Re: Dispersion parameter

# Re: Dispersion parameter

```Dear Bruce,

> I had understood the general concept of using the dispersion factor (S^2) in
> the Poisson model (as in McCullagh and Nelder's book) as:
> Var(Y) = S^2 E[Y]
> such that allowance can be made for under- or over-dispersion.
> However, from Schall's paper, this 'extra component of dispersion' appears to
> act like a residual on the transformed/underlying scale.

There are two approaches here - in the case of overdispersion at least.

If you know the within group variance you can use it to generate a weight.
If there is extra variation at the group level then it is natural to fit it
as a extra variance component.  The alternative is to scale the weight matrix.

So for a gven weight matrix W with mean value 10 [for the sake of argument]
we can say  Var(o) = W^-1  ~= .1I
Var(o) = W^-1 + aI  ~=  (a+0.1)I
or Var(o) = bW^-1 ~= 0.1bI = (a+0.1)I if b = 1+10a

If W is a scaled I matrix then the last two forms are exactly equal.

For underdispersion, 'a' would have a negative sign and it becomes
possible for some observations to have a negative variance
[when the known variance implicit in W is less than the magnitude of 'a';
which is far less acceptable than simply having a negative variance
component].  So for underdispersion, as your case, I prefer the third
form.

The grdc example in the manual [Sec 9.5] is an example of the second
form [for a Normal] where the scale is fixed.  An alternative
but less plausible analysis would be to float the !S2= parameter
and drop the 'units' term.

> > or !POI !LOG ~ mu breed year age !r animal perm
> >
> >
> > With Asreml, this eventually sends both animal and perm variance
> > components to zero (tornd.asr).
>

suggests there is no extra poisson variation in the data.  I.e.
the data is underdispersed before you start.

Robin Thompson has several times expressed unease with what I am
doing but I have not yet understood his problem.  I said
> NOTE that the dispersion factor estimated by ASREML is estimated
> as S^2 in the Normal REML and so is based on the working reiduals.
>
> Conventially, the dispersion parameter in GLM models has been
> estimated from the residual DEVIANCE [i.e. Deviance
> residuals]  ASREML does print this deviance dispersion value as a
>  heterogeneity factor.
>
>
> I trust this is clear.
>
> While ASREML lets you calculate the variance components as
> above[below],
> the interpretation of them is not as easily rationalised to my mind
> as the binomial case when we use the Normal or Logistic distributions
> for the implicit residual variance [threshold model].
He replied
> *****i cannot see how
> if we are scaling this underlying distribution
> then i cannot see how we can estimate the scale
> in both cases we are arguing that the assumed variance model may be
> out by a factor
> so we use s* mean or s*Npq

I do know know whether Robin's 'How?' refers to computation
or to the philosophical logic of the operation.  I'll see him
in a fortnight and hope to understand his meaning.

If there is serious over or under dispersion, it means the data
does not really support our assumption that it is say poisson.
I suppose it is an open question and a value judgement whether
to make the analysis 'fit' by scaling the variance, by adding another component,
or by assuming a different mean variance relationship.

In the GLM case, [single variance component], the only effect of
the scaling parameter is to change the 'standard errors' of the
fitted effects.  However, in the GLMM case, it allows the whole
structure of V=R+ZGZ' to change as demonstrated in this case.
So this will effect the fitted effects as well as their standard errors.

Thats enough rambling.

Arthur

<><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>
Arthur Gilmour PhD                  mailto:Arthur.Gilmour@agric.nsw.gov.au
Principal Research Scientist (Biometrics)            fax: <61> 2 6391 3899
NSW Agriculture                                           <61> 2 6391 3922
Orange Agricultural Institute             telephone work: <61> 2 6391 3815
Forest Rd, ORANGE, 2800, AUSTRALIA                  home: <61> 2 6362 0046

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