I have formulated three objectives that I hope this book will achieve for the reader.
These objectives are based on long experience teaching a course in multivariate
methods, consulting on multivariate problems with researchers in many fields, and
guiding statistics graduate students as they consulted with similar clients.
The first objective is to gain a thorough understanding of the details of various
multivariate techniques, their purposes, their assumptions, their limitations, and so
on. Many of these techniques are related; yet they differ in some essential ways. We
emphasize these similarities and differences.
The second objective is to be able to select one or more appropriate techniques for
a given multivariate data set. Recognizing the essential nature of a multivariate data
set is the first step in a meaningful analysis. We introduce basic types of multivariate
data in Section 1.4.
The third objective is to be able to interpret the results of a computer analysis
of a multivariate data set. Reading the manual for a particular program package is
not enough to make an intelligent appraisal of the output. Achievement of the first
objective and practice on data sets in the text should help achieve the third objective.
This chapter introduces the basic elements of matrix algebra used in the remainder
of this book. It is essentially a review of the requisite matrix tools and is not intended
to be a complete development. However, it is sufficiently self-contained so that those
with no previous exposure to the subject should need no other reference. Anyone
unfamiliar with matrix algebra should plan to work most of the problems entailing
numerical illustrations. It would also be helpful to explore some of the problems
involving general matrix manipulation.
With the exception of a few derivations that seemed instructive, most of the results
are given without proof. Some additional proofs are requested in the problems. For
the remaining proofs, see any general text on matrix theory or one of the specialized
matrix texts oriented to statistics, such as Graybill (1969), Searle (1982), or Harville
(1997) The transpose operation does not change a scalar, since it has only one row and
one column.
If the transpose operator is applied twice to any matrix, the result is the original
matrix: Informally, a random variable may be defined as a variable whose value depends on
the outcome of a chance experiment. Generally, we will consider only continuous
random variables. Some types of multivariate data are only approximations to this
ideal, such as test scores or a seven-point semantic differential (Likert) scale consisting of ordered responses ranging from strongly disagree to strongly agree. Special
techniques have been developed for such data, but in many cases, the usual methods
designed for continuous data work almost as well.a single observation y. The variance σ2 is defined shortly. The notation E(y)
indicates the mean of all possible values of y; that