# [R-sig-ME] Dispersion parameter in glmmadmb and model selection

Paul Johnson paul.johnson at glasgow.ac.uk
Tue Mar 10 16:25:45 CET 2015

```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
 403.43
> log(fit\$alpha)
 6.000003

On 10 Mar 2015, at 01:39, Zahwa Al Ayyash (Student) <zsa11 at mail.aub.edu> wrote:

> 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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>
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```