The PRINQUAL Procedure

Overview

The PRINQUAL procedure obtains linear and nonlinear transformations of variables by using the method of alternating least squares to optimize properties of the transformed variables' covariance or correlation matrix. Nonoptimal transformations for logarithm, rank, exponentiation, inverse sine, and logit are also available with PROC PRINQUAL.

The PRINQUAL (principal components of qualitative data) procedure is a data transformation procedure that is based on the work of Kruskal and Shepard (1974); Young, Takane, and de Leeuw (1978); Young (1981); and Winsberg and Ramsay (1983). You can use PROC PRINQUAL to

generalize ordinary principal component analysis to a method capable of analyzing data that are not quantitative
perform metric and nonmetric multidimensional preference (MDPREF) analyses (Carroll 1972)
preprocess data, transforming variables prior to their use in other data analyses
summarize mixed quantitative and qualitative data and detect nonlinear relationships
reduce the number of variables for subsequent use in regression analyses, cluster analyses, and other analyses

The PRINQUAL procedure provides three methods of transforming a set of qualitative and quantitative variables to optimize the transformed variables' covariance or correlation matrix. These methods are

maximum total variance (MTV)
minimum generalized variance (MGV)
maximum average correlation (MAC)

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 variance accounted for by a few linear combinations (using the MTV method) locates the observations in a space with 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 sum of correlations also reduces the dimensionality. The transformed qualitative (nominal and ordinal) variables can be thought of as 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 maximal, the generalized variance of the variables is locally minimal, or the average of the correlations is locally maximal.

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

transform nominal variables by scoring the categories to optimize the covariance matrix (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 untie 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, exponential, power, logit, and inverse trigonometric sine are available.
for all transformations, 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). Refer to Gifi (1990) and Young (1981).

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 transformation 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.

The displayed output from PROC PRINQUAL is a listing of the iteration history. However, the primary output from PROC PRINQUAL is an output data set. By default, the procedure creates an output data set that contains variables with _TYPE_='SCORE'. These observations contain original variables, transformed variables, components, or data approximations. If you specify the CORRELATIONS option in the PROC PRINQUAL statement, the data set also contains observations with _TYPE_='CORR'; these observations contain correlations or component structure information.

The Three Methods of Variable Transformation

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