot of the empirical semi-variogram for the residual, in FigureF.7, Model 19 is chosen as the best to explain the data. The chosen model has spline row term, randomrow and genetic effects, fixed check varieties effects, and residual competition effects in row and columndirections modeled with Band(2) in column and sar(2) in row direction. The REML estimates of varianceparameters are in Table F.8. Note that the direct genetic component is small, only 0.3 times the residualcomponent and there are negative correlations in both the column and row directions, which, given thespacing between plots, is not expected in the column direction. One hypothesis to be tested is if thelodging effect interferes in the correlation effects.

02

46

810

12

0

10

20

30

0.0

0.5

1.0

1.5

2.0

Col (lag)

Row (lag)

Figure F.7. Plot of the empirical semi-variogram for the residuals of the Local 58 of Piracicaba. Thisrefers to Model 19.


Variance parameters Ratios (γ) Estimates Standard errorsspl(Row) 0.04 13.53 16.43Row (σ̃2


g) 0.30 95.58 57.13Residual (σ̃2) 1.00 320.24 50.77

Spatial:Column (ρ̃c1) — -0.14 0.07Spatial:Column (ρ̃c2) — 0.14 0.07Spatial:Row (θ̃r1) — 0.39 0.10Spatial:Row (θ̃r2) — -0.22 0.05

Table F.9 shows the Wald F tests for fixed terms and the estimated fixed effects are presentedin Table F.10. Note that in this model for this experiment, there is evidence of significant differencesbetween the group of test lines and the group of checks (Control) and within the checks (Control:Checks).However the denominator residual degrees of freedom are an approximation and for Control appears tobe faulty for this data.

150

Table F.9. Wald F test for fixed terms from Model 19. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom.

Fixed terms Df denDF F test p-value(Intercept) 1.00 109.70 9404.00 0.00Control 1.00 0.00 3.95 0.00lin(Row) 1.00 97.50 2.74 0.10Control:Check 3.00 14.00 28.10 0.00

Table F.10. Estimates of the fixed effects for Model 19 and their respective standard errors.

Effects Estimates Standard errorsINTERC 0.00 —PAD1 -33.28 6.26PAD5 -50.73 6.30PAD6 -9.45 6.30lin(Row) -0.19 0.11overall Checks 0.00 —overall Test lines -29.51 10.08overall mean (µ) 158.29 10.29

The 30 best test lines with predicted values and respective standard errors from Model 19 arepresented in Table F.11. Comparing the selection did at the Model 19 and the traditional model (withrandom row, column and direct genetic effects), there are some best test lines different for both models.The correlation between the EBLUPs is also high, 0.966, as shown in Figure F.8.


Test lines Pred. values Stand. errors Test lines Pred. values Stand. errorsPI370 147.98 8.81 PI162 140.10 8.81PI340 143.56 8.85 PI150 139.87 8.82PI239 142.72 8.86 PI395 139.12 8.81PI38 142.67 8.76 PI104 139.08 8.81PI18 142.37 8.83 PI400 139.01 8.82PI82 142.18 8.86 PI138 138.62 8.76PI17 141.48 8.77 PI96 138.34 8.79PI51 141.42 8.81 PI37 137.95 8.80PI387 140.79 8.83 PI320 137.82 8.81PI215 140.70 8.79 PI318 137.74 8.83PI183 140.45 8.84 PI414 137.44 8.82PI269 140.42 8.78 PI300 137.25 8.81PI196 140.30 8.84 PI167 137.09 8.78PI293 140.28 8.76 PI121 137.03 8.83PI165 140.20 8.85 PI368 136.91 8.84

151

−10 −5 0 5 10 15 20

−10

05

1015

EBLUPs − traditionalE

BLU

Ps

− s

elec

ted

mod

el cor = 0.966


Analyzing Local 76, the random column component was boundary for the fitted models. Ob-serve in Table F.12 the sequence of models that was tested. While the genetic competition componentwas significant, AR(1) was used for the local residual effects (models 2 and 3), it was not significant oncethe more appropriate local effects model of Id × Band(2) was fitted (models 18 and 19). Hence, Model19 is selected to represent the data.

152Ta

ble

F.1

2.Su

mm

ary

ofth

efit

ted

mod

els

toLo

cal7

6of

Pira

cica

baw

ithre

stric

ted

log-

likel

ihoo

d(lo

g-lik

.)an

dth

ep-v

alue

ofth

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153

Figure F.9 shows the plots of the faces of the empirical variogram for the residuals from Model 18and 19. Using reduced rank models, the empirical variogram plot had some problems with the simulatedenvelopes. However, it is possible to observe the shape of residuals for both models. The REML estimatesof variance parameters from Model 19 are in Table F.13.

0

2

4

6



(a)

0.0

0.5

1.0

1.5



(b)

0

2

4



(c)

0.0

0.5

1.0

1.5

2.0



(d)

Figure F.9. Plots of the column and row faces of the empirical variogram for the residuals for Local 54of Piracicaba experiment for models 18 (panels (a) and (c)) and 19 (panels (b) and (d)). The panels (a)and (b) are column direction and the others are row direction.


Variance parameters Ratios (γ) Estimates Std. errorsRow (σ̃2


g) 0.66 208.99 62.41Residual (σ̃2) 1.00 317.95 53.45

Spatial:Row (ρ̃r1) — -0.24 0.08Spatial:Row (ρ̃r2) — 0.26 0.07

Note that the genetic component is small than the residual (ratio is 0.66) and the row corre-lations are opposite in magnitude (-0.24 and 0.26) indicating there are competition at the residual leveland spatial dependence. Table F.14 presents the Wald F tests for the fixed terms in Model 19 and TableF.15 shows the estimated fixed effects. As it can be seen, there is no evidence of a significant differ-ence between the check and test lines groups. This contrast is denoted as Control. However, there aresignificant differences between the checks (Control:Check).

154

Table F.14. Wald F test for fixed terms from Model 19. Df means degrees of freedom and DenDF isthe approximate denominator degrees of freedom.

Fixed terms Df denDF F test p-value(Intercept) 1.00 496.00 4128.00 0.00Control 1.00 496.00 1.16 0.28Control:Check 3.00 496.00 39.68 0.00

Table F.15. Estimates of the fixed effects from Model 19 and their respective standard errors.

Effects Estimates Standard errorsINTERC 0.00 —PAD1 -12.20 6.08PAD5 -67.14 6.18PAD6 -3.53 6.14overall Checks 0.00 —overall Test lines -24.50 14.66overall mean (µ) 173.89 14.76

The 30 best test lines with predicted values and respective standard errors from Model 19 arepresented in Table F.16. Comparing the genetic EBLUPs from models 19 and the traditional (withrandom row, column and direct genetic effects), the correlation is strong, 0.956, but the selected testlines are not the same between the models. See Figure F.10.


