Publication Description
We review the problem of obtaining a distribution-free confidence interval for the probability (p) that one random variable is less than another independent random variable based on the two-sample uncensored Wilcoxon-Mann-Whitney statistic. The pivotal quantities used by Sen (1967, Sankhya, Series A 29, 95-102) and Govindarajulu (1968, Annals of the Institute of Statistical Mathematics 20, 229-238) are modified to take into account that the variance of estimated p depends on p. A number of simulations were conducted to compare the various methods. The results suggest that the proposed modification is superior in the sense that over the entire range of p the modification generally yields values of coverage probability much closer to nominal coverage than the methods of Sen (S) and Govindarajulu (G). The coverage values for the S and G methods are very close to each other; both share the characteristics for one-sided lower confidence limits of being conservative for small values of p and nonconservative (sometimes extremely so) for large values of p. One-sided upper limits have coverages that are essentially the mirror image of lower-limit coverages. The contrasts between the modified method and the S and G methods diminish in magnitude as sample size increases but are still nontrivial for a sample size of 80. Our tentative conclusion is that the modified procedure is preferable to that of Sen and Govindarajulu and can be used when the sample size in both groups is equal to or greater than 20. Limitations of our simulations are briefly discussed. An example illustrating the use of our modified pivotal quantity and comparing it with the S and G results is presented.