Modelling nonindependent random effects in multilevel models

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Modelling non-independent random effects in
multilevel models

• William Browne
• Harvey Goldstein
• University of Bristol

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A standard multilevel (VC) model

Fixed Random level 2 level 1
residual
The are assumed independent.

• But this is sometimes unrealistic
• Repeated measures growth models with closely
  spaced occasions
• Schools competing for resources in a zero-sum
  environment

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Repeated measures growth curves

• A simple model of linear growth with random
  slopes

A model for (non-independent) level 1 residuals
might be written
Leading to an exponential decay function.
(Goldstein and Healy 1994)
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Schools in competition

where this correlation is inversely proportional
to the (resource) distance between the schools
. If we can specify a suitable (set of )
distance functions then we can estimate the
relevant parameters. One possibility is to use
the extent of overlap between appropriately
defined catchment areas. Work using the ALSPAC
cohort is currently underway.
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Other link functions

Logit Log
These have the following forms
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Link functions
Link function f(s). From left to right
hyperbolic, logit, log
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Parameters and estimation

• We need to estimate the parameters of the
  correlation function, the variances and the fixed
  effects.
• We propose an MCMC algorithm and have programmed
  this for general 2- level models where
  correlations can exist at either or both levels
  and responses can be normal or binary.
• Steps are a mixture of Gibbs and MH sampling with
  adaptive proposal distributions and suitable
  diffuse priors

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Example 1 Growth data

• The data are 9 measurements on 20 boys around
  age 13, approximately 3 months apart
• Fitting a 2-level model with random linear and
  quadratic coefficients does not remove residual
  autocorrelation among level 1 residuals.
• We model the correlation as a negative
  exponentially decreasing function of the time
  difference
• We use a log (exponential) link since
  correlations should be positive
• In discrete time (equal intervals) this is a
  standard first order autoregressive model
• We fit a 4-th degree polynomial with and without
  random linear coefficient

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Results
random slope
Intercept only
For model A the correlation between measurements
0.25 years apart is 0.73 and for model B is 0.996.
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An equivalence

• For a 2-level variance components model the full
  covariance matrix among the level 1 units in a
  level 2 unit can be written in the form

where in this case there are 4 level 1 units. For
the model with an equal correlation structure at
level 1 and no level 2 variation the
corresponding covariance matrix is equivalent,
namely
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A level 2 example dependence based on distance
apart

• We have a three level model consisting of
  schools at level 3, cohorts or year groups at
  level 2 (2004,2005,2006) and students at level 1.
  The data are taken from the PLASC/NPD database
  response is GCSE score and predictors include 11
  year KS test score
• We fit as a 2-level model (school cohorts at
  level 2) with a correlation structure between
  cohorts within schools and dummies for years

So that the correlation is modelled by a constant
decreasing function of time difference
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RESULTS

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Conclusions

• These models provide a useful generalisation to
  standard independence models and are readily
  extended to non-normal responses, cross
  classifications etc.
• They allow us to more realistically describe the
  behaviour of institutions that are interactive
  rather than independently behaving units