[R-sig-ME] Parametrization of the beta-binomial model in glmmadmb {glmmADMB}

[Torbjørn Ergon]

On 22.12.2013 00:32, Ben Bolker wrote:
> On 13-12-18 01:28 PM, Torbjørn Ergon wrote:
>> Dear list,
>>
>> I want to do some simulation based goodness-of-fit assessment of a
>> (zero-inflated) beta-binomial model fitted with glmmadmb, but I realize
>> I don't quite understand how this model is parameterized.
>>
>> I understand that the linear predictor is
>> inverse-logit(alfa/(alpha+beta)) where alpha and beta are the parameters
>> of the beta-distribution (right?). But what is the exact definition of
>> "Beta-binomial dispersion parameter" (fit.glmmadmb$alpha) expressed in
>> terms of the parameters of the beta distribution (alpha and beta)?
>
>      The ultimate answer is in the glmmadmb.tpl file included in the package:
>
>   system.file("tpl","glmmadmb.tpl",package="glmmADMB")
>
>
>      case 9: // beta-binomial
> 	Ni = sum(y(_i));
> 	tmpl = log_comb(Ni,y(_i,1)) + // log(C(Ni,y(_i,1)))
>                 gammln(y(_i,2)+alpha*(1-lambda))+
>                 gammln(y(_i,1)+alpha*lambda)-
>                 gammln(Ni+alpha) + // lbeta(...)
> 	    -(gammln(alpha*(1-lambda))+
>                  gammln(alpha*lambda)-gammln(alpha)); // lbeta(...)
>
>    Have you looked at the Morris, W. 1997. American Naturalist
> 150:299-327 reference given?  This is also discussed in Bolker 2008:
>
> Morris (1997) suggests a different parameterization that
> uses an overdispersion parameter θ , like the k parameter of the
> negative binomial distribution. In this case the parameters are N, the
> per-trial probability p (= a/(a + b)), and θ (= a + b).
>
>    θ in Bolker 2008 corresponds to "alpha" here.
>    p in Bolker 2008 corresponds to "lambda" here, and is the
> inverse-logit / logistic transformation of the linear predictor (beta_0
> + beta_1*x_1 + ....)
>
>

Thanks!

It is confusing that the help file (?glmmadmb version 0.7.7) uses alpha 
in the text (as in the glmmadmb object) but theta in the expression of 
the density function.

I think it would help a lot if you, after the first sentence referring 
to Morris, wrote something like "p = a/(a+b) and alpha = a+b, where a 
and b are the two parameters of the beta-distribution. Hence, the 
variance of the beta-binomial distribution is 
V_{B}(N)*(alpha+N)/(alpha+1) where V_{B}(N) is the in the binomial 
variance Np(1-p)".

Cheers,

Torbjørn