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:
performs simple and multiple correspondence analyses, with 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. The results are displayed in plots and tables and are also available in output data sets.
performs a principal component analysis and outputs standardized or unstandardized principal component scores. The results are displayed in plots and tables and are also available in output data sets.
performs a principal component analysis of qualitative data and multidimensional preference analysis. The results are displayed in plots and are also available in output data sets.
performs principal component and common factor analyses with rotations and outputs component scores or estimates of common factor scores. The results are displayed in plots and tables and are also available in output data sets.
performs a canonical correlation analysis and outputs canonical variable scores. The results are displayed in tables and are also available in output data sets for plotting.
Many other SAS/STAT procedures can also analyze multivariate data—for example, the CATMOD, GLM, REG, CALIS, and TRANSREG procedures as well as the procedures for clustering and discriminant analysis.
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 31: 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 by 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) and produce plots of the results.