  Recent work has focused on the potential and pitfalls of causal
identification in observational studies with multiple simultaneous treatments.
Building on previous work, we show that even if the conditional distribution of
unmeasured confounders given treatments were known exactly, the causal effects
would not in general be identifiable, although they may be partially
identified. Given these results, we propose a sensitivity analysis method for
characterizing the effects of potential unmeasured confounding, tailored to the
multiple treatment setting, that can be used to characterize a range of causal
effects that are compatible with the observed data. Our method is based on a
copula factorization of the joint distribution of outcomes, treatments, and
confounders, and can be layered on top of arbitrary observed data models. We
propose a practical implementation of this approach making use of the Gaussian
copula, and establish conditions under which causal effects can be bounded. We
also describe approaches for reasoning about effects, including calibrating
sensitivity parameters, quantifying robustness of effect estimates, and
selecting models that are most consistent with prior hypotheses.
