Multivariate Analysis in The Human Services

Transcription

1 Multivariate Analysis in The Human Services

2 INTERNATIONAL SERIES IN SOCIAL WELFARE Series Editor: William J. Reid State University of New York at Albany Advisory Editorial Board: Weiner W. Boehm Rutgers, The State University New Brunswick, N.J., USA Naomi Golan University of Haifa ISRAEL Tilda Goldberg Policy Studies Institute London, England, UK Francis J. Turner Laurentian University Ontario, CANADA Previously Published Books in the Series: Crane, J.A., The Evaluation of Social Policies, 1982.

3 Multivariate Analysis in the Human Services John R. Schuerman University of Chicago Kluwer Nijhoff Publishing Boston The Hague Dordrecht Lancaster

4 Distributors for North America: Kluwer Boston, Inc. 190 Old Derby Street Hingham, MA 02043, U.S.A. Distributors Outside North America: Kluwer Academic Publishers Group Distribution Center P.O. Box AH Dordrecht, The Netherlands Library of Congress Cataloging in Publication Data Schuerman, John R. Multivariate analysis in the human services. Bibliography: p. Includes index. ISBN-13: e-isbn-13: DOl: / Social service-research-statistical methods. 2. Multivariate analysis. I. Title. HV11.S ' SAS is the registered trademark of SAS Institute Inc., Gary. NC SPSS is a registered trademark of SPSS Inc. of Chicago, Illinois for its proprietary computer software. No materials describing such software may be produced or distributed without the written permission of SPSS Inc. Copyright 1983 by Kluwer-Nijhoff Publishing. No part of this book may be reproduced in any form by print, photoprint, microfilm, or any other means without written permission of the publisher.

5 Contents List of Tables and Figures Acknowledgments viii xii Introduction 2 Mathematical Preliminaries Functions of Variables Matrices Matrix Algebra Some Matrices We Will Encounter Singularity of Matrices and Determinants I nverse of Matrices Problems 3 Multiple Regression I The Model in Matrix Terms Review of Analysis of Variance Two-Way Analysis of Variance The Analysis of Variance of Regression Interpretation of Regression Coefficients Residuals 4 Multiple Regression II Bu ilding a Regression Equation Coding of Categorical Variables for Regression Analysis Part and Partial Correlation-Statistical Control v

6 vi 5 More on Matrices Vectors Transformation of a Vector by a Matrix Projections Problems 6 Principal Components Analysis Two Variables, Three Cases Two Variables, n Cases Three Variables p Variables Scaling of Principal Components Reducing the Number of Principal Components Naming the Principal Components Example 7 Factor Analysis Points as Variables Instead of Individuals Subspaces The Decomposition of Variables The Correlation Matrix and Its Factors Extraction Methods Rotation Factor Scores Example 8 Multivariate Tests of Means Single-Sample Mean Test Two-Sample Mean Test Three or More Samples Example 9 Discriminant Analysis Geometric Representation Algebra of Discriminant Analysis The Discriminant Coefficients Significance Testing Classification 10 Other Multivariate Techniques Multivariate Multiple Regression Canonical Correlation Mu Itivariate Analysis of Covariance CONTENTS "l

7 CONTENTS vii 11 Repeated Measures Analysis Single-Group Designs N-Sample Case Appendixes A. The Greek Alphabet B. Random Variables, Expected Values, and Variance C. A Little Calculus D. A Little Trigonometry E. Still More on Matrices F. Logarithms G. Matrix Routines in SAS Bibliography Index

8 List of Tables and Figures Tables 3-1. Scatterplot Regression Analysis Produced by the SPSS Computer System Histogram of Standardized Residuals Normal Probability Plot of Standardized Residuals Scatterplot of Standardized Residuals Questions Designed to Tap Students' Attitudes toward Research Means and Standard Deviations for Questions Listed in Table Eigenvalues First Four Columns of Principal Components Matrix Beginning Communalities and Eigenvalues Principal Factor Matrix Final Communalities and Factor Eigenvalues Varimax Rotated Factor Matrix Factor Score Coefficients Cell Means and Standard Deviations Multivariate Tests of Significance Within Groups and Between Groups SSCP Matrix Tests of Significance Multivariate Multiple Regression Univariate and Step down Tests Regression Equations for Dependent Variables Orthonormalized Transformation Matrix 204 viii

