## 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.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.