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Over the last three decades, case-crossover designs have found manyapplications in health sciences, especially in air pollution epidemiology. Theyare typically used, in combination with partial likelihood techniques, todefine a conditional logistic model for the responses, usually health outcomes,conditional on the exposures. Despite the fact that conditional logistic modelshave been shown equivalent, in typical air pollution epidemiology setups, tospecific instances of the well-known Poisson time series model, it is oftenclaimed that they cannot allow for overdispersion. This paper clarifies therelationship between case-crossover designs, the models that ensue from theiruse, and overdispersion. In particular, we propose to relax the assumption ofindependence between individuals traditionally made in case-crossover analyses,in order to explicitly introduce overdispersion in the conditional logisticmodel. As we show, the resulting overdispersed conditional logistic modelcoincides with the overdispersed, conditional Poisson model, in the sense thattheir likelihoods are simple re-expressions of one another. We further providethe technical details of a Bayesian implementation of the proposedcase-crossover model, which we use to demonstrate, by means of a largesimulation study, that standard case-crossover models can lead to dramaticallyunderestimated coverage probabilities, while the proposed models do not. Wealso perform an illustrative analysis of the association between air pollutionand morbidity in Toronto, Canada, which shows that the proposed models are morerobust than standard ones to outliers such as those associated with publicholidays.

Case-crossover designs and overdispersion with application in air pollution epidemiology