## Overview: PRINQUAL Procedure

The PRINQUAL procedure performs principal component analysis (PCA) of qualitative, quantitative, or mixed data. PROC PRINQUAL is based on the work of Kruskal and Shepard (1974); Young, Takane, and de Leeuw (1978); Young (1981); Winsberg and Ramsay (1983). PROC PRINQUAL finds linear and nonlinear transformations of variables, using the method of alternating least squares, that optimize properties of the transformed variables’ correlation or covariance matrix. Nonoptimal transformations such as logarithm and rank are also available. You can use ODS Graphics to display the results. You can use PROC PRINQUAL to do the following:

• fit metric and nonmetric principal component analyses

• perform metric and nonmetric multidimensional preference (MDPREF) analyses (Carroll, 1972)

• transform data prior to their use in other analyses

• reduce the number of variables for subsequent use in regression analyses, cluster analyses, and other analyses

• detect nonlinear relationships

PROC PRINQUAL provides three methods, each of which seeks to optimize a different property of the transformed variables’ covariance or correlation matrix. These methods are as follows:

• maximum total variance, or MTV

• minimum generalized variance, or MGV

• maximum average correlation, or MAC

The MTV method is based on a PCA model, and it is the most commonly used method. All three methods attempt to find transformations that decrease the rank of the covariance matrix computed from the transformed variables. Transforming the variables to maximize the total variance accounted for by a few linear combinations locates the observations in a space with a dimensionality that approximates the stated number of linear combinations as much as possible, given the transformation constraints. Transforming the variables to minimize their generalized variance or maximize the average correlations also reduces the dimensionality, but without a stated target for the final dimensionality. See the section The Three Methods of Variable Transformation for more information about all three methods.

The data can contain variables measured on nominal, ordinal, interval, and ratio scales of measurement (Siegel, 1956). Any mix is allowed with all methods. PROC PRINQUAL can do the following:

• transform nominal variables by optimally scoring the categories (Fisher, 1938)

• transform ordinal variables monotonically by scoring the ordered categories so that order is weakly preserved (adjacent categories can be merged) and the covariance matrix is optimized. You can undo ties optimally or leave them tied (Kruskal, 1964). You can also transform ordinal variables to ranks.

• transform interval and ratio scale of measurement variables linearly, or transform them nonlinearly with spline transformations (de Boor, 1978; van Rijckevorsel, 1982) or monotone spline transformations (Winsberg and Ramsay, 1983). In addition, nonoptimal transformations for logarithm, rank, exponential, power, logit, and inverse trigonometric sine are available.

• estimate missing data without constraint, with category constraints (missing values within the same group get the same value), and with order constraints (missing value estimates in adjacent groups can be tied to preserve a specified ordering). See Gifi (1990) and Young (1981).

The transformed qualitative (nominal and ordinal) variables can be thought of as being quantified by the analysis, with the quantification done in the context set by the algorithm. The data are quantified so that the proportion of variance accounted for by a stated number of principal components is locally maximized, the generalized variance of the variables is locally minimized, or the average of the correlations is locally maximized.

The PROC PRINQUAL iterations produce a set of transformed variables. Each variable’s new scoring satisfies a set of constraints based on the original scoring of the variable and the specified transformation type. First, all variables are required to satisfy standardization constraints; that is, all variables have a fixed mean and variance. The other constraints include linear constraints, weak order constraints, category constraints, and smoothness constraints. The new set of scores is selected from the sets of possible scorings that do not violate the constraints so that the method criterion is locally optimized.