The average within-group correlation is estimated as:
_xy_WG=_xy_Mix(d_x^2p(1-p)+1)(d_y^2p(1-p)+1)-d_x^2d_y^2p^2(1-p)^2r_wg = r_mix * sqrt((dx^2 * p * (1 - p) + 1) * (dy^2 * p * (1 - p) + 1)) - sqrt(dx^2 * dy^2 * p^2 * (1 - p)^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.