Test lines Pred. values Stand. errors Test lines Pred. values Stand. errorsPI163 184.06 11.44 PI171 166.24 11.26PI318 173.91 11.33 PI27 166.20 11.37PI269 173.74 11.32 PI395 164.63 11.43PI385 171.58 11.32 PI42 164.36 11.31PI53 171.43 11.30 PI336 164.17 11.30PI388 171.23 11.28 PI356 164.07 11.32PI7 169.36 11.25 PI348 163.90 11.32PI17 168.86 11.29 PI206 163.88 11.28PI220 168.82 11.31 PI204 163.71 11.31PI359 168.00 11.27 PI95 163.63 11.26PI207 167.67 11.29 PI180 163.59 11.28PI267 167.05 11.28 PI83 163.22 11.25PI115 166.43 11.31 PI422 162.98 11.30PI340 166.38 11.37 PI416 162.97 11.27PI405 166.34 11.28 PI191 162.77 11.32

155

−30 −20 −10 0 10 20 30

−30

−10

010

2030


EB

LUP

s −

sel

ecte

d m

odel cor = 0.956


F.3 Joint analysis

There is low correlation, around 0.3, between the genetic EBLUPs from each Local obtainedusing the models selected in the individual analyses. As shown in Figure F.11, there are few test linesconsidered the best in pairs of Locals; only five or six.

−20 −10 0 10 20 30

−10

05

1015


Gen

etic

EB

LUP

s −

Loc

al 5

8 cor = 0.314

(a)

−20 −10 0 10 20 30

−30

−10

010

2030


Gen

etic

EB

LUP

s −

Loc

al 7

6 cor = 0.306

(b)

−10 −5 0 5 10 15

−30

−10

010

2030


Gen

etic

EB

LUP

s −

Loc

al 7

6 cor = 0.282

(c)

Figure F.11. Dispersion plots for the genetic EBLUPs values from selected model for each PiracicabaLocal. Panels refer the relation between Local (a) 54 and 58; (b) 54 and 76; (c) 58 and 76. The cut-offsfor the 30 best test lines in each Local are indicated by the dotted line.

Based on the separate analysis for each Local, a joint analysis of all three Locals was formulated

156

in order to investigate the magnitudes of the genetic variance at each Local and to assess the geneticcovariance between Locals. To do this the models shown in Table F.17 were tested and the results of theREML tests are also given in Table F.17.

Table F.17. Summary of the fitted models to joint analysis of the Piracicaba experiments carried outin the Local 54 (1), 58 (2) and 76 (3), with REML log (log-lik.) and the p-value of the REML ratio test.All the models have the same effects modeled in the individual analysis plus the fixed Local effects. Hereσ2gi is the genetic variance at the ith Local, σgij is the genetic correlation between the ith and jth Locals

and σg∗ is the genetic correlation between two locals which has been hypothesis to be equal for two ormore pairs of Locals.

Model Genetic Effects Log-lik. Test p-value

1

σ2g1 0 00 σ2

g2 00 0 σ2

g3

-5353.00

2

σ2g1 σg12 σg13

σg12 σ2g2 σg23

σg13 σg23 σ2g3

-5302.79 M2 vs M1 0.00

3

σ2g1 ρσg1σg2 ρσg1σg3

ρσg1σg2 σ2g2 ρσg2σg3

ρσg1σg3 ρσg2σg3 σ2g3

-5303.10 M2 vs M3 0.73

4

σ2g ρσgσg2 ρσ2

g

ρσgσg2 σ2g2 ρσgσg2

ρσ2g ρσgσg2 σ2

g

-5303.11 M3 vs M4 0.44

5 σ2g

1 ρ ρρ 1 ρρ ρ 1

-5303.64 M4 vs M5 0.15

Using the REML ratio test, Model 5 is chosen as the best to explain the effects that occurredin these experiments. The model has the same nuisance effects found in individual analysis added fixedlocal effects, a single direct genetic variance for all Locals and genetic correlation between Locals thatwas equal for all pairs of Locals. Table F.18 shows the REML estimates of the variance parameters fromselect model.

Table F.18. REML estimates of the variance parameters for fitted Model 5 in the joint analysis of theLocals in Piracicaba.

Variance parameters Estimates Standard errorsLocal: Direct genetic (σ̃2

g) 155.10 32.28cor(Local):Direct genetic (ρ̃g) 0.84 0.17Local54:Row (σ̃2

r1) 60.14 20.24Local58:Row (σ̃2

r2) 23.10 12.78Local76:Row (σ̃2

r3) 130.07 39.60Local58:spl(Row) 12.53 14.73Local 54: Residual (σ̃2

1) 251.17 31.98Spatial:Row (ρ̃1r) -0.23 0.06

Local 58:Residual (σ̃22) 275.36 31.96

Spatial:Column (ρ̃2c1) -0.12 0.07Spatial:Column (ρ̃2c2) 0.15 0.07Spatial:Row (θ̃2r1) 0.39 0.10Spatial:Row (θ̃2r2) -0.21 0.05

Local 76: Residual (σ̃23) 354.68 38.54

Spatial:Row (θ̃3r1) -0.20 0.06Spatial:Row (θ̃3r2) 0.19 0.05

Comparing the predicted values from model 5, given in Table F.19, it is observed that theseresults had small standard errors, when compared to the separate analysis of the Locals, and a large

157

number of clones in common among the Locals within the top 30 test lines.

Table F.19. Predicted values (pred.) and standard errors (error) of the 30 best test lines from Model 5for each Local.

Local 54 Local 58 Local 76Test lines Pred. Error Test lines Pred. Error Test lines Pred. ErrorPI318 138.75 8.46 PI318 160.99 8.67 PI318 182.22 8.77PI340 130.75 8.48 PI340 158.44 8.69 PI340 176.23 8.80PI269 127.20 8.46 PI269 154.56 8.63 PI269 174.54 8.78PI356 125.43 8.46 PI395 150.99 8.67 PI395 169.73 8.81PI395 124.20 8.47 PI370 150.64 8.63 PI388 168.53 8.77PI320 123.31 8.43 PI38 149.54 8.60 PI356 168.48 8.78PI388 122.31 8.46 PI18 148.55 8.66 PI320 166.85 8.75PI203 121.59 8.45 PI320 148.30 8.64 PI385 166.74 8.78PI342 121.17 8.44 PI196 147.09 8.67 PI220 166.57 8.76PI136 120.97 8.44 PI356 147.08 8.63 PI7 165.90 8.76PI38 120.68 8.45 PI82 146.62 8.70 PI38 165.53 8.76PI220 120.15 8.45 PI17 146.33 8.62 PI18 165.26 8.76PI37 120.01 8.45 PI388 146.08 8.69 PI17 165.23 8.78PI408 119.70 8.45 PI37 145.97 8.64 PI136 164.70 8.75PI196 119.41 8.46 PI203 145.90 8.63 PI191 164.61 8.79PI160 119.26 8.44 PI150 145.67 8.66 PI394 164.04 8.77PI370 119.16 8.44 PI96 145.43 8.63 PI218 163.89 8.76PI385 119.05 8.47 PI385 145.35 8.67 PI203 163.87 8.75PI7 118.87 8.45 PI215 145.30 8.64 PI82 163.84 8.78PI18 118.86 8.45 PI160 144.91 8.68 PI348 163.79 8.76PI100 118.71 8.48 PI7 144.87 8.68 PI196 163.59 8.77PI191 118.59 8.50 PI138 144.86 8.58 PI370 163.58 8.76PI150 118.47 8.44 PI220 144.78 8.61 PI192 163.31 8.79PI296 118.30 8.44 PI191 144.69 8.70 PI160 163.28 8.76PI192 118.22 8.50 PI192 144.57 8.68 PI115 163.11 8.77PI218 118.19 8.46 PI51 144.56 8.65 PI37 162.98 8.77PI394 118.14 8.46 PI136 144.46 8.64 PI408 162.90 8.75PI297 118.11 8.45 PI342 144.19 8.63 PI95 162.87 8.75PI348 118.03 8.45 PI165 144.10 8.69 PI237 162.87 8.75PI115 117.68 8.45 PI408 143.92 8.65 PI287 162.72 8.74

