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

A deviance goodness-of-fit test compares a fitted generalized linear model with its saturated statistical model. With $n$ independent response groups, an identifiable rank-$p$ mean model and known common dispersion parameter $\varphi$, the candidate statistic is the scaled devianceIn an appropriate large-information regime, compare it with a chi-squared distribution on $n-p$ statistical degrees of freedom. For a normal linear model this is exact when $\varphi ={\sigma }^2$ is known: $D$ is the residual sum of squares. For Poisson regression, $\varphi =1$; adequate expected cell counts are still needed for a useful approximation.^[ref]

More individual Bernoulli observations do not automatically justify comparing their residual deviance with ${\chi }_{n-p}^2$. Small cell counts, dependence and overdispersion can invalidate that calibration. An unknown dispersion parameter also needs an explicit estimation and testing convention; estimating it from $D/(n-p)$ and then treating $D/\hat{\varphi }$ as an independent goodness-of-fit check is circular.

A logistic regression fitted to ungrouped Bernoulli observations has residual deviance $D=-2\ell (\hat{\beta })$. Comparing it automatically with ${\chi }_{n-p}^2$ is generally invalid: the saturated dimension grows with $n$, with one observation per fitted probability. If the true model has only an intercept and success probability $1/2$, then $D/n\to 2\log 2$, whereas ${\chi }_{n-1}^2/n\to 1$. Fixed-dimensional nested-model likelihood-ratio tests can still use Wilks theorem under regularity. For absolute fit, use meaningful grouping, residual diagnostics, or a fitted-model parametric bootstrap.