# Generalized Linear Mixed Models

## The issues

There is the capacity to fit a wider class of
models which include additional random effects for non-normal error
distributions. The inclusion of random terms in a GLM is usually
referred to as a Generalized Linear Mixed Model (GLMM). For GLMMs,
ASReml uses what is commonly referred to as penalized
quasi-likelihood or PQL (Breslow and Clayton, 1993). The technique is also
known by other names, including Schall's technique (Schall, 1991),
IRREML (Engel and Keen, 1994), pseudo-likelihood (Wolfinger and O'Connell, 1993) and
joint maximisation (Harville and Mee, 1984, Gilmour et al, 1985). It is implemented in many
statistical packages, for instance, in MLwiN
(Goldstein et al, 1998), the IRREML procedure
of Genstat (Keen, 1994), the GLMMIXED macro in SAS and in the GLMMPQL
function in R, to name a few.
The PQL technique is based on a first order Taylor series
approximation to the likelihood. It has been shown to perform poorly
for certain types of GLMMs. In particular, for binary GLMMs where
the number of random effects is large compared to the number of
observations, it can underestimate the variance components severely
(50\ 1995, Goldstein and Rasbash, 1996, Rodriguez and Goldman, 2001).
For other types of GLMMs, such as poisson data with many observations
per random effect, it has been reported to perform quite well (e.g.
Breslow, 2003).
As well as the above references, users can consult
McCulloch and Searle (2001) for more information about GLMMs.
Most studies investigating PQL have focussed on
estimation bias. Much less attention has been given to the wider
inferential issues such as hypothesis testing. In addition, the
performance of this technique has only been assessed on a small set
of relatively simple GLMMs. Anecdotal evidence from users suggests
that this technique can give very misleading results in certain
situations.
Therefore we cannot recommend the use of this technique for general
use. It is included in the current version of ASReml for advanced \warn
users. It is highly recommended that its use be accompanied by some
form of cross-validatory assessment for the specific dataset
concerned. For instance, one way of doing this would be by
simulating data using the same design and using parameter values
similar to the parameter estimates achieved, such as used in
Millar and Willis (1999).
The standard GLM Analysis of Deviance ( !AOD) should not be used
when there are random terms in the model
as the variance components are reestimated for each submodel.
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