Sensitivity analysis for matching with multiple controls

In observational studies, treated and control subjects may difler in ways that have not been observed, so that reference distributions for statistics that would be appropriate in randomized experiments may lead to incorrect inferences. A previous paper (Rosenbaum, 1987a) proposed a method for displaying the sensitivity of permutation inferences to bias due to an unobserved covariate in two cases: (i) the case of matched pairs, with arbitrary response variables, and (ii) the case of matching with multiple controls in the special case of binary responses. The current note shows that an analogous argument applies in the case of matching with multiple controls with arbitrary responses, providing the test statistic used is a member of a certain family of generalizations of Wilcoxon's signed-rank statistic, a family that includes methods for binary and censored data.