Re: Dispersion parameter

# Re: Dispersion parameter

```Hi,
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.

> Dear Bruce.
>
> The GLMM procedure defines a diagonal weight matrix based on the
> derivative of the link and the variance that applies to a family.
>
> The weight is just the inverse of the scaled variance.
>
> Adding the !DISP qualifier allows ASREML to estimate a scaling
> parameter to put with this weight matrix.
>
> In you example, the dispersion parameter S^2 is 0.166.
>
> For the purposes of discussion, let W = I.
>
> The total variance is the   0.166[1 + .042 + .092] = .166[1.134] = .188
>
> Without the !DISP qualifier, we are saying the total variance = 1 + a + b
> which forces a and b to zero [given we say they are +ve]

Basically, I am trying to put this into a context I can understand and use.
Is the following interpretation correct?

Under the assumed model without using the !DISP qualifier, there is
underdispersion because the variance is less than the mean.  Since a and b are
sources of overdispersion, they must be zero due to the relationship between
the mean and the variance.  With the !DISP qualifier this restriction between
the mean and the variance is removed.

In terms of estimating a heritability and repeatability, I would think that I
can use the definition of Foulley and Im (1993 GSE):
Heritability = a/(a+b+E[Y])
where E[Y] is the expectation of Y.  My reasoning is that the dispersion
factor is across the variance of Y.

However, if the dispersion factor is treated like a residual, then, following
the Normal case, a definition on the underlying scale would:
heritability =a/[a+b + S^2]

In either case, I am happy with the underlying scale as my overall aim is not
the estimation heritability.

Does anyone have any thoughts, concerns or other
options that they would like to share?  I would appeciate it very much.

Bruce

>
> 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].
>
> Arthur
>
> > Date: Wed, 12 Apr 2000 17:44:59 -0500
> > From: Bruce Southey <southey@uiuc.edu>
> > X-Accept-Language: en
> > MIME-Version: 1.0
> > To: arthur.gilmour@agric.nsw.gov.au, "asreml@chiswick.anprod.csiro.au"
> <asreml@chiswick.anprod.csiro.au>
> > Subject: Dispersion parameter
> >
> > Hi,
> > I am working with the fitting a Poisson repeatability model to ovulation
> > rate records.  The model is
> >
> > 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).
> >  Source                Model  terms     Gamma     Component    Comp/SE
> > % C
> >  animal                  854    854  0.258454E-02  0.258454E-02   0.16
> > -28 P
> >  perm                    358    358  0.888961E-07  0.888961E-07   0.00
> > -99 P
> >  Variance                937    921   1.00000       1.00000       0.00
> > 0 F
> >
> >
> > However, if I include the dispersion option (!DISP) I get reasonable
> > values (torn2.asr):
> >
> > Source                Model  terms     Gamma     Component    Comp/SE
> > % C
> >  animal                  854    854  0.421773E-01  0.702392E-02   1.38
> > 0 P
> >  perm                    358    358  0.921755E-01  0.153503E-01   2.98
> > 0 P
> >  Variance                937    921   1.00000      0.166533      17.95
> > 0 U
> >
> > I am hoping that would be able to explain this to me, especially the
> > intrepretation of the dispersion parameter in this context and providing
> > estimates of heritability and repeatability.
> >
> > Bruce
>
> <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>
> 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
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>
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