ABSTRACT
This chapter discusses rank based tests, including the Wilcoxon signed rank and Wilcoxon ranked sum tests. The inclusion of these topics is in part due to their applicability in an age when computing is the expected method for interacting with data, but even more importantly because it provides students an opportunity to explore test assumptions and power by comparing these tests with parametric methods. Rank based tests allow the student to carry out an entire test without any requiring deep theory (the CLT), since the test statistics are approachable and it is straightforward to simulate or even compute their distributions under the null hypothesis.
The chapter shows a complete example of a Wilcoxon signed rank test with all possible arrangements of the positive ranks. The next section, on two sample tests, applies the signed rank test to paired data. It then introduces the Wilcoxon rank sum test for independent samples. We discuss the application of rank based tests to ordinal data, with an example. Using simulations, we compare the power of t-test and Wilcoxon test and discuss how to determine sample sizes needed for testing at a desired power. The effect size A is suggested as a way to present the results of Wilcoxon rank sum tests. We discuss the consistency of a hypothesis test, and give simulation evidence that the t-test is consistent, while the Wilcoxon test may not be. A vignette discusses the receiver operating characteristic (ROC) curve, and relates the Wilcoxon test, the effect size A, and the area under the ROC curve to each other.