F.4 Conclusions

With each individual analysis it can be noted the peculiarity of the Local, in which was modeledrandom row effects and residual competition in this direction, but spatial effects were not found in Local54. The models in Local 58 showed a negative correlation in the column direction. A possible explanationis that there is an operational effect such that the sugarcane for one plot ends up being erroneouslyincluded in that for a plot in a neighbouring column. The ratios between genetic and residual componentswere approximately 0.77, 0.30 and 0.66 for Locals 54, 58 and 76 respectively. This results in betterselection for Local 54, because it has the largest genetic variance relative to the residual variance. Whencomparing the 30 best predicted values for the test lines from each Local there were few similarities amongthe lines selected; there were around three to six common test lines across Locals and the correlationbetween the EBLUPs from pairs of Locals was around 0.30. However, in the joint analysis better accuracyin the selection of the test lines is obtained, with predictions having lower standard errors. The modelchosen for the genetic variation indicates the same genetic component (155.10) for all three Locals anda positive strong correlation (0.84) between the genetic effects from different Locals. In this case, thebetter selection is in Local 54, because it has the lowest residual variance. The number of common testlines in the best 30 lines is better; around half of the 30 best predicted values for the test lines for each

158

Local occur in all three Locals.

159

G ARAÇATUBA - GROUP OF EXPERIMENTS

There were three sites, nominated: Local 101, 130 and 551. However, each site had small areaswith different test lines (New) and rectangular arrays. For Local 101, there are three areas with 3 × 18,15 × 21 and 3 × 15 columns by rows. In Local 130 and 551 were two areas each and dimensions 26 × 12,5 × 15, 9 × 21 and 11 × 24 columns by rows. The number of test lines assessed were 325 for Local 101;321 for Local 130; and 318 for Local 551. There were 4 commercial varieties nominated “PAD1”, “PAD2”,“PAD3” and “Interc”, each having different numbers of replicates for each Local. These experiments wereplanted in 2014 and the yield in TCH was obtained in 2015, in a single cut.

In subsection G.1 the results of the descriptive analysis for Local 101 will be described andthe model testing and the EBLUPs obtained using the selected model for this Local will be presentedin subsubsection G.1.1. Subsection G.2 and subsubsection G.2.1 contain the same information for Local130 and that for Local 551 is in subsection G.3 and subsubsection G.3.1. Subsection G.4 contains theconclusions for these trials.

G.1 Local 101

A summary for the TCH of the groups of clones carried out in Local 101 of Araçatuba ispresented in Table G.1. It can be observed that in areas 1 and 3 there are no replications of “PAD”checks. The value of the average for PAD1 found in area 3 is smaller than in other sites. Observationsof Figure G.1, that shows the boxplot of TCH of the groups of clones in each area, reveals some possibleoutliers, but these are test lines and PAD3 and the values are similar to those in the other experiments.

Table G.1. Descriptive analysis of the groups of clones carried out in Local 101 of Araçatuba.

Groups of clones Minimum Maximum Mean Variance Number of plots

Are

a1

PAD1 - - 135.70 - 1PAD2 - - 123.80 - 1PAD3 - - 123.80 - 1Inerspersed (INTERC) 117.90 151.20 134.13 222.67 6Test line (New) 76.20 152.40 114.09 373.03 42

Are

a2

PAD1 119.00 129.80 125.24 24.05 5PAD2 89.30 133.30 109.76 387.13 5PAD3 110.70 156.00 124.76 318.66 5Interspersed (INTERC) 72.60 154.80 111.80 421.28 35Test line (New) 66.70 173.80 114.35 360.41 252

Are

a3

PAD1 - - 69.00 - 1PAD2 - - 133.30 - 1PAD3 - - 114.30 - 1Interspersed (INTERC) 72.60 129.80 99.30 550.82 5Test line (New) 82.10 150.00 115.70 290.98 31

160T

CH

80

100

120

140

160

INTERC New PAD1 PAD2 PAD3

1


2


3

Figure G.1. Boxplot of TCH in each area for Local 101 of Araçatuba.

See in Figure G.2 the heat maps relating to TCH in each area.

Column

Row

INTERC

AR309

AR310

AR311

AR308

AR313

PAD1

AR317

AR319

INTERC

AR323

AR324

AR315

AR326

AR98

X

X

X

AR149

INTERC

AR153

AR140

PAD3

AR139

AR65

AR138

AR144

AR3

INTERC

AR66

AR148

AR72

AR69

AR160

AR90

AR91

AR159

AR154

INTERC

AR157

AR158

AR163

AR82

AR83

AR64

PAD2

AR67

INTERC

AR143

AR141

AR142

AR68

AR63

AR145

80

90

100

110

120

130

140

150

(a)

Column

Row

INTERC

AR9

PAD3

AR8

AR7

AR11

AR26

AR14

AR31

INTERC

AR19

AR20

AR33

AR34

AR25

AR10

INTERC

AR6

AR13

PAD2

AR16

AR15

AR30

AR24

AR28

INTERC

AR36

AR29

AR32

AR18

AR12

PAD1

INTERC

AR17

AR27

AR35

AR22

AR23

AR21

X

X

X

X

X

X70

80

90

100

110

120

130

140

150

(b)

