Springer Series in Statistics Advisors: J. Berger, S. Fienberg, J. Gani K. Krickeberg, I. Oikin, B. Singer

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Andersen/BlJrgan/Gill/Keiding: Statistical Models Based on Counting Processes. Anderson: Continuous-Time Markov Chains: An Applications-Oriented Approach. Andrews/Herzberg: Data: A Collection of Problems from Many Fields for the

Student and Research Worker. Anscombe: Computing in Statistical Science Through APL. Berger: Statistical Decision Theory and Bayesian Analysis, 2nd edition. BoLJarine/Zacks: Prediction Theory for Finite Populations. Bremaud: Point Processes and Queues: Martingale Dynamics. Brockwell/Davis: Time Series: Theory and Methods, 2nd edition. Choi: ARM A Model Identification. Daley/Vere-Jones: An Introduction to the Theory of Point Processes. Dzhaparidze: Parameter Estimation and Hypothesis Testing in Spectral Analysis of

Stationary Time Series. Farrell: Multivariate Calculation. Federer: Statistical Design and Analysis for Intercropping Experiments: Volume I. Fienberg/Hoaglin/Kruskal/Tanur (Eds.): A Statistical Model: Frederick Mosteller's

Contributions to Statistics, Science, and Public Policy. Goodman/Kruskal: Measures of Association for Cross Classifications. Grandell: Aspects of Risk Theory. Hall: The Bootstrap and Edgeworth Expansion. Hardie: Smoothing Techniques: With Implementation in S. Hartigan: Bayes Theory. Heyer: Theory of Statistical Experiments. Jolliffe: Principal Component Analysis. Kotz/Johnson (Eds.): Breakthroughs in Statistics Volume I. Kotz/ Johnson (Eds.): Breakthroughs in Statistics Volume II. Kres: Statistical Tables for Multivariate Analysis. Leadbetter/Lindgren/Rootzen: Extremes and Related Properties of Random

Sequences and Processes. Le Cam: Asymptotic Methods in Statistical Decision Theory. Le Cam/Yang: Asymptotics in Statistics: Some Basic Concepts. Manoukian: Modern Concepts and Theorems of Mathematical Statistics. Manton: Forecasting the Health of Elderly Populations. Miller, Jr.: Simultaneous Statistical Inference, 2nd edition. Mosteller/Wallace: Applied Bayesian and Classical Inference: The Case of The

Federalist Papers. Pollard: Convergence of Stochastic Processes. Pratt/Gibbons: Concepts of Nonparametric Theory. Read/Cressie: Goodness-of-Fit Statistics for Discrete Multivariate Data. Reiss: A Course on Point Processes. Reiss: Approximate Distributions of Order Statistics: With Applications to

Nonparametric Statistics. Ross: Nonlinear Estimation.

(continued after index)

Walter T. Federer

Statistical Design and Analysis for Intercropping Experiments

Volume I: Two Crops

With 35 Illustrations

Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest

Walter T. Federer Biometrics Unit Cornell University Ithaca, NY 14853-7801 USA

Library of Congress Cataloging-in-Publication Data Federer, Walter Theodore, 1915-

Statistical design and analysis for intercropping experimentsfby Walter T. Federer.

p. cm.-(Springer series in statistics) Includes bibliographical references and index. Contents: v. 1. Two crops

ISBN-13: 978-1-4613-9307-8 e-ISBN-13: 978-1-4613-9305-4 DOl: 10.1007/978-1-4613-9305-4 1. Intercropping-Experiments. 2. Experimental design.

I. Title. II. Series. S603.5.F43 1993 631.5'8-dc20 92-29586

Printed on acid-free paper.

© 1993 Springer-Verlag New York, Inc.

Softcover reprint of the hardcover 1st edition 1993

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA) except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as under­stood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Production coordinated by Brian Howe and managed by Francine Sikorski; manufacturing supervised by Vincent Scelta. Typeset by Asco Trade Typesetting Ltd., Hong Kong.

