  Over the last three decades, case-crossover designs have found many
applications in health sciences, especially in air pollution epidemiology. They
are typically used, in combination with partial likelihood techniques, to
define a conditional logistic model for the responses, usually health outcomes,
conditional on the exposures. Despite the fact that conditional logistic models
have been shown equivalent, in typical air pollution epidemiology setups, to
specific instances of the well-known Poisson time series model, it is often
claimed that they cannot allow for overdispersion. This paper clarifies the
relationship between case-crossover designs, the models that ensue from their
use, and overdispersion. In particular, we propose to relax the assumption of
independence between individuals traditionally made in case-crossover analyses,
in order to explicitly introduce overdispersion in the conditional logistic
model. As we show, the resulting overdispersed conditional logistic model
coincides with the overdispersed, conditional Poisson model, in the sense that
their likelihoods are simple re-expressions of one another. We further provide
the technical details of a Bayesian implementation of the proposed
case-crossover model, which we use to demonstrate, by means of a large
simulation study, that standard case-crossover models can lead to dramatically
underestimated coverage probabilities, while the proposed models do not. We
also perform an illustrative analysis of the association between air pollution
and morbidity in Toronto, Canada, which shows that the proposed models are more
robust than standard ones to outliers such as those associated with public
holidays.
