Design and Analysis of Experiments: Introduction to Experimental Design, Volume 1, Second Edition

653 Pages·2008·12.53 MB·English

by  Klaus Hinkelmann

Design and Analysis
of Experiments
THEW lLEY BICENTENNIAL-KNOWLEDGFEO R GENERATIONS
G a c hg eneration has its unique needs and aspirations. When Charles Wiley first
opened his small printing shop in lower Manhattan in 1807, it was a generation
of boundless potential searching for an identity. And we were there, helping to
define a new American literary tradition. Over half a century later, in the midst
of the Second Industrial Revolution, it was a generation focused on building the
future. Once again, we were there, supplying the critical scientific, technical, and
engineering knowledge that helped frame the world. Throughout the 20th
Century, and into the new millennium, nations began to reach out beyond their
own borders and a new international community was born. Wiley was there,
expanding its operations around the world to enable a global exchange of ideas,
opinions, and know-how.
For 200 years, Wiley has been an integral part of each generation’s journey,
enabling the flow of information and understanding necessary to meet their needs
and fulfill their aspirations. Today, bold new technologies are changing the way
we live and learn. Wiley will be there, providing you the must-have knowledge
you need to imagine new worlds, new possibilities, and new opportunities.
Generations come and go, but you can always count on Wiley to provide you the
knowledge you need, when and where you need it!
4
WILLIAM J. PESCE PETER BOOTH WILEY
PRESIDENT AND CHIEF EXECUTIVEO mCER CHAIRMAN OF THE BOARD
Design and Analysis
of Experiments
Volume 1
Introduction to Experimental Design
Second Edition
Klaus Hinkelmann
Virginia Polytechnic Institute and State University
Department of Statistics
Blacksburg, VA
Oscar Kempthorne
Iowa State University
Department of Statistics
Ames, IA
BICENTENNIAL
BICENTENNIAL
WILEY-INTERSCIENCE
A John Wiley & Sons, Inc., Publication
Copyright 0 2008 by John Wiley & Sons. Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
KOp art of this publication may be reproduced, stored in a retrieval system. or transmitted in any form
or by any means, electronic, mechanical, photocopying, recording, scanning. or otherwise, except as
permitted under Section 107 or 108 of the 1976 United States Copyright Act. without either the prior
written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to
the Copyright Clearance Center, Inc.; 222 Rosewood Drive, Danvers, MA 01923, [phone], fax
[phone], or on the web at www.copyright.com. Requests to the Publisher for permission should
be addressed to the Permissions Department, John Wiley & Sons, Inc., 11 1 River Street, Hoboken, ZJ
07030, (201) 748-601 1. fax [phone]. or online at http://www.wiley.com/go/permission.
Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in
preparing this book. they make no representations or warranties with respect to the accuracy or
completeness of the contents of this book and specifically disclaim any implied warranties of
merchantability or fitness for a particular purpose. No warranty may be created or extended by sales
representatives or written sales materials. The advice and strategies contained herein may not be
suitable for your situation. You should consult with a professional where appropriate. Neither the
publisher nor author shall be liable for any loss of profit or any other commercial damages, including
but not limited to special, incidental. consequential, or other damages.
For general information on our other products and services or for technical support, please contact our
Customer Care Department within the United States at [phone], outside the United States at
[phone] or fax [phone].
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may
not be available in electronic format. For information about Wiley products. visit our web site at
www.wiley.com.
Wiley Bicentennial Logo: Richard J. Pacific0
Library of Congress Cataloging-in-Publication Data:
Hinkelmann, Klaus, 1932-
Design and analysis of experiments / Klaus Hinkelmann, Oscar
Kempthome. 2nd ed.
-
v. cm. - (Wiley series in probability and statistics)
Includes index.
