Dispersion parameter in glmmadmb and model selection

Hi Zahwa, 

In answer to the first question, the dispersion parameter, alpha, is inversely proportional to the amount of additional variance due to overdispersion...

Y ~ Poisson(lambda)
Var(Y) = lambda

Y ~ NB(lambda, alpha)
Var(Y) = lambda + lambda^2 / alpha

...so your error distribution appears not to be very overdispersed. In fact, if probably isn?t overdispersed at all, as I think glmmadmb puts an upper limit on the alpha estimate of exp(6) = 403.43, presumably to prevent it wondering off towards infinity when there is no evidence of overdispersion. This would explain why you get the same estimate from different (non-overdispersed) data sets. Try simulating some Poisson data and fitting an NB model (see code below). 

I guess for low lambda, e.g. around 10, lambda^2 / 403 will be reasonable approximation of zero addition variance, but not for higher lambda, e.g. > 100. Not sure how glmmadmb copes with that, or if it?s possible to raise this ceiling.

Best wishes,
Paul

set.seed(1234);
y <- rpois(100, 5)
dummy.group <- factor(rep(1:10, each = 10))
fit <- glmmadmb(y ~ (1 | dummy.group), family = "nbinom2")
summary(fit)

Call:
glmmadmb(formula = y ~ (1 | dummy.group), family = "nbinom2")

AIC: 422.6 

Coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)    1.520      0.047    32.3   <2e-16 ***
---
Signif. codes:  0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1

Number of observations: total=100, dummy.group=10 
Random effect variance(s):
Group=dummy.group
             Variance    StdDev
(Intercept) 2.696e-08 0.0001642

Negative binomial dispersion parameter: 403.43 (std. err.: 0.72454)

Log-likelihood: -208.295

fit$alpha

[1] 403.43

log(fit$alpha)

[1] 6.000003

Dear list,

1) I am fitting a negative binomial and zero-inflated negative binomial models using glmmadmb. I am getting a very high dispersion parameter:

Negative binomial dispersion parameter: 403.43 (std. err.: 0.39244)

I am aware that my data might be over-dispersed, but what does the very high value indicate? Could there be an error in estimation?
Also, surprisingly, I am using two different data sets to estimate the neg. bin. models, and I am getting the same value (403.43) but with different std. errors. Any clues?

2) My second question is rather general; What could be the best ways to compare glmmadmb models and select the best amongst Poisson, Neg. Bin, Zero-inflated Poisson, Zero-inflated Neg. Bin., Hurdle Poisson and Hurdle Neg. Bin?

PS: My models employ a random effect to capture correlation among individuals (IDs).

Many thanks to your help,
Zahwa Al-Ayyash






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