Bayesian and likelihood methods for fitting multilevel models with complex level-1 variation. (English) Zbl 1132.62312
Summary: 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.
MSC:
| 62F15 | Bayesian inference |
| 65C40 | Numerical analysis or methods applied to Markov chains |
| 62P25 | Applications of statistics to social sciences |
Keywords:
Adaptive Metropolis-Hastings sampling; Educational data; Heteroscedasticity; Hierarchical modelling; IGLS; Markov chain Monte Carlo (MCMC); Maximum-likelihood methods; MCMC efficiency; Multilevel modelling; RIGLSReferences:
| [1] | Best, N.G., Cowles, M.K., Vines, S.K., 1995. CODA Manual version 0.30. MRC Biostatistics Unit, Cambridge, UK. |
| [2] | Browne, W.J., 1998. Applying MCMC Methods to Multilevel Models. Ph.D. dissertation, Department of Mathematical Sciences, University of Bath, UK. |
| [3] | Browne, W. J.; Draper, D.: Implementation and performance issues in the Bayesian and likelihood Fitting of multilevel models. Comput. statist. 15, 391-420 (2000) · Zbl 1037.62013 |
| [4] | Browne, W.J., Draper, D., 2001. A comparison of Bayesian and likelihood methods for fitting multilevel models. Under review. · Zbl 1331.62125 |
| [5] | Bryk, A. S.; Raudenbush, S. W.: Hierarchical linear models: applications and data analysis methods.. (1992) |
| [6] | Bryk, A.S., Raudenbush, S.W., Seltzer, M., Congdon, R., 1988. An Introduction to HLM: Computer Program and User’s Guide, 2nd Edition. University of Chicago, Department of Education, Chicago. |
| [7] | Dempster, A.; Laird, N.; Rubin, D.: Maximum likelihood from incomplete data via the EM algorithm (with discussion). J. royal statist. Soc. ser. B 39, 1-88 (1977) · Zbl 0364.62022 |
| [8] | Draper, D., 2002. Bayesian Hierarchical Modeling. Springer, New York, forthcoming. |
| [9] | Gelfand, A. E.; Ghosh, S. K.: Model choice: a minimum posterior predictive loss approach. Biometrika 85, 1-11 (1998) · Zbl 0904.62036 |
| [10] | Gelman, A.; Carlin, J. B.; Stern, H. S.; Rubin, D. B.: Bayesian data analysis.. (1995) · Zbl 1279.62004 |
| [11] | Gelman, A.; Roberts, G. O.; Gilks, W. R.: Efficient metropolis jumping rules.. Bayesian statistics 5, 599-607 (1995) |
| [12] | Gilks, W. R.; Wild, P.: Adaptive rejection sampling for Gibbs sampling. J. royal statist. Soc. ser. C 41, 337-348 (1992) · Zbl 0825.62407 |
| [13] | Goldstein, H.: Multilevel mixed linear model analysis using iterative generalised least squares. Biometrika 73, 43-56 (1986) · Zbl 0587.62143 |
| [14] | Goldstein, H.: Restricted unbiased iterative generalised least squares estimation. Biometrika 76, 622-623 (1989) · Zbl 0677.62064 |
| [15] | Goldstein, H.: Multilevel statistical models. (1995) · Zbl 1014.62126 |
| [16] | Goldstein, H.; Rasbash, J.; Yang, M.; Woodhouse, G.: A multilevel analysis of school examination results. Oxford rev. Education 19, 425-433 (1993) |
| [17] | Raftery, A. E.; Lewis, S. M.: How many iterations in the Gibbs sampler?. Bayesian statistics 4, 763-773 (1992) |
| [18] | Rasbash, J., Browne, W.J., Goldstein, H., Yang, M., Plewis, I., Healy, M., Woodhouse, G., Draper, D., Langford, I., Lewis, T., 2000. A User’s Guide to MLwiN (Version 2.1). Institute of Education, University of London, London. |
| [19] | Spiegelhalter, D.J., Thomas, A., Best, N.G., Gilks, W.R., 1996. BUGS 0.5 Examples (Version ii). Medical Research Council Biostatistics Unit, Cambridge. |
| [20] | Spiegelhalter, D.J., Thomas, A., Best, N.G., Gilks, W.R., 1997. BUGS: Bayesian Inference Using Gibbs Sampling (Version 0.60). Medical Research Council Biostatistics Unit, Cambridge. |
| [21] | Weisberg, S.: Applied linear regression. (1985) · Zbl 0646.62058 |
| [22] | Yang, M., Rasbash, J., Goldstein, H., Barbosa, M., 2000. MLwiN Macros for Advanced Multilevel Modelling (Version 2.0). Institute of Education, University of London, London. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
