STATISTICAL PRINCIPLES IN EXPERIMENTAL DESIGN Second Edition B. J. WINER Professor of Psychology Purdue University INTERNATIONAL STUDENT EDITION McGRAW-HILL KOGAKUSHA, LTD. TOKYO DUSSELDORF JOHANNESBURG LONDON MEXICO NEW DELHI PANAMA RIO DE JANEIRO SINGAPORE SYDNEY
Preface Introduction 1 xiii Chapter 1 INFERENCE WITH RESPECT TO MEANS AND VARIANCES 4 1.1 Basic terminology in sampling 4 1.2, Basic terminology in statistical estimation 7 1.3 Basic terminology in testing statistical hypotheses 10 1.4 Testing hypotheses on means a assumed known. 14 1.5 Testing hypotheses on means a unknown 21 1.6 Testing hypotheses about the difference between two means assuming homogeneity of variance 26 1.7 Computational formulas for the t statistic 35 1.8 Test for homogeneity of variance 37 1.9 Testing hypotheses about the difference between two means assuming population variances not equal 41 1.10 Testing hypotheses about the difference between two means correlated observations 44 1.11 Combining several independent tests on the same hypothesis 49 1.12 Outliers and winsorized t statistic. 51 1.13 Multivariate analog of test on differences between two means Hotelling's T 2 '54 vii
VU1,. CONTENTS Chapter 2, LINEAR MODELS 58 2.1 Linear model no distribution assumptions. 58 2.2 Linear model^estimation in univariate case 85 2.3 Linear model multivariate case with distribution assumptions 94 2.4 Correlations 105 ' m 2.5 Dwyer and SWP algorithms for the inverse of a symmetric matrix 120 2.6 Transformations yielding uncorrelated variables 126 2.7 Two sets of predictors ' 137 2.8 Testing statistical hypotheses fixed model 139 2.9 Regression of regression coefficients on supplementary variables 145 Chapter 3 DESIGN AND ANALYSIS OF SINGLE-FACTOR EXPERIMENTS 149 3.1 Introduction 149 3.2 Definitions and numerical example 152 3.3 Structural model for single-factor experiment model I 160 3.4 Structural model for single-factor experiment model II (variancecomponent model) '.. 167 3.5 Methods for deriving estimates and their expected values - 168 3.6 Comparisons among treatment means 170 3.7 Use of orthogonal components in tests for trend 177 3.8 Use of studentized range statistic 185 3.9 Alternative procedures for making a posteriori tests 196 3.10 Comparing all means with a control 201 3.11 Tests for homogeneity of variance. 205 3.12 Unequal sample sizes 210 3.13 Power and determination of sample size fixed model 220 3.14 Linear model with fixed variables 228 3.15 Multivariate analysis of variance 232 3.16 Randomized complete-block designs 240 3.17 Some special features of the variance-component model 244.3.18 Maximum-likelihood estimation and likelihood-ratio test 251 3.19 General principle in hypothesis testing 255 3.20 Testing the hypothesis of equality of a subset of T 3 (fixed model) 257 Chapter 4 SINGLE-FACTOR EXPERIMENTS HAVING REPEATED MEASURES ON THE SAME ELEMENTS 261 4.1 Purpose 261 4.2 Notation and computational procedures 262 4.3 Numerical example. 267 4.4 Statistical basis for the analysis 273 4.5 Use of analysis of variance to estimate reliability of measurements 283 4.6 Tests for trend 296'
4.7 Analysis of variance for ranked data 301 4.8 Dichotomous data 303 4.9 Hotelling's T 2. 305 IX Chapter 5 DESIGN AND ANALYSIS OF FACTORIAL EXPERIMENTS 309 5.1 General purpose 309 5.2 Terminology and notation 311 ' 5.3 Main effects *.316 5.4 Interaction effects 318 5.5 Experimental error and its estimation 320 5.6 Estimation of mean squares due to main effects and interaction effects 321-5.7 Principles for constructing F ratios 332 5.8 Higher-order factorial experiments 335 5.9 Estimation and tests of significance for three-factor experiments 343 5.10 Simple effects and their tests 347 5.11 Geometric interpretation of higher-order interactions 351 5.12 Nested factors (hierarchal designs) 359 5.13 Split-plot designs 366 5.14 Rules for deriving the expected values of mean squares 371 5.15 Quasi F ratios 375 5.16 Preliminary tests on the model and pooling procedures 378 5.17 Individual comparisons 384 5.18 Partition of main effects and interaction into trend components 388 5.19 Replicated experiments ', 3 9 1 5.20 The case n = 1 and a test for nonadditivity 394 5.21 The choice of a scale of measurement and transformations 397 5.22 Unequal cell frequencies 402 5.23 Unequal cell frequencies least-squares estimation. 