Bayesian sensitivity analysis for causal effects from $2\times2$ tables in the presence of unmeasured confounding with application to presidential campaign visits

Abstract

Presidents often campaign on behalf of candidates during elections. Do these campaign visits increase the probability that the candidate will win? While one might attempt to answer this question by adjusting for observed covariates, such an approach is plagued by serious data limitations. In this paper we pursue a different approach. Namely, we ask: what, if anything, should one infer about the causal effect of a presidential campaign visit using a simple cross-tabulation of the data? We take a Bayesian approach to this problem and show that if one is willing to use substantive information to make some (possibly weak) assumptions about the nature of the unmeasured confounding, sharp posterior estimates of causal effects are easy to calculate. Using data from the 2002 midterm elections, we find that, under a reasonable set of assumptions, a presidential campaign visit on the behalf of congressional candidates helped those candidates win elections.