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A basic query. We are doing a meta analysis where we are gathering estimated relationships from a series of papers. Each of these studies estimates two parameters of interest say b1 and b2 and reports their standard errors as usual. We are interested in the quantity (b1 - b2). My understanding is that we cannot derive a standard error for (b1 - b2) simply from the standard errors for b1 and b2 - we would also need cov(b1, b2) (which are not reported of course). Correct? And just taking the square root of the sum of squared standard errors would not be an acceptable approximation, yes?

Thanks for any insight, and apologies for the basic nature of this query!

Thanks, Bhavani

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    $\begingroup$ You are right. If b1 and b2 are independent, you can take the square root of the sum of squared standard errors. For example, b1 is the sample mean age in trt A and b2 is the sample mean age in trt B. $\endgroup$ – user158565 Dec 11 '18 at 15:26
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    $\begingroup$ Thank you! I should clarify that b1 and b2 both derive from a multivariate regression. And so independence would be hard to claim. $\endgroup$ – Bhavani Shankar Dec 11 '18 at 15:42
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Yes, you need information about the (individual) standard deviations and the (mutual) covariance of $b_1$ and $b_2$.

To see this, notice first that the standard deviation $\sigma(b)=\sqrt{Var(b)}$. Now, given that $b=b_1-b_2$ holds, then

$$\sigma^2(b)=Var(b)=Var(b_1)+Var(b_2)-2Cov(b_1,b_2)$$,

with $\sigma(b_1)=\sqrt{Var(b_1)}$ and $\sigma(b_2)=\sqrt{Var(b_2)}$. Hence, all three terms are needed.

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