From: Spencer Graves <spencer.graves_at_pdf.com>

Date: Fri 23 Jun 2006 - 10:23:52 EST

*> compare the relative magnitude of fixed and random effects, e.g. to
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*> prioritize efforts to better understand and possibly manage processes
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*> being studied. I will offer some thoughts on this, and I hope if there
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*> are errors in my logic or if someone else has a better idea, we will
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*> both benefit from their comments.
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*>
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*> The ideal might be an estimate of something like a mean square for
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*> a particular effect to compare with an estimated variance component.
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*> Such mean squares were a mandatory component of any analysis of variance
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*> table prior to the (a) popularization of generalized linear models and
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*> (b) availability of software that made it feasible to compute maximum
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*> likelihood estimates routinely for unbalanced, mixed-effects models.
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*> However, anova(lme(...)) such mean squares are for most purposes
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*> unnecessary cluster in a modern anova table.
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*>
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*> To estimate a mean square for a fixed effect, consider the
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*> following log(likelihood) for a mixed-effects model:
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*>
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*> lglk = (-0.5)*(n*log(2*pi*var.e)-log(det(W)) +
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*> t(y-X%*%b)%*%W%*%(y-X%*%b)/var.e),
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*>
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*> where n = the number of observations,
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*>
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*> b = the fixed-effect parameter variance,
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*>
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*> and the covariance matrix of the residuals, after integrating out the
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*> random effects is var.e*solve(W). In this formulation, the matrix "W"
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*> is a function of the variance components. Since they are not needed to
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*> compute the desired mean squares, they are suppressed in the notation here.
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*>
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*> Then, the maximum likelihood estimate of
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*>
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*> var.e = SSR/n,
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*>
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*> where SSR = t(y-X%*%b)%*%W%*%(y-X%*%b).
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*>
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*> Then
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*>
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*> mle.lglk = (-0.5)*(n*(log(2*pi*SSR/n)-1)-log(det(W))).
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*>
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*> Now let
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*>
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*> SSR0 = this generalized sum of squares of residuals (SSR) without
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*> effect "1",
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*>
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*> and
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*>
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*> SSR1 = this generalized SSR with this effect "1".
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*>
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*> If I've done my math correctly, then
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*>
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*> D = deviance = 2*log(likelihood ratio)
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*> = (n*log(SSR0/SSR1)+log(det(W1)/det(W0)))
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*>
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*> For roughly half a century, a major part of "the analysis of
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*> variance" was the Pythagorean idea that the sum of squares under H0 was
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*> the sum of squares under H1 plus the sum of squares for effect "1":
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*>
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*> SSR0 = SS1 + SSR1.
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*>
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*> Whence,
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*>
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*> exp((D/n)-log(det(W1)/det(W0))) = 1+SS1/SSR1.
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*>
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*> Thus,
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*>
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*> SS1 = SSR1*(exp((D/n)-log(det(W1)/det(W0)))-1).
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*>
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*> If the difference between deg(W1) and det(W0) can be ignored, we get:
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*>
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*> SS1 = SSR1*(exp((D/n)-1).
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*>
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*> Now compute MS1 = SS1/df1, and compare with the variance components.
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*>
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*> If there is a flaw in this logic, I hope someone will disabuse me
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*> of it.
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*>
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*> If this seems too terse or convoluted to follow, please provide a
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*> simple, self-contained example, as suggested in the posting guide!
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*> "www.R-project.org/posting-guide.html". You asked a theoretical
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*> question, you got a theoretical answer. If you want a concrete answer,
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*> it might help to pose a more concrete question.
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*>
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*> Hope this helps.
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*> Spencer Graves
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*>
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*> Bruno L. Giordano wrote:
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https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html Received on Fri Jun 23 11:40:56 2006

Date: Fri 23 Jun 2006 - 10:23:52 EST

Comments? Hope this helps. Spencer Graves Spencer Graves wrote:

> You have asked a great question: It would indeed be useful to

>> Hello, >> Is there a way to compare the relative relevance of fixed and random >> effects in mixed models? I have in mind measures of effect size in >> ANOVAs, and would like to obtain similar information with mixed models. >> >> Are there information criteria that allow to compare the relevance of >> each of the effects in a mixed model to the overall fit? >> >> Thank you, >> Bruno >> >> ______________________________________________ >> R-help@stat.math.ethz.ch mailing list >> https://stat.ethz.ch/mailman/listinfo/r-help >> PLEASE do read the posting guide! >> http://www.R-project.org/posting-guide.html > ______________________________________________R-help@stat.math.ethz.ch mailing list

https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html Received on Fri Jun 23 11:40:56 2006

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