Asymptotic separability in sensitivity analysis

In an observational study in which each treated subject is matched to sever al untreated controls by using observed pretreatment covariates, a sensitiv ity analysis asks how hidden biases due to unobserved covariates might alte r the conclusions. The bounds required for a sensitivity analysis are the s olution to an optimization problem. In general, this optimization problem i s not separable, in the sense that one cannot find the needed optimum by pe rforming a separate optimization in each matched set and combining the resu lts. We show, however, that this optimization problem is asymptotically sep arable, so that when there are many matched sets a separate optimization ma y be performed in each matched set and the results combined to yield the co rrect optimum with negligible error. This is true when the Wilcoxon rank su m test or the Hodges-Lehmann aligned rank test is applied in matching with multiple controls. Numerical calculations show that the asymptotic approxim ation performs well with as few as 10 matched sets. In the case of the rank sum test, a table is given containing the separable solution. With this ta ble, only simple arithmetic is required to conduct the sensitivity analysis . The method also supplies estimates, such as the Hodges-Lehmann estimate, and confidence intervals associated with rank tests. The method is illustra ted in a study of dropping out of US high schools and the effects on cognit ive test scores.