A test of independence based on a sign covariance related to Kendall's tau

The standard two-variable chi-square test is typically consistent for all alternatives to independence, but effectively treats the data as nominal which may lead to loss of power for ordinal data. Alternatively, a test based on Kendall's tau does take ordinality into account, but only has power against a narrow set of alternatives. This paper introduces a new test aimed at filling this gap, i.e., it is designed for ordinal data and to have omnibus asymptotic power. Our test is a permutation test based on a modification of Kendall's tau, denoted $\tau^*$. Based on partial proofs and numerical evidence, we conjecture $\tau^*$ to be nonnegative, and zero if and only if independence holds. An interpretation of $\tau^*$ in terms of concordance and discordance for sets of four observations is given. The new coefficient is a sign version of a correlation coefficient introduced by \citeA{bergsma06}.