A goodness-of-fit test for regression models with discrete outcomes

Regression models are often used to analyze discrete outcomes, but classical goodness-of-fit tests such as those based on the deviance or Pearson's statistic can be misleading or have little power in this context. To address this issue, we propose a new test, inspired by the work of Czado et al. (Biometrics, 65(4):1254–1261, 2009), which involves no randomization, tuning parameter, or binning of covariates. The statistic's large-sample distribution under the null hypothesis is determined; as it involves unknown parameter values, one must resort to a bootstrap procedure to compute (Formula presented.) -values. Simulations are conducted to investigate the ability of the test to detect a broad range of model misspecifications commonly seen in practice. The proposed procedure is seen to perform well in all the scenarios considered as well as on real data.