This paper is concerned with nonparametric methods for comparing medians of paired data with unpaired values on both responses. A new nonparametric test statistic is proposed in this paper based on a Mann-Whitney U test making comparisons across complete and incomplete pairs. A method of finding the null hypothesis distribution for this statistic is presented using a permutation approach. A Monte Carlo simulation study is described to make power comparisons among four already-existing nonparametric test statistics and this new test statistic. It is concluded that this new test statistic is fairly powerful in handling this kind of data compared to the other four test statistics. Finally, all five test statistics are applied to a real dataset for comparing the proportions of certain T cell receptor gene families in a cancer study. The introduction of this new nonparametric test statistic is of public health importance because it is a powerful statistical method for dealing with a pattern of missing data that may be encountered in clinical and public health research
