Re: GLMMs

# Re: GLMMs

Dear Kim,
>
> Dear Arthur,
>
> could you please provide a reference on the methodology used for your
> Binomial GLMM? I've just read a paper written by Engel et al (1995)
> discussing your Gilmour et al. (1985) procedure, but aren't sure if this is
> what you use in ASREML? I can't find reference to your exact procedure in
> the manual, but I guess by now you will not be using your early approach....?
>
The apptoach used in ASREML is that of Schall\item
Schall, R. (1991).
Estimation in generalized linear models with random effects.
{\it Biometrika,} {\bf 78}, 719-727.

It is also what I call Joint expectation in my Biometrika 85 paper.

I have not implemented the marginal method which is the main focus
of my Biometrika paper, in ASREML.

> I have also read the various answers regarding aspects of running GLMMs
> (on-line and in the manual), and have a fairly general understanding of
> what's going on (I think). However, I am a bit confused as to exactly what
> information the Deviance value is telling you (and why you calculate it
> twice per iteration used to estimate the VC) and how this is related to
> dispersion. This probably just reflects my appalling understanding of GLMMs
> in general, but would appreciate it if you could help out with a laymans
> description of what exactly is going on (and I can read your reference
> material at leisure until I better understand it :-)).
>
The quantity reported as 'Deviance' is a measure of lack of fit
from the binomial distribution and is defined in table 4.1 of the current
ASREML manual.  i.e. Sigma 2n(y ln(y/m) + (1-y) ln ((1-y)/(1-m)))
where y is the observed binomial proportion for the sample
m is the predicted mean binomial proportion for the sample
n is the sample size

It is calculated twice because ASREML does 2 iterations of
prediction under the linear model for each update of the variance parameters.

A GLM book will explain the Deviance to you and the iteration
procedure which involves using a working variable.  i.e.
we analysys  Y = XB + (y-m)/ dm  where  XB is the predicted value
from the previous iteration, which is used to calculate m and dm which
is the derivative of m with respect to Y (i.e. change from the y-m scale
to the Y scale.

In the mixed model case, the Deviance does not capture all the likelihood
and I do not know how to get the rest easily.  The LogL value
reports the likelihood for the working variable (assuming normallity)
and so contains the extra bit but contains other stuff as well).

Hence the solution from ASREML is not at a maximum of the LogL value
or of the Deviance value.

Basically the Schall method takes the well behaved GLM method and
extends it by adding random effects in the linear model.  It is
then no longer so well behaved.

> For example, it is my understanding that when your variance heterogenity
> factor (deviance/DF) is approximately 1 (as occurs under and animal model)
> there is no need to allow for overdispersion because it doesn't exist when
> cluster size=1 (ie don't re-run using !disp). Further, that the deviance is
> supposed to represent the goodness of fit of replacing y values with fitted
> values. I would have thought this also equated with maximising the
> information contained in the working variables for parameter estimation,
> but guess it doesn't account for the larger measurement error associated
> with the small cluster size (ie an animal) compared to larger cluster sizes
> (eg. a sire)? Hence, even though the deviance is smallest and disp is
> approx. 1 under an animal model, this in no way tells you that the estimate
> of additive variance (for example) is going to be the best of alternative
> models. In fact, the additive variance under an animal model is strongly
> biased downwards - not very helpful if you are wanting to fit an
> can be picked up by additional random effects which have inherently high
> sampling correlations with the additive effects (eg maternal). ie resulting
> in spurious maternal effects for example. It is also not possible to
> (my simulations often show the incorrect direction of change under more
> parameters). Is this something to do with  not being able to use deviance
> as an appropriate measure of fit when cluster size=1 (although cluster size
> presumably varies for the different effects: ie., = 1 for animal effects
> but is large for sire effects)?
>
The downwards bias is related to sample size, the smaller the sample the bigger
the bias
so the animal model is the worst case.
It arises because the actual data con only take two values 0 ot 1
and we use a first order approximation to an underlying scale.

> Have I got the right understanding of what deviance, variance heterogenity
> and dispersion mean in the context of interpreting output from ASREML? Is
> the deviance adjust a scaled deviance - or is your variance heterogenity
> (VH) value a scaled deviance (D/disp)? What if the implied dispersion is
> incorrect? It seems to me that after running with !disp (sire model), the
> error variance was altered but the VH value was pretty much the same anyway
> (ie, no improvement in fit - just rescaling?). Is this always the case with
> using !disp, and if so, why bother rescaling weights if the fit of the
> model is not improved?
>

In the GLM case, the usual estimate of dispersion is Deviance/df

So in ASREML you can fit the model, estimate dispersion this way
and refit the model nominating the dispersion.

Alternatively, !DISP without a value estimates the dispersion as a variance
parameter in the linear model (of the working variable) which is
quite a different thing.

> Also, regarding models where we want to estimate maternal effects (for
> example), what are the best options for 0/1 data?
>
My feeling is that you are brave to attempt an anima;l model and to estimate
maternal effects would be foolhardy.  I do not know what is likely to happen
but I would wary of the results.  The machinery will run but I am
not convinced it is appropriate to the task.  I do not have any
other suggestions though.

Arthur
> Anyway, no doubt I have just displayed how little I understand about GLMMs,
>
> Hope you are enjoying your visit OS (or are you back now?)...
>
I'm back
> Cheers
>
> Kim
>
> Kim Bunter (M.Rur.Sc)
> PhD Student
> Animal Genetics and Breeding Unit
> University of New England
> Armidale, NSW, 2351
> AUSTRALIA
>
> Ph (ISD): -61-2-67733788
> Fax (ISD): -61-2-67733266
> email: kbunter@metz.une.edu.au
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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
Forest Rd, ORANGE, 2800, AUSTRALIA                  home: <61> 2 6362 0046

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