Two-sample location problem is one of the most encountered problems in statistical practice. The two most commonly studied subtypes of two-sample location problem involve observations from two populations that are either independent or completely paired, but a third subtype can oftentimes occur in practice when some observations are paired and some are not. Partially paired two-sample problems, also known as paired two-sample problems with missing data, often arise in biomedical fields when it is difficult for some invasive procedures to collect data from an individual at both conditions we are interested in comparing. Existing rank-based two-sample comparison procedures for partially paired data, however, do not make efficient use of all available data. In order to improve the power of testing procedures for this problem, we propose several new rank-based test statistics and study their asymptotic distributions and, when necessary, exact variances. Through extensive numerical studies, we show that the best overall power come from the proposed tests based on weighted linear combinations of the test statistics comparing paired data and the test statistics comparing independent data, using weights inversely proportional to their variances. We illustrate the proposed methods with a real data example from HIV research for prevention.

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