Contents: v. 1. Introduction to experimental design
ISBN 978-0-471-72756-9 (cloth)
1. Experimental design. I. Kempthome, Oscar. 11. Title.
QA279.K45 2008
519.5'7-dc22 2007017347
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
Contents
Preface to the Second Edition xvii
Preface to the First Edition xxi
1 The Processes of Science 1
1.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. 1 Observations in Science . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Two Types of Observations . . . . . . . . . . . . . . . . . . . 2
1.1.3 From Observation to Law . . . . . . . . . . . . . . . . . . . 3
1.2 DEVELOPMENT OF THEORY . . . . . . . . . . . . . . . . . . . . 5
1.2.1 The Basic Syllogism . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Induction, Deduction, and Hypothesis . . . . . . . . . . . . . 6
1.3 THE NATURE AND ROLE OF THEORY IN SCIENCE . . . . . . . 8
1.3.1 What Is Science? . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 Two Types of Science . . . . . . . . . . . . . . . . . . . . . 9
1.4 VARIETIES OF THEORY . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.1 Two Types of Theory . . . . . . . . . . . . . . . . . . . . . . 11
1.4.2 What Is a Theory? . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 THE PROBLEM OF GENERAL SCIENCE . . . . . . . . . . . . . . 14
1.5.1 Two Problems . . . . . . . . . . . . . . . . . . . . . . . . . 15
1 S.2 The Role of Data Analysis . . . . . . . . . . . . . . . . . . . 15
1 S.3 The Problem of Inference . . . . . . . . . . . . . . . . . . . 16
1.6 CAUSALITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6.1 Defining Cause. Causation. and Causality . . . . . . . . . . . 17
1.6.2 The Role of Comparative Experiments . . . . . . . . . . . . . 19
1.7 THEUPSHOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.8 WHAT IS AN EXPERIMENT? . . . . . . . . . . . . . . . . . . . . 21
1.8.1 Absolute and Comparative Experiments . . . . . . . . . . . . 22
1.8.2 Three Types of Experiments . . . . . . . . . . . . . . . . . . 23
1.9 STATISTICAL INFERENCE . . . . . . . . . . . . . . . . . . . . . . 24
1.9.1 Drawing Inference . . . . . . . . . . . . . . . . . . . . . . . 24
1.9.2 Notions of Probability . . . . . . . . . . . . . . . . . . . . . 25
1.9.3 Variability and Randomization . . . . . . . . . . . . . . . . . 26
vi CONTENTS
2 Principles of Experimental Design 29
2.1 CONFIRMATORY AND EXPLORATORY EXPERIMENTS . . . . 29
2.2 STEPS OF DESIGNED INVESTIGATIONS . . . . . . . . . . . . . 30
2.2.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . 31
2.2.2 Subject Matter Model . . . . . . . . . . . . . . . . . . . . . 32
2.2.3 Three Aspects of Design . . . . . . . . . . . . . . . . . . . . 33
2.2.4 Modeling the Response . . . . . . . . . . . . . . . . . . . . . 35
2.2.5 Choosing the Response . . . . . . . . . . . . . . . . . . . . . 36
2.2.6 Principles of Analysis . . . . . . . . . . . . . . . . . . . . . 36
2.3 THE LINEAR MODEL . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.1 Three Types of Effects . . . . . . . . . . . . . . . . . . . . . 37
2.3.2 Experimental and Observational Units . . . . . . . . . . . . . 38
2.3.3 Outline of a Model . . . . . . . . . . . . . . . . . . . . . . . 40
2.4 ILLUSTRATING INDIVIDUAL STEPS: STUDY 1 . . . . . . . . . 41
2.4.1 The Questions and Hypotheses . . . . . . . . . . . . . . . . . 41
2.4.2 The Experiment and a Model . . . . . . . . . . . . . . . . . . 41
2.4.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4.4 Alternative Experimental Setup . . . . . . . . . . . . . . . . 44
2.5 THREE PRINCIPLES OF EXPERIMENTAL DESIGN . . . . . . . . 45
2.6 THE STATISTICAL TRIANGLE: STUDY 2 . . . . . . . . . . . . . 46
2.6.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . 46
2.6.2 Four Experimental Situations . . . . . . . . . . . . . . . . . 46
2.7 PLANNING THE EXPERIMENT: THINGS TO THINK ABOUT . . 5 1
2.8 COOPERATION BETWEEN SCIENTIST AND STATISTICIAN . . 53
2.9 GENERAL PRINCIPLE OF INFERENCE AND TYPES OF
STATISTICAL ANALYSES . . . . . . . . . . . . . . . . . . . . . . 56
2.9.1 General Model . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.9.2 Outline of the ANOVA . . . . . . . . . . . . . . . . . . . . . 56
