Wilcoxon-Mann-Whitney test for two independent samples

Question

This is my first question on this side, so please don't mind if not everything is correct :)

I'm currently trying to understand the Wilcoxon-Rank-Test/Mann-Whitney-Wilcoxon Test, but it got me kind of confused. As far as I understood it is to test whether the null hypothesis $F_X = F_Y$ can be rejected at level $\alpha$, where $F_X$ is the cumulative distribution of the random variable X and same for Y.

Now here's the problem: Mostly it is assumed that the cumulative distributions are continuous. The test statistic, the distribution of the rank vector and so on are derived based on that assumption. So I don't quite get why the test can be used if the data is "merely" of ordinal scale? How is the test statistic derived in this case? I couldn't really find a proper source explaining me my problem, so could anyone recommend me a good book,script etc. or explain briefly why the test works on ordinal data ?

Thank you!

Answer 1

The test is indeed originally based on an assumption of continuous cdf and that is how the tables for it are derived.

However:

1. You can use the same test statistic and conduct a permutation test; this will deal with the pattern of ties; in large samples you can sample from the permutation distribution rather than use complete evaluation (and get a good estimate of the uncertainty in the p-value).

2. You can work out the variance of the test statistic under the pattern of ties and use a large sample normal approximation (if the sample sizes are large enough)

In either case the test will work on merely ordinal data - in the sense that you will attain - either exactly or approximately - a test with a desired significance level (from the available possible significance levels).