> 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
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.
Your original comment
> > 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
> 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].
> *****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 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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