  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.
