Weighted <italic>M</italic>-statistics With Superior Design Sensitivity in Matched Observational Studies With Multiple Controls

In a nonrandomized or observational study, a weak association between receipt of the treatment and an outcome may be explained not as effects caused by the treatment but rather by a small bias in the assignment of individuals to treatment or control; however, a strong association may be explained as noncausal only by a large bias. The strength of the association between treatment and outcome is not uniform across the data from a study, and this motivates giving greater weight where the association is stronger. In an observational study with treated-control matched pairs, it is known that results are less sensitive to unmeasured biases if pairs with small absolute differences in outcomes are given little weight in the analysis; more precisely, such a test statistic has superior design sensitivity. How should outcomes be weighted if an observational study is matched in sets with one treated subject and several controls? An <italic>M</italic>-statistic is the quantity equated to zero in defining Huber's <italic>M</italic>-estimates, including the mean, and it is used in testing hypotheses and setting confidence limits. In matched sets, a weighted <italic>M</italic>-statistic increases the weight of some matched sets and decreases the weight of others. Not unlike the case of matched pairs, weighted <italic>M</italic>-statistics with suitable weights have larger design sensitivities, and hence greater power in a sensitivity analysis, than unweighted statistics for symmetric unimodal errors, such as Normal, logistic, or <italic>t</italic>-distributed errors. This issue is examined using an asymptotic measure, the design sensitivity, and using simulation. For one Normal sampling situation, weighting the matched sets increased the power of a 0.05 level sensitivity analysis from 0.05 without weights to 0.75 with weights. An example from NHANES 2009-2010 concerning methylmercury in the blood of people who consume large amounts of fish is used to illustrate.