  The most popular ways to test for independence of two ordinal random
variables are by means of Kendall's tau and Spearman's rho. However, such tests
are not consistent, only having power for alternatives with ``monotonic''
association. In this paper, we introduce a natural extension of Kendall's tau,
called $\tau^*$, which is non-negative and zero if and only if independence
holds, thus leading to a consistent independence test. Furthermore,
normalization gives a rank correlation which can be used as a measure of
dependence, taking values between zero and one. A comparison with alternative
measures of dependence for ordinal random variables is given, and it is shown
that, in a well-defined sense, $\tau ^*$ is the simplest, similarly to
Kendall's tau being the simplest of ordinal measures of monotone association.
Simulation studies show our test compares well with the alternatives in terms
of average $p$-values.
