Multivariate Analysis in the Human Services

276 Pages·1983·5.65 MB·English

by  John R. Schuerman (auth.)

Multivariate Analysis in
The Human Services
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.
Multivariate Analysis in
the Human Services
John R. Schuerman
University of Chicago
Kluwer·Nijhoff Publishing
Boston The Hague Dordrecht Lancaster
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 322
3300AH 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: 978-94-009-6663-5 e-ISBN-13: 978-94-009-6661-1
DOl: 10.1007/978-94-009-6661-1
1. Social service-Research-Statistical methods.
2. Multivariate analysis. I. Title.
HV11.S382 1983 361'.0072 82-20328
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 with
out written permission of the publisher.
Contents
List of Tables and Figures viii
Acknowledgments xii
Introduction
2
Mathematical Preliminaries 5
Functions of Variables 5
Matrices 9
Matrix Algebra 13
Some Matrices We Will Encounter 17
Singularity of Matrices and Determinants 20
I nverse of Matrices 21
Problems 22
3
Multiple Regression I 25
The Model in Matrix Terms 29
Review of Analysis of Variance 33
Two-Way Analysis of Variance 35
The Analysis of Variance of Regression 38
Interpretation of Regression Coefficients 47
Residuals 50
4
Multiple Regression II 63
Bu ilding a Regression Equation 63
Coding of Categorical Variables for Regression Analysis 65
Part and Partial Correlation-Statistical Control 67
v
vi CONTENTS
5
More on Matrices 73
Vectors 73
Transformation of a Vector by a Matrix 86
Projections 90
Problems 91
6
Principal Components Analysis 93
Two Variables, Three Cases 94
Two Variables, n Cases 103
Three Variables 105
p Variables 107
Scaling of Principal Components 110
Reducing the Number of Principal Components 111
Naming the Principal Components 112
Example 113
7
Factor Analysis 121
Points as Variables Instead of Individuals 121
Subspaces 123
The Decomposition of Variables 124
The Correlation Matrix and Its Factors 128
Extraction Methods 133
Rotation 135
Factor Scores 138
Example 140
8
Multivariate Tests of Means 147
Single-Sample Mean Test 147
Two-Sample Mean Test 153
Three or More Samples 155
Example 160
9
Discriminant Analysis 167
Geometric Representation 167
Algebra of Discriminant Analysis 171
The Discriminant Coefficients 172
Significance Testing 1"l3
Classification 174
10
Other Multivariate Techniques 177
Multivariate Multiple Regression 177
Canonical Correlation 186
Mu Itivariate Analysis of Covariance 187
CONTENTS vii
11
Repeated Measures Analysis 189
Single-Group Designs 190
N-Sample Case 199
Appendixes 209
A. The Greek Alphabet 211
B. Random Variables, Expected Values, and Variance 213
C. A Little Calculus 221
D. A Little Trigonometry 243
E. Still More on Matrices 249
F. Logarithms 259
G. Matrix Routines in SAS 261
Bibliography 267
Index 271
List of Tables and Figures
Tables
3-1. Scatterplot 28
3-2. Regression Analysis Produced by the
SPSS Computer System 44
3-3. Histogram of Standardized Residuals 57
3-4. Normal Probability Plot of Standardized Residuals 58
3-5. Scatterplot of Standardized Residuals 59
6-1. Questions Designed to Tap Students'
Attitudes toward Research 114
6-2. Means and Standard Deviations for
Questions Listed in Table 6-1 115
6-3. Eigenvalues 116
6-4. First Four Columns of Principal Components Matrix 117
7-1. Beginning Communalities and Eigenvalues 141
7-2. Principal Factor Matrix 142
7-3. Final Communalities and Factor Eigenvalues 143
7-4. Varimax Rotated Factor Matrix 144
7-5. Factor Score Coefficients 145
8-1. Cell Means and Standard Deviations 161
