Statistical evidence for GLM regression parameters: a robust likelihood approach

When a likelihood ratio is used to measure the strength of evidence for one hypothesis over another, its reliability (i.e. how often it produces misleading evidence) depends on the specification of the working model. When the working model happens to be the 'true' or 'correct' model, the probability of observing strong misleading evidence is low and controllable. But this is not necessarily the case when the working model is misspecified. Royall and Tsou (J. R. Stat. Soc., Ser. B 2003; 65:391-404) show how to adjust working models to make them robust to misspecification. Likelihood ratios derived from their 'robust adjusted likelihood' are just as reliable (asymptotically) as if the working model were correctly specified in the first place. In this paper, we apply and extend these ideas to the generalized linear model (GLM) regression setting. We provide several illustrations (both from simulated data and real data concerning rates of parasitic infection in Philippine adolescents), show how the required adjustment factor can be obtained from standard statistical software, and draw some connections between this approach and the 'sandwich estimator' for robust standard errors of regression parameters. This substantially broadens the availability and the viability of likelihood methods for measuring statistical evidence in regression settings.