Multilevel Multivariate Models with responses at several levels

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Multilevel Multivariate Models with responses at
several levels

• Harvey Goldstein
• Centre for Multilevel Modelling
• University of Bristol

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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.

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

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Two issues

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

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Latent Normal variables1. Ordered categories

• Suppose we have a p-category response, numbered
  1,.p. Consider the probit link proportional
  odds model

are thresholds reduces to familiar
binary Probit model when p2
Note that we condition on other (adjusted)
responses in model
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MCMC steps

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

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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.

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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.
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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.

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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.

Coeff Listwise delete Normal Model Latent 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)
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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)
• Goldstein, H. (2003). Multilevel Statistical
  Models. Third edition. London, Edward Arnold