Using quantile averages in matched observational studies

In two observational studies, one investigating the effects of minimum wage laws on employment and the other of the effects of exposures to lead, an e stimated treatment effect's sensitivity to hidden bias is examined. The est imate uses the combined quantile averages that were introduced in 1981 by B . M. Brown as simple, efficient, robust estimates of location admitting bot h exact and approximate confidence intervals and significance tests. Closel y related to Gastwirth's estimate and Tukey's trimean, the combined quantil e average has asymptotic efficiency for normal data that is comparable with that of a 15% trimmed mean, and higher efficiency than the trimean, but it has resistance to extreme observations or breakdown comparable with that o f the trimean and better than the 15% trimmed mean. Combined quantile avera ges provide consistent estimates of an additive treatment effect in a match ed randomized experiment. Sensitivity analyses are discussed for combined q uantile averages when used in a matched observational study in which treatm ents are not randomly assigned. In a sensitivity analysis in an observation al study, subjects are assumed to differ with respect to an unobserved cova riate that was not adequately controlled by the matching, so that treatment s are assigned within pairs with probabilities that are unequal and unknown . The sensitivity analysis proposed here uses significance levels, point es timates and confidence intervals based on combined quantile averages and ex amines how these inferences change under a range of assumptions about biase s due to an unobserved covariate. The procedures are applied in the studies of minimum wage laws and exposures to lead. The first example is also used to illustrate sensitivity analysis with an instrumental variable.