In this article, we discuss the effect of removing the independence assumptions between the residuals in two-level random effect models. We first consider removing the independence between the Level 2 residuals and instead assume that the vector of all residuals at the cluster level follows a general multivariate normal distribution. We demonstrate how this assumption can allow us to fit higher levels of clustering and school competition effects via an example from education. We then consider removing the assumption of independence between Level 1 residuals within clusters. We show how this extension can allow time series type models. Both normal and binary responses are considered.
BesagJ.KooperbergC. L. (1995). On conditional and intrinsic autoregressions. Biometrika, 82, 733–746.
2.
BesagJ.YorkJ.MollieA. (1991). Bayesian image restoration with two applications in spatial statistics (with discussion). Annals of the Institute of Statistical Mathematics, 43, 1–59.
3.
BrowneW. J. (2006). MCMC algorithms for constrained variance matrices. Computational Statistics and Data Analysis, 50, 1655–1677.
4.
BrowneW. J.DraperD. (2000). Implementation and performance issues in the Bayesian and likelihood fitting of multilevel models. Computational Statistics, 15, 391–420.
5.
ClaytonD. G.KaldorJ. (1987). Empirical Bayes estimates of age standardised relative risks for use in disease mapping. Biometrics, 43, 671–681.
6.
GoldsteinH. (2003). Multilevel statistical models
(3rd ed.).
London: Edward Arnold.
7.
GoldsteinH.BurgessS.McConnellB. (2007). Modelling the effect of pupil mobility on school differences in educational achievement. Journal of the Royal Statistical Society, Series A, 170, 941–954.
GoldsteinH.HealyM. J. R.RasbashJ. (1994). Multilevel time series models with applications to repeated measures data. Statistics in Medicine, 13, 1643–1655.
RasbashJ.BrowneW. J.HealyM.CameronB.CharltonC. (2000). The MLwiN software package version 1.10.
London: Institute of Education, University of London.
12.
RasbashJ.SteeleF.BrowneW. J.ProsserB. (2004). A user’s guide to MLwiN.
London: Institute of Education, University of London.
13.
SpiegelhalterD. J.BestN. G.CarlinB. P.van der LindeA. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society, Series B, 64, 583–640.
