Suppose that we observe a p-dimensional vector of traits Y i = (Y i (1),…, Y i (p))T, genotype G i, and a q-dimensional vector of covariates Z i = (Z i (1),…, Z i (q))T for the ith subject in a population-based study with n subjects, and {(Y i, G i, Z i) : i = 1,…, n} are independent samples. For subjects i and j, let Y i and Y j be their vectors of traits, respectively, and analogously, G i and G j and Z i and Z j are their genotypes and covariates. Generalized from Kendall's tau, a U statistic is defined to measure the association between Y and G as follows: (2)U=(n2)−1∑i<j(Yi−Yj)(Gi−Gj). Without considering the covariates and conditioning on all phenotypes, U follows an asymptotically normal distribution in the absence of association [63]. To accommodate covariates, a weighted U statistic has been developed [64, 65]. We refer to Jiang and Zhang [64] for a detailed description of the method. For the purpose of comparison, we present the results with and without considering age as the covariate. Recall that our analysis is stratified by ethnicity and gender.
