# Re: [R] overdispersion

From: Peter Dalgaard <p.dalgaard_at_biostat.ku.dk>
Date: Fri 12 Jan 2007 - 23:27:30 GMT

John Maindonald wrote:
> 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!
>
I don't really disagree, of course. I was mainly being provocative.

However, these models play tricks on our intuition. When people speak of overdispersion, they usually imply just what you said: independent data with the correct mean, but somehow a different variance - a mathematical impossibility for binary data.

One particular thing to notice is that if the individual means are heterogeneous but sampled independently from the same underlying distribution; you still end up with a marginal binomial distribution. If they are not sampled independently, then you get departures from the binomial, but it may well be in the direction of underdispersion. For an extreme case, take a sample of 50 men and 50 women and count the number of people with breasts.

(If you do the same thing with a random sample of 100 _people_, you get the binomial distribution again. Unless you're counting the number of breasts...)
> 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@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@stat.math.ethz.ch wrote:
>
>
>> From: Peter Dalgaard <p.dalgaard@biostat.ku.dk>
>> Date: 12 January 2007 5:04:26 AM
>> To: evaiannario <evaiannario@libero.it>
>> Cc: "r-help@stat.math.ethz.ch" <r-help@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.
>>
>
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