2.10 OTHER CONSIDERATIONS FOR EXPERIMENTAL DESIGNS . . 58
3 Survey of Designs And Analyses 61
3.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2 ERROR-CONTROL DESIGNS . . . . . . . . . . . . . . . . . . . . 62
3.3 TREATMENT DESIGNS . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4 COMBINING IDEAS FROM ERROR-CONTROL AND
TREATMENT DESIGNS . . . . . . . . . . . . . . . . . . . . . . . . 65
3.5 SAMPLING DESIGNS . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.6 ANALYSIS AND STATISTICAL SOFTWARE . . . . . . . . . . . . 68
3.7 SUMMARY., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4 Linear Model Theory 71
4.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.1.1 The Concept of a Model . . . . . . . . . . . . . . . . . . . . 71
4.1.2 Comparative and Absolute Experiments . . . . . . . . . . . . 73
4.2 REPRESENTATION OF LINEAR MODELS . . . . . . . . . . . . . 73
4.3 FUNCTIONAL AND CLASSIFICATORY LINEAR MODELS . . . 74
CONTENTS vii
4.3.1 Functional Models . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.2 Classificatory Models . . . . . . . . . . . . . . . . . . . . . 74
4.3.3 Models with Classificatory and Functional
Components . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4 THE FITTING OF y = Xp . . . . . . . . . . . . . . . . . . . . . . 76
4.4.1 The Notion of Identifiability . . . . . . . . . . . . . . . . . . 76
4.4.2 The Notion of Estimability . . . . . . . . . . . . . . . . . . . 77
4.4.3 The Method of Least Squares . . . . . . . . . . . . . . . . . 77
4.4.4 Theory of Linear Equations . . . . . . . . . . . . . . . . . . 81
4.5 MOORE-PENROSE GENERALIZED INVERSE . . . . . . . . . . . 84
4.6 CONDITIONED LINEAR MODEL . . . . . . . . . . . . . . . . . . 85
4.6.1 Affine Linear Model . . . . . . . . . . . . . . . . . . . . . . 85
4.6.2 Normal Equations for the Conditioned Model . . . . . . . . . 87
4.6.3 Different Types of Conditions . . . . . . . . . . . . . . . . . 88
4.6.4 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.7 TWO-PART LINEAR MODEL . . . . . . . . . . . . . . . . . . . . 90
4.7.1 Ordered Linear Models . . . . . . . . . . . . . . . . . . . . . 90
4.7.2 Using Orthogonal Projections . . . . . . . . . . . . . . . . . 91
4.7.3 Orthogonal ANOVA . . . . . . . . . . . . . . . . . . . . . . 93
4.8 SPECIAL CASE OF A PARTITIONED MODEL . . . . . . . . . . . 94
4.9 THREE-PART MODELS . . . . . . . . . . . . . . . . . . . . . . . . 94
4.10 TWO-WAY CLASSIFICATION WITHOUT INTERACTION . . . . 95
4.1 1 K-PART LINEAR MODEL . . . . . . . . . . . . . . . . . . . . . . . 97
4.11.1 The General Model and Its Sums of Squares . . . . . . . . . . 97
4.1 1.2 The Means Model . . . . . . . . . . . . . . . . . . . . . . . 99
4.12 BALANCED CLASSIFICATORY STRUCTURES . . . . . . . . . . 100
4.12.1 Factors, Levels, and Partitions . . . . . . . . . . . . . . . . . 101
4.12.2 Nested, Crossed, and Confounded Factors . . . . . . . . . . . 101
4.12.3 The Notion of Balance . . . . . . . . . . . . . . . . . . . . . 102
4.12.4 Balanced One-way Classification . . . . . . . . . . . . . . . 102
4.12.5 Two-way Classification with Equal Numbers . . . . . . . . . 103
4.12.6 Experimental versus Observational Studies . . . . . . . . . . 104
4.12.7 General Classificatory Structure . . . . . . . . . . . . . . . . 106
4.12.8 The Well-Formulated Model . . . . . . . . . . . . . . . . . . 109
4.13 UNBALANCED DATA STRUCTURES . . . . . . . . . . . . . . . . 112
4.13.1 Two-Fold Nested Classification . . . . . . . . . . . . . . . . 112
4.13.2 Two-way Cross-Classification . . . . . . . . . . . . . . . . . 113
4.13.3 Two-way Classification without Interaction . . . . . . . . . . 116
4.14 ANALYSIS OF COVARIANCE MODEL . . . . . . . . . . . . . . . 118
4.14.1 The Question of Explaining Data . . . . . . . . . . . . . . . 118
4.14.2 Obtaining the ANOVA Table . . . . . . . . . . . . . . . . . . 120
4.14.3 The Case of One Covariate . . . . . . . . . . . . . . . . . . . 121
4.14.4 The Case of Several Covariates . . . . . . . . . . . . . . . . 121
4.15 FROM DATA ANALYSIS TO STATISTICAL INFERENCE . . . . . 122
4.16 SIMPLE NORMAL STOCHASTIC LINEAR MODEL . . . . . . . . 123
4.16.1 The Notion of Estimability . . . . . . . . . . . . . . . . . . . 123
...
