A tree based lack-of-fit test for multiple logistic regression.

Several omnibus tests have been developed to assess the fit of a regression model. But many of these lack-of-fit tests focus on the simple regression setting. Here, we focus on multiple logistic regression. Pearson's well-known chi-square test statistic and the deviance statistic are no longer valid in the case that the model contains one or more continuous covariates. To overcome this difficulty, Hosmer and Lemeshow proposed a Pearson type statistic based on groups defined by the so-called deciles of risk. We propose a test statistic that is similar in approach to the Hosmer and Lemeshow statistic in that the observations are classified into distinct groups. In the procedure proposed here however, the grouping is not according to probabilities fitted under the null model. We use a recursive partitioning algorithm to divide the sample space into different groups. This generally allows for a more powerful assessment of the model fit. Simulations are carried out to compare the results of the proposed test to that of Hosmer and Lemeshow. Three data examples illustrate the performance of the tree based lack-of-fit test, in comparison to several other tests.