1 / 10

Modelling non-independent random effects in multilevel models

Modelling non-independent random effects in multilevel models. Harvey Goldstein and William Browne University of Bristol NCRM LEMMA 3. The standard multilevel model. The usual 2-level (variance components) model as used e.g. for studying school effects can be written as:

dian
Télécharger la présentation

Modelling non-independent random effects in multilevel models

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.

More Related