Comment on “A Case for Nonparametrics” by Bower et al.

While we welcome Bower et al.’s (2022) exploration of how teaching need not rely on parametric methods alone, we wish to raise some issues with their presentation. First, “nonparametric”—meaning methods that do not rely on an assumed form of frequency distribution—is not synonymous with “rank-based.” When, as in Bower et al.’s (2022) examples, interest lies in differences in mean or median between groups, we could nonparametrically use permutation tests with test statistics that—straightforwardly—describe differences in sample mean or median between groups. These exact approaches use no assumptions other than independence of the outcomes (Berry, Johnston, and Mielke 2019, sec. 3.3) and that the test statistic being used captures deviations from the null that are of some scientific interest. So there is no need to switch to less-relevant rank-based methods, much less present them as the natural alternative to being parametric. We do agree with Bower et al. (2022) that user-friendly implementations are important, but permutation tests are available via simple R commands (e.g., the coin package’s oneway_test() function (Hothorn et al. 2008)) and shiny applications that are built on them. Second, the Kruskal-Wallis and Wilcoxon tests are not tests for population mean rank in the same sense that ANOVA and the t-test are tests for the mean, or Mood’s test is a test for the median. The issue is that the population mean and median for a subgroup are defined by the distribution of the response in just that subgroup. The mean rank, in contrast, depends on the distribution of responses in all subgroups and their sample sizes, so whether group 1 has higher mean rank than group 2 can depend on which other groups are also in the dataset. When the groups are not stochastically ordered, this leads to surprisingly complicated behavior of the Kruskal-Wallis test (Brown and Hettmansperger 2002). Finally, with regard to pedagogy we recommend the Data Problem-Solving Cycle (Wild and Pfannkuch 1999) in which the first step, “Problem,” identifies the question to address. This