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*To*: bsouthey@bigfoot.com*Subject*: Re: Dispersion parameter*From*: Arthur Gilmour <gilmoua@apollo.agric.nsw.gov.au>*Date*: Fri, 14 Apr 2000 09:53:04 +1000 (EST)*Cc*: asreml@chiswick.anprod.csiro.au, robin.thompson@bbsrc.ac.uk*Reply-To*: Arthur Gilmour <gilmoua@apollo.agric.nsw.gov.au>*Sender*: asreml-owner@lamb.chiswick.anprod.csiro.au

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. 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 > 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 ASREML is still free by anonymous ftp from pub/aar on ftp.res.bbsrc.ac.uk or point your web browser at ftp://ftp.res.bbsrc.ac.uk/pub/aar/ To join the asreml discussion list, send the message subscribe mailto:asreml-request@chiswick.anprod.CSIRO.au To send messages to the list, mailto:asreml@chiswick.anprod.CSIRO.au Asreml list archive: http://www.chiswick.anprod.csiro.au/lists/asreml <> <> <> <> <> <> <> "Christ Jesus came into the world to save sinners" I Timothy 1:15. <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><> -- Asreml mailinglist archive: http://www.chiswick.anprod.csiro.au/lists/asreml

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