Re: Dispersion parameter

Re: Dispersion parameter

```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]

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.
>
> Thanks in advance,
> 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
Forest Rd, ORANGE, 2800, AUSTRALIA                  home: <61> 2 6362 0046

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