[R] Effect size in mixed models

[Andrew Robinson]

Hi Spencer,

what you're describing now seems to me to be similar to the idea of
intra-class correlation.  I have previously used the relative and
absolute amounts of variance represented at different hierarchical
levels of random effects, and they seemed to me to be easy to
interpret and useful, although I used method of moments estimators
instead of ML.  This ease of interpretation and utility is especially
true when the random effects structure matches the physical layout of
a sample, e.g., multiple measurements within trees within plots within
stands within forests etc.
 
Sadly I cannot see how to cleanly generalize the principle to
situations of crossed random effects.  Also, as Goldstein et al point
out in a nice paper in Understanding Statistics (2002): "Partitioning
variation in multilevel models", random covariates also complicate the
picture substantially.

Cheers

Andrew

On Thu, Jun 22, 2006 at 05:23:52PM -0700, Spencer Graves wrote:
> 	  I just learned that my earlier suggestion was wrong.  It's better to 
> compute the variance of the predicted or fitted values and compare those 
> with the estimated variance components.
> 
> 	  To see how to do this, consider the following minor modification of 
> an example in the "lme" documentation:
> 
> fm1. <- lme(distance ~ age, data = Orthodont, random=~1)
> fm2. <- lme(distance ~ age + Sex, data = Orthodont, random = ~ 1)
> 
> # str(fm1.) suggested the following:
>  > var(fm2.$fitted[, "fixed"]-fm1.$fitted[, "fixed"])
> [1] 1.312756
>  > VarCorr(fm1.)[, 1]
> (Intercept)    Residual
>   "4.472056"  "2.049456"
>  > VarCorr(fm2.)[, 1]
> (Intercept)    Residual
>   "3.266784"  "2.049456"
> 
> 	  In this example, the subject variance without considering "Sex" was 
> 4.47 but with "Sex" in the model, it dropped to 3.27, while the Residual 
> variance remained unchanged at 2.05.  The difference between 
> fm2.$fitted[, "fixed"] and fm1.$fitted[, "fixed"] is the change in the 
> predictions generated by the addition of "Sex" to the model.  The 
> variance of that difference was 1.31.  Note that 3.27 + 1.31 = 4.58, 
> which is moderately close to 4.47.
> 
> 	  In sum, I think we can get a reasonable estimate of the size of an 
> effect from the variance of the differences in the "fixed" portion of 
> the fitted model.
> 
> 	  Comments?
> 	  Hope this helps.
> 	  Spencer Graves
> 
> Spencer Graves wrote:
> >       You have asked a great question:  It would indeed be useful to 
> > compare the relative magnitude of fixed and random effects, e.g. to 
> > prioritize efforts to better understand and possibly manage processes 
> > being studied.  I will offer some thoughts on this, and I hope if there 
> > are errors in my logic or if someone else has a better idea, we will 
> > both benefit from their comments.
> > 
> >       The ideal might be an estimate of something like a mean square for 
> > a particular effect to compare with an estimated variance component.
> > Such mean squares were a mandatory component of any analysis of variance 
> > table prior to the (a) popularization of generalized linear models and 
> > (b) availability of software that made it feasible to compute maximum 
> > likelihood estimates routinely for unbalanced, mixed-effects models. 
> > However, anova(lme(...)) such mean squares are for most purposes 
> > unnecessary cluster in a modern anova table.
> > 
> >       To estimate a mean square for a fixed effect, consider the 
> > following log(likelihood) for a mixed-effects model:
> > 
> >       lglk = (-0.5)*(n*log(2*pi*var.e)-log(det(W)) + 
> > t(y-X%*%b)%*%W%*%(y-X%*%b)/var.e),
> > 
> > where n = the number of observations,
> > 
> >       b = the fixed-effect parameter variance,
> > 
> > and the covariance matrix of the residuals, after integrating out the 
> > random effects is var.e*solve(W).  In this formulation, the matrix "W" 
> > is a function of the variance components.  Since they are not needed to 
> > compute the desired mean squares, they are suppressed in the notation here.
> > 
> >       Then, the maximum likelihood estimate of
> > 
> >       var.e = SSR/n,
> > 
> > where SSR = t(y-X%*%b)%*%W%*%(y-X%*%b).
> > 
> >       Then
> > 
> >       mle.lglk = (-0.5)*(n*(log(2*pi*SSR/n)-1)-log(det(W))).
> > 
> >       Now let
> > 
> >       SSR0 = this generalized sum of squares of residuals (SSR) without 
> > effect "1",
> > 
> > and
> > 
> >       SSR1 = this generalized SSR with this effect "1".
> > 
> >       If I've done my math correctly, then
> > 
> >       D = deviance = 2*log(likelihood ratio)
> >         = (n*log(SSR0/SSR1)+log(det(W1)/det(W0)))
> > 
> >       For roughly half a century, a major part of "the analysis of 
> > variance" was the Pythagorean idea that the sum of squares under H0 was 
> > the sum of squares under H1 plus the sum of squares for effect "1":
> > 
> >       SSR0 = SS1 + SSR1.
> > 
> >       Whence,
> > 
> >       exp((D/n)-log(det(W1)/det(W0))) = 1+SS1/SSR1.
> > 
> > Thus,
> > 
> >       SS1 = SSR1*(exp((D/n)-log(det(W1)/det(W0)))-1).
> > 
> >       If the difference between deg(W1) and det(W0) can be ignored, we get:
> > 
> >       SS1 = SSR1*(exp((D/n)-1).
> > 
> >       Now compute MS1 = SS1/df1, and compare with the variance components.
> > 
> >       If there is a flaw in this logic, I hope someone will disabuse me 
> > of it.
> > 
> >       If this seems too terse or convoluted to follow, please provide a 
> > simple, self-contained example, as suggested in the posting guide! 
> > "www.R-project.org/posting-guide.html".  You asked a theoretical 
> > question, you got a theoretical answer.  If you want a concrete answer, 
> > it might help to pose a more concrete question.
> > 
> >       Hope this helps.
> >       Spencer Graves   
> > 
> > Bruno L. Giordano wrote:
> >> 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
> >>
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> >
> 
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-- 
Andrew Robinson  
Department of Mathematics and Statistics            Tel: +61-3-8344-9763
University of Melbourne, VIC 3010 Australia         Fax: +61-3-8344-4599
Email: a.robinson at ms.unimelb.edu.au         http://www.ms.unimelb.edu.au