Estimation and prediction for spatial generalized linear mixed models using high order Laplace approximation

Evangelos Evangelou, Zhengyuan Zhu, Richard L Smith

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Estimation and prediction in generalized linear mixed models are often hampered by intractable high dimensional integrals. This paper provides a framework to solve this intractability, using asymptotic expansions when the number of random effects is large. To that end, we first derive a modified Laplace approximation when the number of random effects is increasing at a lower rate than the sample size. Second, we propose an approximate likelihood method based on the asymptotic expansion of the log-likelihood using the modified Laplace approximation which is maximized using a quasi- Newton algorithm. Finally, we define the second order plug-in predictive density based on a similar expansion to the plug-in predictive density and show that it is a normal density. Our simulations show that in comparison to other approximations, our method has better performance. Our methods are readily applied to non-Gaussian spatial data and as an example, the analysis of the rhizoctonia root rot data is presented.
Original languageEnglish
Pages (from-to)3564-3577
Number of pages14
JournalJournal of Statistical Planning and Inference
Issue number11
Early online date17 May 2011
Publication statusPublished - Nov 2011



  • spatial statistics
  • maximum likelihood estimation
  • predictive inference
  • Laplace approximation
  • generalized linear mixed models

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