Poisson Regression

Definition

Poisson Regression

Log-linear model is a Generalized Linear Model with log link function. Log-linear model models the log expected counts of an event as a linear combination of independent variables. $\ln (\mu )=\ln [E(y)]={\mathbf{x}}^{⊺}\mathbf{\beta }$

Log-Linear Model for Contingency Table

Consider a $J\times K$ Contingency Table, then the model is defined as ${\eta }_{jk}=\ln ({\mu }_{jk})=\ln [E(y_{jk})]=\mu +{\alpha }_j+{\beta }_k+(\alpha \beta {)}_{jk}$ where $\mu$ is the overall effect, ${\alpha }_j$ and ${\beta }_k$ are main effect, and $(\alpha \beta {)}_{jk}$ are interaction effect.

The Deviance of the model is defined as $D=2\left[\sum _{j=1}^J\sum _{k=1}^K\left(y_{jk}\ln \left(\frac{y_{jk}}{{\hat{\mu }}_{jk}}\right)-(y_{jk}-{\widehat{mu}}_{jk})\right)\right]$ and the Pearson’s Chi-squared Statistic is defined as $X^2=\sum _{j=1}^J\sum _{k=1}^K\frac{(y_{jk}-{\hat{\mu }}_{jk}{)}^2}{{\hat{\mu }}_{jk}}$