Variations on Split Plot and SplitBlock Experiment Designs

Variations on Split Plot and SplitBlock Experiment Designs

WALTER T. FEDERERCornell University

Departments of Biological Statistics and Computational Biology and Statistical

Sciences

Ithaca, NY

FREEDOM KINGCornell University

Department of Biological Statistics and Computational Biology

Ithaca, NY

WILEY-INTERSCIENCE

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Library of Congress Cataloging-in-Publication Data

Federer, Walter Theodore, 1915-Variations on split plot and split block experiment

designs / Walter T. Federer, Freedom King.

p. cm.

Includes bibliographical references and index.

ISBN-13: 978-0-470-08149-5 (acid-free paper)

ISBN-10: 0-470-08149-X (acid-free paper)

1. Experimental design. 2. Blocks (Group theory) I. King, Freedom, 1955- II. Title.

QA279.F427 2007

519.5’7–dc22 2006049672

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

To Edna, my lovely wife and helpmate

To Fidela, my loved wife and our sonsEmmanuel, Willy, Fabrice and Yves.

Contents

Preface xiii

Chapter 1. The standard split plot experiment design 1

1.1. Introduction 1

1.2. Statistical design 3

1.3. Examples of split-plot-designed experiments 6

1.4. Analysis of variance 9

1.5. F-tests 12

1.6. Standard errors for means and differences between means 14

1.7. Numerical examples 16

1.8. Multiple comparisons of means 23

1.9. One replicate of a split plot experiment design

and missing observations 26

1.10. Nature of experimental variation 28

1.11. Repeated measures experiments 29

1.12. Precision of contrasts 29

1.13. Problems 31

1.14. References 32

Appendix 1.1. Example 1.1 code 34

Appendix 1.2. Example 1.2 code 36

Chapter 2. Standard split block experiment design 39

2.1. Introduction 39

2.2. Examples 41

2.3. Analysis of variance 43

2.4. F-tests 44

2.5. Standard errors for contrasts of effects 45

2.6. Numerical examples 46

2.7. Multiple comparisons 52

2.8. One replicate of a split block design 53

2.9. Precision 53

vii

2.10. Comments 54

2.11. Problems 55

2.12. References 55

Appendix 2.1. Example 2.1 code 56

Appendix 2.2. Example 2.2 code 56

Appendix 2.3. Problems 2.1 and 2.2 data 60

Chapter 3. Variations of the split plot experiment design 61

3.1. Introduction 61

3.2. Split split plot experiment design 62

3.3. Split split split plot experiment design 67

3.4. Whole plots not in a factorial arrangement 73

3.5. Split plot treatments in an incomplete block

experiment design within each whole plot 74

3.6. Split plot treatments in a row-column arrangement

within each whole plot treatment and in different

whole plot treatments 75

3.7. Whole plots in a systematic arrangement 76

3.8. Split plots in a systematic arrangement 77

3.9. Characters or responses as split plot treatments 77

3.10. Observational or experimental error? 79

3.11. Time as a discrete factor rather than as a continuous factor 80

3.12. Inappropriate model? 86

3.13. Complete confounding of some effects and split plot

experiment designs 90

3.14. Comments 91

3.15. Problems 91

3.16. References 93

Appendix 3.1. Table 3.1 code and data 94

Chapter 4. Variations of the split block experiment design 97

4.1. Introduction 97

4.2. One set of treatments in a randomized complete block

and the other in a Latin square experiment design 98

4.3. Both sets of treatments in split block arrangements 100

4.4. Split block split block or strip strip block experiment design 100

4.5. One set of treatments in an incomplete block design and

the second set in a randomized complete block design 106

4.6. An experiment design split blocked across

the entire experiment 107

4.7. Confounding in a factorial treatment design and in a

split block experiment design 108

4.8. Split block experiment design with a control 111

4.9. Comments 114

viii contents

4.10. Problems 114

4.11. References 115

Appendix 4.1. Example 4.1 code 115

Chapter 5. Combinations of SPEDs and SBEDs 120

5.1. Introduction 120

5.2. Factors A and B in a split block experiment design and

factor C in a split plot arrangement to factors A and B 120

5.3. Factor A treatments are the whole plot treatments and

factors B and C treatments are in a split block arrangement

within each whole plot 125

5.4. Factors A and B in a standard split plot experiment design

and factor C in a split block arrangement over both

factors A and B 127

5.5. A complexly designed experiment 130

5.6. Some rules to follow for finding an analysis for complexly

designed experiments 135

5.7. Comments 138

5.8. Problems 139

5.9. References 139

Appendix 5.1. Example 5.1 code 139

Appendix 5.2. Example 5.2 data set, code, and output 144

Chapter 6. World records for the largest analysis of variance table(259 lines) and for the most error terms (62) in oneanalysis of variance 147

