Diagnostic Tools for Distributional Negative Binomial Regression: Residuals, Component Plots, and Model Comparison

Abstract

Distributional negative binomial (NB) regression lets both the conditional mean and the overdispersion parameter depend on covariates, but applied work still lacks a compact, model-based diagnostic workflow for this additional flexibility. We develop three complementary post-fit tools. First, randomized quantile residuals (RQRs) provide a single normal-scale check of the fitted count distribution and mean equation: with known parameters they are exactly standard normal under correct specification, while with estimated parameters they are best interpreted as approximate graphical diagnostics. In a targeted simulation, the RQR QQ plot detects omitted mean structure (\(\mathrm{KS}\;D=0.042\), \(p=0.002\) at \(n=2000\)) and returns to the reference line when the mean is correctly specified (\(\mathrm{KS}\;D=0.010\), \(p=0.988\)). Second, we introduce score-based component-plus-residual plots for the dispersion equation, allowing analysts to identify slope misfit, nonlinear effects, and stratum-level departures in the log-dispersion predictor. Third, we compare AIC, BIC, and likelihood-ratio checks for choosing among candidate dispersion specifications. In the simulation design considered here, BIC and likelihood-ratio checks perform best once \(n\geq300\); at \(n=1000\) both select the true dispersion specification in 99% of replications, whereas AIC selects an overfitted specification in 15%. An application to NMES1988 physician-visit counts illustrates the full workflow and shows why global RQR diagnostics should be combined with targeted checks of the dispersion equation.