[R] Effect size in mixed models

[Spencer Graves]

	  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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