Deviance goodness-of-fit test ($D\approx {\chi }_{n-p}^2$)

For a suitably regular generalized linear model, a residual deviance $D$ can be compared approximately with a chi-squared distribution on $n-p$ degrees of freedom to assess fit relative to the saturated model. Sparse or very small expected cell counts can invalidate this reference distribution. A large p-value means no detected lack of fit, not proof of correctness; dependence, wrong mean structure or overdispersion still require diagnostics.