# Assessment of Regression Models with Discrete Outcomes Using Quasi-Empirical Residual Distribution Functions

## TL;DR

This paper introduces a new residual distribution function for assessing regression models with discrete outcomes, providing a more reliable and principled tool than traditional methods, especially for highly discrete data.

## Contribution

It proposes a quasi-empirical residual distribution function that improves model assessment for discrete outcomes without needing noise injection, and proves its asymptotic properties.

## Key findings

- The proposed function converges to the identity under correct model specification.
- It outperforms traditional residuals in simulation studies for model assessment.
- It effectively detects model misspecification in discrete outcome models.

## Abstract

Making informed decisions about model adequacy has been an outstanding issue for regression models with discrete outcomes. Standard assessment tools for such outcomes (e.g. deviance residuals) often show a large discrepancy from the hypothesized pattern even under the true model and are not informative especially when data are highly discrete (e.g. binary). To fill this gap, we propose a quasi-empirical residual distribution function for general discrete (e.g. ordinal and count) outcomes that serves as an alternative to the empirical Cox-Snell residual distribution function. The assessment tool we propose is a principled approach and does not require injecting noise into the data. When at least one continuous covariate is available, we show asymptotically that the proposed function converges uniformly to the identity function under the correctly specified model, even with highly discrete outcomes. Through simulation studies, we demonstrate empirically that the proposed quasi-empirical residual distribution function outperforms commonly used residuals for various model assessment tasks, since it is close to the hypothesized pattern under the true model and significantly departs from this pattern under model misspecification, and is thus an effective assessment tool.