# Computing repeatability from overdispersed zero-inflated negative binomial GLMMM in R

I'm trying to compute repeatability of a count response variable from a Generalized linear mixed model with multiple fixed effects and individual ID as a random effect. I'm dealing with both overdispersion and zero-inflation, and am therefore looking to use a negative binomial distribution and zero-inflated GLMM, which has lead me to use the glmmADMB package. My response variable is the number of times that a subdominant male reindeer is chased by a dominant male reindeer, used as a proxy for what I'm calling "boldness" for the time being (i.e. number of times that the subdominant male instigated aggression from dominant male). I am interested to see if this behaviour can be examined in the animal personality paradigm, and therefore am trying to calculate repeatability.

glmm.nb.zi <- glmmadmb(Times.chased.dis ~ factor(Year) + Obs.time + Age + Day_num +
ASR + Weight.adj + DomAg ,
random =~1|ID, data=data.r, zeroInflation=TRUE, family ="nbinom")

Family: nbinom
alpha = 1.9336
Zero inflation: p = 0.18252

Fixed effects:
Log-likelihood: -924.442
AIC: 1874.884
Formula: Times.chased.dis ~ factor(Year) + Obs.time + Age + Day_num + ASR +
(Intercept) factor(Year)2010 factor(Year)2013 factor(Year)2014         Obs.time
2.14602382       0.32409301       1.11688488       1.05475592       0.01505949
0.05685858      -0.07570367      -1.96255801      -1.76393450       1.41894334

Random effects:
Structure: Diagonal matrix
Group=ID
Variance StdDev
(Intercept)  0.03242   0.18

Number of observations: total=683, ID=46


I'm having trouble calculating repeatability as I cannot figure out the total variance associated with the random effect (specifically I don't know how to find the residual variance). Does anyone know how to calculate repeatability in this situation?

Would it be easier if I accounted for overdispersion by including an observation level random effect and allowed it to assume a Poisson distribution?