Modelling non-independent random effects in multilevel models Harvey Goldstein and William Browne University of Bristol NCRM LEMMA 3

The standard multilevel model Response Covariate School residual Student residual

Lack of residual independence

Estimation and extension

Examples: 1. longitudinal exam results for schools Data are GCSE results for 54 schools, 29,506 students for 3 years ( ) from NPD. First a ‘saturated’ model which is 2-level (school, student) where each school has 3 random effects, one for each year that are correlated: ParameterEstimateStandard error Intercept Year Year Pretest Level 1 variance Level 2 covariance matrix: standard errors in brackets DIC (PD) (128.2)

longitudinal exam results for schools ParameterEstimateStandard error Intercept Year Year Pretest Level 1 variance Level 2 covariance matrix DIC (PD) (126.4)

Examples: 2. child growth data A sample of 9 repeated height measures on 21 boys aged every approx. 3 months. Age centered on years Autocorrelation structure is Results on next slide

ParameterEstimate (SE) Intercept148.9 Age6.16 (0.35) 2.16 (0.47) 0.39 (0.16) (0.45) Level 2 covariance mtx Intcpt65.9 Age Level 1 variance0.21 (0.03) (0.20) DIC (PD)344.0 (58.1) Fourth order polynomial. Burnin = Iterations=10,000 Correlations off-diagonal

Further extensions Discrete, e.g. binary, response. Use latent normal (probit) model Multivariate models Allow level 1 variance to depend on explanatory variables Allow level 2 random effects in level 1 variance and correlation functions

Reference and acknowledgements Work supported by ESRC NCRM Reference: Browne, W. and Goldstein, H. (2010). MCMC sampling for a multilevel model with non-independent residuals within and between cluster units. J. of Educational and Behavioural Statistics, 35,