  Matched observational studies are commonly used to study treatment effects in
non-randomized data. After matching for observed confounders, there could
remain bias from unobserved confounders. A standard way to address this problem
is to do a sensitivity analysis. A sensitivity analysis asks how sensitive the
result is to a hypothesized unmeasured confounder U. One method, known as
simultaneous sensitivity analysis, has two sensitivity parameters: one relating
U to treatment assignment and the other to response. This method assumes that
in each matched set, U is distributed to make the bias worst. This approach has
two concerning features. First, this worst case distribution of U in each
matched set does not correspond to a realistic distribution of U in the
population. Second, sensitivity parameters are in absolute scales which are
hard to compare to observed covariates. We address these concerns by
introducing a method that endows U with a probability distribution in the
population and calibrates the unmeasured confounder to the observed covariates.
We compare our method to simultaneous sensitivity analysis in simulations and
in a study of the effect of second-hand smoke exposure on blood lead levels in
U.S. children.
