Sensitivity Analysis for Multiple Comparisons in Matched Observational Studies through Quadratically Constrained Linear Programming

A sensitivity analysis in an observational study assesses the robustness of significant findings to unmeasured confounding. While sensitivity analyses in matched observational studies have been well addressed when there is a single outcome variable, accounting for multiple comparisons through the existing methods yields overly conservative results when there are multiple outcome variables of interest. This stems from the fact that unmeasured confounding cannot affect the probability of assignment to treatment differently depending on the outcome being analyzed. Existing methods allow this to occur by combining the results of individual sensitivity analyses to assess whether at least one hypothesis is significant, which in turn results in an overly pessimistic assessment of a study's sensitivity to unobserved biases. By solving a quadratically constrained linear program, we are able to perform a sensitivity analysis while enforcing that unmeasured confounding must have the same impact on the treatment assignment probabilities across outcomes for each individual in the study. We show that this allows for uniform improvements in the power of a sensitivity analysis not only for testing the overall null of no effect, but also for null hypotheses on \textit{specific} outcome variables while strongly controlling the familywise error rate. We illustrate our method through an observational study on the effect of smoking on naphthalene exposure

Optimal Tests of Treatment Effects for the Overall Population and Two Subpopulations in Randomized Trials, using Sparse Linear Programming

We propose new, optimal methods for analyzing randomized trials, when it is suspected that treatment effects may differ in two predefined subpopulations. Such sub-populations could be defined by a biomarker or risk factor measured at baseline. The goal is to simultaneously learn which subpopulations benefit from an experimental treatment, while providing strong control of the familywise Type I error rate. We formalize this as a multiple testing problem and show it is computationally infeasible to solve using existing techniques. Our solution involves a novel approach, in which we first transform the original multiple testing problem into a large, sparse linear program. We then solve this problem using advanced optimization techniques. This general method can solve a variety of multiple testing problems and decision theory problems related to optimal trial design, for which no solution was previously available. In particular, we construct new multiple testing procedures that satisfy minimax and Bayes optimality criteria. For a given optimality criterion, our new approach yields the optimal tradeoff? between power to detect an effect in the overall population versus power to detect effects in subpopulations. We demonstrate our approach in examples motivated by two randomized trials of new treatments for HIV

On the Equivalence of f-Divergence Balls and Density Bands in Robust Detection

The paper deals with minimax optimal statistical tests for two composite hypotheses, where each hypothesis is defined by a non-parametric uncertainty set of feasible distributions. It is shown that for every pair of uncertainty sets of the f-divergence ball type, a pair of uncertainty sets of the density band type can be constructed, which is equivalent in the sense that it admits the same pair of least favorable distributions. This result implies that robust tests under $f$-divergence ball uncertainty, which are typically only minimax optimal for the single sample case, are also fixed sample size minimax optimal with respect to the equivalent density band uncertainty sets.Comment: 5 pages, 1 figure, accepted for publication in the Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP) 201