9 8 7 6 5 4 3 2 1

Dedicated to

Edna

Preface

Two volumes are being published on the topic of the title of this book. Vol­ume I, the present one, deals with the statistical design and analysis of inter­cropping experiments in which there are mixtures (intercrops) of two crops and/or the individual (sole) crops. Volume II will deal with the statistical design and analysis of three or more crops in the mixture (intercrop), together with sole crops and possible mixtures of two crops. It is necessary to compre­hend fully the concepts and analyses for mixtures of two crops prior to con­sidering three or more crops in the mixture. The utility, concepts, comprehen­sion, and application of techniques for two crops are an order of magnitude more difficult than for sole crops only. The degree of difficulty in these as­pects for three or more crops in the mixture is an order of magnitude greater than when considering only two crops. Hence, the reader is cautioned to comprehend fully Volume I before proceeding to Volume II. Most published literature deals with two crops in a mixture. In practice, the number of crops in a cropping system may be quite large. Mixtures of three or more crops are quite common in practice; e.g., pastures. The last chapter of Volume I con­siders design concepts and experiment designs that may be of use for inter­cropping experiments. The last two chapters of Volume II will contain a bibliography of publications on intercropping, which are not cited at the end of each chapter of Volumes I and II, and a discussion of applications of the material for intercropping experiments to other areas. Some of the areas are survey sampling, chemistry, hay crop mixtures, repeated block designs, dietary studies, and recreational and educational programs.

In presenting the statistical design and analysis for intercropping experi­ments, involving mixtures of two crops with or without the sole crops, we have attempted to present the topics in an order of increasing difficulty. First, the situation involving one main crop and one supplementary crop is

vii

viii Preface

considered in Chapter 2. Here we add little over that appearing in standard statistical methods books. Then, in Chapter 3, we consider both crops to be main crops and analyze the individual crop responses. Again, little is added on statistical methodology that is not standard. In Chapter 4, both crops are considered to be main crops and a combined response for the yields of both crops is required. This involves creating variables as in a multivariate analysis. The forms used will not ordinarily be those from standard multi­variate analyses. Ratios of yields, prices, or other variables are used. This is an innovation over other procedures appearing in the literature. We show that several analyses are desirable, as opposed to one when only sole crops are in the experiment. Density of crops is held constant up to here. In Chap­ter 5, density is a variable for the two main crops and yield is modeled as a function of density. In Chapter 6, we model responses in much the same way as they are for diallel crossing systems in breeding investigations, except that yields from both crops in the mixture are available. In Chapter 7, we do the same type of modeling for the case when the individual crop yields are not available. This is closer to the ordinary diallel crossing situation. In Chapter 8, spatial arrangements of two crops are discussed, with many arrangements being considered. In the ninth chapter of Volume I, analyses for replacement series experiments and a linear programming approach, for considering two responses simultaneously in a replacement series, are discussed. The last chapter contains a discussion of design concepts and experiment designs that are considered to be of use for intercropping experiments.

As of this date, most of the theoretical work for Volume II, Chapters 11 to 20, has been completed. Chapters 12 and 13 have already been written and are in the process of being put in final form. A search for appropriate examples is being made. A bibliography on intercropping experiments, Chap­ter 19, has been made but will require updating.

WALTER T.FEDERER

Acknowledgments

Many thanks are due to Norma Phalen for her patience and painstaking efforts in the preparation of this manuscript through its many revisions over the last 15 years or more. Thanks are also due to Colleen Bushnell for preparation of the figures.

Several individuals made comments on the various versions of the book, and their contributions are greatly appreciated. L.N. Balaam, Anila Wijesinha, Marta Zanelli, and others were involved in many discussions with the author. The data furnished by T. de Aquino-Portes, J.R.P. Carvalho, H.C. Ezumah, and Roger Mead were most helpful for examples to illustrate the procedures. Various students, in several classes conducted over the years, also made many valuable comments, which were helpful in prepar­ing a final version of this book. Special acknowledgment goes to Anila Wijesinha. She and the author spent many days discussing various aspects of the statistical analyses given in Chapters 2 to 5. The density relationships in Chapter 5 are mainly due to her. Because of her other interests, distance, and commitments, Anila decided not to continue in the coauthorship role. All her efforts, insights, and contributions are gratefully appreciated.

Last, appreciation is expressed to my wife, Edna, for her patience and encouragement during the preparation of the manuscript for the book.

