# [R] meta-analysis question

Remko Duursma den.duurs at lycos.com
Tue Sep 2 01:07:21 CEST 2003

```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 <spencer.graves at pdf.com>
To: den.duurs at lycos.com
Cc: rhelp <r-help at r-project.org>

>	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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>
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