[R-sig-ME] ICC estimates for overdispersed proportion or count data

[Sophie Picq]

Dear all,

I am trying to compute repeatability estimates for proportion/count data
 (intra-class correlation coefficients ICC) following Nakagawa & Schielzeth
(2010) on agressive mimicry data from reef fish.

My data: I observed 19 tagged fish directly on a reef, at repeated times
over 4 months (between 5 to 12 times each). Each observation period was 45
minutes long. The response variable is proportion of time spent following
(tracking) their model species. This has a known denominator/maximum
possible value corresponding to the total time of the observation period
(45mns).

I am dealing with overdispersion: my data has a lot of 0s and is skewed to
lower values, the max being 0,15 (Some fish just don't show that behaviour;
the ones who do don't do it a lot). I started using a binomial GLMM with
either additive or multiplicative overdispersion on this tracking
proportion using fish ID as the random effect in the rptR package of
Nakagawa & Schielzeth. My estimated overdispersion parameter using the
multiplicative overdispersion model is 71 (using glmmPQL and logit link,
rpt.binomGILMM.multi)

This makes me think that using a beta-binomial model would be more correct
to deal with this overdispersion, but I can't find any information on ICC
computations for beta-binomial GLMMs.

My questions are:

1. First, how do I know whether using a multiplicative or additive
overdispersion model is more correct? I get really different repeatability
results using either (really low with multiplicative, and around 0,5 with
additive). I've gone through Browne et al (2005) but still don't get how to
choose which one makes more sense.

2. Is there any way to compute ICC for a beta-binomial model and would that
be the way to go?

3. Since all total observation times were 45 minutes, I could also just use
the count data (total number of seconds tracking per observation) instead
of proportion of time spent tracking  (I have 2 observation events that had
to be shorter than 45 mns due to field work problems which I would have to
throw out...)

If that is more correct, I could use a Poisson GLMM. I did that using the
rptR package, but I still deal with a lot of overdispersion.  The estimated
overdispersion using the multiplicative model is 70. But the repeatability
estimates are actually really close using either additive or multiplicative
(around 0.5).

4.Given this overdispersion, is it more correct to compute ICC form a
negative binomial GLMM? I've seen 2 discussions on this topic but don't
know the current state:

http://stats.stackexchange.com/questions/166699/how-compute-the-intra-class-correlation-for-a-negative-binomial-mixed-model-in-l/169722#169722

and

https://github.com/lme4/lme4/issues/329

Please feel free to ask me if this was not clear enough...

I would really appreciate any help!

Thank you so much,

Sophie Picq

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