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.
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 ofy=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
On 1 Sep 2003 at 13:04, Remko Duursma wrote: I guess NO. The R-squared's doesn? give information about the relative infoemation content in each study. Kjetil Halvorsen> 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
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. >> >> ______________________________________________ >> R-help at stat.math.ethz.ch mailing list >> https://www.stat.math.ethz.ch/mailman/listinfo/r-help > >______________________________________________ >R-help at stat.math.ethz.ch mailing list >https://www.stat.math.ethz.ch/mailman/listinfo/r-help >