Education 231C

Applied Categorical & Nonnormal Data Analysis

Multinomial Logistic Regression Models

Multinomial logistic regression involves nominal response variables more than two categories. Multinomial logit models are multiequation models. A response variable with k categories will generate k-1 equations. Each of these k-1 equations is a binary logistic regression comparing a group with the reference group. Multinomial logistic regression simultaneously estimates the k-1 logits. Further, it is also the case, that the model tests all possible combinations among the k groups although it only displays coefficients for the k-1 comparisons.

Say that we had a response variable with three levels, the probabilites for each of the levels could be obtained as follows:

This system of equations is unidentified, that is, there is more than one solution to the coefficients that lead to the same probabilities. To make the system identifiable, one of the coefficients is set to 0. It doesn't matter which one since they each yield the same probabilities. We will set the probability for β1 to 0, yielding: This, in turn, leads to the following probabilities relative to the reference group, in this case, group 1. Thus, the two coefficients, β2 and β3 represent the log odds of being in the target groups relative to the reference group.

In multinomial logistic regression the exponentiated coefficients are not odds ratios per se. The coefficients can be interpreted as relative risk ratios (RRR). Recall from the unit on contingency tables that relative risk in the 2x2 table was defined as

In multinomial logistic regression the relative risk can be defined as, Thus, the relative risk ratio for multinomial logit would be Example 1

Example 2 Simplified mlogview Example

Example Using prgen

prgen allows you to generate predicted probabilities for the response groups while holding other variables constant at specific values. These predicted probabilities can then used in scatterplots and other graphs.

Categorical Data Analysis Course

Phil Ender