  In matched observational studies, the inferred causal conclusions pretending
that matching has taken into account all confounding can be sensitive to
unmeasured confounding. In such cases, a sensitivity analysis is often
conducted, which investigates whether the observed association between
treatment and outcome is due to effects caused by the treatment or it is due to
hidden confounding. In general, a sensitivity analysis tries to infer the
minimum amount of hidden biases needed in order to explain away the observed
association between treatment and outcome, assuming that the treatment has no
effect. If the needed bias is large, then the treatment is likely to have
significant effects. The Rosenbaum sensitivity analysis is a modern approach
for conducting sensitivity analysis for matched observational studies. It
investigates what magnitude the maximum of the hidden biases from all matched
sets needs to be in order to explain away the observed association, assuming
that the treatment has no effect. However, such a sensitivity analysis can be
overly conservative and pessimistic, especially when the investigators believe
that some matched sets may have exceptionally large hidden biases. In this
paper, we generalize Rosenbaum's framework to conduct sensitivity analysis on
quantiles of hidden biases from all matched sets, which are more robust than
the maximum. Moreover, we demonstrate that the proposed sensitivity analysis on
all quantiles of hidden biases is simultaneously valid and is thus a free lunch
added to the conventional sensitivity analysis. The proposed approach works for
general outcomes, general matched studies and general test statistics. Finally,
we demonstrate that the proposed sensitivity analysis also works for bounded
null hypotheses as long as the test statistic satisfies certain properties. An
R package implementing the proposed method is also available online.
