Chapter Contents |
Previous |
Next |

Introduction to Multivariate Procedures |

The procedures discussed in this chapter investigate relationships among variables without designating some as independent and others as dependent. Principal component analysis and common factor analysis examine relationships within a single set of variables, whereas canonical correlation looks at the relationship between two sets of variables. The following is a brief description of SAS/STAT multivariate procedures:

- CORRESP
- performs simple and multiple correspondence analyses, using a
contingency table, Burt table, binary table, or raw categorical data as
input. Correspondence analysis is a weighted form of principal
component analysis that is appropriate for frequency data.
- PRINCOMP
- performs a principal component analysis and outputs
standardized or unstandardized principal component scores.
- PRINQUAL
- performs a principal component analysis of qualitative data
and multidimensional preference analysis.
- FACTOR
- performs principal component and common factor
analyses with rotations and outputs component
scores or estimates of common factor scores.
- CANCORR
- performs a canonical correlation analysis and outputs canonical variable scores.

The purpose of *principal component analysis* (Rao 1964)
is to derive a small number of linear combinations (principal
components) of a set of variables that retain as much of
the information in the original variables as possible.
Often a small number of principal components
can be used in place of the original variables
for plotting, regression, clustering, and so on.
Principal component analysis can also be viewed as an attempt
to uncover approximate linear dependencies among variables.

The purpose of *common factor analysis* (Mulaik 1972) is to
explain the correlations or covariances among a set of variables
in terms of a limited number of unobservable, latent variables.
The latent variables are not generally computable
as linear combinations of the original variables.
In common factor analysis, it is assumed that the variables
are linearly related if not for uncorrelated random error
or *unique variation* in each variable; both the linear
relations and the amount of unique variation can be estimated.

Principal component and common factor analysis are
often followed by rotation of the components or factors.
*Rotation* is the application of a
nonsingular linear transformation to components
or common factors to aid interpretation.

The purpose of *canonical correlation analysis* (Mardia,
Kent, and Bibby 1979) is to explain or summarize the
relationship between two sets of variables by finding a small
number of linear combinations from each set of variables
that have the highest possible between-set correlations.
Plots of the canonical variables can be
useful in examining multivariate dependencies.
If one of the two sets of variables consists of dummy
variables generated from a classification variable,
the canonical correlation is equivalent to canonical
discriminant analysis (see Chapter 21, "The CANDISC Procedure").
If both sets of variables are dummy variables, canonical
correlation is equivalent to simple correspondence analysis.

The purpose of *correspondence analysis* (Lebart,
Morineau, and Warwick 1984; Greenacre 1984; Nishisato
1980) is to summarize the associations between a set of
categorical variables in a small number of dimensions.
Correspondence analysis computes scores on each dimension
for each row and column category in a contingency table.
Plots of these scores show the relationships among the categories.

The PRINQUAL procedure obtains linear and nonlinear transformations of variables using the method of alternating least squares (Young 1981) to optimize properties of the transformed variables' covariance or correlation matrix. PROC PRINQUAL nonlinearly transforms variables, improving their fit to a principal component model. The name, PRINQUAL, for principal components of qualitative data, comes from the special case analysis of fitting a principal component model to nominal and ordinal scale of measurement variables (Young, Takane, and de Leeuw 1978). However, PROC PRINQUAL also has facilities for smoothly transforming continuous variables. All of PROC PRINQUAL's transformations are also available in the TRANSREG procedure, which fits regression models with nonlinear transformations. PROC PRINQUAL can also perform metric and nonmetric multidimensional preference (MDPREF) analyses (Carroll 1972). The PRINQUAL procedure produces very little displayed output; the results are available in an output data set.

Chapter Contents |
Previous |
Next |
Top |

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.