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

2. The standard multilevel model Response Covariate School residual Student residual

3. Lack of residual independence

4. Estimation and extension

5. Examples: 1. longitudinal exam results for schools Data are GCSE results for 54 schools, 29,506 students for 3 years (2004-2006) 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 Intercept0.0150.027 Year 2 -0.0430.020 Year 3 -0.0040.019 Pretest0.7190.004 Level 1 variance0.4670.004 Level 2 covariance matrix: standard errors in brackets DIC (PD)61399.8 (128.2)

6. longitudinal exam results for schools ParameterEstimateStandard error Intercept0.0170.026 Year 2-0.0410.017 Year 3-0.0020.018 Pretest0.7190.004 2.1440.612 Level 1 variance0.4670.004 Level 2 covariance matrix DIC (PD)61401.0 (126.4)

7. Examples: 2. child growth data A sample of 9 repeated height measures on 21 boys aged 11- 14 every approx. 3 months. Age centered on 12.25 years Autocorrelation structure is Results on next slide

8. ParameterEstimate (SE) Intercept148.9 Age6.16 (0.35) 2.16 (0.47) 0.39 (0.16) -1.55 (0.45) Level 2 covariance mtx Intcpt65.9 Age0.60 3.0 0.123 0.65 0.64 Level 1 variance0.21 (0.03) -0.20 (0.20) DIC (PD)344.0 (58.1) Fourth order polynomial. Burnin = 1000. Iterations=10,000 Correlations off-diagonal

9. 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

10. 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, 453-473.