Models can be compared statistically using likelihood ratio tests provided that they fit the same fixed effects and their covariance parameters are nested. Nesting of covariance parameters occurs when the covariance parameters in the simpler model can be obtained by restricting some of the parameters in the more complex model (e.g. a compound symmetry pattern is nested within a Toeplitz pattern, but it is not nested within a first-order autoregressive pattern). The likelihood ratio test statistic is given by
where DF = difference in number of covariance parameters fitted.
If the covariance parameters in the models compared are not nested, statistical comparison using a likelihood ratio test is not valid. In this situation comparisons of each model with a simpler model which is nested within both models couldbe made and the model giving the most significant improvement selected. Alternatively, Akaike's or Schwarz's criteria defined above could be used.
Again, we anticipate that these tests will be approximately satisfactory when based on the quasi- or pseudo-likelihood statistics arising out of GLMMs and categorical mixed models, but are unaware of formal justification for this.
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