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

[Ben Bolker]

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 + ....)







> 
> Cheers,
> 
> Torbjørn Ergon
> University of Oslo
> 
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