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Research and evaluation in the human services usually involves a relatively large number of variables. We are interested in phenomena that have many aspects and many causes. The techniques needed to deal with many variables go beyond those of introductory statistics. Elementary procedures in statistics are limited in usefulness to situations in which we have two or three variables. When we have more than that, application of elementary techniques will often yield mis leading results. Why are elementary techniques inadequate when applied to many variables? Why, for example, should we not simply interpret a series of correlations of independent and dependent variables? The answer lies in the fact that these correlations are not independent pieces of information. The correlations of vari ables x and z with yare affected by the association of x with z. Hence, talk about the "effect" of x on y will be somewhat ambiguous, since we will be in cluding in that effect some of the effects of z. We would like to be able to sort out these effects. This is the problem of "estimation," that is, estimating the relationships or effects between variables, taking into account their relationships with other variables.
List of contents
1 Introduction.- 2 Mathematical Preliminaries.- Functions of Variables.- Matrices.- Matrix Algebra.- Some Matrices We Will Encounter.- Singularity of Matrices and Determinants.- Inverse 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.- Building a Regression Equation.- Coding of Categorical Variables for Regression Analysis.- and Partial Correlation-Statistical Control.- 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.- Multivariate Analysis of Covariance.- 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.