CONTENTS
Vlll
4.16.2 Gauss-Markov Linear Model . . . . . . . . . . . . . . . . . . 124
4.16.3 Ordinary Least Squares and Best Linear Unbiased Estimators 126
4.16.4 Expectation of Quadratic Forms . . . . . . . . . . . . . . . . 128
4.17 DISTRIBUTION THEORY WITH GXNLM . . . . . . . . . . . . . 128
4.17.1 Distributional Properties of X'P . . . . . . . . . . . . . . . . 128
4.17.2 Distribution of Sums of Squares . . . . . . . . . . . . . . . . 130
4.17.3 Testing of Hypotheses . . . . . . . . . . . . . . . . . . . . . 131
4.18 MIXED MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.18.1 The Notion of Fixed, Mixed and Random Models . . . . . . . 132
4.18.2 Aitken-like Model . . . . . . . . . . . . . . . . . . . . . . . 133
4.18.3 Mixed Models in Experimental Design . . . . . . . . . . . . 134
5 Randomization 137
5.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.1.1 Observational versus Intervention Studies . . . . . . . . . . . 137
5.1.2 Historical Controls versus Repetitions . . . . . . . . . . . . . 139
5.2 THE TEA TASTING LADY . . . . . . . . . . . . . . . . . . . . . . 139
5.3 TRIANGULAR EXPERIMENT . . . . . . . . . . . . . . . . . . . . 140
5.3.1 Medical Example . . . . . . . . . . . . . . . . . . . . . . . . 141
5.3.2 Randomization. Probabilities. and Beliefs . . . . . . . . . . . 141
5.4 SIMPLE ARITHMETICAL EXPERIMENT . . . . . . . . . . . . . . 142
5.4.1 Noisy Experiments . . . . . . . . . . . . . . . . . . . . . . . 142
5.4.2 Investigative Experiments and Beliefs . . . . . . . . . . . . . 144
5.4.3 Randomized Experiments . . . . . . . . . . . . . . . . . . . 145
5.5 RANDOMIZATION IDEAS . . . . . . . . . . . . . . . . . . . . . . 148
5.6 EXPERIMENT RANDOMIZATION TEST . . . . . . . . . . . . . . 150
5.7 INTRODUCTION TO SUBSEQUENT CHAPTERS . . . . . . . . . 151
6 Completely Randomized Design 153
6.1 INTRODUCTION AND DEFINITION . . . . . . . . . . . . . . . . 153
6.2 RANDOMIZATION PROCESS . . . . . . . . . . . . . . . . . . . . 154
6.2.1 Use of Random Numbers . . . . . . . . . . . . . . . . . . . . 154
6.2.2 Design Random Variables . . . . . . . . . . . . . . . . . . . 154
6.3 DERIVED LINEAR MODEL . . . . . . . . . . . . . . . . . . . . . 157
6.3.1 Conceptual Responses and Observations . . . . . . . . . . . . 157
6.3.2 Distributional Properties . . . . . . . . . . . . . . . . . . . . 159
6.3.3 Additivity in the Broad Sense . . . . . . . . . . . . . . . . . 161
6.3.4 Error Structure . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.3.5 Summary of Results . . . . . . . . . . . . . . . . . . . . . . 164
6.4 ANALYSIS OF VARIANCE . . . . . . . . . . . . . . . . . . . . . . 165
6.4.1 Deriving the ANOVA Table . . . . . . . . . . . . . . . . . . 165
6.4.2 Obtaining Expected Mean Squares . . . . . . . . . . . . . . . 168
6.5 STATISTICAL TESTS . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.5.1 Enumerating Randomizations . . . . . . . . . . . . . . . . . 171
6.5.2 Randomization Test . . . . . . . . . . . . . . . . . . . . . . 172
6.6 APPROXIMATING THE RANDOMIZATION TEST . . . . . . . . . 174
CONTENTS ix
