Test of difference for multiple dependent groups

[NimaDarbari]

I have a sort of more statistical type of question. This is the scenario: I have the composition of board of directors for thousands of firms and I want to compare males and female board members in each firm along several attributes. For instance, lets say I want to compare the age of male and female directors. The age of male and female directors in each firm cannot be considered to be independent of each other, my question is on how to take this into account? For the sake of simplicity, lets assume that there are 1000 firms, each have exactly 8 directors in their board, 4 male and 4 female. How can I test if male directors are older across firms? Would it be appropriate to run a separate test for each firm and then just report a summary of test results? I need to thank your answers in advance specifically as this is more of a Statistics type of question and not explicitly Stata.

[George Ford]

why not regress age on male with a firm fixed effect?

[NimaDarbari]

Thanks for your suggestion. My main concern is that my attribute of interest (age for instance) may have different dispersion for male and female subgroups in each firm. This would perhaps lead to incorrect standard errors in the regression that you suggest

[George Ford]

I'm not sure what you mean.

If you want to know how the means differ across male/female, then regression will do the trick.

If you want to ask "are female board members younger when male board members are younger", then you'd have some regression or tabulation of female age and male age.

[NimaDarbari]

I think if the distribution around the corresponding mean is too different in the two subgroups, this regression is not useful. For instance, the estimated coefficient may turn to be 1.5 and significant. This means females are 1.5 years younger for sure but it doesn't necessarily mean that the difference is significant, as for instance there may be more variability in the women ages than in mean ages. In other words the homoscedasticity assumption doesn't hold. I was thinking about using female and male subgroups in each firm as a cluster (i.e. number of clusters would be equal to the number of firms times two) and then use cluster adjusted standard errors in the regression. This addresses part of the problem I guess but still doesn't take into account that the female and male subgroups in a firm are also not independent of each other

[George Ford]

If you're concerned about heteroskedasticity, use robust or cluster.

The difference in the variance between the two is accounted for in the standard error.

The fixed effect addresses the dependence to a large extent by centering on the average age (or dependent variable).

It all depends on the question of interest. If it's age difference, then use regression. If is agef = f(agem), then different model.