[R] Underdispersion with anova testing methods

[ripley@stats.ox.ac.uk]

On Thu, 21 Mar 2002, Patrick Connolly wrote:

> Using anova of a glm with test = "Chisq", I get this:
>
> Analysis of Deviance Table
>
> Model: poisson, link: log
>
> Response: Days
>
> Terms added sequentially (first to last)
>
>
>                 Df Deviance Resid. Df Resid. Dev P(>|Chi|)
> NULL                              373     370.56
> Block            3    71.05       370     299.51 2.543e-15
> Variety          1    94.04       369     205.47 3.096e-22
> Instar           3   126.21       366      79.26 3.553e-27
> Variety:Instar   3     7.74       363      71.53      0.05
>
>
> Other interactions were much less prominent.  I'm only interested in
> the effect of Variety, and one would expect an effect of Instar, so I
> want to be able to block on Instar and Block.
>
> Evidently, the Chisquare test does not use an estimate of the
> dispersion parameter.  Were it included, the interaction would have a
> much smaller probability.  Is there a good reason why one should or
> should not makes such an adjustment?  In any case, will it matter when
> I'm interested only in the effect of Variety?

1) A poisson family does not, a quasipoisson family does. It's your
choice, but a Poisson distribution does not have a dispersion
parameter so you would be fitting a quasi-model.

2) It's not valid to look at terms in a sequential anova table like this,
as they all interact (and especially at main effects with an interaction
present).  Use drop1().

> Alternatively, could the underdispersion come from my ignoring the
> fact that the insects are measured at the four different instars and
> so the independence assumption is not true.  I could not think of a way
> of taking that lack of independence into account.

Are these small counts?  If so, the residual deviance is a badly biased
estimate of the dispersion parameter, so there might not be
under-dispersion at all.  See the V&R on-line Statistical Complements
for examples.

-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272860 (secr)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595

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