This paper presents a rank-based procedure for parameter estimation and hypothesis testing when the data are a mixture of paired observations and independent samples. Such a situation may arise when comparing two treatments. When both treatments can be applied to a subject, paired data will be generated. When it is not possible to apply both treatments, the subject will be randomly assigned to one of the treatment groups. Our rank-based procedure allows us to use the data from the paired sample and the independent samples to make inferences about the difference in the mean responses. The rank-based procedure uses both types of data by combining the Wilcoxon signed-rank statistic and the Wilcoxon-Mann-Whitney statistic. The exact and asymptotic distributions of the test statistic under the null hypothesis are determined as well as the limiting distribution of the point estimate. We also consider the Pitman efficacy of our rank-based procedure and its efficiency with respect to mean-based procedures.