Causal effect on a target population: a sensitivity analysis to handle missing covariates

Randomized Controlled Trials (RCTs) are often considered as the gold standard to conclude on the causal effect of a given intervention on an outcome, but they may lack of external validity when the population eligible to the RCT is substantially different from the target population. Having at hand a sample of the target population of interest allows to generalize the causal effect. Identifying this target population treatment effect needs covariates in both sets to capture all treatment effect modifiers that are shifted between the two sets. However such covariates are often not available in both sets. Standard estimators then use either weighting (IPSW), outcome modeling (G-formula), or combine the two in doubly robust approaches (AIPSW). In this paper, after completing existing proofs on the complete case consistency of those three estimators, we compute the expected bias induced by a missing covariate, assuming a Gaussian distribution and a semi-parametric linear model. This enables sensitivity analysis for each missing covariate pattern, giving the sign of the expected bias. We also show that there is no gain in imputing a partially-unobserved covariate. Finally we study the replacement of a missing covariate by a proxy. We illustrate all these results on simulations, as well as semi-synthetic benchmarks using data from the Tennessee Student/Teacher Achievement Ratio (STAR), and with a real-world example from critical care medicine.