Bayesian and likelihood methods for fitting multilevel models with complex level-1 variation

Browne, W. J., Draper, D., Goldstein, H. and Rasbash, J.
Computational Statistics and Data Analysis, 39:2, 203-225

In multilevel modelling it is common practice to assume constant variance at level 1 across individuals. In this paper we consider situations where the level-1 variance depends on predictor variables. We examine two cases using a dataset from educational research; in the first case the variance at level 1 of a test score depends on a continuous “intake score” predictor, and in the second case the variance is assumed to differ according to gender. We contrast two maximum-likelihood methods based on iterative generalised least squares with two Markov chain Monte Carlo (MCMC) methods based on adaptive hybrid versions of the Metropolis-Hastings (MH) algorithm, and we use two simulation experiments to compare these four methods. We find that all four approaches have good repeated-sampling behaviour in the classes of models we simulate. We conclude by contrasting raw- and log-scale formulations of the level-1 variance function, and we find that adaptive MH sampling is considerably more efficient than adaptive rejection sampling when the heteroscedasticity is modelled polynomially on the log scale.

Number of levels
Model data structure
Response types
Multivariate response model?
Longitudinal data?
Further model keywords
Substantive discipline
Substantive keywords

Paper describes MCMC algorithms for complex level 1 variation

Paper submitted by
William Browne, Bristol Veterinary School, University of Bristol,
Edit this page