Logistic regression enables you to model the probabilities of the levels of a categorical Y variable based on the values of a continuous X variable. Logistic regression has long been used in a variety of applications such as modeling dose-response data and purchase-choice data. Many texts in categorical statistics cover logistic regression (Agresti 1990), in addition to texts specifically focused on logistic regression (Hosmer and Lemeshow 1989).
In many logistic regression settings, you can also use discriminant analysis, especially if you prefer to think of the continuous variables as Y variables and the categories as X variables and work backward. However, discriminant analysis assumes that the continuous data are normally distributed random responses, rather than fixed regressors. See “Discriminant Analysis” in Multivariate Methods.
Simple logistic regression in the Fit Y by X platform is a more graphical and simplified version of the general models for categorical responses in the Fit Model platform. For examples of more complex logistic regression models, see “Logistic Regression Models” in Fitting Linear Models. For logistic regression using the normal distribution function, also called probit analysis, see “Example of a Probit Model with Binomial Errors” in Predictive and Specialized Modeling.
Nominal logistic regression models estimate the probability of choosing one of the levels of the Y variable as a smooth function of the continuous X variable. The fitted probabilities must be between 0 and 1, and they must sum to 1 across the levels of the Y variable for a given value of the X variable.
In a logistic probability plot, the vertical axis represents probability. For k levels of the Y variable, k - 1 smooth curves partition the total probability (which equals 1) among the levels of the Y variable. The fitting principle for a logistic regression minimizes the sum of the negative natural logarithms of the probabilities fitted to the response events that occur (that is, maximum likelihood).
When the Y variable is ordinal, a modified version of logistic regression is used for fitting. The cumulative probability of being at or below each level of the Y variable is modeled by a curve. The curves for each level have the same shape, but they are shifted to the right or left.
The ordinal logistic model fits a different intercept, but the same slope, for each of r - 1 cumulative logistic comparisons, where r is the number of levels of the Y variable. The ordinal model is preferred to the nominal model when it is appropriate because it has fewer parameters to estimate.