In this paper, we propose a practical computational method to obtain the maximum likelihood estimates (MLE) for mixed models with non-normal random effects. By simply multiplying and dividing a standard normal density, we reformulate the likelihood conditional on the non-normal random effects to that conditional on the normal random effects. Gaussian quadrature technique, conveniently implemented in SAS Proc NLMIXED, can then be used to carry out the estimation process. Our method substantially reduces computational time, while yielding similar estimates to the probability integral transformation method (J. Comput. Graphical Stat. 2006; 15:39-57). Furthermore, our method can be applied to more general situations, e.g. finite mixture random effects or correlated random effects from Clayton copula. Simulations and applications are presented to illustrate our method.

Original languageEnglish
Pages (from-to)3105-3124
Number of pages20
JournalStatistics in medicine
Issue number16
StatePublished - Jul 20 2008


  • Gamma frailty model
  • Gaussian copula
  • Generalized linear mixed model
  • Heterogeneity model
  • Logistic distribution


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