Modelling non-independent random effects in multilevel models

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 • The usual 2-level (variance components) model as used e.g. for studying school effects can be written as: • Residuals typically assumed independent. • Suppose, having adjusted for other factors, schools are in competition: Student residual School residual Response Covariate BUT

3. Lack of residual independence • What one school ‘gains’ another may ‘lose’ – resources, teachers etc. so residuals might be negatively correlated. • This correlation could be a function of ‘distance’ or other joint factors • So let’s assume • ensures • ensures • A possible distance function is

4. Estimation and extension • MCMC with suitable (typically diffuse) priors • MH updating for correlation function and level 2 variance, Gibbs for other parameters • We can extend to level 1 correlated residuals with a similar formulation: • This is useful for time series data such as in repeated measures data, where e.g. we could have: • =)

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: • Parameter Estimate Standard error • Intercept 0.015 0.027 • Year 2 -0.043 0.020 • Year 3 -0.004 0.019 • Pretest 0.719 0.004 • Level 1 variance 0.467 0.004 • Level 2 covariance matrix: standard errors in brackets • DIC (PD) 61399.8 (128.2)

6. longitudinal exam results for schools • Now we fit a simpler model with first order ‘autoregressive inverse tanh’ correlation function and note slightly reduced DIC: • =

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

8. 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.