Deviance

Definition

Deviance is a goodness-of-fit statistic for Generalized Linear Model.

Let ${\hat{\beta }}_{\text{max}}$ be an estimator of $\beta$ under maximal model $(n=p)$. The deviance is defined as $D=2\ln \lambda =2[l{\hat{\beta }}_{\text{max}};y)-l\hat{\beta };y)]\stackrel{a}{\sim }{\chi }^2(n-p)$ where $l$ is a log-likelihood, $p$ is the number of parameter of the current model, and $n$ is the sample size.

Consider a null hypothesis $H_0$: the current model is not good, can be tested with the deviation. We reject $H_0$ if $D>{\chi }_{\alpha }^2(n-p)$

Examples

Distribution	Deviance
Normal	$\sum (y_i-{\hat{\mu }}_i{)}^2/{\sigma }^2$
Poisson	$2\sum [y_i\log (y_i/{\hat{\mu }}_i)-(y_i-{\hat{\mu }}_i)]$
Binomial	$2\sum [y_i\log (y_i/{\hat{\mu }}_i)+(m_i-y_i)\log [(m_i-y_i)/(m_i-{\hat{\mu }}_i)]]$
Gamma	$2\sum [-\log (y_i/{\hat{\mu }}_i)+(y_i-{\hat{\mu }}_i)/{\hat{\mu }}_i]$
Inverse Gaussian	$\sum (y_i-{\hat{\mu }}_i{)}^2/({\hat{\mu }}_i^2{y}_i)$