Assessing the impact of unmeasured confounding for binary outcomes using confounding functions

A critical assumption of causal inference is that of no unmeasured confounding: for estimated exposure effects to have valid causal interpretations, a sufficient set of predictors of exposure and outcome must be adequately measured and correctly included in the respective inference model(s). In an observational study setting, this assumption will often be unsatisfied, and the potential impact of unmeasured confounding on effect estimates should be investigated. The confounding function approach allows the impact of unmeasured confounding on estimates to be assessed, where unmeasured confounding may be due to unmeasured confounders and/or biases such as collider bias or information bias. Although this approach is easy to implement and pertains to the sum of all bias, its use has not been widespread, and discussion has typically been limited to continuous outcomes. In this paper, we consider confounding functions for use with binary outcomes and illustrate the approach with an example. We note that confounding function choice encodes assumptions about effect modification: some choices encode the belief that the true causal effect differs across exposure groups, whereas others imply that any difference between the true causal parameter and the estimate is entirely due to imbalanced risks between exposure groups. The confounding function approach is a useful method for assessing the impact of unmeasured confounding, in particular when alternative approaches, e.g. external adjustment or instrumental variable approaches, cannot be applied. We provide Stata and R code for the implementation of this approach when the causal estimand of interest is an odds or risk ratio.