  Randomized Controlled Trials (RCTs) are often considered the gold standard
for estimating causal effect, but they may lack 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
us to generalize the causal effect. Identifying the treatment effect in the
target population requires covariates to capture all treatment effect modifiers
that are shifted between the two sets. Standard estimators then use either
weighting (IPSW), outcome modeling (G-formula), or combine the two in doubly
robust approaches (AIPSW). However such covariates are often not available in
both sets. In this paper, after proving L1-consistency of these three
estimators, we compute the expected bias induced by a missing covariate,
assuming a Gaussian distribution, a continuous outcome, and a semi-parametric
model. Under this setting, we perform a sensitivity analysis for each missing
covariate pattern and compute the sign of the expected bias. We also show that
there is no gain in linearly imputing a partially-unobserved covariate. Finally
we study the substitution 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 a real-world
example from critical care medicine.
