[R-sig-ME] Variance components in a binomial mixed model

[Ben Bolker]

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On 02/15/2011 12:42 PM, Nicholas Lewin-Koh wrote:
> Howdy,
> I have a question sort of related to Ben's "differing variances within
> different random effects levels".
> I have a data set consisting of 23 mice, 11 Treated and 12 controls. The
> mice are bred to have
> a neurodengerative disease, and their neuromuscular connections degrade over
> time. So the animals were sacrificed and
> a left and right leg muscle were removed and sectioned, fixed on slides and
> stained appropriately. There were 12 slides
> per a muscle, and 6 sections per slide. Two slides per side were chosen, and
> in all sections on the slide the total number of neuro-muscular
> junctions, and the number innervated was counted. Oh and within each muscle,
> there were 5 compartments, so counts are by muscle compartment.
> So I have a binomial response, and for fixed effects I have Sex +
> Treatment*Musclecompartment
> and random (1|Animal/Side/Slide/Section) So the model is
> 
> gm1 <- glmer(cbind(Innervated.count,Total.count-Innervated.count) ~ Sex +
> Treatment*Muscle.Compartment+(1|animal.ID/Side/Slide/Section), data=dat,
> family = binomial)
> 
> So my questions are about interpretation of the variance components in a
> binomial model.
> 1) the variance component for slide is 0, is that because there are only 2
> slides
>      or is something else going on, there is an estimated variance for side,
> and when I last
>      counted there were only 2 of those as well :)

   It basically means that you have so little power to detect variance
among slides within side that your best estimate is zero; the total
observed variation among slides is not much bigger than that expected
from the variation at lower levels (sections within slides).

> 2) do the variances/covariances have a similar interpretation to the
> Gaussian case
>      for which in an lmm everything is easier to understand? Meaning that
> the error term is
>     binomal, glm part, but the random effects are gaussian, so i am looking
> at the variances
>     from a mixture model, or is that just the integrated variance over all
> levels.
> 

  Since you are using a random effects on the intercept only, the
variances are variances of (assumed) Gaussian random variation in the
logit probability (since you are using the default logit link) across
groups at particular grouping levels.

  This is a small data set, be very careful with the p-values (Wald or
LRT)!  If you are happy on the bleeding edge, you can try the parametric
bootstrapping examples that I committed to the r-forge repository.  (It
looks like the Linux builds might be a bit out of date -- if you install
the package and don't get a result for help("simulate-mer"), let me know
...)

   cheers
    Ben Bolker
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