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.

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