Hello,

I am calculating the repeatability of a random effect of individual in a
model using lmer and wanted to check that I'm understanding the residual
variance term correctly. I have seen similar questions previously posted,
but not quite answering my specific question clearly.

The outcome variable is a proportion (of two white blood cell types in a
sample), so I am using a binomial error distribution with logit link. I
have added an observation-level random effect to account for
overdispersion. A simplified version of the model is:

nl <- lmer (prop.test ~          # a proportion created from cbind (n, l)
              (1|obs.id)              # observation-level random effect
            +  (1|uid)                 # individual random effect
            + (1|year)                # random effect of year
            + date + age + sex  # fixed effects
            , data = NL.main
            , family = "binomial"
)

>From the Nakagawa & Shielzeth 2010 Biological Reviews paper on this topic,
the residual variance for an additive overdispersion model of this type
should be the variance of the residual term + (pi^2)/3. From a previous
thread on this mailing list, I read that the observation-level random
effect gives the residual variance term. So two main questions:

1. Is the residual variance then calculated as: (variance explained by
obs.id) + (pi^2) / 3 ?

2. If so, then the repeatability of individual random effect in my model is
very small (0.02681726), yet significant at p = 0.002875. Is it strange to
have such a significant random effect with such a small intraclass
correlation coefficient? I do have a very large sample size (863
observations on 334 individuals), but still didn't know if this outcome was
suspicious...

Many thanks in advance!
Tina

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