Multinomial goodness-of-fit tests for logistic regression models

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 gxc 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 chi(2) 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 chi(2) test (X-2) and its normalized test (z). The null distribution of C-g is well approximated by the chi(2) distribution with (g-2) x (c-1) degrees of freedom. The sampling distribution of X-2 is compared with a chi(2) distribution with n x (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 (c) 2008 John Wiley & Sons, Ltd.