STATISTICAL PRINCIPLES IN EXPERIMENTAL DESIGN

Transcription

1 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

2 Preface Introduction 1 xiii Chapter 1 INFERENCE WITH RESPECT TO MEANS AND VARIANCES Basic terminology in sampling 4 1.2, Basic terminology in statistical estimation Basic terminology in testing statistical hypotheses Testing hypotheses on means a assumed known Testing hypotheses on means a unknown Testing hypotheses about the difference between two means assuming homogeneity of variance Computational formulas for the t statistic Test for homogeneity of variance Testing hypotheses about the difference between two means assuming population variances not equal Testing hypotheses about the difference between two means correlated observations Combining several independent tests on the same hypothesis Outliers and winsorized t statistic Multivariate analog of test on differences between two means Hotelling's T 2 '54 vii

3 VU1,. CONTENTS Chapter 2, LINEAR MODELS Linear model no distribution assumptions Linear model^estimation in univariate case Linear model multivariate case with distribution assumptions Correlations 105 ' m 2.5 Dwyer and SWP algorithms for the inverse of a symmetric matrix Transformations yielding uncorrelated variables Two sets of predictors ' Testing statistical hypotheses fixed model Regression of regression coefficients on supplementary variables 145 Chapter 3 DESIGN AND ANALYSIS OF SINGLE-FACTOR EXPERIMENTS Introduction Definitions and numerical example Structural model for single-factor experiment model I Structural model for single-factor experiment model II (variancecomponent model) ' Methods for deriving estimates and their expected values Comparisons among treatment means Use of orthogonal components in tests for trend Use of studentized range statistic Alternative procedures for making a posteriori tests Comparing all means with a control Tests for homogeneity of variance Unequal sample sizes Power and determination of sample size fixed model Linear model with fixed variables Multivariate analysis of variance Randomized complete-block designs Some special features of the variance-component model Maximum-likelihood estimation and likelihood-ratio test General principle in hypothesis testing 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 Purpose Notation and computational procedures Numerical example Statistical basis for the analysis Use of analysis of variance to estimate reliability of measurements Tests for trend 296'

4 4.7 Analysis of variance for ranked data Dichotomous data Hotelling's T IX Chapter 5 DESIGN AND ANALYSIS OF FACTORIAL EXPERIMENTS General purpose Terminology and notation 311 ' 5.3 Main effects * Interaction effects Experimental error and its estimation Estimation of mean squares due to main effects and interaction effects Principles for constructing F ratios Higher-order factorial experiments Estimation and tests of significance for three-factor experiments Simple effects and their tests Geometric interpretation of higher-order interactions Nested factors (hierarchal designs) Split-plot designs Rules for deriving the expected values of mean squares Quasi F ratios Preliminary tests on the model and pooling procedures Individual comparisons Partition of main effects and interaction into trend components Replicated experiments ', The case n = 1 and a test for nonadditivity The choice of a scale of measurement and transformations Unequal cell frequencies Unequal cell frequencies least-squares estimation Estimability in a general sense Estimation of variance components ' Estimation of the magnitude'of experimental effects 428 Chapter 6 FACTORIAL EXPERIMENTS COMPUTATIONAL PROCEDURES AND NUMERICAL EXAMPLES General purpose p x q factorial experiment having n observations per cell p x q factorial experiment unequal cell frequencies Effect of scale of measurement on interaction p x q x 'r factorial experiment having n observations per cell Computational procedures for nested factors Factorial experiment with a single control group Test for nonadditivity Computation of trend components General computational formulas for main effects and interactions Missing data Special computational procedures when all factors have two levels 490

5 6.13 Illustrative applications Unequal cell frequencies least-squares solution, Analysis of variance in terms of polynomial regression 505 Chapter 7 MULTIFACTOR EXPERIMENTS HAVING REPEATED MEASURES THE SAME ELEMENTS General purpose Two-factor experiment with repeated measures on one factor Three-factor experiment with repeated measures (case I) Three-factor experiment with repeated measures (case II) Other multifactor repeated-measure plans Tests on trends T est ' n g equality and symmetry of covariance matrices Unequal group size 599 Chapter 8 FACTORIAL EXPERIMENTS IN WHICH SOME OF THE INTERACTIONS ARE CONFOUNDED General purpose ' Modular arithmetic ' Revised notation for factorial experiments Method for obtaining the components of, interactions Designs for 2x2x2 factorial experiments in blocks of size Simplified computational procedures for 2 k factorial experiments Numerical example of 2 x 2 x 2 factorial experiment in blocks of size Numerical example.of 2x2x2 factorial experiment in blocks of size 4 (repeated measures) Designs for 3x3 factorial experiments Numerical example of 3 x 3 factorial experiment in blocks of size Designs for 3 x 3 x 3 factorial experiments Balanced 3x2x2 factorial experiment in blocks of size Numerical example of 3x2x2 factorial experiment in blocks of size x3x3x2 factorial experiment in blocks of size Fractional replication. 676 Chapter 9 LATIN SQUARES AND RELATED DESIGNS Definition of Latin square _ Enumeration of Latin squares Structural relation between Latin squares and three-factor factorial experiments Uses of Latin squares Analysis of Latin-square designs no repeated measures Analysis of Greco-Latin squares Analysis of Latin squares repeated measures 711

6 Chapter 10 ANALYSIS OF COVARIANCE General purpose, Single-factor experiment Numerical example of single-factor experiment Factorial experiment Computational procedures for factorial experiment Factorial experiment repeated measures 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 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

7 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