6.6.1 Moments of the Test Statistic . . . . . . . . . . . . . . . . . . 174
6.6.2 Approximation by the F-Test . . . . . . . . . . . . . . . . . . 177
6.6.3 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . 177
6.7 CRD WITH UNEQUAL NUMBERS OF REPLICATIONS . . . . . . 179
6.7.1 Randomization . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.7.2 The Model and ANOVA . . . . . . . . . . . . . . . . . . . . 180
6.7.3 Comparing Randomization Test and F-Test . . . . . . . . . . 180
6.8 NUMBER OF REPLICATIONS . . . . . . . . . . . . . . . . . . . . 180
6.8.1 Power of the F-Test . . . . . . . . . . . . . . . . . . . . . . . 182
6.8.2 Smallest Detectable Difference . . . . . . . . . . . . . . . . . 184
6.8.3 Practical Considerations . . . . . . . . . . . . . . . . . . . . 185
6.9 SUBSAMPLING IN A CRD . . . . . . . . . . . . . . . . . . . . . . 191
6.9.1 Subsampling Model . . . . . . . . . . . . . . . . . . . . . . 191
6.9.2 Inferences with Subsampling . . . . . . . . . . . . . . . . . . 193
6.9.3 Comparison of CRDs without and with Subsampling . . . . . 193
6.10 TRANSFORMATIONS . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.10.1 Nonadditivity in the General Sense . . . . . . . . . . . . . . 196
6.10.2 Nonconstancy of Variances . . . . . . . . . . . . . . . . . . . 197
6.10.3 Choice of Transformation . . . . . . . . . . . . . . . . . . . 198
6.10.4 Power Transformations . . . . . . . . . . . . . . . . . . . . . 200
6.1 1 EXAMPLES USING SAS@ . . . . . . . . . . . . . . . . . . . . . . 201
6.12 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
7 Comparisons of Treatments 213
7.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
7.2 COMPARISONS FOR QUALITATIVE TREATMENTS . . . . . . . 213
7.2.1 Treatment Contrasts . . . . . . . . . . . . . . . . . . . . . . 214
7.2.2 Orthogonal Contrasts . . . . . . . . . . . . . . . . . . . . . . 214
7.2.3 Partitioning the Treatment Sum of Squares . . . . . . . . . . 215
7.3 ORTHOGONALITY AND ORTHOGONAL COMPARISONS . . . . 218
7.4 COMPARISONSFORQUANTITATIVETREATMENTS . . . . . . 219
7.4.1 Comparisons for Treatments with Equidistant Levels . . . . . 219
7.4.2 Use of Orthogonal Polynomials . . . . . . . . . . . . . . . . 220
7.4.3 Contrast Sums of Squares and the ANOVA . . . . . . . . . . 223
7.5 MULTIPLE COMPARISON PROCEDURES . . . . . . . . . . . . . 224
7.5.1 Multiple Comparisons and Error Rates . . . . . . . . . . . . . 224
7.5.2 Least Significant Difference Test . . . . . . . . . . . . . . . . 225
7.5.3 Bonferroni t-Statistics . . . . . . . . . . . . . . . . . . . . . 225
7.5.4 Studentized Range Procedure . . . . . . . . . . . . . . . . . 226
7.5.5 Duncan’s Multiple Range Test . . . . . . . . . . . . . . . . . 226
7.5.6 Scheffk’s Procedure . . . . . . . . . . . . . . . . . . . . . . 227
7.5.7 Comparisons with a Control . . . . . . . . . . . . . . . . . . 227
7.5.8 Alternatives to Tests Based on Normality . . . . . . . . . . . 228
7.6 GROUPING TREATMENTS . . . . . . . . . . . . . . . . . . . . . . 229
7.7 EXAMPLES USING SAS@ . . . . . . . . . . . . . . . . . . . . . . 230
