[R] Effect size in mixed models

Bruno L. Giordano bruno.giordano at music.mcgill.ca
Tue Jun 20 06:19:34 CEST 2006

OK and thanks!! I will try to apply the approach you suggest to my data.

I found a possibly interesting reference on this topic (if my fast reading 
is error-free):

E.  Demidenko, 2004, Mixed Models: Theory and Applications (John Wiley & 
Sons), Chapter 9 (Diagnostics and influence analysis)

outlines a "perturbation-based" methodology called Infinitesimal influence 

which analyzes the stability of the solution as a function of perturbations 
in the predictors (or in the isolated observations). In this case, 
predictors whose perturbation leads to greater changes in the solution are 
assumed to be more relevant. In the same chapter the idea is extended to 
mixed models, although it focuses on the influence of clusters of 
observations on the solution, and not on the influence of random effects.

A really rough idea I had, more in line with my current mathematical 
background, is to simply examine the percent increase in the residual 
variance when one (or more) fixed/random effect(s) is(are) removed from the 

With this brute approach I found, for example, that for the outcome I am 
analyzing (4 random + 4 fixed effects with interactions) the entire fixed 
portion is less relevant than any of the random effects in isolation.


----- Original Message ----- 
From: "Spencer Graves" <spencer.graves at pdf.com>
To: "Bruno L. Giordano" <bruno.giordano at music.mcgill.ca>
Cc: <r-help at stat.math.ethz.ch>
Sent: Monday, June 19, 2006 10:52 PM
Subject: Re: [R] Effect size in mixed models

>   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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