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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