All Times ET
Keywords: Model diagnostics, Functional residuals, Discrete data
Model diagnostics is an indispensable component of regression analysis, yet it is under addressed in standard textbooks on generalized linear models. This lack of exposition is attributed to the fact that when the outcome data are discrete, classical residuals, such as Pearson’s residual and deviance residual, have limited utility in diagnostics. We establish a novel framework for model diagnostics of discrete data regression. Unlike the literature defining a single-valued quantity as the residual, we propose to use a function as a vehicle to retain the residual information. In the presence of discreteness, we show that such a functional residual is appropriate for summarizing the residual randomness that cannot be captured by the structural part of the model. We establish its theoretical properties, which leads to the innovation of new diagnostic tools including the functional-residual-vs-covariate plot and Function- to-Function (Fn-Fn) plot. Our numerical studies demonstrate that the use of these tools can reveal a variety of model misspecifications, such as not properly including a higher-order term, an explanatory variable, an interaction effect, a dispersion parameter, or a zero-inflation component. As a general notion, it considerably broadens the diagnostic scope as it applies to virtually all parametric models for binary, ordinal, and count data, all in a unified diagnostic scheme.