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Table 1 Pearson residuals for commonly used regression models for count data

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

Model

Pearson residuals

Poisson

\(r^{P}_{i} = \frac {y_{i}-\hat {\lambda }_{i}}{\sqrt {\hat {\lambda }_{i}}}\)

NB

\(r^{P}_{i} = \frac {y_{i}-\hat {\lambda }_{i}}{\sqrt {\hat {\lambda }_{i}+\hat {\lambda }_{i}^{2}/k}}\)

ZIP

\(r^{P}_{i} = \frac {y_{i}-\left (1-\hat {p}_i \right)\hat {\lambda }_i}{\sqrt {\left (1-\hat {p}_i \right)\hat {\lambda }_i \left [ 1+\hat {p}_i \hat {\lambda }_i \right ]}}\)

ZINB

\(r^{P}_{i} = \frac {y_{i}-(1-\hat p_{i})\hat \lambda _{i}}{\sqrt {(1-\hat p_{i})\left (\hat \lambda _{i}+\frac {{\hat \lambda _{i}}^{2}}{k}\right)+{\hat \lambda _{i}}^{2} \left ({\hat p_{i}}^{2}+\hat p_{i}\right)}}\)