Multinomial goodness‐of‐fit tests for logistic regression models

Abstract

We examine the properties of several tests for goodness‐of‐fit for multinomial logistic regression. One test is based on a strategy of sorting the observations according to the complement of the estimated probability for the reference outcome category and then grouping the subjects into g equal‐sized groups. A g × c contingency table, where c is the number of values of the outcome variable, is constructed. The test statistic, denoted as C_g, is obtained by calculating the Pearson χ² statistic where the estimated expected frequencies are the sum of the model‐based estimated logistic probabilities. Simulations compare the properties of C_g with those of the ungrouped Pearson χ² test (X²) and its normalized test (z). The null distribution of C_g is well approximated by the χ² distribution with (g−2) × (c−1) degrees of freedom. The sampling distribution of X² is compared with a χ² distribution with n × (c−1) degrees of freedom but shows erratic behavior. With a few exceptions, the sampling distribution of z adheres reasonably well to the standard normal distribution. Power simulations show that C_g has low power for a sample of 100 observations, but satisfactory power for a sample of 400. The tests are illustrated using data from a study of cytological criteria for the diagnosis of breast tumors. Copyright © 2008 John Wiley & Sons, Ltd.