  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.
