Confidence intervals for P (Y₁ > Y₂) with normal outcomes in linear models

Recently, there is an emerging interest in the inference of P (Y₁ > Y₂) where Y₁ and Y₂ stand for two independent continuous random variables. So far, most of the research in this field focuses on simply comparing two outcomes without adjusting for covariates. This paper mainly presents a large sample approach based on a noncentral t distribution for the confidence interval estimation of P (Y₁ > Y₂) with normal outcomes in linear models. Furthermore, the performance of the proposed large sample approach is compared with that of a generalized variable approach and a bootstrap approach. Simulation studies demonstrate that for small-to-medium sample sizes, both the large sample approach and the generalized variable approach provide confidence intervals with satisfying coverage probabilities whereas the bootstrap approach can be slightly liberal for certain scenarios. The proposed approaches are applied to three real-life data sets.