Residual deviance ($D=2\varphi \{{\ell }_{\mathrm{saturated}}(\varphi )-{\ell }_{\mathrm{fitted}}(\varphi )\}$)

The unscaled residual deviance of an exponential dispersion family compares its fitted mean with the saturated statistical model at a common dispersion parameter. Its normalization removes the factor $1/\varphi$ in the log-likelihood. In Poisson regression and binomial regression, $\varphi =1$, so it is twice the saturated-minus-fitted log-likelihood. Under suitable regularity, $D/\varphi$ has approximately a chi-squared distribution with the residual degrees of freedom; this is not an automatic accurate approximation for sparse responses. Nested deviance reductions with estimated dispersion lead to approximate F-tests after scaling by the dispersion estimate.

The null deviance is the unscaled deviance of the intercept-only mean model relative to the saturated model at a common dispersion parameter $\varphi$, retaining any known generalized linear model offset. It supplies a baseline against which predictor terms improve the fit. Its difference from a fitted model's residual deviance, divided by $\varphi$, is the statistic for a likelihood-ratio test of those added terms, with nuisance parameters handled consistently in both fits. In Poisson and binomial families, $\varphi =1$.