Hi,
There are many issues involved here. Some of it pertains to the use of
generalized linear mixed models in general. I hope the following clears
certain aspects up or at least points the discussion towards the main
issues.
Haja - many distributions (e.g. binomial and Poisson) depend on the mean.
Hence why some people recommend variance stablizing transformations.
However, I doubt that these are valid for GLMM's.
The potential lack of information in an animal model relative a sire model
is well known and not just limited to non-normal distributions. Failure
to get estimates from one method and not the other is not a problem of the
model but the algorithm and it's implementation. The frequentist and most
Bayesian approaches are approximations of varying degrees! Some of the
MCMC relates to improper posteriors. See Hoeschele and Tier (1995 Genet
Sel Evol 27:519) about MCMC 'blowing up'.
The basic definition of the sire model, the sire-dam model and the animal
model, assuming a pure additive model means that these are NOT linearly
equivalent! This is the major error that occurs in Mayer's 1995 paper
where it was concluded that the so-called equivalent sire and animals were
in fact not equivalent. This should of been clear since in order to make
linearly equivalent models, you need the same first two moments. That
is you have to account for the 3/4 of the additive variance in the sire
model otherwise it is overdispersed relative to the animal model. The
'residual' variance is the variance of some function - e.g. the
logistic or probit functions in the binary case. These are the same
'residual' variances used in these models so you must get different
answers unless there is no additive genetic variance.
In the sire-dam models, it was often observed that estimated genetic
variance was different between the sire and dam variances. Often
attributed to maternal effects, common environmental effects, dominance
effects etc., which indicate the initial assumptions do not hold. In addition
animal models also use all the genetic links that are present where the
other models assume independence between all sires and dams.
>
> But first of all, it is not quite clear to me, if animal models applied to
> discrete data lead to proper estimates?
>
What do you mean by 'proper'?
Under the Bayesian sense the posteriors should be (in theory),
implementation issues (such as approximations made) aside.
All Frequentist methods rely on some approximation and usually involve
maximizing a joint likelihood following Henderson et al. (1959) approach.
Others end with similar algorithms. Templeman (1998) reviewed
generalized linear mixed models in the Journal of Dairy Science and
discussed some of the issues involved. McCulloch and Feng (1996 Technical
report, Cornell University) showed that these methods have two undesirable
properties of inconsistency and lack of invariance under equivalent
specification of the statistical problem. They also show why the
joint-maximization works of the normal distribution with identity link and
hence, when it will not work for other approaches. This would agree with
Patrik's email.
With the options in ASREML, there is no excuse not to try potentially more
correct assumptions. My experience is that it very much depends on the
data, algorithm and software and the assumed model. Some of the failures
reported have been due to the incorrect use of the assumed distribution -
particularly associated with the failure to correctly account for possible
dispersion. It may also relate to inappropriate link functions. Just
because the data is binary or counts, it does not mean that binomial with
probit link or Poisson with log link, respectively, are the correct
distributional assumptions. It is clear that it is the amount of
information about the different parameters that is important rather than
the size of the data set.
Several authors, e.g. Hoeschele, Templeman, have shown that if the data is
binary or Poisson distributed, then methods based on these distributions
are superior to normality with identity link. I have also seen this with
real data.
Bruce
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