Chapter 59: 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. Experimental graphics are now available with the PRINQUAL procedure. For more information, see the ODS Graphics section on page 3677.

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

The three methods of variable transformation provided by PROC PRINQUAL are discussed in the following sections.

The Maximum Total Variance (MTV) Method

The MTV method (Young, Takane, and de Leeuw 1978) is based on the principal component model, and it attempts to maximize the sum of the first r eigenvalues of the covariance matrix. This method transforms variables to be (in a least-squares sense) as similar to linear combinations of r principal component score variables as possible, where r can be much smaller than the number of variables. This maximizes the total variance of the first r components (the trace of the covariance matrix of the first r principal components). Refer to Kuhfeld, Sarle, and Young (1985).

On each iteration, the MTV algorithm alternates classical principal component analysis (Hotelling 1933) with optimal scaling (Young 1981). When all variables are ordinal preference ratings, this corresponds to Carroll s (1972) MDPREF analysis. You can request the dummy variable initialization method suggested by Tenenhaus and Vachette (1977), who independently proposed the same iterative algorithm for nominal and interval scale-of-measurement variables.

The Minimum Generalized Variance (MGV) Method

The MGV method (Sarle 1984) uses an iterated multiple regression algorithm in an attempt to minimize the determinant of the covariance matrix of the transformed variables. This method transforms each variable to be (in a least-squares sense) as similar to linear combinations of the remaining variables as possible. This locally minimizes the generalized variance of the transformed variables, the determinant of the covariance matrix, the volume of the parallelepiped defined by the transformed variables, and the sphericity (the extent to which a quadratic form in the optimized covariance matrix defines a sphere). Refer to Kuhfeld, Sarle, and Young (1985).

On each iteration for each variable, the MGV algorithm alternates multiple regression with optimal scaling. The multiple regression involves predicting the selected variable from all other variables. You can request a dummy variable initialization using a modification of the Tenenhaus and Vachette (1977) method that is appropriate with a regression algorithm. This method can be viewed as a way of investigating the nature of the linear and nonlinear dependencies in, and the rank of, a data matrix containing variables that can be nonlinearly transformed. This method tries to create a less-than -full-rank data matrix. The matrix contains the transformation of each variable that is most similar to what the other transformed variables predict.

The Maximum Average Correlation (MAC) Method

The MAC method (de Leeuw 1985) uses an iterated constrained multiple regression algorithm in an attempt to maximize the average of the elements of the correlation matrix. This method transforms each variable to be (in a least-squares sense) as similar to the average of the remaining variables as possible.

On each iteration for each variable, the MAC algorithm alternates computing an equally weighted average of the other variables with optimal scaling. The MAC method is similar to the MGV method in that each variable is scaled to be as similar to a linear combination of the other variables as possible, given the constraints on the transformation. However, optimal weights are not computed. You can use the MAC method when all variables are positively correlated or when no monotonicity constraints are placed on any transformations. Do not use this method with negatively correlated variables when some optimal transformations are constrained to be increasing because the signs of the correlations are not taken into account. The MAC method is useful as an initialization method for the MTV and MGV methods.