Josh,

For the multiple-degree-of-freedom effects whose sizes you are interested in summarizing, I guess the appropriate effect size would have to be some sort of R-square-like statistic. Unfortunately, there is no widely agreed upon R-square-like effect size statistic for mixed models in general. But there are plenty of suggestions. See the linked section in the FAQ below for more info:

http://glmm.wikidot.com/faq#toc15

The following recent paper reviews several of the past suggestions for R-square-like statistics for mixed models and then the authors advance one of their own:

http://onlinelibrary.wiley.com/doi/10.1111/j.2041-210x.2012.00261.x/abstract

I guess you should just pick whichever one seems most sensible to you and be sure to cite its origin.

Jake

> To: r-sig-mixed-models@r-project.org
> From: bbolker@gmail.com
> Date: Mon, 14 Oct 2013 01:37:42 +0000
> Subject: Re: [R-sig-ME] effect sizes in lmer
> 
> Joshua Hartshorne <jkhartshorne@...> writes:
> 
> > 
> > Hi Henrik,
> > 
> > I'm not sure if I follow your reply. Using fixef will give me 24 effects,
> > including the intercept. An ANOVA would give 7. What I need to report are
> > the 7 ANOVA-style effects, not the 24 me-style ones.
> > 
> > To make this more concrete, I have 4 types of stimuli, two different types
> > of tests, and 3 conditions. What the readers are going to want to know is
> > whether there is an omnibus interaction. Fixef reports 6 omnibus
> > interactions -- one for every level of the interaction. 
> >I will also need to
> > report the lower-level interactions and main effects. (Doesn't matter
> > whether these are truly interpretable in the face of a significant
> > higher-order interaction: It's standard practice to report them.)
> 
>   I sympathize with this point of view but it also worries me ("oh,
> this doesn't make sense, but in my field we do it anyway ...")
>  
> > I can measure the significance of the ANOVA-style omnibus interaction by
> > using model comparison. But that doesn't give me an effect size exactly.
> > (One suggestion I heard recently was to use the change in AIC as an effect
> > size.)
> 
>   I would go ahead and use -2*log-likelihood difference (deviance 
> difference). This is
> asymptotically chi-squared distributed, and is the analogue of the
> F statistic in an ANOVA table, so if the F-statistic is what you 
> define as "the effect size" in an ANOVA context then by extension
> it makes sense to use the log-likelihood diff.
> 
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