Estimating Crude or Common Odds Ratios in Case-Control Studies with Informatively Missing Exposure Data

In case-control studies, the crude odds ratio derived from a 2 x 2 table and the common odds ratio adjusted for stratification variables are staple measures of exposure-disease association. While missing exposure data are encountered in the majority of such studies, formal attempts to deal with them are rare, and a complete-case analysis is the norm. Furthermore, the probability that exposure is missing may depend on true exposure status, so the missing-at-random assumption is often unreasonable. In this paper, the authors present an adjustment to the usual product binomial likelihood to properly account for missing data. Estimation of model parameters without restrictive assumptions requires an additional data collection effort akin to a validation study. Closed-form results are provided to facilitate point and confidence interval estimation of crude and common odds ratios after properly accounting for informatively missing data. Simulations assess performance of the likelihood-based estimates and inferences, and they display the potential for bias in complete-case analyses. An example is presented to illustrate the approach.