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

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 scoreâ€� 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.