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Currently I add Var1+Var2=VarSUM12 and then perform a linear regression on VarSUM12~x to obtain test statistics for x and get insight into weak associations that are present across Var1 and Var2 with x.

I would like to combine summary statistics (coefficient and standard error) from the individual regressions (Var1~x and Var2~x) to obtain the coefficient/se of VarSUM12~x. I guess the coefficient is easy, but is it possible to combine the standard errors?

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You can't add them.

If the two were independent, or even just uncorrelated, you could add the variances (the squares of the standard errors), but if they're correlated you need to incorporate the covariances:

$\text{Var}(Y_1+Y_2) = \text{Var}(Y_1)+\text{Var}(Y_2)+2\text{Cov}(Y_1,Y_2)$

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I'm not sure I understand correctly the OP's question. But if $Y_1$ and $Y_2$ are jointly Gaussian, you should be able to push this a tad further: $cov(Y_1,Y_2)=(\mbox{Var}(Y_1+Y_2)-\mbox{Var}(Y_1-Y_2))/4$ – user603 Apr 14 '14 at 15:18
I noticed that the Beta(Y1+Y2)=Beta(Y1)+Beta(Y2) is for the 'significant' associations (ie with a small P value) but is often off for associations with large P values, is this correct? – tafelplankje Sep 19 '14 at 8:29
The claim in the first line is not true in general. – Glen_b Sep 19 '14 at 8:56

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