The Appropriateness of the Wilcoxon Test in Ordinal Data

In two-sample studies with ordinal responses, the Wilcoxon rank-sum test is generally chosen to test equality of the distributions, in spite of it being a specific test of location shift. I compared the power of the exact tests based on the Wilcoxon statistic, O'Brien's generalized Wilcoxon statistic, and the omnibus Smirnov statistic in the presence of location shift and scale alternatives. All three tests can gain power as a function of the scale parameter, depending on its magnitude relative to the shift parameter. As the relative influence of the scale parameter grows, the O'Brien test is the most powerful. I also compared the power of the asymptotic Wilcoxon test with its exact version, and found that it either inflates or deflates power if there are scale changes present. I present results of a simulation study.