  Unobserved confounding presents a major threat to causal inference from
observational studies. Recently, several authors suggest that this problem may
be overcome in a shared confounding setting where multiple treatments are
independent given a common latent confounder. It has been shown that under a
linear Gaussian model for the treatments, the causal effect is not identifiable
without parametric assumptions on the outcome model. In this paper, we show
that the causal effect is indeed identifiable if we assume a general binary
choice model for the outcome with a non-probit link. Our identification
approach is based on the incongruence between Gaussianity of the treatments and
latent confounder, and non-Gaussianity of a latent outcome variable. We further
develop a two-step likelihood-based estimation procedure.
