WINER Professor of Psychology and Statistics Purdue University McGRAW-HILL BOOK COMPANY 1962 New York San Francisco Toronto 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 mathe- matics 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. 1 and 2 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 one- semester 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 8 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.
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. 6 with the corresponding material in Chap.
Selected topics from Chaps. 7 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. Pearson and the trustees of Biometrika for permission to reproduce parts of Tables B.9 from Biometrika Tables for Statisticians, vol. The author is indebted to H. Guthrie for permission to reproduce Table B.4, which was taken from WADC Technical Report 58-484, vol.
The author is indebted to C. Dunnett and the editor of the Journal of the American Statistical Association for permission to reprint Table B. The author is also indebted to C. Wallis for permission to reprint Table B.8, which appears in Techniques of Statistical Analysis, 1947.
The author is also indebted to L. Mahmoud as well as the editor of Psychometrika for permission to reprint Table B. Special thanks are due to Mrs. Lehman and Mrs.
Smith for excellent secretarial assistance in preparing the manuscript. The author is particularly grateful to Dr. Wood for many reasons, and to Dr. Lubin, whose critical reading of the manuscript did much to help the author prepare the present version of this book.
Winer Contents Preface v Introduction 1 Chapter 1. Basic Concepts in Statistical Inference .1 Basic terminology in sampling .2 Basic terminology in statistical estimation .3 Basic terminology in testing statistical hypotheses. Testing Hypotheses about Means and Variances .1 Testing hypotheses on means—a assumed known .2 Tests of hypotheses on means—a estimated from sample data .3 Testing hypotheses about the difference between two means— assuming homogeneity of variance .4 Computational formulas for the t statistic .5 Test for homogeneity of variance 33 2.6 Testing hypotheses about the difference between two means— assuming that population variances are not equal .7 Testing hypotheses about the difference between two means— .8 Combining several independent tests on the same hypothesis. Design and Analysis of Single-factor Experiments .2 Definitions and numerical example .3 Structural model for single-factor experiment—model I .4 Structural model for single-factor experiment—model I I (variance component model) .5 Methods for deriving estimates and their expected values .6 Comparisons among treatment means .7 Use of orthogonal components in tests for trend .8 Use of the studentized range statistic .9 Alternative procedures for making a posteriori tests .10 Comparing all means with a control 89 3.11 Tests for homogeneity of variance .12 Unequal sample sizes .13 Determination of sample size.
104 Viii CONTENTS Chapter 4. Single-factor Experiments Having Repeated Measures on the Same Elements .2 Notation and computational procedures .4 Statistical basis for the analysis 116 4.5 Use of analysis of variance to estimate reliability of measurements 124 4.6 Tests for trend .7 Analysis of variance for ranked data 136 4. Design and Analysis of Factorial Experiments .2 Terminology and notation .5 Experimental error and its estimation 150 5.6 Estimation of mean squares due to main effects and interaction effects .7 Principles for constructing Fratios .8 Higher-order factorial experiments 162 5.9 Estimation and tests of significance for three-factor experiments .10 Simple effects and their tests 174 5.11 Geometric interpretation of higher-order interactions .13 Split-plot designs .14 Rules for deriving the expected values of mean squares 195 5.16 Preliminary tests on the model and pooling procedures 202 5.18 Partition of main effects and interaction into trend components 211 5.20 The case n = 1 and a test for nonadditivity .21 The choice of a scale of measurement and transformations 218 5.22 Unequal cell frequencies .23 Unequal cell frequencies—least-squares solution. Factorial Experiments—Computational Procedures and Numerical Examples .2 p • a factorial experiment having n observations per cell .3 p x q factorial experiment—unequal cell frequencies .4 Effect of scale of measurement on interaction .•; <• factorial experiment having n observations per cell 248 6.6 Computational procedures for nested factors .7 Factorial experiment with a single control group .8 Test for nonadditivity .9 Computation of trend components 273 6.10 General computational formulas for main effects and interactions 278 6.12 Special computational procedures when all factors have two levels 283 6.14 Unequal cell frequencies—least-squares solution 291 Chapter 7.
