Sensitivity analysis for m-estimates, tests, and confidence intervals in matched observational studies

Huber's m-estimates use an estimating equation in which observations are permitted a controlled level of influence. The family of m-estimates includes least squares and maximum likelihood, but typical applications give extreme observations limited weight. Maritz proposed methods of exact and approximate permutation inference for ni-tests, confidence intervals, and estimators, which can be derived from random assignment of paired subjects to treatment or control. In contrast, in observational studies, where treatments are not randomly assigned, subjects matched for observed covariates may differ in terms of unobserved covariates, so differing outcomes may not be treatment effects. In observational studies, a method of sensitivity analysis is developed for m-tests, m-intervals, and m-estimates: it shows the extent to which inferences would be altered by biases of various magnitudes due to nonrandom treatment assignment. The method is developed for both matched pairs, with one treated subject matched to one control, and for matched sets, with one treated subject matched to one or more controls. The method is illustrated using two studies: (i) a paired study of damage to DNA from exposure to chromium and nickel and (ii) a study with one or two matched controls comparing side effects of two drug regimes to treat tuberculosis. The approach yields sensitivity analyses for: (i) m-tests with Huber's weight function and other robust weight functions, (ii) the permutational t-test which uses the observations directly, and (iii) various other procedures such as the sign test, Noether's test, and the permutation distribution of the efficient score test for a location family of distributions. Permutation inference with covariance adjustment is briefly discussed.