8-2. Multivariate Tests of Significance 162
8-3. Within·Groups and Between·Groups SSCP Matrix 163
8-4. Tests of Significance 164
10-1. Multivariate Multiple Regression 183
10-2. Univariate and Step down Tests 184
10-3. Regression Equations for Dependent Variables 185
11-1. Orthonormalized Transformation Matrix 204
viii
LIST OF TABLES ix
11-2. Estimates for Transformed Overall Variables 205
11-3. Tests of Within-Subjects Part of Design 206
Figures
2-1. Probability Function of a Continuous Variable. 7
3-1. Scatterplot with Prediction Line. 29
3-2. Distribution of y's at Two Values of x. 40
3-3. Cumulative Normal Distribution. 51
3-4. Normal Probability Plot. 52
3-5. Residual Plot with Equal Variance. 53
3-6. Residual Plot without Equal Variance. 54
3-7. Residual Plot without Equal Variance. 55
3-8. Curved Residual Plot. 56
4-1. Distinction between the Partial and the Part Correlation 71
5-1. Plane with Coordinate Axes. 74
5-2. Vectors as Directed Line Segment. 75
5-3. Determination of Vector Length. 76
5-4. Two Vectors and Their Angle. 77
5-5. Sum of Two Vectors. 78
5-6. Product of a Vector and a Scalar. 79
5-7. Product of a Vector and a Scalar When
Scalar Is Less Than One. 80
5-8. Product of a Vector and a Scalar When
Scalar Is a Negative Number. 81
5-9. Decomposing Vectors. 82
5-10. Two Vectors on the Same Line of a Plane. 83
5-11. Two Dependent Vectors. 84
5-12. Vectors of Length One Along Coordinate Axes. 85
5-13. A Simple One Vector Situation. 88
5-14. An Example of Transforming Vectors. 89
5-15. An Example of Rotating Axes. 90
5-16. Example of Projection of a Vector. 91
5-17. Example of Projection of a Vector. 92
6-1. Plot of Deviation Values. 95
6-2. Rotation of Coordinate Axes. 96
6-3. Principal Components as New Coordinate Axes. 102
6-4. Plot of Bivariate Normal Distribution. 104
6-5. Principal Components of Bivariate Normal Distribution. 105
6-6. Figure 6-5 Oriented Horizontally. 106
x LIST OF FIGURES
6-7. Relation between Regression Lines and
First Principal Component. 107
6-8. Plot of Three Multivariate, Nonnally
Distributed Variables. 108
6-9. Ellipsoid of Figure 6-8 Oriented Horizontally 108
6-10. Location of Three Principal Components. 109
6-11. Scree Test. 112
7-1. Location of One Variable in a Space. 122
7-2. Location of Two Variables in a Space. 124
7-3. Three Variables in a Three Dimensional Space. 125
7-4. One Variable with Two Common Factors. 126
7-5. Initial Factor Solution. 136
7-6. Nonorthogonal Factors. 137
8-1. Bivariate Normal Distribution. 149
8-2. Contour Map of Bivariate Normal Distribution. 151
9-1. Scatterplot of Two Groups on Two Variables. 168
9-2. Three Groups and Two Variables. 169
9-3. Discriminant Function for Three Groups
and Two Variables. 170
9-4. Two Discriminant Functions. 171
9-5. One Discriminant Function. 172
9-6. Two Discriminant Functions. 173
9-7. Three Groups Lined Up on One Discriminant Function. 174
11-la. Linear Plot of Occasions. 193
II-lb. Curved Plot of Occasions. 194
11-2. Plot of Four Occasions. 195
11-3. Plot of Means over Time. 197
11-4. Plot of Means for Each of Seven Groups in Time. 200
C-1. Simple Linear Function. 222
C-2. Curved Function. 223
C-3. Function with Two Limits at One Point. 224
C-4. Determination of the Derivative. 226
C-5. Graph of [(x) = (x - 1)2 + 3. 230
C-6. Point of Inflection. 231
C-7. Function of Two Variables. 233
C-8. The Least Squares Solution for Regression Coefficients. 234
C-9. Unit Circle with Ellipses. 241
C-10. Unit Circle with Tangential Ellipse. 242
D-1. Unit Circle. 244
D-2. Unit Circle with Constructed Angle. 244