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Keywords: Asymptotic relative efficiency; Nonparametric; Paired data; Power; Receiver operating characteristic curve.
The paired t-test or Wilcoxon signed rank (WSR) test is a frequently used parametric or nonparametric test for paired data, however, both require that the continuous outcomes are fully observed. In clinical studies of life-threatening diseases, some patients may die or experience other adverse events during the studies which preclude the observation of post-treatment outcomes, then paired t-test and WSR cannot be used for these cases to test the change of the outcomes from baseline to post-treatment. Motivated by the six-minute walk distance data from two cardiovascular clinical trials, in this paper, a new nonparametric testing procedure for paired data, which doesn't require the continuous outcomes to be fully observed as long as the rank is known, was proposed. It was proved that the new nonparametric testing procedure is conditionally distribution free given the ranks in each group and asymptotically distribution free. Extensive Monte Carlo simulations were conducted to compare the power of the new nonparametric testing procedure with paired t-test, WSR, and sign test under different bivariate distributions, and receiver operating characteristic (ROC) curve was used to visualize the relationship between Type-I error and power since it calibrates the Type I error automatically and compares the power under controlling Type I error at the same level. The simulation results confirmed that the paired t-test is the most powerful, and the power of the new nonparametric testing procedure is comparable with the WSR test at different Pearson correlation coefficients under bivariate normal distribution when complete observations exist. However, in the setting of bivariate lognormal distribution with equal variance and complete observations, the new nonparametric testing procedure is the most powerful at different Pearson correlation coefficients. The new nonparametric testing procedure for paired data is always more powerful than sign test, which is the only competitor when the continuous outcomes are not fully observed because of adverse events but the rank is known. Furthermore, the asymptotic relative efficiency of the new nonparametric testing procedure with respect to paired t-test was derived and compared with WSR and sign test, which further confirmed the power comparison.