# [R-sig-ME] glmmTMB: phi in betabinomial dispersion model

John Maindonald john@m@|ndon@|d @end|ng |rom @nu@edu@@u
Thu Jan 14 09:30:25 CET 2021

```This relates to the betabinomial error family.

Note that, in the way that the model is formulated in glmmTMB,
the binomial is a limiting case of the betabinomial as phi goes to
infinity, not a special case.  This seems to me somewhat unfortunate.
Was the parameterization used chosen in preference to other possibilities
for any particular reason?  The sigma parameter in the gamlss implementation
is 1/phi, with a log link as the default, where phi is the betabinomial sigma
parameter.  I am not sure what the limiting lower value of intra-class correlation
rho is, if the link is set as “identity”, but in theory it might do mildly negative.

Is there any particular reason for the glmmTMB parameterization?

The theory, if the sigma parameter is suitable parameterized, allows the
intra-class correlation rho to go negative.  See:
Prentice, RL. 1986. “Binary Regression Using an Extended Beta-Binomial Distribution,
with Discussion of Correlation Induced by Covariate Measurement Errors.” Journal of
the American Statistical Association 81 (394): 321–27.
Does anyone know of a readily usable implementation of that more general
error model?

John Maindonald             email: john.maindonald using anu.edu.au<mailto:john.maindonald using anu.edu.au>

If you specify  "dispformula=~1”, the output is the same as when a dispformula is not
specified.  Certainly, that is the case for models with a betabinomial error.  One can
extract phi as “phi=sigma(obj)

If you specify
a dispformula with >1 coefficients, then calculate either

coef(HCbb.cll)[['disp’]]  ## Gives the coefficients
or
coef(summary(HCbb.cll))[['disp’]]  ## Adds SE, etc, information

In both cases a logarithmic link function is used.

The following is a function that I have found useful:

getRho <-
function (obj)
{
mm <- model.matrix(obj\$modelInfo\$allForm\$dispformula, data = obj\$frame)
fixdisp <- fixef(obj)[["disp"]]
1/(1 + exp(mm %*% fixdisp))
}

If you just want phi, rather than the intra-class correlation rho (which makes more
intuitive sense to me), just set “phi =  exp(mm %*% fixdisp)”.  I think this works
in just the same way with nbinom2 errors, but I have not checked the details.

John Maindonald             email: john.maindonald using anu.edu.au<mailto:john.maindonald using anu.edu.au>

On 13/01/2021, at 21:16, matthias.suter--- via R-sig-mixed-models <r-sig-mixed-models using r-project.org<mailto:r-sig-mixed-models using r-project.org>> wrote:

When fitting a negative binomial model in glmmTMB() using "family=nbinom2", but without specifying the dispersion model, the summary() gives an estimate for the dispersion parameter as "Overdispersion parameter for nbinom2 family (): XX", which represents - as far I understand - the phi value described in the manual (also termed theta in other environments).

If a dispersion model is specified with "dispformula=~ ...", the summary() gives the estimates for the "Dispersion model". How do these relate to the phi value? I read in the manual that "phi=exp(eta) (where eta is the linear predictor from the dispersion model)". Does this indicate that these estimates of the Dispersion model appear on the log scale, and to get phi, they must be back-transformed?

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