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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.

Nathaniel E. Helwig develops statistical methods at the intersection of psychology and computational statistics. His University of Minnesota research biography emphasizes multivariate and nonparametric statistics for complex observations.^[ref]

His Generalized Linear Models in R by Nathaniel E. Helwig explains how a link function connects a response's conditional expected value with a linear predictor. For logistic regression, the predictor represents log odds; for Poisson regression, a logarithmic link ensures a positive fitted mean. This separates the model for the mean from the response's probability distribution, while making deviance and dispersion part of model assessment.^[ref]