[R] overdispersion

[John Maindonald]

I would say rather that for binary data (binomial data with n=1) it  
is not possible to detect overdispersion from examination of the  
Pearson chi-square or the deviance.   Overdispersion may be, and  
often is, nevertheless present.  I am arguing that overdispersion is  
properly regarded as a function of the variance-covariance structure,  
not as a function of the sample data.

The variance of a two-point distribution is a known function of the  
mean, providing that independence and identity of distribution can be  
assumed, or providing that the correlation structure is otherwise  
known and the mean is constant. That proviso is crucial!

If there is some sort of grouping, it may be appropriate to aggregate  
data over the groups, yielding data that have a binomial form with  
n>1.  Over-dispersion can now be detected from the Pearson chi-square  
or from the deviance.  Note that the quasi models assume that the  
multiplier for the binomial or other variance is constant with p;  
that may or may not be realistic.  Generalized linear mixed models  
make their own different assumptions about how the variance changes  
as a function of p; again these may or may not be realistic.

It is then the "error" structure that is crucial. To the extent that  
distracts from careful thinking about that structure, the term  
"overdispersion is unsatisfactory.

There's no obvious way that I can see to supply glm() with an  
estimate of the dispersion that has been derived independently of the  
current analysis.  Especially in the binary case, this would  
sometimes be useful.

John Maindonald             email: john.maindonald at anu.edu.au
phone : +61 2 (6125)3473    fax  : +61 2(6125)5549
Centre for Mathematics & Its Applications, Room 1194,
John Dedman Mathematical Sciences Building (Building 27)
Australian National University, Canberra ACT 0200.


On 12 Jan 2007, at 10:00 PM, r-help-request at stat.math.ethz.ch wrote:

> From: Peter Dalgaard <p.dalgaard at biostat.ku.dk>
> Date: 12 January 2007 5:04:26 AM
> To: evaiannario <evaiannario at libero.it>
> Cc: "r-help at stat.math.ethz.ch" <r-help at stat.math.ethz.ch>
> Subject: Re: [R] overdispersion
>
>
> evaiannario wrote:
>> How can I eliminate the overdispersion for binary data apart the  
>> use of the quasibinomial?
> There is no such thing as overdispersion for binary data. (The  
> variance of a two-point distribution is a known function of the  
> mean.) If what you want to do is include random effects of some  
> sort of grouping then you might look into generalized linear mixed  
> models via lmer() from the lme4 package or glmmPQL from MASS.