The mixture correlation for two groups is estimated as:
r_xy_Mix_xy_WG+d_x^2d_y^2p^2(1-p)^2(d_x^2p(1-p)+1)(d_y^2p(1-p)+1)r_mix = (r_wg + sqrt((p - 1)^2 * p^2 * dx^2 * dy^2)) / sqrt((1 - (p - 1) * p * dx^2) * (1 - (p - 1) * p * dy^2))
where _xy_WGr_wg is the average within-group correlation, _xy_Mixr_mix is the overall mixture correlation,
d_xdx is the standardized mean difference between groups on X, d_ydy is the standardized mean difference between groups on Y, and
p is the proportion of cases in one of the two groups.