  Recent work has focused on the potential and pitfalls of causal
identification in observational studies with multiple simultaneous treatments.
On the one hand, a latent variable model fit to the observed treatments can
identify essential aspects of the distribution of unobserved confounders. On
the other hand, it has been shown that even when the latent confounder
distribution is known exactly, causal effects are still not point identifiable.
Thus, the practical benefits of latent variable modeling in multi-treatment
settings remain unclear. We clarify these issues with a sensitivity analysis
method that can be used to characterize the 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 which are most consistent with prior hypotheses.
