A measure of association between two sets of multivariate responses

Let X and Y denote two distinct sets of multivariate responses, each of which can be a mix of continuous and ordinal data. Suppose that it is of interest to quantify the overall strength of association between X and Y. Our two approaches to this problem are to construct correlation coefficients similar to Kendall's tau in the bivariate case. The first approach uses the average concordance of each pair of individuals, where the average concordance is obtained across all possible pairings of variables between the two sets, as the kernel of the U-statistic. Then the overall correlation between the two sets of multivariate data is expressed in terms of the expected value of the kernel. The second approach estimates pairwise Kendall's taus between the two sets, then uses a summary measure to quantify the overall association. One option for the summary measure is to take the average of taus, and another option is to take the distance of taus. The two approaches yield the same results when the summary measure of the latter is the average. However, using the distance of taus can sometimes detect relationships not seen in the average. We illustrate the proposed methods using data from a national asthma clinical trial in which the first set of variables comprises 12 binary responses of skin allergen tests and the second set of variables comprises three continuous outcomes of pulmonary function and one clinical outcome.