This Article Full Text (PDF) References Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to Saved Citations Download to citation manager Add to My Marked Citations Citing Articles Citing Articles via Google Scholar Citing Articles via Scopus Google Scholar Articles by Ng, E. S. Articles by Rasbash, J. Search for Related Content Social Bookmarking                         What's this?

Estimation in generalised linear mixed models with binary outcomes by simulated maximum likelihood

Centre for Multilevel Modelling, Graduate School of Education, University of Bristol, UK, e.ng{at}bristol.ac.uk

James R Carpenter

Medical Statistics Unit, London School of Hygiene and Tropical Medicine, University of London, UK

Harvey Goldstein

Centre for Multilevel Modelling, Graduate School of Education, University of Bristol, UK

Jon Rasbash

Centre for Multilevel Modelling, Graduate School of Education, University of Bristol, UK

Fitting multilevel models to discrete outcome data is problematic because the discrete distribution of the response variable implies an analytically intractable log-likelihood function. Among a number of approximate methods proposed, second-order penalised quasi-likelihood (PQL) is commonly used and is one of the most accurate. Unfortunately, even the second-order PQL approximation has been shown to produce estimates biased toward zero in certain circumstances. This bias can be marked especially when the data are sparse. One option to reduce this bias is to use Monte-Carlo simulation. A bootstrap bias correction method proposed by Kuk has been implemented in MLwiN. However, a similar technique based on the Robbins-Monro (RM) algorithm is potentially more efficient. An alternative is to use simulated maximum likelihood (SML), either alone or to refine estimates identified by other methods. In this article, we first compare bias correction using the RM algorithm, Kuk’s method and SML. We find that SML performs as efficiently as the other two methods and also yields standard errors of the bias-corrected parameter estimates and an estimate of the log-likelihood at the maximum, with which nested models can be compared. Secondly, using simulated and real data examples, we compare SML, second-order Laplace approximation (as implemented in HLM), Markov Chain Monte-Carlo (MCMC) (in MLwiN) and numerical integration using adaptive quadrature methods (in Stata’s GLLAMM and in SAS’s proc NLMIXED). We find that when the data are sparse, the second-order Laplace approximation produces markedly lower parameter estimates, whereas the MCMC method produces estimates that are noticeably higher than those from the SML and quadrature methods. Although proc NLMIXED is much faster than GLLAMM, it is not designed to fit models of more than two levels. SML produces parameter estimates and log-likelihoods very similar to those from quadrature methods. Further our SML approach extends to handle other link functions, discrete data distributions, non-normal random effects and higher-level models.

Key Words: Bias correction • Kuk’s method • Monte-Carlo integration • numerical integration • Robbins-Monro algorithm • simulated maximum likelihood

Statistical Modelling, Vol. 6, No. 1, 23-42 (2006)
DOI: 10.1191/1471082X06st106oa


CiteULike    Complore    Connotea    Del.icio.us    Digg    Reddit    Technorati    Twitter    What's this?