WALTER T. FEDERER

ix

Contents

Preface Acknowledgments List of Tables List of Figures

CHAPTER 1 Introduction and Definitions

1.1. Introduction and Objectives 1.2. Statistical Design 1.3. Types of Treatment 1.4. Treatment Designs in Category 4 for Intercropping Experiments 1.5. Some Statistical Problems Associated with Response Model

Equations, Statistical Analyses, and Inferences 1.6. Some Concepts and Definitions 1. 7. Literature Cited

CHAPTER 2

Vll

ix xv xix

1 4 6 8

14 16 17

One Main Crop Grown with a Supplementary Crop 20

2.1. Objectives and Introduction 20 2.2. Some Examples of Main Crops Grown with a Supplemental Crop 21 2.3. Statistical Designs and Univariate Analyses of Main Crop Yields

for Experiments on Main and Supplemental Crops 22 2.4. Some Additional Statistical Analyses for Yields of Main Crop

Intercropped with a Supplemental Crop 30 2.5. Multivariate Analyses for a Main Crop Grown with

Supplemental Crops and Grown as a Sole Crop 36 2.6. Summary and Discussion 36 2.7. Problems 37 2.8. Literature Cited 37

Xl

xii

CHAPTER 3 Both Crops Main Crops-Density Constant­Analyses for Each Crop Separately

3.1. Objectives and Introduction 3.2. Treatment and Experiment Designs 3.3. Univariate Analyses on Observed Variables for Each of

the Two Main Crops 3.4. Multivariate Analyses on a Vector of Observed Variables

of One Crop 3.5. Summary and Discussion 3.6. Problems 3.7. Literature Cited

CHAPTER 4 Both Crops Main Crops-Density Constant­Combined Crop Responses

4.1. Objectives and Introduction 4.2. Univariate Analyses on Functions of Combined Variables

of Both Crops 4.3. Land Equivalent Ratio or Relative Yield Total 4.4. Multivariate Analysis on an Observation Vector of

the Intercropping Systems 4.5. Effect of Varying Proportions on Created Functions 4.6. Summary and Discussion 4.7. Problems 4.8. Literature Cited

Appendix 4.1

CHAPTER 5

Contents

39

39 40

42

51 64 65 66

67

67

69 74

77 85 87 90 97 97

Both Crops of Major Interest with Varying Densities 99

5.1. Objectives and Introduction 99 5.2. Treatment Design and Statistical Analysis for Comparisons Among

Density Combinations for Individual Crop Responses 100 5.3. Statistical Analyses for Responses from Both Crops for

a Single Pair of Crops at Varying Densities 109 5.4. Models and Analyses for Monocultures and Mixtures of

Two Crops over a Range of Densities 110 5.5. Analyses for c 1 Lines of Crop 1 and c 2 Lines of Crop 2 119 5.6. Summary and Discussion 121 5.7. Problems 122 5.8. Literature Cited 128

Appendix 5.1. Derivation of Parameter Estimates and Their Variances Using a Generalized Least Squares Method for the Linear Model of Section 5.4 (Prepared by Anila Wijesinha) 128 Appendix 5.2. Distribution of Estimators and Hypothesis Tests (Prepared by Anila Wijesinha) 131

Contents xiii

CHAPTER 6 Monocultures and Their Pairwise Combinations when Responses Are Available for Each Member of the Combination 134

6.1. Introduction 6.2. Treatment Design 6.3. A Response Model and Some Statistical Analyses 6.4. Summary and Discussion 6.5. Problems 6.6. Literature Cited

Appendix 6.1 Appendix 6.2

CHAPTER 7 Monocultures and Their Pairwise Combinations when Separate

134 136 138 150 154 156 157 158

Crop Responses Are Not Available 160

7.1. Introduction 160 7.2. Treatment Designs 161 7.3. Response Model Equations and Some Statistical Analyses 164 7.4. Response Models for a Crop Competition Experiment 170 7.5. Summary and Discussion 186 7.6. Problems 188 7.7. Literature Cited 190

Appendix 7.1. Derivation of Solutions for Response Model for Design 3 191 Appendix 7.2. Derivation of Solutions and Variances for Model 4,

Example 7.5 192

CHAPTER 8 Spatial and Density Arrangements

8.1. Introduction 8.2. Spatial Arrangements-Density Constant 8.3. Spatial Arrangements-Density Variable 8.4. Statistical Analyses for the Arrangements in Section 8.3 8.5. Summary and Discussion 8.6. Problems 8.7. Literature Cited

CHAPTER 9 Some Analytical Variations for Intercropping Studies

9.1. Introduction 9.2. Replacement Series 9.3. Other Indices 9.4. Linear Programming 9.5. Summary and Discussion 9.6. Problems 9.7. Literature Cited

196

196 196 210 220 221 222 223

225

225 225 231 234 239 240 240

XIV

CHAPTER 10

Experiment Designs for Intercropping Experiments

10.1. Introduction 10.2. Principles of Design of Experiments 10.3. Zero-Way Elimination of Heterogeneity 10.4. One-Way Elimination of Heterogeneity 10.5. Two-Way Elimination of Heterogeneity 10.6. Split Plot and Split Block Designs 10.7. More Complex Experiment Designs 10.8. Plot Technique 10.9. Stability Concepts and Parsimonious Experiment Design 10.10. Problems 10.11. Literature Cited