Multifactor Experiments Having Repeated Measures on the Same Elements .2 Two-factor experiment with repeated measures on one factor .3 Three-factor experiment with repeated measures (case I) .4 Three-factor experiment with repeated measures (case II) .5 Other multifactor repeated-measure plans 349 7.6 Tests on trends .7 Testing equality and symmetry of covariance matrices .8 Unequal group size. Factorial Experiments in Which Some of the Interactions Are Confounded .3 Revised notation for factorial experiments .4 Method for obtaining the components of interactions .5 Designs for 2 x 2 x 2 factorial experiments in blocks of size 4 394 8.6 Simplified computational procedures for 2k factorial experiments 399 8.7 Numerical example of 2 x 2 x 2 factorial experiment in blocks of size 4 .8 Numerical example of 2 x 2 x 2 factorial experiment in blocks of size 4 (repeated measures) .9 Designs for 3 x 3 factorial experiments .10 Numerical example of 3 x 3 factorial experiment in blocks of size 3 .11 Designs for 3 x 3 x 3 factorial experiments .12 Balanced 3 x 2 x 2 factorial experiment in blocks of size 6 .13 Numerical example of 3 x 2 x 2 factorial experiment in blocks of size 6 .143 x 3 x 3 x 2 factorial experiment in blocks of size 6. Balanced Lattice Designs and Other Balanced Incomplete-block Designs 456 9.2 Balanced simple lattice .3 Numerical example of balanced simple lattice .4 Balanced lattice-square designs .5 Balanced incomplete-block designs • 477 9.6 Numerical example of balanced incomplete-block design .8 Numerical example of Youden square .9 Partially balanced designs .10 Numerical example of partially balanced design .11 Linked paired-comparison designs 511 X CONTENTS Chapter 10. Latin Squares and Related Designs .1 Definition of Latin square .2 Enumeration of Latin squares .3 Structural relation between Latin squares and three-factor factorial experiments .4 Uses of Latin squares .5 Analysis of Latin-square designs—no repeated measures 524 10.6 Analysis of Greco-Latin squares 536 10.7 Analysis of Latin squares—repeated measures.
Analysis of Covariance .2 Single-factor experiments 581 11.3 Numerical example of single-factor experiment 588 11.5 Computational procedures for factorial experiment 599 11.6 Factorial experiment—repeated measures .7 Multiple covariates 618 Appendix A. Topics Closely Related to the Analysis of Variance .l Kruskal-Wallis H test .2 Contingency table with repeated measures .3 Comparing treatment effects with a control .4 General partition of degrees of freedom in a contingency table 629 A.5 Hotelling's T2 test for the equality of k means .6 Least-squares estimators—general principles .l Unit normal distribution.4 Distribution of the studentized range statistic 648 B.6 Distribution of t statistic in comparing treatment means with a control .7 Distribution of FmiiK statistic .8 Critical values for Cochran's test for homogeneity of variance .9 Chi-square distribution .10 Coefficients of orthogonal polynomials .l 1 Curves of constant power for the test on main effects .l2 Random permutations of 16 numbers. , 661 References to Experiments. 667 Introduction The design of an experiment may be compared to an architect's plans for a structure, whether it be a giant skyscraper or a modest home.
The basic requirements for the structure are given to the architect by the prospective owner. It is the architect's task to fill these basic requirements; yet the architect has ample room for exercising his ingenuity. Several different plans may be drawn up to meet all the basic requirements. Some plans may be more costly than others; given two plans having the same cost, one may offer potential advantages that the second does not.
I n the design of an experiment, the designer has the role of the architect, the experimenter the role of the prospective owner. These two roles are not necessarily mutually exclusive—the experimenter may do a considerable portion of the design work. The basic requirements and primary objectives of the experiment are formulated by the experimenter; the experimenter may or may not be aware of the possible alternative approaches that can be followed in the conduct of his experiment. It is the designer's function to make the experimenter aware of these alternatives and to indicate the poten- tial advantages and disadvantages of each of the alternative approaches.
It is, however, the experimenter's task to reach the final decision about the conduct of the experiment. The individual best qualified to design an experiment is the one who is (1) most familiar with the nature of the experimental material, (2) most familiar with the possible alternative methods for designing the experiment, (3) most capable of evaluating the potential advantages and disadvantages of the alternatives. Where an individual possesses all these qualifications, the roles of experimenter and designer are one. On some research problems in many experimental fields, the experimenter is capable of making all the necessary decisions without seeking extensive assistance.