Diagnostics for Regression Models with Discrete Outcomes Using Surrogate Empirical Residual Distribution Functions
Making informed decisions about model adequacy has been an outstanding issue for regression models with discrete outcomes. Standard residuals for such outcomes show a large discrepancy from the hypothesized pattern even under the true model and are often not informative especially when data are highly discrete. To fill this gap, we propose a surrogate 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 diagnostic tool we propose is a principled approach and does not require injecting noise to 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 (e.g. binary) outcomes. Through simulation studies, we demonstrate empirically that the proposed surrogate empirical residual distribution function outperforms other commonly used residuals for various diagnostic tasks, since it is close to the hypothesized pattern under the true model and significantly departs from this pattern with model misspecification, and is thus an effective diagnostic tool.