Column

Row

INTERC

AR80

AR92

AR94

AR102

AR88

AR161

AR183

AR86

INTERC

AR226

AR178

AR228

AR316

AR318

PAD3

AR306

AR307

INTERC

AR314

AR312

AR96

INTERC

AR156

AR77

AR79

PAD2

AR85

AR155

AR84

AR89

INTERC

AR332

AR327

AR328

AR329

AR184

AR185

AR182

AR227

INTERC

AR230

AR173

AR71

INTERC

AR174

AR81

AR103

AR107

AR105

AR179

AR175

AR333

INTERC

AR331

PAD1

AR171

AR330

AR168

AR177

AR181

AR87

INTERC

AR73

AR75

AR150

INTERC

AR76

AR152

AR147

AR162

AR70

AR169

AR106

AR101

INTERC

AR229

AR104

AR78

AR95

PAD2

AR325

AR322

AR321

AR167

AR165

AR164

AR170

INTERC

AR176

AR146

PAD1

AR151

AR93

AR74

AR320

AR180

INTERC

AR100

AR99

AR2

AR37

AR44

AR46

AR40

AR166

AR206

AR196

AR248

AR245

INTERC

AR247

AR201

AR199

AR208

AR119

AR113

PAD3

AR112

INTERC

AR55

AR54

AR50

AR38

AR56

AR41

AR198

AR237

PAD2

AR236

AR202

AR255

INTERC

AR4

AR233

AR231

AR190

AR250

AR45

AR186

AR249

INTERC

AR246

AR187

AR126

AR189

AR58

AR1

AR258

AR253

AR62

AR213

AR262

AR256

INTERC

AR259

AR244

AR121

AR110

AR108

AR47

PAD3

AR43

INTERC

AR42

AR120

AR49

AR39

AR264

AR263

AR266

AR212

PAD1

AR267

AR216

AR219

INTERC

AR286

AR289

AR123

AR115

AR114

AR117

PAD3

AR118

INTERC

AR116

AR111

AR188

INTERC

AR288

AR287

AR284

AR280

AR223

PAD2

AR291

AR285

INTERC

AR294

AR48

AR51

AR52

AR53

AR125

AR127

AR124

INTERC

AR57

AR122

AR265

INTERC

AR293

AR298

AR299

AR60

AR61

AR134

AR133

AR131

INTERC

AR239

AR242

AR243

AR252

PAD1

AR251

AR217

AR220

INTERC

AR218

AR130

AR129

INTERC

AR279

AR278

AR222

AR271

AR272

PAD2

AR215

AR210

INTERC

AR207

AR254

AR197

AR234

AR232

AR194

AR240

AR241

INTERC

AR214

AR281

AR283

INTERC

AR224

AR261

AR225

AR260

AR109

AR137

AR193

AR191

INTERC

AR192

AR136

AR135

AR200

PAD3

AR235

AR238

AR195

AR257

AR203

AR204

AR211

INTERC

AR268

PAD1

AR205

AR270

AR275

AR221

AR296

AR132

INTERC

AR302

AR301

AR290

AR282

AR292

AR295

AR300

AR274

AR276

AR273

AR269

AR209

INTERC

AR304

AR305

X

X

X

X

X

X

X

X

X

X

X

X

X 60

80

100

120

140

160

180

(c)

Figure G.2. Heat maps relating to TCH in Local 101 of Araçatuba. The panel refers to (a) area 1; (b)area 3 and (c) area 2. The names PAD1, PAD2, PAD3 and INTERC are the check plots and the testlines are labeled alpha-numerically (AR plus number).

161

G.1.1 Analysis

In area 1, there is no evidence of spatial dependence or competition effect. Table G.2 presentsthe summary of the fitted models. When comparing the models, it can be observed that the log likelihoodfor Model 15 is smallest and was significantly different from Model 14; hence it was selected as the bestmodel.

162Ta

ble

G.2

.Su

mm

ary

ofth

efit

ted

mod

els

toLo

cal1

01ar

ea1

with

rest

ricte

dlo

g-lik

elih

ood

(log-

lik.)

and

thep-v

alue

ofth

eR

EML

ratio

test

.A

llth

em

odel

sin

clud

ea

rand

omdi

rect

gene

ticeff

ects

for

the

test

lines

(G)

and

fixed

chec

ksva

rietie

seff

ects

.T

heot

her

effec

tsar

ere

pres

ente

dw

ithth

esy

mbo

ls:H

(har

vest

);C

o(c

one)

;S(s

owin

g);C

(col

umn)

;R(r

ow);

N(n

eigh

bour

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tic);

spl(.

)(s

plin

ein

som

edi

rect

ion

indi

cate

din

pare

nthe

ses)

and

lin(.)

(line

artr

end

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me

dire

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nin

dica

ted

inpa

rent

hese

s).

The

stru

ctur

esca

nbe

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S(.)

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truc

ture

d;R

R(.)

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uced

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(1)

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oreg

ress

ive;

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(.)-b

ande

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rrel

atio

n,th

enu

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rin

dica

tes

the

orde

rof

corr

elat

ion;

and

Id-i

dent

ity.

Mod

elEff

ects

Log-

lik.

Test

p-v

alue

Glo

bal/

extr

aneo

usG

enet

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cal

Fixe

dR

ando

m(c

olum

n×

row

)1

C+

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S(G

:N)

AR

(1)×

AR

(1)

-159

.54

2C

US(

G:N

)A

R(1

)×

AR

(1)

-159

.54

M1

vsM

21.

003

CR

R(G

:N)

AR

(1)×

AR

(1)

-159

.52

M2

vsM

31.

004

CG

AR

(1)×

AR

(1)

-159

.65

M3

vsM

40.

305

CG

AR

(1)×

Band

(3)

not

conv

erge

d-s

ingu

larit

y6

CG

AR

(1)×

Band

(2)

-159

.56

7C

GA

R(1

)×

Band

(1)

-159

.56

M6

vsM

71.

008

CG

AR

(1)×

Id-1

59.9

2M

7vs

M8

0.39

9C

GId

×Id

-160

.05

M8

vsM

90.

6110

HC

GId

×Id

-158

.38

not

signi

fican

t11

Co

CG

Id×

Id-1

58.4

1no

tsig

nific

ant

12S

CG

Id×

Id-1

58.4

0no

tsig

nific

ant

13lin

(C)+

lin(R

)C

+sp

l(R)

GId

×Id

-157

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14lin

(C)+

lin(R

)G

Id×

Id-1

57.2

4M

13vs

M14

1.00

15lin

(C)

GId

×Id

-156

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signi

fican

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lin(R

)G

Id×

Id-1

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tsig

nific

ant

17lin

(C)

RR

(G:N

)Id

×Id

-161

.21

M17

vsM

150.

44

163

In Figure G.3, there are the plots of the row and column faces of the empirical variogram forthe residuals referring to Models 4 and 15. The values obtained are inside the expected range, this showsthat there is no lack of adjustment on the spatial dependence.

0.0

0.5

1.0

1.5

0.0 0.5 1.0 1.5 2.0Col differences


(a)

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0Col differences


(b)

0

1

2



(c)

0

1

2

3



(d)

Figure G.3. Plots of the row and column faces of the empirical variogram for the residuals for Local101 area 1 of Araçatuba experiment for models 4 (panels (a) and (c)) and 15 (panels (b) and (d)). Thepanels (a) and (b) are column direction and the others are row direction.

Table G.3 presents the Wald F tests for the fixed terms for Model 15 and Table G.4 shows theestimated fixed effects. As it can be seen, there is no evidence of significant difference between check andthe mean of the test lines (Control), nor among the checks (Control:Check).

Table G.3. Wald F test for fixed effects for Model 15. Df means degrees of freedom and DenDF is theapproximate denominator degrees of freedom.