Index

Contents

242

242 247 250 252 262 272 279 280 281 289 291

295

List of Tables

Table 1.1. Extent of polyculture culture for selected crops in Brazil (Kass, 1978). 3

Table 1.2. Different aspects of competition experiments (Mead, 1979). 10

Table 1.3. Types of long-term experiments. 12 Table 2.1. Yields of maize for ten treatments in a randomized

complete block design for number of ears per plant and for grain weight in grams per square meter. 24

Table 2.2. Analyses of variance for ears per plant and for grain weight per square meter. 25

Table 2.3. Means of yields and number of ears per plant for maize. 28 Table 2.4. Probabilities of a greater F-value for mean squares in

analysis of variance from Table 2.2. 30 Table 2.5. Percentage of sole crop yields for various intercropping

systems. 32 Table 2.6. Alternative partitioning of treatment sum of squares for

ears per plant and for grain weight per square meter. 34 Table 3.1. Analysis of variance for mn + m treatments in r blocks

of a randomized complete block design. 43 Table 3.2. Yields of observed responses for beans for twelve

treatments in a randomized complete block design with four blocks. 45

Table 3.3. Means of yields and yield components of four bean varieties under three different cropping systems. 48

Table 3.4. Analyses of variance on bean yields and yield components. 50 Table 3.5. Multivariate analysis of variance for variables Xl' X 2'

and X 3 of Table 3.2. 55

xv

XVI List of Tables

Table 3.6. MANOV A and associated statistics for the four bean varieties with variables Xl' X 2, and X3 using estimated missing plots and GENSTAT. 61

Table 3.7. MANOV A and associated statistics for three bean variables with Xl' X 2, and X3 using unequal number analysis of SAS GLM. 62

Table 4.1. Estimates of crop values for a 3: 1 ratio of bean to maize prices and an analysis of variance on this variable. 70

Table 4.2. Yields for variable X4 = Bhij and for maize grain weight, M hij , and derived values of yields. 72

Table 4.3. Mean values of relative LERs and price, and an analysis of variance on the variable Mh(i)j + 2Bhi(j). 77

Table 4.4. Means for beans and for maize yields and functions of yields. 81

Table 4.5. Bivariate analysis of variance for mixtures of maize and beans with crop yields as variables. 81

Table 5.1. Mean values from three blocks and analysis of variance for variables husk weight per plot and grain weight per hectare for maize (from Aidar, 1978). 103

Table 5.2. A bivariate analysis of variance on biblend responses. 110 Table 5.3. An analysis on the yields of crop i in mono culture

and biblend. 114 Table 5.4. Key-out of degrees of freedom of crop 1 responses for

the VI = cl(ml - 1) + c l c2(ml - l)(m2 - 1) treatments in a randomized complete block design. 120

Table 6.1. An analysis of variance for rn monoculture responses and rn(n - 1) biblend responses in a randomized complete block design with r blocks and n(n + 1)/2 treatments. 141

Table 6.2. Yields of monocultures and component yields (g) of biblends for each of three bean cultivars from four experiments. 143

Table 6.3. Analysis of variance, parameter estimates, and associated standard deviations for each of four experiments. 148

Table 6.4. Analysis of variance on monoculture and component yields of mixtures of three bean cultivars from four experiments. 149

Table 6.5. Parameter estimates and associated standard deviations for the combined analysis of the four experiments. 150

Table 7.1. Analysis of variance for sole crop and biblend responses from a randomized complete block design (Design 2). 165

Table 7.2. Analysis of variance for sole crop, biblend, and reciprocal biblend responses from a randomized complete block design for Design 3 for Models (7.1) and (7.11). 167

List of Tables XVll

Table 7.3. Analysis of variance for Design 3 in a randomized complete block design for Models (7.1) and (7.23). 168

Table 7.4. Combined yields of mixtures from Table 6.2, Ithaca, 1966. 169 Table 7.5. Analysis of variance and F-tests for data of Table 7.4. 170 Table 7.6. Yields of grain in grams for a crop competition

experiment with four wheat varieties. 171 Table 7.7. Model (7.28) solutions and variances for data of Table 7.6. 173 Table 7.8. An analysis of variance for the data of Table 7.6 using

response model equations (7.23). 174 Table 7.9. Solutions for effects and a partitioning of the

combinations sum of squares using response model equation (7.29). 178

Table 7.10. Solutions for effects and a partitioning of the combinations sum of squares using response model equation (7.30). 180

