Testing the significance of the RV coefficient

The relationship between two sets of variables defined for the same individuals can be evaluated by the RV coefficient. However, it is impossible to assess by the RV value alone whether or not the two sets of variables are significantly correlated, which is why a test is required. Asymptotic tests do exist but fail in many situations, hence the interest in permutation tests. However, the main drawbacks of the permutation tests are that they are time consuming. It is therefore interesting to approximate the permutation distribution with continuous distributions (without doing any permutation). The current approximations (normal approximation, a log-transformation and Pearson type III approximation) are discussed and a new one is described: an Edgeworth expansion. Finally, these different approximations are compared for both simulations and for a sensory example.