[R-sig-ME] INterpretation and use of parameters from beta glmmadmb - apology

[Ben Bolker]

Biuw, Martin <Martin.Biuw at ...> writes:

> Hi all, I¡¯m using glmmADMB to fit a mixed model with a beta
> famiily, and I would like some clarification on how to interpret the
> parameter estimates. Specifically, I would like to use the results
> as input into the R function ¡®dbeta¡¯, but I¡¯m not sure what in
> the glmmadmb output corresponds to the shape1 and shape2 parameters
> of dbeta. I¡¯m assuming the ¡®alpha¡¯, ¡®b¡¯ and ¡®stdbeta¡¯
> components are the ones to go for. But which is which?

> According to the dbeta help page, shape1 and shape2 corresponds to
> alpha and beta respectively in the beta density function:
> gamma(alpha+beta)/(gamma(alpha)gamma(beta))x^(alpha-1)(1-x)^(beta-1)
> But how do these correspond to the ¡®alpha¡¯ (dispersion parameter)
> and ¡®b¡¯ in the glmmadmb output? I get the most realistic results
> (when comparing to the distribution of my data) if I let the the
> ¡®alpha¡¯ corresponds to shape2 (i.e. ¥â in the above expression)
> and ¡®b¡¯ to shape1 (¥á in the expression). Maybe I¡¯m only being
> confused by the nomenclature? But if I¡¯ve understood the maths
> behind the beta distribution properly, when shape1 is greater than
> shape2 the distribution is left-skewed, and the dispersion is
> determined by the ratio shape2/shape1. And vice versa for a
> right-skewed distribution. So how can I determine which goes in as
> shape1 and shape2 when calculating my beta densities?

We need a parameterization that separates the mean and the
variance, for example as in the parameterization underlying
the beta-binomial in Morris (1997), American Naturalist 150:299-327
(not quite the same thing but a parallel situation).

Looking at the guts of the TPL (model definition) file we find:

file.show(system.file("tpl","glmmadmb.tpl",package="glmmADMB"))

  df1b2variable ln_beta_density(double y,const df1b2variable & mu,
    const df1b2variable& phi)
  {
    df1b2variable omega=mu*phi;
    df1b2variable tau=phi-mu*phi;
    df1b2variable lb=betaln(omega,tau);
    df1b2variable d=(omega-1)*log(y)+(tau-1)*log(1.0-y)-lb;
    return d;
  }

Where mu is the mean (as determined from a logit-linked
linear model) and phi is the dispersion parameter.  Thus
omega and tau are the shape parameters ...