# The PRINQUAL Procedure

### 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). See SAS Technical Report R-108.

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 MDPREF analysis (Carroll, 1972). You can request the dummy variable initialization method suggested by Tenenhaus and Vachette (1977), who independently propose 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). See SAS Technical Report R-108.

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 by 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.