Generalized Linear Mixed Models

The issues

This section was written by Damian Collins

A Generalized Linear Mixed Model (GLMM) is an extension of a GLM to include random terms in the linear predictor. Inference concerning GLMMs is impeded by the lack of a closed form expression for the likelihood. ASReml currently uses an approximate likelihood technique called penalized quasi-likelihood, or PQL (Breslow and Clayton, 1993), which is based on a first order Taylor series approximation. This technique is also known as Schall�s technique (Schall, 1991), pseudo-likelihood (Wolfinger and O�Connell, 1993) and joint maximisation (Harville and Mee, 1984, Gilmour etal, 1985). Implementations of PQL are found in many statistical packages, for instance, in the GLMM (Welham, 2005) and the IRREML procedures of Genstat (Keen, 1994), the MLwiN package (Goldstein ηl, 1998), the GLMMIX macro in SAS (Wolfinger, 1994), and in the GLMMPQL function in R.

The PQL technique is well-known to suffer from estimation biases for some types of GLMMs. For grouped binary data with small group sizes, estimation biases can be over 50% (e.g. Breslow and Lin, 1995, Goldstein and Rasbash, 1996, Rodriguez and Goldman, 2001, Waddington et al, 1994). For other GLMMs, PQL has been reported to perform adequately (e.g. Breslow, 2003). McCulloch and Searle (2001) also discuss the use of PQL for GLMMs.

The performance of PQL in other respects, such as for hypothesis testing, has received much less attention, and most studies into PQL have examined only relatively simple GLMMs. Anecdotal evidence suggests that this technique may give misleading results in certain situations. Therefore we cannot recommend the use of this technique for general use, and it is included in the current version of ASReml for advanced users. If this technique is used, we recommend the use of cross-validatory assessment, such as applying PQL to simulated data from the same design (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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