[R] meta-analysis question

Spencer Graves spencer.graves at pdf.com
Mon Sep 1 22:50:04 CEST 2003

	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.
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
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