Problems with pearson and deviance residuals

For a normal linear regression model, the Pearson and deviance residuals are identical and have an approximate normal distribution under the true model. However, their distributions are often skewed and non-normally distributed for counts regression models [8, 20]. It is argued that the deviance residuals typically follow more closely a normal distribution than the Pearson residuals; nevertheless, as μ_i/ϕ→∞, both Pearson and deviance residuals from an exponential family model approach to the normal distribution due to the distribution for the response variable converging to normality. However, the asymptotic normal distribution only holds when the mean of the response variable is relatively large. Further, the residual plots often exhibit parallel curves according to distinct response values, imposing great challenges for visual inspection. Hence, Pearson and deviance residuals are difficult to use for graphically assessing the GOF of count regression models.

Further, the overall GOF of a regression model is often assessed based on the sum squares of the Pearson and deviance residuals, i.e., X2=∑i=1nriP2 and D2=∑i=1nriD2, respectively. Asymptotically, under a correctly specified normal regression model, we can expect X² and D² to have a chi-square distribution χn−p2, where n is the sample size, and p is the number of parameters. In practice, we often fail to achieve large samples, which renders the null distribution of this statistic invalid. It is also recognized that this approximation for diagnosing count regression models can be very poor even for large sample sizes [9, 21].