The permutation distribution of matrix correlation statistics

Many statistics used to test for association between pairs (y_i, z_i) of multivariate observations, sampled from n individuals in a population, are based on comparing the similarity a_ij of each pair (i, j) of individuals, as evidenced by the values y_i and y_j, with their similarity b_ij based on the values z_i, and Z_j. A common strategy is to compute the sample correlation between these two sets of values. The appropriate null hypothesis distribution is that derived by permuting the z_i's at random among the individuals, while keeping the y_i's fixed. In this paper, a Berry–Esseen bound for the normal approximation to this null distribution is derived, which is useful even when the matrices a and b are relatively sparse, as is the case in many applications. The proofs are based on constructing a suitable exchangeable pair, a technique at the heart of Stein's method.