Testing for concordance between several criteria

The problem of whether the rankings of some objects given by a set of criteria (or judges) show any agreement or are more or less independent is addressed. The most familiar measure for concordance is the Kendall W coefficient. Classical tests for concordance are the Friedman test and the F test. Legendre Species associations: the Kendall coefficient of concordance revisited. J. Agric. Biol. Environ. Stat. 2005;10(2):226–245 compared via simulation the Friedman test and its permutation version. Unfortunately, the simulation study of Legendre was very limited because it considered neither the copula aspect nor the F test. Kendall W is a rank-based correlation measure, and therefore it is not affected by the marginal distributions of the underlying variables, but only by the copula of the multivariate distribution. In this article, the simulation study of Legendre is deeply extended by considering the copula aspect as well as the F test. It is shown that the Friedman test is too conservative and less powerful than both the F test and the permutation test for concordance which always have a correct size and behave alike. The F test should be preferred because it is computationally much easier. Surprisingly, the power function of the tests is not much affected by the type of copula.