6.1. Introduction 147

6.2. Description of the experiment 148

6.3. Preliminary analyses for the experiment 152

6.4. A combined analysis of variance partitioning of the

degrees of freedom 157

6.5. Some comments 163

6.6. Problems 163

6.7. References 163

Appendix 6.1. Figure 6.1 to Figure 6.6 164

Chapter 7. Augmented split plot experiment design 169

7.1. Introduction 169

7.2. Augmented genotypes as the whole plots 170

7.3. Augmented genotypes as the split plots 174

7.4. Augmented split split plot experiment design 176

7.5. Discussion 180

7.6. Problems 180

7.7. References 181

contents ix

Appendix 7.1. SAS code for ASPED, genotypes as whole plots,

Example 7.1 182

Appendix 7.2. SAS code for ASPEDT, genotypes as split plots,

Example 7.2 185

Appendix 7.3. SAS code for ASSPED, Example 7.3 186

Chapter 8. Augmented split block experiment design 188

8.1. Introduction 188

8.2. Augmented split block experiment designs 188

8.3. Augmented split blocks for intercropping experiments 193

8.4. Numerical example 8.1 194

8.5. Comments 197

8.6. Problems 197

8.7. References 198

Appendix 8.1. Codes for numerical Example 8.1 198

Chapter 9. Missing observations in split plot and split blockexperiment designs 202

9.1. Introduction 202

9.2. Missing observations in a split plot experiment design 203

9.3. Missing observations in a split block experiment design 204

9.4. Comments 204

9.5 Problems 204

9.6. References 206

Appendix 9.1. SAS code for numerical example in Section 9.2 206

Appendix 9.2. SAS code for numerical example in Section 9.3 209

Chapter 10. Combining split plot or split block designed experimentsover sites 213

10.1. Introduction 213

10.2. Combining split plot designed experiments over sites 213

10.3. Combining split block designed experiments over sites 217

10.4. Discussion 219

10.5. Problems 219

10.6. References 219

Appendix 10.1. Example 10.1 219

Appendix 10.2. Example 10.2 229

Chapter 11. Covariance analyses for split plot and split blockexperiment designs 239

11.1. Introduction 239

11.2. Covariance analysis for a standard split plot design 240

11.3. Covariance analysis for a split block experiment design 250

11.4. Covariance analysis for a split split plot experiment design 255

x contents

11.5. Covariance analysis for variations of designs 259

11.6. Discussion 260

11.7. Problems 260

11.8. References 262

Appendix 11.1. SAS code for Example 11.1 263

Appendix 11.2. SAS code for Example 11.2 264

Appendix 11.3. SAS code for Example 11.3 265

Index 267

contents xi

Preface

When it comes to designed experiments, researchers often end up creating complex

designs without having sufficient analytical expertise to handle. Researchers in plant

breeding, animal science, health sciences and so forth, come to statistical consulting

with data from rather very complex designs from time to time. Unfortunately,

statistical courses taken by these researchers may not have covered these

sophisticated designs. To make matters even more severe, there is an alarming

shortage of textbooks covering complex designs. To help alleviate the analytical

challenges of researchers dealing with complex designs, we have decided to write

this book and we do hope that it will be helpful to a lot of researchers. Understanding

and mastery of the designs covered here, assume a prior exposure to the basic

experimental designs such as: one-way completely randomized design, completely

randomized factorial experiment designs, randomized complete blocks with one or

more factors, incomplete blocks, row-column designs, Latin-square designs and so

forth. These basic designs are easy to analyze since one is dealing with one

experimental error given one has a single level of randomization of the treatment

combinations between the levels of various factors to the experimental units.

Nonetheless, this type of randomization might be rather simplistic and inappropriate

depending on the existing experimental conditions along with the constraints

imposed by limited resources. As a result, the experimenter might be forced to have

different randomizations and therefore experimental units of unequal sizes at

different levels of randomization, to overcome logistical and/or technological

constraints of an experiment. This opens up a class of more complex designs called

split plot designs or split block designs with at least two types of experimental errors.

In either case, several variations can occur with a possibility of a further partitioning

of the experimental units, leading to smaller and smaller experimental units

paralleled with more error terms used to test the significance of various factors’

effects. Furthermore, an experiment design might consist of a combination of these

two types of designs, along with treatments arranged following the basic designs for

some of the factors under investigation. A textbook on variations of split plot and

split block designs points in the right direction by addressing the urgent need of

researchers dealing with complex designs for which no reference is available to the

xiii

best of our knowledge. We have encountered a few researchers in this type of

situation through our statistical consulting activities. We are therefore convinced that

this book will be a valuable resource not only to researchers but also to instructors

teaching experiment designs courses. It is also important to adequately equip

graduate students with the important skills in complex designs for a better readiness

to real life situation challenges as far as designed experiments are concerned.

Another important innovation of this textbook consists of tackling the issue of error

reduction through blocking, analysis of covariance, or both. While blocking

relatively homogeneous experimental units into groups might help reduce sub-

stantially the experimental error, there are situations where it is neither sufficient by

itself nor feasible at all. Thus, use of available auxiliary information on the

experimental units has proven to significantly reduce the experimental error through

analysis of covariance. Analysis of covariance enables one to better control the

experimental error when covariates are judiciously chosen. We have added a chapter

on analysis of covariance to specifically provide researchers with helpful analytical

tools needed when dealing with covariates in complex designs.

WALTER T. FEDERER

FREEDOM KING

May 2006

xiv preface