9 LIST OF TABLES ix Estimates for Transformed Overall Variables Tests of Within-Subjects Part of Design 206 Figures 2-1. Probability Function of a Continuous Variable Scatterplot with Prediction Line Distribution of y's at Two Values of x Cumulative Normal Distribution Normal Probability Plot Residual Plot with Equal Variance Residual Plot without Equal Variance Residual Plot without Equal Variance Curved Residual Plot Distinction between the Partial and the Part Correlation Plane with Coordinate Axes Vectors as Directed Line Segment Determination of Vector Length Two Vectors and Their Angle Sum of Two Vectors Product of a Vector and a Scalar Product of a Vector and a Scalar When Scalar Is Less Than One Product of a Vector and a Scalar When Scalar Is a Negative Number Decomposing Vectors Two Vectors on the Same Line of a Plane Two Dependent Vectors Vectors of Length One Along Coordinate Axes A Simple One Vector Situation An Example of Transforming Vectors An Example of Rotating Axes Example of Projection of a Vector Example of Projection of a Vector Plot of Deviation Values Rotation of Coordinate Axes Principal Components as New Coordinate Axes Plot of Bivariate Normal Distribution Principal Components of Bivariate Normal Distribution Figure 6-5 Oriented Horizontally. 106

10 x LIST OF FIGURES 6-7. Relation between Regression Lines and First Principal Component Plot of Three Multivariate, Nonnally Distributed Variables Ellipsoid of Figure 6-8 Oriented Horizontally Location of Three Principal Components Scree Test Location of One Variable in a Space Location of Two Variables in a Space Three Variables in a Three Dimensional Space One Variable with Two Common Factors Initial Factor Solution Nonorthogonal Factors Bivariate Normal Distribution Contour Map of Bivariate Normal Distribution Scatterplot of Two Groups on Two Variables Three Groups and Two Variables Discriminant Function for Three Groups and Two Variables Two Discriminant Functions One Discriminant Function Two Discriminant Functions Three Groups Lined Up on One Discriminant Function. 11-la. Linear Plot of Occasions. II-lb. Curved Plot of Occasions Plot of Four Occasions Plot of Means over Time C-1. C-2. C-3. C-4. C-5. C-6. C-7. C-8. C-9. C-10. D-1. D-2. Plot of Means for Each of Seven Groups in Time. Simple Linear Function. Curved Function. Function with Two Limits at One Point. Determination of the Derivative. Graph of [(x) = (x - 1) Point of Inflection. Function of Two Variables. The Least Squares Solution for Regression Coefficients. Unit Circle with Ellipses. Unit Circle with Tangential Ellipse. Unit Circle. Unit Circle with Constructed Angle

11 LIST OF FIGURES xi D-3. D-4. D-S. D-6. D-7. Unit Circle Angle of More than 90. Finding the Cosine of an Angle of More than 90. Unit Circle with Angle. Sine and Cosine. Sine and Cosine in a Right Triangle

12 Acknowledgments Several classes at the School of Social Service Administration at the University of Chicago made use of various versions of this book over the past few years. I am grateful for their forbearance and their suggestions. I appreciate helpful advice from a number of colleagues who reviewed parts of this manuscript: Jeanne Marsh, William Reid, John Tropman, Lynn Videka-Sherman, and Lynn Harold Vogel. Gwen Graham meticulously typed early drafts of this material with great care. Deborah Siegel and Lynn Videka-Sherman graciously consented to use data from their dissertations for examples used in the book. All of the figures in this book except for figure 8-1 were produced with the Tell-A-Graf graphics system, a product of Integrated Software Systems Corporation (ISSCO). Figure 8-1 was drawn using the graphing facilities in the SAS computer system. The examples of computer output (the tables) were produced with the SPSS system. Finally, I am indebted to my family, Charlotte, Gabrielle, and Matthew for their support and understanding in the course of this project. xii

13 Multivariate Analysis in The Human Services