7.8 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
X CONTENTS
8 Use of Supplementary Information 239
8.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
8.2 MOTIVATION OF THE PROCEDURE . . . . . . . . . . . . . . . . 240
8.3 ANALYSIS OF COVARIANCE PROCEDURE . . . . . . . . . . . . 242
8.3.1 Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . 242
8.3.2 Least Squares Analysis . . . . . . . . . . . . . . . . . . . . . 242
8.3.3 Least Squares Means . . . . . . . . . . . . . . . . . . . . . . 244
8.3.4 Formulation in Matrix Notation . . . . . . . . . . . . . . . . 245
8.3.5 ANOVA Table . . . . . . . . . . . . . . . . . . . . . . . . . 246
8.4 TREATMENT COMPARISONS . . . . . . . . . . . . . . . . . . . . 250
8.4.1 Preplanned Comparisons . . . . . . . . . . . . . . . . . . . . 250
8.4.2 Multiple Comparison Procedures . . . . . . . . . . . . . . . 251
8.5 VIOLATION OF ASSUMPTIONS . . . . . . . . . . . . . . . . . . . 252
8.5.1 Linear Relationship between x and y . . . . . . . . . . . . . . 252
8.5.2 Common Slope . . . . . . . . . . . . . . . . . . . . . . . . . 253
8.5.3 Covariates Affected by Treatments . . . . . . . . . . . . . . . 256
8.5.4 Normality Assumption . . . . . . . . . . . . . . . . . . . . . 257
8.6 ANALYSIS OF COVARIANCE WITH
SUBS AMPLING . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
8.7 CASE OF SEVERAL COVARIATES . . . . . . . . . . . . . . . . . 2 59
8.7.1 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . 260
8.7.2 Two Covariates . . . . . . . . . . . . . . . . . . . . . . . . . 262
8.8 EXAMPLES USING SAS@ . . . . . . . . . . . . . . . . . . . . . . 264
8.9 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
9 Randomized Block Designs 277
9.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
9.2 RANDOMIZED COMPLETE BLOCK DESIGN . . . . . . . . . . . 278
9.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
9.2.2 Derived Linear Model . . . . . . . . . . . . . . . . . . . . . 278
9.2.3 Estimation of Treatment Contrasts . . . . . . . . . . . . . . . 282
9.2.4 Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . 282
9.2.5 Randomization Test and F-Test . . . . . . . . . . . . . . . . 285
9.2.6 Additivity in the Broad Sense . . . . . . . . . . . . . . . . . 286
9.2.7 Subsampling in an RCBD . . . . . . . . . . . . . . . . . . . 288
9.3 RELATIVE EFFICIENCY OF THE RCBD . . . . . . . . . . . . . . 288
9.3.1 Question of Effectiveness of Blocking . . . . . . . . . . . . . 288
9.3.2 Use of Uniformity Trials . . . . . . . . . . . . . . . . . . . . 290
9.3.3 Interpretation and Use of Relative Efficiency . . . . . . . . . 291
9.4 ANALYSIS OF COVARIANCE . . . . . . . . . . . . . . . . . . . . 292
9.4.1 TheModel . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
9.4.2 Least Squares Analysis . . . . . . . . . . . . . . . . . . . . . 293
9.4.3 The ANOVA Table . . . . . . . . . . . . . . . . . . . . . . . 294
9.5 MISSING OBSERVATIONS . . . . . . . . . . . . . . . . . . . . . . 295
9.5.1 Estimating a Missing Observation . . . . . . . . . . . . . . . 295
9.5.2 Using the Estimated Missing Observation . . . . . . . . . . . 297