404 5.24 Estimability in a general sense 422 5.25 Estimation of variance components ' 425 5.26 Estimation of the magnitude'of experimental effects 428 Chapter 6 FACTORIAL EXPERIMENTS COMPUTATIONAL PROCEDURES AND NUMERICAL EXAMPLES 431 6.1 General purpose - 431 6.2 p x q factorial experiment having n observations per cell 431 6.3 p x q factorial experiment unequal cell frequencies 445 6.4 Effect of scale of measurement on interaction 449 6.5 p x q x 'r factorial experiment having n observations per cell 452 6.6 Computational procedures for nested factors 464 6.7 Factorial experiment with a single control group 468 6.8 Test for nonadditivity 473 6.9 Computation of trend components 478 6.10 General computational formulas for main effects and interactions 484 6.11 Missing data 487 6.12 Special computational procedures when all factors have two levels 490
6.13 Illustrative applications 494 6.14 Unequal cell frequencies least-squares solution, 498 6.15 Analysis of variance in terms of polynomial regression 505 Chapter 7 MULTIFACTOR EXPERIMENTS HAVING REPEATED MEASURES THE SAME ELEMENTS 514 7.1 General purpose 514 7.2 Two-factor experiment with repeated measures on one factor 518 7.3 Three-factor experiment with repeated measures (case I) 539 7.4 Three-factor experiment with repeated measures (case II) 559 7.5 Other multifactor repeated-measure plans 571 7.6 Tests on trends 577 7.7 T est ' n g equality and symmetry of covariance matrices 594 7.8 Unequal group size 599 Chapter 8 FACTORIAL EXPERIMENTS IN WHICH SOME OF THE INTERACTIONS ARE CONFOUNDED 604.8.1 General purpose ' 604 8.2 Modular arithmetic ' 606 8.3 Revised notation for factorial experiments 608 8.4 Method for obtaining the components of, interactions 609 8.5 Designs for 2x2x2 factorial experiments in blocks of size 4 619 8.6 Simplified computational procedures for 2 k factorial experiments 626 8.7 Numerical example of 2 x 2 x 2 factorial experiment in blocks of size 4 630 8.8 Numerical example.of 2x2x2 factorial experiment in blocks of size 4 (repeated measures) 635 8.9 Designs for 3x3 factorial experiments 639 8.10 Numerical example of 3 x 3 factorial experiment in blocks of size 3 646 8.11 Designs for 3 x 3 x 3 factorial experiments 650 8.12 Balanced 3x2x2 factorial experiment in blocks of size 6 661 8.13 Numerical example of 3x2x2 factorial experiment in blocks of size 6 668 8.14 3x3x3x2 factorial experiment in blocks of size 6 671 8.15 Fractional replication. 676 Chapter 9 LATIN SQUARES AND RELATED DESIGNS 685 9.1 Definition of Latin square _ 685 9.2 Enumeration of Latin squares 688 9.3 Structural relation between Latin squares and three-factor factorial experiments 691 9.4 Uses of Latin squares 693 9.5 Analysis of Latin-square designs no repeated measures 696 9.6 Analysis of Greco-Latin squares 709 9.7 Analysis of Latin squares repeated measures 711
Chapter 10 ANALYSIS OF COVARIANCE.. 752. 10.1 General purpose, 752 10.2 Single-factor experiment 755 10.3 Numerical example of single-factor experiment 775 10.4 Factorial experiment 781 40.5 Computational procedures for factorial experiment 787 10.6 Factorial experiment repeated measures 796 10.7 Multiple covariates ' 809 Xi Appendix A RANDOM VARIABLES 813 A.I Random variables and probability distributions 814 A.2 Normal distribution 822 A.3 Gamma and chi-square distributions 824 A.4 Beta and F distributions 828 A.5 Student's t distribution 834 A.6 Bivariate normal distribution 837 A.7 Multivariate normal distribution 839 A.8 Distribution of quadratic forms - 845 Appendix B TOPICS CLOSELY RELATED TO THE ANALYSIS OF VARIANCE 848 B.I Kruskal-Wallis H test 848 B.2 Contingency table with repeated measures 849 B.3 Comparing treatment effects with a control 854 B.4 General partition of. degrees of freedom in a contingency table. 855 Appendix C TABLES ' 860 C.I Unit normal distribution 861 C.2 Student's / distribution 863 C.3 ' F distribution 865 C.3a F distribution (supplement) 868 C.4 Distribution of the studentized range statistic 870 C.5 Arcsin transformation 872. C.6 Distribution of t statistic in comparing treatment means with a control 873 C.7 Distribution of F max statistic 875 C.8 Critical values for Cochran's test for homogeneity of variance 876 C.9 Chi-square distribution. 877 C. 10 Coefficients of orthogonal polynomials 878
XU CONTENTS C.ll Curves of constant power for tests on main effects 879 C.I2 Random permutations of 16 numbers 881 C.I3 Noncentral t distribution 883 C.14 Noncentral F distribution 886 Content References 888 References to Experiments 893 Index., 897