Fixed terms Df denDF F test p-value(Intercept) 1 38.90 1548.00 0.00Control 1 13.70 1.14 0.30lin(Col) 1 19.20 7.69 0.01Control:Check 3 4.50 0.35 0.79

164

Table G.4. Estimates of the fixed effects and their respective standard errors for Model 15.

Effects Estimates Standard errorsINTERC 0.00 —PAD1 -6.44 11.51PAD2 -2.33 11.51PAD3 -10.33 11.11lin(Col) -8.00 3.01overall Test lines 0.00 —overall checks 19.47 16.99overall mean (µ) 130.67 6.90

The 15 predicted best test lines with standard errors are presented in Table G.5.

Table G.5. The 15 best test lines with predicted values (pred. values) and respective standard errors(stand. errors) for Model 15.

Test lines Pred. values Stand. errors Test lines Pred. values Stand. errorsAR 154 144.70 9.00 AR 158 111.63 9.00AR 143 142.99 9.00 AR 323 111.26 8.97AR 65 139.00 8.72 AR 157 110.77 9.00AR 324 135.86 8.97 AR 83 110.77 9.00AR 319 132.43 8.97 AR 148 110.13 8.72AR 91 132.22 8.72 AR 311 109.55 8.97AR 68 132.01 9.00 AR 315 109.55 8.97AR 142 126.03 9.00 AR 317 108.70 8.97AR 82 126.03 9.00 AR 138 107.63 8.72AR 326 123.10 8.97 AR 72 107.63 8.72AR 3 122.89 8.72 AR 139 106.78 8.72AR 313 122.24 8.97 AR 98 105.28 8.97AR 159 121.82 9.00 AR 149 103.36 8.72AR 69 121.18 8.72 AR 67 103.14 9.00AR 153 119.46 8.72 AR 64 102.29 9.00AR 90 115.26 8.72 AR 160 101.64 8.72AR 145 114.98 9.00 AR 310 100.22 8.97AR 163 114.19 9.00 AR 141 98.87 9.00AR 140 113.55 8.72 AR 66 96.58 8.72AR 144 112.69 8.72 AR 63 92.95 9.00AR 308 112.05 8.97 AR 309 82.40 8.97

In area 2, an analysis of the data using different models was tried, but none were satisfactory.The problem was that the residual variance was high and the direct genetic component was close tozero. Thus, two models were compared: Model 1 considered competition effects at the residual level; andModel 2 as the simplest model with only variety effects (checks fixed and test lines random). The resultsare given in Table G.6.

Model1<- asreml(tch ~ Control/Check, random=~ Reg +Col + Row,rcov=~ ar1(Col):corb(Row,k=3),data=place2,na.method.Y = "include",na.method.X = "include")

Model2<- asreml(tch ~ Control/Check, random=~ Reg, rcov=~ (Col):(Row),data=place2,na.method.Y = "include",na.method.X = "include")

165

Table G.6. REML estimates of variance parameters from models 1 and 2 for the experiment in Local101 of Araçatuba in area 2. The values in brackets are the standard errors of the estimates.

Variance parameters Model 1 Model 2Estimates Standard errors Estimates Standard errors

Column (σ̃2c ) 20.51 (13.15) — —

Row (σ̃2r) 0.00 (0.00) — —

Direct genetic (σ̃2g) 0.00 (0.00) 0.00 (0.00)

Residual (σ̃2) 342.71 (29.05) 362.65 (29.76)Spatial (ρ̃c) 0.05 (0.06 ) — —Spatial (ρ̃r1) -0.03 (0.06 ) — —Spatial (ρ̃r2) 0.01 (0.06 ) — —Spatial (ρ̃r3) -0.12 (0.06 ) — —

The same models were fitted for area 3 and the results are presented in Table G.7.

Table G.7. REML estimates of variance parameters from models 1 and 2 for the experiment in Local101 of Araçatuba in area 3. The values in brackets are the standard errors of the estimates.


Column (σ̃2c ) 0.00 (0.00) — —

Row (σ̃2r) 106.98 (67.96) — —


Residual (σ̃2) 226.08 (77.48) 321.55 (77.99)Spatial (ρ̃c) -0.20 (0.42) — —Spatial (ρ̃r1) -0.08 (0.26) — —Spatial (ρ̃r2) 0.28 (0.17) — —Spatial (ρ̃r2) -0.38 (0.15) — —

G.2 Local 130

For Local 130, high values of variance were found in the groups of clones in area 1, mainly for“PAD2”, as can be seen in Table G.8. In Figure G.4, the dispersion of the data can be observed for eachgroup. Some plots are considered outliers and comparing the yield between sites seems to confirm this.Hence, it can expected that the yield of a clone can be affected by the position of the plot to which it issown. Figure G.5 illustrates the heat maps relating to TCH in each area of the Local 130 of Araçatuba.

166



area

1

PAD1 95.20 128.60 112.86 217.02 5PAD2 88.10 166.70 117.86 914.87 5PAD3 82.10 129.80 115.00 367.02 5Inerspersed (INTERC) 69.00 169.00 113.51 515.85 35Test lines (New) 73.80 191.70 120.52 451.16 262

area

2

PAD1 - - 104.80 - 1PAD2 - - 169.00 - 1PAD3 - - 158.30 - 1Inerspersed (INTERC) 98.80 140.50 120.72 223.87 10Test lines (New) 85.70 164.30 125.05 350.97 62

