# Categorical Regression

The use of Categorical Regression is most appropriate when the goal of your analysis is to predict a dependent (response) variable from a set of independent (predictor) variables. As with all optimal scaling procedures, scale values are assigned to each category of every variable such that these values are optimal with respect to the regression. The solution of a categorical regression maximizes the squared correlation between the transformed response and the weighted combination of transformed predictors.

**Relation to other
Categories procedures. **Categorical regression with optimal
scaling is comparable to optimal scaling canonical correlation analysis
with two sets, one of which contains only the dependent variable.
In the latter technique, similarity of sets is derived by comparing
each set to an unknown variable that lies somewhere between all of
the sets. In categorical regression, similarity of the transformed
response and the linear combination of transformed predictors is assessed
directly.

**Relation to standard
techniques. **In standard linear regression, categorical
variables can either be recoded as indicator variables or be treated
in the same fashion as interval level variables. In the first approach,
the model contains a separate intercept and slope for each combination
of the levels of the categorical variables. This results in a large
number of parameters to interpret. In the second approach, only one
parameter is estimated for each variable. However, the arbitrary nature
of the category codings makes generalizations impossible.

If some of the variables are not continuous, alternative analyses are available. If the response is continuous and the predictors are categorical, analysis of variance is often employed. If the response is categorical and the predictors are continuous, logistic regression or discriminant analysis may be appropriate. If the response and the predictors are both categorical, loglinear models are often used.

Regression with optimal scaling offers three scaling levels for each variable. Combinations of these levels can account for a wide range of nonlinear relationships for which any single "standard" method is ill-suited. Consequently, optimal scaling offers greater flexibility than the standard approaches with minimal added complexity.

In addition, nonlinear transformations of the predictors usually reduce the dependencies among the predictors. If you compare the eigenvalues of the correlation matrix for the predictors with the eigenvalues of the correlation matrix for the optimally scaled predictors, the latter set will usually be less variable than the former. In other words, in categorical regression, optimal scaling makes the larger eigenvalues of the predictor correlation matrix smaller and the smaller eigenvalues larger.