Table 1.

Pearson residuals for commonly used regression models for count data

Model	Pearson residuals
Poisson	$r_i^P=\frac{y_i-{\widehat{\lambda }}_i}{\sqrt{{\widehat{\lambda }}_i}}$
NB	$r_i^P=\frac{y_i-{\widehat{\lambda }}_i}{\sqrt{{\widehat{\lambda }}_i+{\widehat{\lambda }}_i^2/k}}$
ZIP	$r_i^P=\frac{y_i-\left(1-{\widehat{p}}_i\right){\widehat{\lambda }}_i}{\sqrt{\left(1-{\widehat{p}}_i\right){\widehat{\lambda }}_i\left[1+{\widehat{p}}_i{\widehat{\lambda }}_i\right]}}$
ZINB	$r_i^P=\frac{y_i-(1-{\widehat{p}}_i){\widehat{\lambda }}_i}{\sqrt{(1-{\widehat{p}}_i)\left({\widehat{\lambda }}_i+\frac{{{\widehat{\lambda }}_i}^2}{k}\right)+{{\widehat{\lambda }}_i}^2\left({{\widehat{p}}_i}^2+{\widehat{p}}_i\right)}}$