# [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
> https://www.stat.math.ethz.ch/mailman/listinfo/r-help

```

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