Table 2 Deviance residuals for commonly used regression models for count data

Poisson

\(r^{D}_{i} ={sign}(y_{i}-\hat {\lambda _{i}}) \left \{2\left [ y_{i} \log \frac { y_{i}}{\hat {\lambda _{i}}} - (y_{i} - \hat {\lambda _{i}}) \right ] \right \}^{1/2} \)

NB

\(r^{D}_{i} = {sign}(y_{i}-\hat \lambda _{i})\left \{2\left [y_{i} \log \frac { y_{i}}{\hat \lambda _{i}} - (y_{i} +k)\log \frac {y_{i}+k}{\hat \lambda _{i}+k} \right ] \right \}^{1/2} \)

ZIP

\( r^{D}_{i} = {sign}(y_{i}-\hat {\mu }_{i}) \left (2\left \{ -y_{i} +y_{i} \log y_{i} -\log y_{i} !\right.\right. \)

\(\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \ \ - I(y_{i}=0)\log \left [ \hat {p}_{i} +(1-\hat {p}_{i}) e^{-\hat {\lambda }_i} \right ] \)

\(\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \ \left.\left. - I(y_{i} > 0)\left [\log (1-\hat {p}_{i}) - \hat {\lambda }_{i} +y_{i} \log \hat {\lambda }_{i} -\log y_{i} ! \right ] \right \}\right)^{1/2} \)

ZINB

\(r^{D}_{i} ={sign}(y_{i}-\hat {\mu }_i) \left (2\left \{{log}\frac {\Gamma (y_{i}+k)}{\Gamma (k)\Gamma (y_{i}+1)} +y_{i} {log}\left (\frac {y_{i}}{y_{i}+k}\right)+k{log}\left (\frac {k}{y_{i}+k}\right)\right.\right.\)

\(\quad \quad -I(y_{i}=0) {log} \left [p_{i}+(1-p_{i})\left (\frac {k}{\lambda _{i}+k}\right)^{k}\right ]\)

\(\quad \ \ \left.\left.-I(y_{i}>0) \left [{log}(1-p_{i})+{log}\frac {\Gamma (y_{i}+k)}{\Gamma (k)\Gamma (y_{i}+1)} +y_{i} {log}\left (\frac {y_{i}}{y_{i}+k}\right)+k{log}\left (\frac {k}{y_{i}+k}\right)\right ] \right \}\right)^{1/2} \)