A comparison of residual diagnosis tools for diagnosing regression models for count data

Examining residuals is a crucial step in statistical analysis to identify the discrepancies between models and data, and assess the overall model goodness-of-fit. In diagnosing normal linear regression models, both Pearson and deviance residuals are often used, which are equivalently and approximately standard normally distributed when the model fits the data adequately. However, when the response vari*able is discrete, these residuals are distributed far from normality and have nearly parallel curves according to the distinct discrete response values, imposing great challenges for visual inspection.

Randomized quantile residuals (RQRs) were proposed in the literature by Dunn and Smyth (1996) to circumvent the problems in traditional residuals. However, this approach has not gained popularity partly due to the lack of investigation of its performance for count regression including zero-inflated models through simulation studies. Therefore, we assessed the normality of the RQRs and compared their performance with traditional residuals for diagnosing count regression models through a series of simulation studies. A real data analysis in health care utilization study for modeling the number of repeated emergency department visits was also presented.

Our results of the simulation studies demonstrated that RQRs have low type I error and great statistical power in comparisons to other residuals for detecting many forms of model misspecification for count regression models (non-linearity in covariate effect, over-dispersion, and zero inflation). Our real data analysis also showed that RQRs are effective in detecting misspecified distributional assumptions for count regression models.

Our results for evaluating RQRs in comparison with traditional residuals provide further evidence on its advantages for diagnosing count regression models.