TC

H

80

100

120

140

160

180


1


2


167

Column

Row

INTERC

AR94

AR95

AR75

AR88

AR76

AR100

AR98

AR91

INTERC

AR69

AR86

AR65

INTERC

AR90

AR73

AR63

AR49

AR87

AR2

AR50

AR44

INTERC

AR104

AR99

AR84

INTERC

AR77

PAD1

PAD3

PAD2

AR92

AR67

AR103

AR68

INTERC

AR93

AR62

AR3

INTERC

AR61

AR219

AR38

AR43

AR89

AR72

AR79

AR81

AR74

AR60

AR57

AR37

INTERC

AR58

AR82

AR53

AR110

AR56

AR112

AR115

AR113

AR48

AR64

AR85

AR83

INTERC

AR46

AR51

AR42

AR1

AR39

AR47

AR71

AR78

AR117

AR111

AR52

AR41

INTERC

AR116

AR114

AR109

AR118

AR54

AR9

PAD1

PAD3

PAD2

AR280

AR13

AR282

INTERC

AR281

AR11

AR304

AR300

AR10

AR302

X

AR284

AR294

AR16

AR292

AR295

INTERC

AR8

AR20

AR23

INTERC

AR283

AR287

AR176

AR291

AR45

AR288

AR296

AR305

INTERC

AR285

AR168

AR276

INTERC

AR22

AR12

AR298

AR7

AR33

AR15

AR303

AR14

INTERC

AR24

AR34

AR27

INTERC

AR30

AR297

AR301

AR299

X

AR293

AR286

AR290

INTERC

AR278

AR18

AR205

INTERC

AR279

AR25

AR29

AR17

PAD1

PAD3

PAD2

AR32

AR21

AR35

AR36

AR31

INTERC

AR19

AR28

AR289

AR210

AR214

AR208

AR209

AR213

AR249

AR204

AR332

AR333

INTERC

AR322

AR326

AR321

AR320

AR315

AR328

AR222

AR218

AR199

AR202

AR193

AR203

INTERC

AR206

AR331

AR324

AR317

AR313

AR316

AR314

AR329

AR330

AR307

AR309

AR312

INTERC

AR310

AR177

AR263

AR327

AR319

PAD1

PAD3

PAD2

AR318

AR306

AR273

AR257

INTERC

AR255

AR264

AR261

INTERC

AR253

AR259

AR272

AR275

AR267

AR271

AR262

AR266

INTERC

AR308

AR311

AR325

INTERC

AR323

AR243

AR268

AR250

AR260

AR120

AR277

AR270

INTERC

AR236

AR265

AR242

INTERC

AR201

AR258

AR269

AR256

AR241

AR231

AR251

AR254

INTERC

AR252

AR122

AR235

INTERC

AR123

AR212

AR237

PAD1

PAD3

PAD2

AR174

AR247

AR248

AR119

AR185

AR238

INTERC

AR240

AR233

AR180

AR246

AR245

AR244

AR171

AR239

AR232

AR182

AR179

AR181

INTERC

AR173

AR166

AR188

AR196

AR197

AR187

AR190

AR101

AR192

AR106

AR105

AR200

INTERC

AR217

AR186

AR195

AR107

AR183

AR121

AR189

AR4

AR207

X

AR96

AR194

INTERC

AR170

AR198

AR191

AR226

80

100

120

140

160

180

200

(a)

Column

Row

INTERC

AR178

AR184

AR97

AR169

AR175

AR167

AR163

AR165

INTERC

AR159

AR162

AR161

AR158

AR157

AR154

INTERC

AR142

AR150

AR149

AR148

AR147

AR145

AR70

AR80

INTERC

AR164

AR160

AR153

AR155

AR144

AR141

INTERC

AR137

PAD2

PAD3

PAD1

AR136

AR133

AR135

AR151

INTERC

AR146

AR139

AR156

AR124

AR125

AR132

INTERC

AR128

AR129

AR131

AR134

AR140

AR143

AR223

AR138

INTERC

AR126

AR224

AR215

AR229

AR221

AR225

INTERC

AR130

AR230

AR228

AR108

AR216

AR220

AR227

AR127

INTERC

AR21180

90

100

110

120

130

140

150

160

170

(b)

Figure G.5. Heat maps relating to TCH in Local 130 of Araçatuba. The panel refers to (a) area 1and (b) area 2. The names PAD1, PAD2, PAD3 and INTERC are the check plots and the test lines arelabeled alpha-numerically (AR plus number). “x” represents the empty plots.

G.2.1 Analysis

As for the Local 101, the direct genetic variance was estimated as zero for all fitted mod-els. Consequently, the following two models were compared and the resulting estimates of the varianceparameters are in Table G.9.

Model1<- asreml(tch ~ Control/Check, random=~ Reg +Col + Row,rcov=~ ar1(Col):corb(Row,k=3),data=place21,na.method.Y = "include",na.method.X = "include")

Model2<- asreml(tch ~ Control/Check, random=~ Reg,rcov=~ (Col):(Row),data=place21,na.method.Y = "include",na.method.X = "include")

168

Table G.9. REML estimates of variance parameters for models 1 and 2 for the experiment in Local 130of Araçatuba in area 1. The values in brackets are the standard errors of the estimates.


Column (σ̃2c ) 46.04 (25.08) — —

Row (σ̃2r) 0.00 (0.00) — —


Residual (σ̃2) 417.59 (36.95) 460.31 (37.34)Spatial (ρ̃c) -0.04 (0.06) — —Spatial (ρ̃r1) -0.05 (0.08) — —Spatial (ρ̃r2) 0.08 (0.07) — —Spatial (ρ̃r3) -0.02 (0.08) — —

For area 2, the models capture a direct genetic effect, however models with competition at thegenetic level did not converge and all the models had a high standard error for the variance components.Table G.10 shows a sequence of fitted models. Using REML ratio test and diagnostic graphs of theresiduals, Model 8 was considered to be the best fit; it included only fixed varieties and random directgenetic effects. See the empirical variogram for this model in Figure G.6.

169

Tabl

eG

.10.

Sum

mar

yof

the

mod

els

fitte

dto

area

2of

the

Loca

l130

with

rest

ricte

dlo

g-lik

elih

ood

(log-

lik.)

and

thep-v

alue

ofth

eR

EML

ratio

test

.A

llth

em

odel

sin

clud

ea

rand

omdi

rect

gene

ticeff

ects

for

the

test

lines

(G)

and

fixed

chec

ksva

rietie

seff

ects

.T

heot

her

effec

tsar

ere

pres

ente

dw

ithth

esy

mbo

ls:H

(har

vest

);C

o(c

one)

;S(s

owin

g);C

(col

umn)

;R(r

ow);

N(n

eigh

bour

gene

tic);

spl(.

)(s

plin

ein

som

edi

rect

ion

indi

cate

din

pare

nthe

ses)

and

lin(.)

(line

artr

end

inso

me

dire

ctio

nin

dica

ted

inpa

rent

hese

s).

The

stru

ctur

esca

nbe

:U

S(.)

-uns

truc

ture

d;A

R(1

)-fi

rst-

orde

rau

tore

gres

sive;

Band

(.)-b

ande

dco

rrel

atio

n,th

enu

mbe

rin

dica

tes

the

orde

rof

corr

elat

ion;

and

Id-i

dent

ity.

Mod

elEff

ects

Log-

lik.

Test

p-v

alue

Glo

bal/

extr

aneo

usG

enet

icLo

cal

Fixe

dR

ando

m(c

olum

n×

row

)1

R+

CU

S(G

:N)

AR

(1)×

AR

(1)

not

conv

erge

d-s

ingu

larit

y2

US(

G:N

)A

R(1

)×

AR

(1)

not

conv

erge

d-s

ingu

larit

y3

R+

CG

AR

(1)×

Band

(3)

-240

.63

4G

AR

(1)×

Band

(3)

-240

.63

M3

vsM

41.

005

GA

R(1

)×

Band

(2)

-240

.65

M4

vsM

50.

836

GA

R(1

)×

Band

(1)

-241

.07

M5

vsM

60.

