Bahadur Efficiency of Observational Block Designs

To be convincing, an observational or nonrandomized study of causal effects must demonstrate that its conclusions cannot be readily explained by a small unmeasured bias in the way individuals were assigned to treatment or control. The Bahadur relative efficiency of a sensitivity analysis compares the performance of different test statistics or different research designs when sensitivity to unmeasured bias is appraised: better statistics and better designs exhibit insensitivity to larger biases. Bahadur efficiency is relevant here because, unlike Pitman efficiency, it can depict efficiency with an effect that is not trivially small: every trivially small treatment effect is sensitive to trivially small biases. The Bahadur efficiency of a sensitivity analysis has been used by various authors in the simple case of matched pairs, but the technical issues are more complex in the case of blocks larger than pairs, and they are developed here for the first time. Choosing a better test statistic for a block design, or choosing a better block size—larger than pairs—can result in enormous increases in the efficiency of a sensitivity analysis. An adaptive choice of test statistic can ensure the better Bahadur efficiency of two competing statistics. An R package weightedRank implements the methods, contains the example and reproduces its analysis. Supplementary materials for this article are available online.