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
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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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