367

GA

R(1

)×

Id-2

41.3

0M

6vs

M7

0.50

8G

Id×

Id-2

41.3

1M

7vs

M8

0.88

9H

GId

×Id

-238

.92

not

signi

fican

t10

SG

Id×

Id-2

39.7

8no

tsig

nific

ant

11C

oG

Id×

Id-2

39.8

4no

tsig

nific

ant

12lin

(C)+

lin(R

)sp

l(C)+

spl(R

)G

Id×

Id-2

40.3

313

lin(C

)+lin

(R)

spl(R

)G

Id×

Id-2

40.3

3M

12vs

M13

1.00

14lin

(C)+

lin(R

)G

Id×

Id-2

40.5

3M

13vs

M14

0.27

15lin

(C)

GId

×Id

-239

.94

not

signi

fican

t16

lin(R

)G

Id×

Id-2

41.9

4no

tsig

nific

ant

17U

S(G

:N)

Id×

Idno

tco

nver

ged

-sin

gula

rity

170

0.0

0.5

1.0

1.5

2.0

0 1 2 3 4Col differences


(a)

0

1

2

3

0 5 10Row differences


(b)

Figure G.6. Plots of the column and row faces of the empirical variogram for the residuals for Local 130area 2 of Araçatuba experiment for Model 8. The panels (a) is column direction and (b) is row direction.

The REML estimates of variance parameters, with ratios (γ) and standard errors, from fittedModel 8 are presented in the Table G.11. As said before, the standard errors are very high when comparedwith their respective components, mainly for direct genetic term. Note that the parameter estimate was127.10, while the standard error was 123.19.

Table G.11. REML estimates of variance parameters for fitted Model 8 to the experiment in the Local130 of the Araçatuba.

Variance parameters Ratios (γ) Estimates Standard errorsDirect genetic (σ̃2

g) 0.57 127.10 123.19Residual (σ̃2) 1.00 223.87 105.53

Table G.12 presentes the Wald F tests for the fixed terms of the Model 8 and, in the TableG.13, there are the estimated fixed effects. See that there is no significant difference between check andthe mean of the test lines, nor among the checks.

Table G.12. Wald F tests for the fixed effects for Model 8. Df means degrees of freedom and DenDF isthe approximate denominator degrees of freedom.

Fixed terms Df denDF F test p-value(Intercept) 1 53.90 2873.00 0.00Control 1 3.30 0.01 0.94Control:Check 3 9.00 5.39 0.02

Table G.13. Estimated fixed parameters and their respective standard errors for Model 8.

Effects Estimates Standard errorsINTERC 0.00 —Check - PAD1 -15.92 15.69Check - PAD2 48.28 15.69Check - PAD3 37.58 15.69overall Test lines 0.00 —overall checks -4.33 12.45overall mean (µ) 125.05 2.38

Table G.14 presents the predicted values and standard errors of test lines for Model 8.

171

Table G.14. Predicted values and standard errors of the test lines for Model 8.Test lines Pred. value Stand. error Test lines Pred. value Stand. errorAR 178 139.26 9.13 AR 230 125.47 9.13AR 215 137.09 9.13 AR 136 125.03 9.13AR 150 135.82 9.13 AR 228 125.03 9.13AR 163 135.39 9.13 AR 131 124.16 9.13AR 125 134.52 9.13 AR 221 124.16 9.13AR 161 134.09 9.13 AR 70 123.73 9.13AR 158 133.22 9.13 AR 148 123.29 9.13AR 135 132.78 9.13 AR 164 123.29 9.13AR 220 132.78 9.13 AR 129 122.86 9.13AR 162 132.35 9.13 AR 139 121.59 9.13AR 127 131.91 9.13 AR 146 121.59 9.13AR 138 131.91 9.13 AR 216 121.59 9.13AR 140 131.91 9.13 AR 169 120.72 9.13AR 165 131.91 9.13 AR 157 120.29 9.13AR 145 130.65 9.13 AR 229 120.29 9.13AR 156 130.21 9.13 AR 142 119.42 9.13AR 133 129.78 9.13 AR 153 119.42 9.13AR 154 129.78 9.13 AR 128 118.98 9.13AR 143 128.92 9.13 AR 149 118.98 9.13AR 97 128.92 9.13 AR 175 118.98 9.13AR 130 128.47 9.13 AR 167 118.55 9.13AR 155 128.47 9.13 AR 134 118.15 9.13AR 124 128.04 9.13 AR 159 118.15 9.13AR 132 127.20 9.13 AR 227 118.15 9.13AR 137 127.20 9.13 AR 141 116.85 9.13AR 80 126.34 9.13 AR 108 115.54 9.13AR 147 125.90 9.13 AR 184 115.11 9.13AR 223 125.90 9.13 AR 160 112.97 9.13AR 224 125.90 9.13 AR 144 111.23 9.13AR 126 125.47 9.13 AR 211 111.23 9.13AR 151 125.47 9.13 AR 225 110.80 9.13

G.3 Local 551

In the descriptive analysis of the groups of clones for Local 551, one high value for the commercialvariety “PAD3” in area 2 is observed, but this seems to be incorrect, because a high yield for this clonewas never obtained. The variance for this group is bigger than the other groups, although “PAD2” alsohas a large variance in area 2 (Table G.15). It is possible to observe, in Figure G.7, that only some plotswith test lines are considered outliers, however the dispersion of commercial variety in area 2 is not verysymmetrical.



area

1


area

2


172

TC

H

50

100

150

200

250


1


2


Figure G.8 has the heat maps relating to TCH in Local 551 for each area.

Column

Row

INTERC

AR328

AR329

AR326

PAD3

AR327

AR333

AR330

AR331

INTERC

X

X

X

X

X

X

X

X

X

X

X

X

INTERC

AR325

AR322

AR319

AR320

AR332

AR321

AR323

AR324

INTERC

AR232

PAD2

AR233

AR193

AR252

AR205

AR190

AR269

INTERC

AR248

X

X

INTERC

AR242

AR4

AR194

AR197

AR243

AR191

AR239

AR247

INTERC

AR241

AR246

AR231

AR249

AR251

AR192

AR245

AR244

INTERC

X

X

X

INTERC

AR238

AR234

AR199

AR201

AR250

AR285

AR271

AR289

INTERC

AR268

AR290

AR212

AR17

AR298

PAD1

AR286

AR208

X

X

X

X

INTERC

AR253

AR222

PAD3

AR219

AR220

AR218

AR266

AR263

INTERC

AR272

AR267

AR210

AR265

AR213

AR211

AR281

X

X

X

X

X

INTERC

AR255

AR202

AR221

AR291

AR262

PAD2

AR296

AR270

INTERC

AR259

AR256

AR254

AR264

AR44

AR45

X

X

X

X

X

X

INTERC

AR126

PAD1

AR58

AR43

AR68

AR116

AR51

AR124

INTERC

AR57

AR110

AR111

AR53

AR144

X

X

X

X

X

X

X

INTERC

AR312

AR310

AR315

PAD3

AR313

AR316

AR317

AR314

INTERC

AR318

AR125

AR50

AR128

X

X

X

X

X

X

X

AR308

INTERC

AR311

AR309

AR307

PAD1

AR306

PAD2

X

X

X

X

X

X

60

80

100

120

140

160

180

200

(a)

