  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 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.