Table 7.11. Solutions for effects and a partitioning of the combinations sum of squares using response model equation (7.42). 184

Table 9.1. Replacement series wheat experiment in a randomized complete block design, Caldwell # 5,8/29/62; grain yield in grams. 229

Table 9.2. Various indices computed for the mixture yields from Examples 2.1 and 3.1. 232

Table 9.3. Kilograms of yield, starch, and protein per hectare for two crops as sole crops and in a 1: 1 mixture, and hectares required for 10,000 kg of starch and 400 kg of protein. 235

Table 10.1. ANOVA table using (10.4.3). 260 Table 10.2. ANOVA for F-rectangle in r rows and c columns. 271 Table 10.3. ANOVAs for response equation (10.6.1) and a design of

type (i). 276 Table 10.4. ANOV As for analyses for s.p. within each w.p. using

equation (10.4.3) and design of type (ii). 277 Table 10.5. ANOVA for split block design of type (iv) using an

appropriate form of equation (10.6.5). 278

List of Figures

Figure 2.1. Mean yields of maize for varieties X and Y in the various cropping systems. 31

Figure 2.2. Varieties ordered according to response of X (solid line) and for number of ears per plant (dashed line). 35

Figure 3.1. Means of bean yields for varieties A, B, C, and D in the various cropping systems. 51

Figure 3.2. Univariate ANOV A on each of p variates of multivariate vector and q degrees of freedom for treatments (Rao, 1973). 54

Figure 4.1. Relative crop values for various ratios of bean prices to maize prices (current price was approximately 4: 1). 74

Figure 4.2. Diagrammatic scheme for representing effects of treatments (adapted from Pearce and Gilliver, 1979). 79

Figure 4.3. Diagrammatic representation of effects for data from Tables 4.3 and 4.4. 84

Figure 4.4. Values of Ym + R Yb plotted against R. 88 Figure 4.5. Treatment/(treatment + error) sums of squares for

O:s;; R:s;; 100. 89 Figure 5.1. Husk weights of maize for varying maize and

bean densities. 104 Figure 5.2. Grain weights of maize for varying maize and

bean densities. 107 Figure 5.3. Grain weights of maize for varying bean and maize

densities 20, 40, and 60. 108 Figure 5.4. 12(1)(/1,12 ) values from equation (5.1) versus bean

densities for each of three maize densities. 117

X1X

xx List of Figures

Figure 5.5. Y2(1)(/I' 12 ) values from equation (5.2) versus bean densities for each of three maize densities. 118

Figure 6.1. Monoculture cultivar effects, ii' and of the ii + Ji - J. effects versus location and year. 151

Figure 6.2. Interaction effects, Yi(j)' as they interact with location and years. 151

Figure 8.1. Balanced arrangement from Veevers and Zafar-Yab (1982). 198

Figure 8.2. NeIder's fan designs. 214 Figure 8.3. Rectangular fan-type designs. 215 Figure 8.4. Circle designs by Okigbo. 216 Figure 8.5. Snail designs. 216 Figure 8.6. Okigbo circles for intercrop maize (m) and beans (b). 217 Figure 8.7. Double rows of maize (m) interspersed with four rows

of beans (b); maize density constant, bean density variable, row spacing variable. 218

Figure 8.8. Snail designs for intercrops maize (m) and beans (b). 219 Figure 9.1. Competition effects between cultivars. 226 Figure 9.2. Data for maize cultivar Y and bean cultivars Band D

from Examples 2.1 and 3.1. 227 Figure 9.3. Replacement series yields for two locations, Caldwell

(e) and McGowan (x). 230 Figure 9.4. Replacement series for starch and protein needs from

two sole crops. 235 Figure 9.5. Replacement series for starch and protein hectare

requirements for sole crops (solid line) and a mixture (dashed lines) in a 0.5 :0.5 ratio of the two crops. 237

Figure 10.1. Fisher's principles of design of experiments. 247 Figure 10.2. An expanded version of Fisher's diagram. 248 Figure 10.3. Cultivar responses to changing environments. 282 Figure 10.4. Types of desirable responses for cultivars. 284 Figure 10.5. Experimental units with a range of environments in

each s.p.e.u. 285 Figure 10.6. Experimental unit for one cultivar and varying density

and planting date. 286