A comparison of Bayesian and likelihood-based methods for fitting multilevel models

Browne, W. J. and Draper, D.
Bayesian Analysis, 1:3, 473-514

We use simulation studies, whose design is realistic for educational and medical research (as well as other fields of inquiry), to compare Bayesian and likelihood-based methods for fitting variance-components (VC) and random-effects logistic regression (RELR) models. The likelihood (and approximate likelihood) approaches we examine are based on the methods most widely used in current ap- plied multilevel (hierarchical) analyses: maximum likelihood (ML) and restricted ML (REML) for Gaussian outcomes, and marginal and penalized quasi-likelihood (MQL and PQL) for Bernoulli outcomes. Our Bayesian methods use Markov chain Monte Carlo (MCMC) estimation, with adaptive hybrid Metropolis-Gibbs sampling for RELR models, and several diffuse prior distributions ( 1(, and U(0, 1 of point estimates and nominal versus actual coverage of interval estimates in re- peated sampling. In two-level VC models we find that (a) both likelihood-based and Bayesian approaches can be made to produce approximately unbiased esti- mates, although the automatic manner in which REML accomplishes this is an advantage, but (b) both approaches had difficulty achieving nominal coverage in small samples and with small values of the intraclass correlation. With the three- level RELR models we examine we find that (c) quasi-likelihood methods for esti- mating random-effects variances perform badly with respect to bias and coverage in the example we simulated, and (d) Bayesian diffuse-prior methods lead to well- calibrated point and interval RELR estimates. While it is yes that the likelihood- based methods we study are considerably faster computationally than MCMC, (i) steady improvements in recent years in both hardware speed and efficiency of Monte Carlo algorithms and (ii) the lack of calibration of likelihood-based methods in some common hierarchical settings combine to make MCMC-based Bayesian fit- ting of multilevel models an attractive approach, even with rather large data sets. Other analytic strategies based on less approximate likelihood methods are also possible but would benefit from further study of the type summarized here.

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Multivariate response model?
Longitudinal data?
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Highly cited paper that looks at the bias and coverage properties of methods for fitting random intercept and random effect logistic regression models

Paper submitted by
William Browne, Bristol Veterinary School, University of Bristol, william.browne@bristol.ac.uk
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