1. Multilevel Multivariate Models with responses at several levels Harvey Goldstein Centre for Multilevel Modelling University of Bristol

2. Some examples of Multiple Response Level (MRL) models A level 2 explanatory variable correlated with level 2 random effects – add second equation with variable as response to avoid bias (+ identifiability constraints) A growth curve model with repeated measures level 1 & adult height level 2 – used for prediction system (Goldstein, 2003). Multiple imputation for missing data with variables at several levels & all variables treated as responses.

3. Linear growth curve model See Henderson et al, 2002 for related example with survival data

4. Imputation for missing data Multiple (random) imputation for multilevel data treats all variables as responses, then: –Multivariate multilevel model with intercepts (or additional predictors) –Random sampling of p sets of residuals to form p complete data sets (via MCMC) –P models fitted and combined See www.missingdata.org.ukwww.missingdata.org.uk

5. Two issues: 1.Response variables for imputation are at several levels – hence MRL models 2.Responses are not all Normal – need to handle ordered and unordered categorical variables: a.Replace these variables with a set of multivariate (latent) Normal variates b.Fit multivariate Normal response model c.Randomly impute missing data on Normal scale d.Convert imputed values to corresponding categories on original scales e.Fit original complete data model etc.

6. Latent Normal variables 1. Ordered categories Suppose we have a p-category response, numbered 1,….p. Consider the probit link proportional odds model Note that we condition on other (adjusted) responses in model are thresholds – reduces to familiar binary Probit model when p=2

7. MCMC steps 1.For use a MH step with suitable (Normal) proposal (more efficient than Albert & Chibbs Gibbs procedure) 2. For a category p response we sample from the standard Normal distribution 3.Leaves us to sample from multilevel multivariate Normal response model (see below).

8. Multicategory Responses The multiple indicant model (Aitchison & Bennett, 1970): consider the multinomial vector with p categories, where the response, y is (0,1) in each category. We select a sample from v distribution and if maximum corresponds to category with a 1 then accept, otherwise choose another sample.

9. Multiple Normal responses at different levels: MCMC steps A 2-level model can be written (i indexes response, j,k levels) The level 2 residuals from level 2 response are obtained by subtraction The level 2 residuals for level 1 responses are conditioned on level 2 residuals for level 2 responses.

10. are multivariate Normal with structured covariance matrix (e.g. zero correlations for p-category variables). Sampled using MH element by element. Level 2 random effects and fixed effects are sampled using Gibbs as in standard MVN case.

11. Missing data data example Class size model with missing data for all variables except class size: total sample size 6611. Listwise deletion of missing data yields 4786 cases. 10 imputations every 500 iterations, 2000 burnin. CoeffListwise deleteNormal ModelLatent Normal model Intercept 0.047 (0.043)-0.010 (0.050)-0.010 (0.045) Prior Maths 0.612 (0.011)0.623 (0.011)0.622 (0.011) Girl -0.042 (0.020)-0.041 (0.019)-0.037 (0.019) FSM -0.144 (0.030)-0.140 (0.029)-0.136 (0.032) Class size -0.008 (0.008)-0.014 (0.009)-0.012 (0.008)

12. References Aitchison, J. and Bennett, J. A. (1970). Polychotomous quantal response by maximum indicant. Biometrika 57: 253-262. Albert, J. H. and Chibb, S. (1993). Bayesian analysis of binary and polychotomous response data. Journal of the American Statistical Association. 88: 669-679 Blatchford, P., Goldstein, H., Martin, C. and Browne, W. (2002). A study of class size effects in English school reception year classes. British Educational Research Journal 28: 169-185. (download from www.mlwin.com/hgpersonal) www.mlwin.com/hgpersonal Goldstein, H. (2003). Multilevel Statistical Models. Third edition. London, Edward Arnold: