[R] meta-analysis question

Spencer Graves spencer.graves at pdf.com
Tue Sep 2 01:55:05 CEST 2003

	  Have you made normal probability plots and scatterplots of the b0's 
and b1's?  Especially if the numbers seem reasonably normally 
distributed, possibly with some outliers, I'd throw out the outliers, 
compute mean, standard deviation and correlation from what I had left, 
and hope that this all was adequate for the purpose at hand.

	  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 

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