  Historically used in settings where the outcome is rare or data collection is
expensive, outcome-dependent sampling is relevant to many modern settings where
data is readily available for a biased sample of the target population, such as
public administrative data. Under outcome-dependent sampling, common effect
measures such as the average risk difference and the average risk ratio are not
identified, but the conditional odds ratio is. Aggregation of the conditional
odds ratio is challenging since summary measures are generally not identified.
Furthermore, the marginal odds ratio can be larger (or smaller) than all
conditional odds ratios. This so-called non-collapsibility of the odds ratio is
avoidable if we use an alternative aggregation to the standard arithmetic mean.
We provide a new definition of collapsibility that makes this choice of
aggregation method explicit, and we demonstrate that the odds ratio is
collapsible under geometric aggregation. We describe how to partially identify,
estimate, and do inference on the geometric odds ratio under outcome-dependent
sampling. Our proposed estimator is based on the efficient influence function
and therefore has doubly robust-style properties.
