  A fundamental limitation of causal inference in observational studies is that
perceived evidence for an effect might instead be explained by factors not
accounted for in the primary analysis. Methods for assessing the sensitivity of
a study's conclusions to unmeasured confounding have been established under the
assumption that the treatment effect is constant across all individuals. In the
potential presence of unmeasured confounding, it has been argued that certain
patterns of effect heterogeneity may conspire with unobserved covariates to
render the performed sensitivity analysis inadequate. We present a new method
for conducting a sensitivity analysis for the sample average treatment effect
in the presence of effect heterogeneity in paired observational studies. Our
recommended procedure, called the studentized sensitivity analysis, represents
an extension of recent work on studentized permutation tests to the case of
observational studies, where randomizations are no longer drawn uniformly. The
method naturally extends conventional tests for the sample average treatment
effect in paired experiments to the case of unknown, but bounded, probabilities
of assignment to treatment. In so doing, we illustrate that concerns about
certain sensitivity analyses operating under the presumption of constant
effects are largely unwarranted.
