# Can quintile/dispersion ratios be decomposed according to subgroups?

I'm try to decompose inequality/dispersion ratios (top 10% to bottom 10% or top 20% to bottom 20%) according to subgroups. I would like to say something like e.g. males are responsible for x% and females y% of the inequality as measured by e.g. top to bottom decile. I know GINI and Theil indices can be decomposed according to between and within group inequalities. Of course, dispersion ratios ignore the within component.

The decile ratio, R, could be described thus: R = (xa + xb + .. xalpha + ya + yb + ... ybeta)/(x1 + x2 + ..xn + y1 + y2 + ....ym) ;

where xa-to-xalpha, ya-to-ybeta are men and women in the top decile and x1-to-xn, y1-to-ym are men and women in the bottom decile

R = (xa + xb + ..xalpha)/(x1 + ..ym) + (ya + yb + ..ym)/(x1 + ..ym)

That is, I can decompose according to the contribution to men and women who make it into the top decile, but I cannot decompose the the overall contribution of of males and females. Can this be done? Thanks