The likelihood and quasilikelihood functions

As in normal mixed models a popular way of fitting the GLMM is based on maximising the likelihood function for the model parameters. However, a difficulty with this is that true likelihood functions can only be defined for random effects and random coefficients models. A true likelihood function is not available for covariance pattern models since a general multivariate distributional form does not exist for non-normal data (for normal data the multivariate normal distribution was used). However, we will show how it is possible to get around this difficulty by defining an alternative function known as the quasi-likelihood function, which has very similar properties to the likelihood function. In this section we will specify the likelihood function for random effects and random coefficients models, define a quasi-likelihood function for covariance pattern models, and then give a general form of the quasi-likelihood function that is appropriate for all types of mixed model.

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