The TRANSREG Procedure

# Overview

The TRANSREG (transformation regression) procedure fits linear models, optionally with spline and other nonlinear transformations, and it can be used to code experimental designs prior to their use in other analyses.

The TRANSREG procedure fits many types of linear models, including

• ordinary regression and ANOVA
• metric and nonmetric conjoint analysis (Green and Wind 1975; de Leeuw, Young, and Takane 1976)
• metric and nonmetric vector and ideal point preference mapping (Carroll 1972)
• simple, multiple, and multivariate regression with variable transformations (Young, de Leeuw, and Takane 1976; Winsberg and Ramsay 1980; Breiman and Friedman 1985)
• redundancy analysis (Stewart and Love 1968) with variable transformations (Israels 1984)
• canonical correlation analysis with variable transformations (van der Burg and de Leeuw 1983)
• response surface regression (Meyers 1976; Khuri and Cornell 1987) with variable transformations

The data set can contain variables measured on nominal, ordinal, interval, and ratio scales (Siegel 1956). Any mix of these variable types is allowed for the dependent and independent variables. The TRANSREG procedure can transform

• nominal variables by scoring the categories to minimize squared error (Fisher 1938), or they can be expanded into dummy variables
• ordinal variables by monotonically scoring the ordered categories so that order is weakly preserved (adjacent categories can be merged) and squared error is minimized. Ties can be optimally untied or left tied (Kruskal 1964). Ordinal variables can also be transformed to ranks.
• interval and ratio scale of measurement variables linearly or nonlinearly with spline (de Boor 1978; van Rijckevorsel 1982) or monotone spline (Winsberg and Ramsay 1980) transformations. In addition, smooth, logarithmic, exponential, power, logit, and inverse trigonometric sine transformations are available.

Transformations produced by the PROC TRANSREG multiple regression algorithm, requesting spline transformations, are often similar to transformations produced by the ACE smooth regression method of Breiman and Friedman (1985). However, ACE does not explicitly optimize a loss function (de Leeuw 1986), while PROC TRANSREG always explicitly optimizes a squared-error loss function.

PROC TRANSREG extends the ordinary general linear model by providing optimal variable transformations that are iteratively derived using the method of alternating least squares (Young 1981). PROC TRANSREG iterates until convergence, alternating

• finding least-squares estimates of the parameters of the model given the current scoring of the data (that is, the current vectors)
• finding least-squares estimates of the scoring parameters given the current set of model parameters
For more background on alternating least-squares optimal scaling methods and transformation regression methods, refer to Young, de Leeuw, and Takane (1976), Winsberg and Ramsay (1980), Young (1981), Gifi (1990), Schiffman, Reynolds, and Young (1981), van der Burg and de Leeuw (1983), Israels (1984), Breiman and Friedman (1985), and Hastie and Tibshirani (1986). (These are just a few of the many relevant sources.)