Kruskal-Wallis and Friedman type tests for nested effects in hierarchical designs

In hierarchical-design experiments researchers need to test for nested effects, but sometimes parametric assumptions are violated. We show that the Kruskal-Wallis (K-W) rank test can be extended to test for such effects. Let B = b1 . . . bh be the tested effect, and A = a1 . . . ag be the effect immediately above it in the hierarchy. If one calculates separate K-W statistics for the effect B at each level of A, then under H0 for B, the sum of all these statistics is asymptotically χh−g. This test is demonstrated using a wellknown experimental economics dataset. We also show that the Friedman test can be extended in a similar fashion for designs that are both nested and crossed. In general, the preferable inference approach is using the label permutation distribution rather than the asymptotic distribution. Codes for carrying out such tests have been implemented and are available.