On two-sample testing for data with arbitrarily missing values

We develop a new rank-based approach for univariate two-sample testing in the presence of missing data which makes no assumptions about the missingness mechanism. This approach is a theoretical extension of the Wilcoxon-Mann-Whitney test that controls the Type I error by providing exact bounds for the test statistic after accounting for the number of missing values. Greater statistical power is shown when the method is extended to account for a bounded domain. Furthermore, exact bounds are provided on the proportions of data that can be missing in the two samples while yielding a significant result. Simulations demonstrate that our method has good power, typically for cases of $10%$ to $20%$ missing data, while standard imputation approaches fail to control the Type I error. We illustrate our method on complex clinical trial data in which patients' withdrawal from the trial lead to missing values.