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

  • Authors:

  • William J. Browne;David Draper;Harvey Goldstein;Jon Rasbash

  • Affiliations:

  • Institue of Education, University of London, 20 Bedford Way, London, UK;Department of Applied Mathematics and Statistics, Baskin School of Engineering, University of California, 1156 High Street, Santa Cruz, CA;Institue of Education, University of London, 20 Bedford Way, London, UK;Institue of Education, University of London, 20 Bedford Way, London, UK

  • Venue:
  • Computational Statistics & Data Analysis
  • Year:
  • 2002

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Abstract

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.