# Multivariate Analysis

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Multivariate analysis

A. Definition. Multivariate analysis is essentially the statistical process of
simultaneously analyzing multiple independent (or predictor) variables with
multiple dependent (outcome or criterion) variables using matrix algebra
(most multivariate analyses are correlational). While these analyses have
been a part of statistics since the early 1900’s, the development of
mainframe and microcomputers and subsequent analytical software has
made the once tedious calculations fairly simple and very fast.

B. Purpose. Human behavior is very complex. Behaviors, emotions,
cognitions, and attitudes can rarely be described in terms of one or two
variables. Furthermore, these traits cannot be measured directly, as say
running speed, but must be inferred from constructs which in turn are
measured by multiple factors or variables. The researcher must construct
his/her view of reality (observations about humans), create multiple
measures to assess the constructs, and conduct an appropriate statistical
analysis of the data, and correctly interpret the outcomes. Basically, the
researcher asks two questions, “Do I have anything important?” and “If
so, what do I have?” Essentially, the first question asks if the specified
model (which is based upon the researcher’s observations of some aspect
second asks what is the best explanation of the relationship between the
variables.
C. Basic Analysis. Multivariate analysis may either be conducted in a classic experimental
design or in non-experimental designs. Whether one is seeking causality or association depends
upon the research question, but both are valid in multivariate analysis. Most of the time, a data
matrix will be analyzed—the form usually consists of rows representing each subject and
columns representing each variable; however, the matrix can also be a correlation matrix, a
variance/covariance matrix, or a sum of-squares/cross-product matrix. The matrix is then solved
simultaneously through matrix (linear) algebra and yields linear composite scores which are
linear combinations of the variables upon which the final solution is based. Most analyses
construct composite scores that maximize the variance associated between one set of variables (X
set, I.V.s, or predictors) and another set (Y set, D.V.s, criteria, or outcomes). The importance of
each variable is determined by its weight (degree of contribution) to its set of variables and is
usually indicated by a numerical coefficient. Fortunately, a good understanding of matrix algebra
is helpful but not necessary in understanding and using multivariate analytic techniques.

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