Double Probability Integral Transform Residuals for Regression Models with Discrete Outcomes

Abstract

The assessment of regression models with discrete outcomes is challenging and has many fundamental issues. With discrete outcomes, standard regression model assessment tools such as Pearson and deviance residuals do not follow the conventional reference distribution (normal) under the true model, calling into question the legitimacy of model assessment based on these tools. To fill this gap, we construct a new type of residuals for regression models with general discrete outcomes, including ordinal and count outcomes. The proposed residuals are based on two layers of probability integral transformation. When at least one continuous covariate is available, the proposed residuals closely follow a uniform distribution (or a normal distribution after transformation) under the correctly specified model. One can construct visualizations such as QQ plots to check the overall fit of a model straightforwardly, and the shape of QQ plots can further help identify possible causes of misspecification such as overdispersion. We provide theoretical justification for the proposed residuals by establishing their asymptotic properties. Moreover, in order to assess the mean structure and identify potential covariates, we develop an ordered curve as a supplementary tool, which is based on the comparison between the partial sum of outcomes and of fitted means. Through simulation, we demonstrate empirically that the proposed tools outperform commonly used residuals for various model assessment tasks. We also illustrate the workflow of model assessment using the proposed tools in data analysis. Supplementary materials for this article are available online.

KEYWORDS:

• Generalized linear models
• Goodness-of-fit
• Model diagnostics

Supplementary Materials

Supplementary material:The supplementary material includes additional simulation results and proofs of the theoretical results in Sections 4 and 2.4. (.pdf file)

R code and package:The R code for simulation and data analysis. The proposed methodology is implemented in the assessor package. See the README contained in the zip file for more details. (code202312.zip, zip file)

Acknowledgments

I thank Dr. James Hodges and the anonymous reviewers for their helpful comments.

Disclosure Statement

The author reports there are no competing interests to declare.

Additional information

Funding

The author gratefully acknowledges the research support from the National Science Foundation (DMS-2210712).