# CHAPTER 11

# RANK-BASED TESTS OF SIGNIFICANCE

## 11.1 THE QUEST FOR POWER

In previous chapters we have said little about the rank-based adaptive tests, preferring instead to rely on tests that used weighted least squares and permutation methods. The adaptive WLS tests were emphasized because they can be used to test any subset of coefficients in a linear model. However, the earliest two-sample adaptive tests, which were based on ranks, had greater power than the *t* test and the Wilcoxon test for many error distributions. Therefore, we will attempt to describe these methods in some detail in this chapter.

Rank-based tests use test statistics that are functions of the data only through the ranks. For example, the Wilcoxon rank-sum test uses the sum of the ranks in the second sample as the test statistic, where the observations are ranked over all observations in both samples. Because the test statistic is a function of the ranks, for small samples it is easy to tabulate the c.d.f. of the permutation distribution and this permutation distribution can be used for any sample taken from a continuous distribution. Hence, the Wilcoxon test can be used with samples from a wide variety of distributions and, because it is based on ranks, it is naturally robust with respect to the presence of an outlier.

The Wilcoxon test statistic, which uses the sum of the ranks in the second sample, is only one of many rank-based two-sample tests. Instead of computing the sum of the ranks in the second sample, ...