Publication date: 11/29/2021

Logistic regression has a long tradition with widely varying applications such as modeling dose-response data and purchase-choice data. Unfortunately, many introductory statistics courses do not cover this fairly simple method. Many texts in categorical statistics cover it (Agresti 1990), in addition to texts on logistic regression (Hosmer and Lemeshow 1989). Some analysts use the method with a different distribution function, the normal. In that case, it is called probit analysis. Some analysts use discriminant analysis instead of logistic regression because they prefer to think of the continuous variables as Ys and the categories as Xs and work backward. However, discriminant analysis assumes that the continuous data are normally distributed random responses, rather than fixed regressors.

Simple logistic regression is a more graphical and simplified version of the general facility for categorical responses in the Fit Model platform. For examples of more complex logistic regression models, see Logistic Regression Models in Fitting Linear Models.

Nominal logistic regression estimates the probability of choosing one of the response levels as a smooth function of the x factor. The fitted probabilities must be between 0 and 1, and must sum to 1 across the response levels for a given factor value.

In a logistic probability plot, the vertical axis represents probability. For k response levels, k - 1 smooth curves partition the total probability (which equals 1) among the response levels. 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 Y is ordinal, a modified version of logistic regression is used for fitting. The cumulative probability of being at or below each response level is modeled by a curve. The curves are the same for each level except that 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 response levels. Each parameter estimate can be examined and tested individually, although this is seldom of much interest.

The ordinal model is preferred to the nominal model when it is appropriate because it has fewer parameters to estimate. In fact, it is practical to fit ordinal responses with hundreds of response levels.

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