 

Craig A Faulhaber wrote:

 

>I am interested in using a generalized linear mixed model with data 

> that best fits a beta distribution (i.e., the data is bounded between 

> 0 and 1 but is not binomial).

..

>For clarification, here's what I'm trying to model:

>I have a beta-distributed response variable (y).  I have a fixed-effect


>explanatory variable (treatment), and I'd like to include a random term


>for individuals used in the experiment.  The model in lmer would be:  y


>~ treatment + (1 | individual).   As far as I can tell, the appropriate


>link function for the model would be the logit.

 

If you want to use a GLM you could use the binomial/logit
quasi-likelihood approach for your ratio. Say the ratio is r=n/N then
use binomial n with binomial total N (these do not have to be integers)
but remember to use prior weights of 1/N and estimate the
over-dispersion parameter. If you use the ratio, r, directly with a
binomial total of 1 then the prior weights are simply 1 and can be
ignored. This quasi-likelihood approach for a ratio was given by
Wedderburn (1974) (see McCullagh and Nelder, 1989, Sec 9.2.4). BTW
random effects with a beta distribution included in the linear predictor
via a link function such as the logit can be fitted as a HGLM
(Hierarchical Generalized Linear Model)(Lee and Nelder, 1996, 2001) for
binomial data (i.e. considered binomial conditional on the random
effects). Only the GenStat package is set up to fit HGLMs (as far as I
know). (L & N, 1996, J.R.Statist.Soc B 58, 619-678; L & N 2001
Biometrika 88, 987-1006).

 

Hope this helps

Steve Candy

 


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