Bootstrap p-values reduce type 1 error of the robust rank-order test of difference in medians

Abstract: The robust rank-order test (Fligner and Policello, 1981) was designed as an improvement of the non-parametric Wilcoxon-Mann-Whitney U-test to be more appropriate when the samples being compared have unequal variance. However, it tends to be excessively liberal when the samples are asymmetric. This is likely because the test statistic is assumed to have a standard normal distribution for sample sizes > 12. This work proposes an on-the-fly method to obtain the distribution of the test statistic from which the critical/p-value may be computed directly. The method of likelihood maximization is used to estimate the parameters of the parent distributions of the samples being compared. Using these estimated populations, the null distribution of the test statistic is obtained by the Monte-Carlo method. Simulations are performed to compare the proposed method with that of standard normal approximation of the test statistic. For small sample sizes (<= 20), the Monte-Carlo method outperforms the normal approximation method. This is especially true for low values of significance levels (< 5%). Additionally, when the smaller sample has the larger standard deviation, the Monte-Carlo method outperforms the normal approximation method even for large sample sizes (= 40/60). The two methods do not differ in power. Finally, a Monte-Carlo sample size of 10^4 is found to be sufficient to obtain the aforementioned relative improvements in performance. Thus, the results of this study pave the way for development of a toolbox to perform the robust rank-order test in a distribution-free manner.