# How to combine regression models?

Say I have three data sets of size $n$ each:

$y_1$ = heights of people from the US only

$y_2$ = heights of men from the whole world

$y_3$ = heights of women from the whole world

And I build a linear model for each with factors $x_i$, $i = 1,..., k$:

$\hat{y}_{j} = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \epsilon_{j}$

with $\epsilon$ having the usual properties for OLS. And I may use a factor $x_i$ in more than one regression.

My question is: How could I combine the regressions such that I can obtain estimates for:

$y_{12}$ = height of men from the US only

$y_{13}$ = height of women from the US only

for which I do not have data

I thought of perhaps some sort of weighting:

$\hat{y}_{12} = w_{1} \hat{y}_{1} + (1 - w_{1}) \hat{y}_{2}$

but then I wouldn't know what to use for $w_1$.

• I don't have anything solid enough to be an answer, but as a comment: the first thing that comes to mind is using a single hierarchical (mixed) regression. But I really can't figure out what would be the random effects, so maybe it wouldn't work. Thought I'd throw the idea out there, though. – Wayne Jan 14 '16 at 14:25
• Thanks for the suggestion. Yes, it would appear that the for hierarchical model you would need to do it on the $y_{12}$ data, no? – J4y Jan 18 '16 at 13:59
• My initial thought was on y with an intercept by sex -- something like height ~ f1 + f2 + f3 + (1 | sex) in the R package lmer – Wayne Jan 18 '16 at 18:33

I recommend simply fitting a mixture of distributions to the height data from the US only, because the distributions of height in men and women are reasonably different. This would estimate the parameters of two distributions that when summed together best describe the variation in the data. The parameters of these distributions (mean and variance, since a Gaussian distribution should work fine) give you the information you are after. The R packages mixtools and mixdist let you do this; I'm sure there are many more as well.