S T A T I S T I C A L I N B J P R I N C I P L E S E X P E R I M E N T A L D E S I G N WINER Professor of Psychology and Statistics Purdue University McGRAW-HILL New York BOOK COMPANY San Francisco Toronto 1962 London STATISTICAL PRINCIPLES IN EXPERIMENTAL DESIGN Copyright © 1962 by McGraw-Hill, Inc Printed in the United States of America All rights reserved This book, or parts thereof, may not be reproduced in any form without permission of the publishers Library of Congress Catalog Card Number 61-13174 70980 10111213 HDMM 7654321069 Preface Written primarily for students and research workers in the area of the behavioral sciences, this book is meant to provide a text and comprehensive reference source on statistical principles underlying experimental design Particular emphasis is given to those designs that are likely to prove useful in research in the behavioral sciences The book primarily emphasizes the logical basis of principles underlying designs for experiments rather than mathematical derivations associated with relevant sampling distributions The topics selected for inclusion are those covered in courses taught by the author during the past several years Students in these courses have widely varying backgrounds in mathematics and come primarily from the fields of psychology, education, economics, sociology, and industrial engineering It has been the intention of the author to keep the book at a readability level appropriate for students having a mathematical background equivalent to freshman college algebra From experience with those sections of the book which have been used as text material in dittoed form, there is evidence to indicate that, in large measure, the desired readability level has been attained Admittedly, however, there are some sections in the book where this readability goal has not been achieved The first course in design, as taught by the author, has as a prerequisite a basic course in statistical inference The contents of Chaps and review the highlights of what is included in the prerequisite material These chapters are not meant to provide the reader with a first exposure to these topics They are intended to provide a review of terminology and notation for the concepts which are more fully developed in later chapters By no means is all the material included in the book covered in a onesemester course In a course of this length, the author has included Chaps 3, 4, parts of 5, 6, parts of 7, parts of 10, and parts of 11 Chapters through 11 were written to be somewhat independent of each other VI PREFACE Hence one may read, with understanding, in these chapters without undue reference to material in the others In general, the discussion of principles, interpretations of illustrative examples, and computational procedures are included in successive sections within the same chapter However, to facilitate the use of the book as a reference source, this procedure is not followed in Chaps and Basic principles associated with a large class of designs for factorial experiments are discussed in Chap Detailed illustrative examples of these designs are presented in Chap For teaching purposes, the author includes relevant material from Chap with the corresponding material in Chap Selected topics from Chaps through 11 have formed the basis for a second course in experimental design Relatively complete tables for sampling distributions of statistics used in the analysis of experimental designs are included in the Appendix Ample references to source materials having mathematical proofs for the principles stated in the text are provided The author is indebted to E S Pearson and the trustees of Biometrika for permission to reproduce parts of Tables B.l, B.3, B.7, and B.9 from Biometrika Tables for Statisticians, vol I, 2d ed The author is indebted to H L Harter, D S Clem, and E H Guthrie for permission to reproduce Table B.4, which was taken from WADC Technical Report 58-484, vol I I , 1959 The author is indebted to C W Dunnett and the editor of the Journal of the American Statistical Association for permission to reprint Table B.6 The author is also indebted to C Eisenhart, M W Hastay, and W A Wallis for permission to reprint Table B.8, which appears in Techniques of Statistical Analysis, 1947 The author is also indebted to L S Feldt and M W Mahmoud as well as the editor of Psychometrika for permission to reprint Table B.I Special thanks are due to Mrs G P Lehman and Mrs R L Smith for excellent secretarial assistance in preparing the manuscript The author is particularly grateful to Dr D A Wood for many reasons, and to Dr A Lubin, whose critical reading of the manuscript did much to help the author prepare the present version of this book B J Winer Contents Preface v Introduction Chapter Basic Concepts in Statistical Inference 1.1 Basic terminology in sampling 1.2 Basic terminology in statistical estimation 1.3 Basic terminology in testing statistical hypotheses 2.1 Testing hypotheses on means—a assumed known 2.2 Tests of hypotheses on means—a estimated from sample data 2.3 Testing hypotheses about the difference between two means— assuming homogeneity of variance 2.4 Computational formulas for the t statistic 2.5 Test for homogeneity of variance 2.6 Testing hypotheses about the difference between two means— assuming that population variances are not equal 2.7 Testing hypotheses about the difference between two means— correlated observations 2.8 Combining several independent tests on the same hypothesis 14 20 Chapter Testing Hypotheses about Means and Variances Chapter Design and Analysis of Single-factor Experiments 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 14 Introduction Definitions and numerical example Structural model for single-factor experiment—model I Structural model for single-factor experiment—model I I (variance component model) Methods for deriving estimates and their expected values Comparisons among treatment means Use of orthogonal components in tests for trend Use of the studentized range statistic Alternative procedures for making a posteriori tests Comparing all means with a control Tests for homogeneity of variance Unequal sample sizes Determination of sample size 24 31 33 36 39 43 46 48 56 62 63 65 70 77 85 89 92 96 104 Viii CONTENTS Chapter Single-factor Experiments Having Repeated Measures on the Same Elements 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Chapter Design and Analysis of Factorial Experiments 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21 5.22 5.23 105 Purpose 105 Notation and computational procedures 106 Numerical example Ill Statistical basis for the analysis 116 Use of analysis of variance to estimate reliability of measurements 124 Tests for trend Analysis of variance for ranked data 136 Dichotomous data 138 140 General purpose 140 Terminology and notation 141 Main effects 146 Interaction effects 148 Experimental error and its estimation 150 Estimation of mean squares due to main effects and interaction effects 151 Principles for constructing Fratios 160 Higher-order factorial experiments 162 Estimation and tests of significance for three-factor experiments 170 Simple effects and their tests 174 Geometric interpretation of higher-order interactions 178 Nested factors (hierarchal designs) 184 Split-plot designs 191 Rules for deriving the expected values of mean squares 195 Quasi F ratios 199 Preliminary tests on the model and pooling procedures 202 Individual comparisons 207 Partition of main effects and interaction into trend components 211 Replicated experiments 213 The case n = and a test for nonadditivity 216 The choice of a scale of measurement and transformations 218 Unequal cell frequencies 222 Unequal cell frequencies—least-squares solution 224 Chapter Factorial Experiments—Computational Procedures and Numerical Examples 6.1 General purpose 6.2 p • a factorial experiment having n observations per cell 6.3 p x q factorial experiment—unequal cell frequencies 6.4 Effect of scale of measurement on interaction 6.5 p x q •;