If this were really important and I didn't have anything else to work with, I might compute some kind of "Mahalinobis distance" of each point from the (trimmed?) mean relative to the (trimmed?) covariance, and then I might try to plot this Mahalinobis distance" vs. the x range and R-squared numbers. However, I'm not certain what I would do with the result.
hope this helps. spencer graves
Remko Duursma wrote:
This is really helpful, however: i do not have the covariance
matrix for each study. In fact, i only have b0, b1, R-squared and the range of x's used for the fit.
Rekmo --------- Original Message ---------
DATE: Mon, 01 Sep 2003 13:50:04 From: Spencer Graves <[EMAIL PROTECTED]> To: [EMAIL PROTECTED] Cc: rhelp <[EMAIL PROTECTED]>
Can you get the covariance matrices of the vectors b = c(b0, b1)? There is a reasonable literature on meta-analysis with which I'm not very familiar. However, a standard thing to do is to compute a weighted average with weights proportional to the inverse of the covariance matrices, while testing to evaluate whether the b's plausibly all estimate the same thing.
The theory is as follows: Suppose b.i ~ N.k(mu, Sig.i), i = 1, 2, ..., n. If you have a covariance matrix for each vector b.i, then you have this set-up. Assuming you do have (or can approximate) Sig.i, then
l.i = log(likelihood(b.i)) = (-0.5)*(k*log(2*pi)+log(det(Sig.i))+t(b.i-mu)%*%solve(Sig.i, (b.i-mu))).
The first derivative of l.i with respect to mu is as follows:
D.l.i = solve(Sig.i, (x.i-mu)).
The solution for mu of sum(D.l.i)=0 is as follows:
mu.hat = solve(sum(Sig.i), sum(solve(Sig.i, (x.i-mu)))).
One could also derive various statistics for evaluating whether it is plausible to believe that these b.i's all come from the same population. I would assume that the literature on meta-analysis would deal with this, but I have not looked much at that literature, and I'll leave that question to others.
hope this helps. spencer graves
Remko Duursma wrote:
Dear R-helpers,
i have the following situation: i have a bunch of
y=b0 + b1*x from different studies, and want to estimate a "general" y=f(x). I only have the b0,b1's and R-squareds. Should i weigh the separate equations by their R-squared?
thanks
Remko
^'~,_,~'^'~,_,~'^'~,_,~'^'~,_,~'^'~,_,~'^'~,_,~' Remko Duursma, Ph.D. student Forest Biometrics Lab / Idaho Stable Isotope Lab University of Idaho, Moscow, ID, U.S.A.
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