Column

Row

INTERCAR287AR32AR7PAD3AR3AR5

AR19AR177

INTERCXXXXXXXXXXXXXX

AR215INTERCAR240AR200AR236AR207AR206AR260AR203AR257

INTERCAR261AR216PAD2

AR204AR258AR209AR275AR237

INTERCAR235AR196AR195AR198

AR9AR101

INTERCAR15

AR301PAD1

AR279AR288AR30AR31

AR294INTERCAR303AR230AR228AR302AR276AR217AR278AR214

INTERCAR227AR229AR293

AR20AR180AR152

INTERCAR105AR176AR184AR174AR164PAD3AR21AR34

INTERCAR76AR33

AR163AR170AR27

AR179AR292AR103

INTERCAR16AR14

AR283AR185AR295AR223

INTERCAR13


AR284AR157

INTERCAR23

AR100AR99AR82AR98

AR158PAD2AR36

INTERCAR77

AR166AR183PAD3AR22

AR167INTERCAR168AR35AR91


INTERCAR89PAD1AR29

AR172AR182AR26AR95

AR156INTERC

AR305AR6

AR88AR93

AR299AR106

INTERCAR282PAD2AR90AR87

AR297AR224AR92

AR107INTERCAR104AR83AR25AR18

AR165AR66AR65

AR130

AR160AR1

AR120AR96

AR162AR85AR38

INTERCAR117AR147AR70

AR148AR71AR41AR2

AR69INTERCAR102AR119AR161PAD1AR62

AR146AR37

AR94AR121PAD3

AR134AR131AR54

AR112AR56

INTERCAR114AR133AR136AR132AR137AR81

AR151AR145

INTERCAR73AR74AR75

AR139AR84AR63

INTERCAR189AR59

AR127AR159AR79AR78PAD2

AR153INTERCAR150AR80

AR154AR86

AR155AR55AR60AR72

INTERCAR140AR67

AR141AR138AR64

AR109INTERCAR188AR108AR186AR113AR118AR187AR52

AR115INTERC

AR48AR47AR49PAD1AR46AR42

AR142AR122

INTERCAR143AR123AR129AR149 50

100

150

200

250

(b)

Figure G.8. Heat maps relating to TCH in the Local 551 of Araçatuba. The panel (a) refers to area 1and (b) area 2. The names PAD1, PAD2, PAD3 and INTERC are the check plots and the test lines arelabeled alpha-numerically (AR plus number). “x” represents the empty plots.

173

G.3.1 Analysis

Again, in area 1, there is a problem in estimating genetic effects. Three models fitted forarea 1 are shown in Table G.16 and the estimates of the variance parameters from these models areshown in Table G.17. Even though the estimated genetic components are not zero, the standard error isconsiderably larger than the estimate and so can be taken to be zero.

Table G.16. Summary of the models fitted to Local 551 area 1 of the Araçatuba experiment withrestricted log (log-lik.). All the models include a random direct genetic effect for the test lines, fixedeffects for checks and there is one model with linear trend in column direction (lin(Col)). AR(1) is thefirst-order autoregressive structure and Id is the identity structure.

Model Effects Log-lik. Test p-valueGlobal/extraneous Genetic Localfixed Random (column × row)

1 G AR(1) × AR(1) -482.372 lin(Col) G AR(1) × AR(1) -480.353 lin(Col) G Id × Id -480.83

Table G.17. REML estimates of variance parameters for models 1, 2 and 3 for the experiment in Local551 of Araçatuba in area 1. The values in brackets are the standard errors (std.error) of the estimates.

Variance parameters Model 1 Model 2 Model 3Estimate Std.error Estimate Std.error Estimate Std.error

Direct genetic (σ̃2g) 51.38 (151.62) 98.72 (139.82) 95.40 (141.74)

Residual (σ̃2) 468.41 (134.29) 416.62 (120.90) 419.65 (122.60)Spatial (ρ̃c) -0.06 (0.11) -0.07 (0.12) — —Spatial (ρ̃r) 0.09 (0.11) 0.08 (0.12) — —

Figure G.9 has the plots of the row and column face of the empirical variogram for the residualsreferring to Model 3, this shows that there is not a strong spatial dependence effect for this site.

0

1

2

3

0 2 4 6 8Col differences


(a)

0

1

2

3

4



(b)

Figure G.9. Plots of the column and row faces of the empirical variogram for the residuals for Local 551area 1 of Araçatuba experiment for Model 3. The panels (a) is column direction and (b) is row direction.

For area 2, the direct genetic component is close to zero for all models tested. Table G.18presents some models and in Table G.19 there are the estimated variance components and their respectivestandard errors.

174

Table G.18. Summary of the models fitted to Local 551 area 2 of the Araçatuba experiment withrestricted log-likelihood (log-lik.). All the models include a random direct genetic effects for the testlines, fixed effects for checks and there is one model with linear trend in column direction (lin(Col)).AR(1) is the first-order autoregressive structure; Band(3) is the third-order banded correlation and Id isthe identity structure.

Model Effects Log-lik. Test p-valueGlobal/extraneous Genetic LocalFixed Random (column × row)

1 Col + Row G AR(1) × Band(3) -892.552 Col G Id × Id -892.883 lin(Col) Col G Id × Id -887.77

Table G.19. REML estimates of variance parameters from models 1 and 3 for the experiment in Local551 of Araçatuba in area 2. The values in brackets are the standard errors (std.error) of the estimates.

Variance parameters Model 1 Model 3Estimate Std.error Estimate Std.error

Column (σ̃2c ) 131.09 (69.39) 37.68 (27.56)

Row (σ̃2r) 0.00 (0.00) — —


Residual (σ̃2) 472.40 (44.05) 470.94 (43.42)Spatial (ρ̃c) -0.04 (0.07) — —Spatial (ρ̃r1) 0.00 (0.07) — —Spatial (ρ̃r2) 0.03 (0.07) — —Spatial (ρ̃r3) -0.01 (0.07) — —

This data was also analysed without the “PAD3´´ outlier (first column 1 and fourth row, witha yield of 246.40), but the genetic component was smaller than its estimated standard error. Hence thereis no adequate model for the selection of test lines.

G.4 Conclusions

The experiments carried out in Araçatuba present high environmental variation (σ2) and small(close to zero) direct genetic components (σ2

g) for Local 101 (areas 2 and 3), Local 130 (area 1) and Local551 (both areas). Hence the models tested for these experiments did not result in definitive direct geneticeffect.

At Local 101 area 1, it was possible to find nonzero EBLUPs for the test lines of this experimentwith random direct genetic component equal to 262.37 and a residual component of 105.71. Local 130area 2 also produced nonzero EBLUPs for the test lines, but the estimated standard error for directgenetic component was approximately the same as that of the variance parameter.

Two observations are made. Firstly, in every Local there was at least one experiment with lessthan 10 rows, which makes it difficult to estimate spatial dependence, and/or less than 100 plots whichlimits the ability to estimate variances. Secondly, because even the larger experiments performed poorly,it could be that the sites in Araçatuba are inherently more variable or there were greater difficultiesencountered in carrying out the operations for the experiment. To improve experiments carried out inthis region will require more uniform areas and/or more careful performance of the operations during theexperiment.

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