An exact test with high power and robustness to unmeasured confounding effects

In observational studies, it is agreed that the sensitivity of the findings to unmeasured confounders needs to be assessed. The issue is that a poor choice of test statistic can result in overstated sensitivity to hidden bias of this kind. In this article, a new adaptive test is proposed, guided by considerations of low sensitivity to hidden bias: it is tailored so that its power is greater than other leading tests, both in finite and infinite samples. One way of defining power in case of possible confounders is as the probability of reporting robustness (ie, insensitivity) of a true discovery to potential bias. In case of finite samples, we compute the power by simulations. When sample size approaches infinity, a meaningful indicator of the power is the design sensitivity, which is computed analytically and found to be better in the new test than in existing tests. Another asymptotic criterion for comparing tests when there is concern for confounders is Bahadur efficiency. The proposed test outperforms commonly used tests in terms of Bahadur efficiency in most sampling situations. The advantages of the new test mainly stem from its adaptivity: it combines two test statistics and consequently achieves the best design sensitivity and the best Bahadur efficiency of the two. As a "real-world" examination, we compare 441 daily smokers to 441 nonsmokers, to test the effect of smoking on periodontal disease. The new test is more robust to unmeasured confounders than both the Wilcoxon signed rank test and the paired t-test.