  The case-crossover design (Maclure, 1991) is widely used in epidemiology and
other fields to study causal effects of transient treatments on acute outcomes.
However, its validity and causal interpretation have only been justified under
informal conditions. Here, we place the design in a formal counterfactual
framework for the first time. Doing so helps to clarify its assumptions and
interpretation. In particular, when the treatment effect is non-null, we
identify a previously unnoticed bias arising from common causes of the outcome
at different person-times. We analytically characterize the direction and size
of this bias and demonstrate its potential importance with a simulation. We
also use our derivation of the limit of the case-crossover estimator to analyze
its sensitivity to treatment effect heterogeneity, a violation of one of the
informal criteria for validity. The upshot of this work for practitioners is
that, while the case-crossover design can be useful for testing the causal null
hypothesis in the presence of baseline confounders, extra caution is warranted
when using the case-crossover design for point estimation of causal effects.
