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MULTIVARIATE STATISTICAL METHODS A First Course ----=--=-- This page intentionally left blank MULTIVARIATE STATISTICAL METHODS A First Course A. Marcoulides G eorge California State University, Fullerton Scott L. Hershberger University of Kansas O Routledge Taylor & Francis Group NEW YORK AND LONDON Copyright © 1997 by Lawrence Erlbaum Associates, Inc. All rights reserved. No part of this book may be reproduced in any form, by photostat, microfilm, retrieval system, or any other means, without the prior written permission of the publisher. First published by Lawrence Erlbaum Associates, Inc., Publishers 10 Industrial A venue Mahwah, New Jersey 07430 This edition published 2012 by Routledge 711 Third Avenue, New York, NY 10017 27 Church Road, Hove, East Sussex BN3 2FA Cover design by Kathryn Houghtaling Ubrary of Congress Cataloging-in-Publication Data Marcoulides, George A. Multivariate statistical methods : a first course / George A. Marcoulides, Scott L. Hershberger. p. cm. Includes bibliographical references and index. ISBN 0-8058-2571-1 (c) - ISBN 0-8058-2572-X (p). 1. Social sciences-Statistical methods. 2. SAS (Computer file). 3. Multivariate analysis-Data processing. I. Hershberger, Scott L. II. Title. HA29.M261233 1997 300'.1'5195--dc21 96-46287 CIP 10 9 8 7 6 5 4 3 Contents Preface ix 1 Introduction 1 The Impact of Computers 2 A Summary of SAS 3 The SAS Package and Multivariate Analysis 5 Exercises 6 2 Basic Matrix Algebra 9 Matrix Definitions 9 Matrix Operations 10 Matrix Addition and Subtraction 11 Scalar Multiplication 12 Matrix Multiplication 12 Matrix Inversion 15 Eigenvalues and Eigenvectors of a Matrix 18 Exercises 21 3 The Multivariate Normal Distribution and Tests of Significance 23 ll1e Univariate Standard Normal Distribution 24 Univariate Sampling Distributions and Statistical Inference 26 Other Sampling Distributions 27 Inferences About Differences Between Group Means 28 The F Statistic and Analysis of Variance 30 The Multivariate Normal Distribution 32 Multivariate Sampling Distributions and Statistical Inference 35 Hotelling's Statistic 36 Inferences About Differences Between Group Means 36 Multivariate Analysis of Variance 39 v vi CONTENTS An Example of Multivariate Analysis of Variance Using SAS 42 Evaluating Multivariate Normality 48 Exercises 52 4 Factorial Multivariate Analysis of Variance 56 The Univariate Two-Way Analysis of Variance 57 Fixed, Random, and Mixed Factorial Designs 63 Two-Way Multivariate Analysis of Variance 64 An Example of Two-Way MANOVA Using SAS 68 More Advanced Factorial MANOVAs 79 Designs With Unequal Observations 80 Exercises 80 5 Discriminant Analysis 85 Descriptive Discriminant Analysis 86 Selecting the Discriminant Criterion 87 Maximizing the Discriminant Criterion 88 Discriminant Functions 89 Interpretation of Discriminant Functions 89 Discriminant Function Plots 97 Tests of Statistical Significance in Discriminant Analysis 97 An Example of Descriptive Discriminant Analysis Using SAS 100 Factorial Descriptive Discriminant Analysis Designs 102 Predictive Discriminant Analysis 104 Classification Based on Generalized Distance 105 Posterior Probability of Group Membership 105 An Overview of the Different Types of Classification Rules 107 Linear/Equal Prior Probability Classification Rules 107 Linear/Unequal Prior Probability Rules 108 Quadratic/Equal Prior Probability Rules 109 Quadratic/Unequal Prior Probability Rules 110 Type of Data Used for Classification 110 The Fisher Two-Group Classification Function 111 Evaluating Classification Accuracy 112 Numerical Classification Example 114 Exercises 129 6 Canonical Correlation 133 Multiple Correlation/Regression 134 An Example of Regression Analysis Using SAS 138 Discriminant Analysis and Regression Analysis 142 Canonical Correlation Analysis 144 Obtaining the Canonical Correlations 147 Interpretation of Canonical Variates 149 Tests of Significance of Canonical Correlations 151 The Practical Importance of Canonical Correlations 152 Numerical Example Using SAS 152 Exercises 158 CONTENTS vii 7 Principal Components and Factor Analysis 162 PCA Versus FA 164 Principal Component Analysis 164 An Example of PCA 165 The Principal Components 167 Principal Component Loadings 168 How Many Components? 1 72 Loose Ends 173 An Example of PCA Using SAS 174 Factor Analysis 176 Methods of Estimation 184 How Many Factors Should Be Used in a Factor Analysis> 187 Rotation of Factors 189 Loose Ends 194 An Example of Factor Analysis Using SAS 195 Exercises 203 8 Confirmatory Factor Analysis and Structural Equation Modeling 208 A Short History of Confirmatory Factor Analysis 209 Why the Term "Structural Equation Model"> 211 Using Diagrams to Represent Models 213 Mathematical Representation of Structural Equation Models 214 The Confirmatory Factor Analysis Model 218 Model Estimation 220 The Problem of Identification 223 A Numerical CFA Example 226 Goodness-of-Fit Indices 243 Multisample Analysis 252 Second-Order Factor Analysis 255 An Extension to a Simple Stmctural Equation Model 260 A SEM With Recursive and Nonrecursive Paths 263 Exercises 269 Appendix A 281 Appendix B 290 References 308 Au~orIndex 315 Subject Index 319 This page intentionally left blank Preface The purpose of this book is to introduce multivariate statistical methods to advanced undergraduate and graduate business students, although students of other disciplines will also find the book useful. The material presented is suitable for a one-semester introductory course and provides coverage of what we believe to be the most commonly used multivariate techniques. The book is not intended to be particularly comprehensive; rather, the in tention is to keep mathematical details to a minimum while conveying the basic principles of multivariate statistical methods. Like most academic authors, our choice of the material in the book and mode of presentation are a joint product of our research and our teaching. One way to articulate the rationale for the mode of presentation is to draw a distinction between mathematical statisticians who gave birth to the field of multivariate statistics, like Hotelling or Wilks, and those who focus on methods for data analysis and the interpretation of results. Possibly the distinction between Pythagoreans (mathematicians) and Archimedeans (sci entists) is useful, as long as one does not assume that Pythagoreans are not interested in data analysis and Archimedeans are not interested in contrib uting to the mathematical foundations of their discipline. We certainly feel more comfortable as Archimedeans, although we occasionally indulge in Pythagorean thinking. Therefore, this book is primarily written for individuals concerned with data analysis, although tme expertise requires familiarity with both approaches. It is assumed that readers already have a working knowledge of intro ductory statistics, particularly tests of significance using the normal, t, F, and chi-square distributions, single and multiple factor analysis of variance, and ix