A measure of goodness of fit of a Poisson regression model is deviance. In comparing two Poisson regression models, if you compute the deviance for each model, then calculate: drop-in-deviance = residual deviance for reduced model - residual deviance for the larger model. A large drop-in-deviance favors the larger model.

Question

A measure of goodness of fit of a Poisson regression model is  deviance.  In comparing two Poisson regression models, if you compute the deviance for each model, then calculate: drop-in-deviance = residual deviance for reduced model - residual deviance for the larger model.  A large drop-in-deviance favors the larger model.

BlackTom AI · Accepted Answer

The question concerns the interpretation of drop-in-deviance when comparing two Poisson regression models and asks whether a large drop-in-deviance favors the larger model.
Option True: A positive drop-in-deviance is computed as residual deviance of the reduced model minus residual deviance of the larger model. If the larger model fits better, its residual deviance is smaller, so RD_reduced > RD_larger and the difference is positive. Therefore, a large drop-in-deviance indeed indicates that moving to the larger model substantially improved fit, which means the larger model is favored by this metric.
Option False: If you were to claim that a large drop-in-deviance favors the smaller model, you would be suggesting that a bigger reduction in unexplained variation supports choosing the simpler model. In practice, a larger positive drop-in-deviance tells you the more complex (larger) model reduces deviance more relative to the simpler (reduced) model, i.e., it supports the larger model. Thus this statement mislabels what the drop-in-deviance is indicating.
Additional nuance: The deviance difference (drop-in-deviance) is analogous to a likelihood ratio test statistic in large samples, where a larger positive value provides evidence in favor of the more complex model. If the drop-in-deviance were small or negative, that would argue less for the larger model or even for the reduced model, respectively.
In summary, understanding the sign and magnitude of the drop-in-deviance helps determine which model explains the data better, with a large positive value favoring the larger model and a negative value or near-zero value suggesting little to no improvement when adding complexity.

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