Goodness-of-Fit Test with Deviance

Definition

Assume that a current model with $p$-parameters $\mathbf{\beta }=({\beta }_0,{\beta }_1,\dots ,{\beta }_q,\dots ,{\beta }_{p-1}{)}^{⊺}$ and consider a null hypothesis $H_0:{\beta }_q=\cdots ={\beta }_{p-1}=0$. Then, the test Statistic is defined as $\Delta D=D_0-D_1=2[l({\hat{\mathbf{\beta }}}_1;\mathbf{y})-l({\hat{\mathbf{\beta }}}_0;\mathbf{y})]\stackrel{a}{\sim }{\chi }^2(p-q)$ where $D$ are the Deviance of the generalized linear models.

We reject $H_0$ if $\Delta D>{\chi }_{\alpha }^2(p-q)$