Deviance
A measure of the discrepancy between a fitted model and the saturated model, playing for a generalized linear model the role the residual sum of squares plays for ordinary regression.
Nelder and Wedderburn, 1972, Journal of the Royal Statistical Society: setting out the generalized linear model, they fit it by maximum likelihood, and the residual sum of squares loses its meaning. Deviance takes its place. It equals twice the difference between the log-likelihood of the saturated model, which reproduces every observation, and that of the fitted model. Its first use is comparing two nested models: the deviance difference asymptotically follows a chi-square distribution with degrees of freedom equal to the number of added parameters, which tests an entire rating variable rather than a single level. Divided by its degrees of freedom, residual deviance also estimates the dispersion parameter; a ratio well above one points to untreated overdispersion. Finally, deviance residuals read like standardized residuals and flag poorly fitted segments. A deviance that keeps falling with every added variable signals overfitting, which is why the choice is settled by a penalized criterion or by validation on held-out data.
Motor frequency model, 2026 financial year, 180,000 policies. Residual deviance 41,200 on 179,970 degrees of freedom. Adding fiscal horsepower as an eight-level variable cuts deviance by 96 points for seven parameters consumed: the chi-square distribution on seven degrees of freedom has a critical value of about 14 at the 5% level, so the variable is kept. Adding vehicle color next cuts deviance by 9 points for six parameters, and the variable is rejected.
deviance, residual deviance, likelihood ratio test
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