A multiple imputation and coarse data approach to residually confounded regression models

Residual confounding is a major problem in analysis of observational data, occurring when a confounding variable is measured coarsely (censored, heaped, missing, etc.) and hence cannot be fully adjusted to obtain a causal estimate by usual means such as multiple regression. The analysis of coarse data has been investigated by Heitjan and Rubin, but methods for coarse covariates are lacking. A fully conditional-specification multiple-imputation approach is possible if we are able to model i) the confounding variable conditional on other information in the dataset and ii) the coarsening mechanism. This provides a very flexible framework for removing residual confounding under our assumptions, including sensitivity analysis. An added complexity over missing data is that it may not be known which observations are coarsened. Programming this method is presented in Stata for various combinations of i) and ii) above, using the ml and mi functions. In the simplest case of a normally distributed confounder subject to known interval-censoring, intreg and mi can be applied. The method is illustrated with simulated data and the true causal effect is recovered in each instance.