Estimation in Marginal Models (GEE and Robust Estimation)

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Estimation in Marginal Models (GEE and Robust
Estimation)
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GEE

• Since there is no convenient specification of the
  joint multivariate distribution of Y for
  marginal models when the responses are discrete,
  we require an alternative to MLE
• GEE is based on the concept of estimating
  equations and provides a very general approach
  for analyzing correlated responses that can be
  discrete or continuous

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GEE

• The essential idea behind GEE is to generalize
  and extend the usual likelihood equations for a
  GLM with a univariate response by incorporating
  the covariance matrix of the vector of responses
  Y
• For the case of linear models, the GLS estimator
  (also called Generalized Least Square estimator)
  for the vector of regression coefficients is a
  special case of the GEE approach

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What we need to specify for implementing GEE
Model for the mean
Known variance function
Working correlation matrix model for the
pariwise correlations among the responses
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Working covariance matrix
V is called the working covariance matrix to
distinguish for the true underlying covariance of
Y
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GEE
minimize
GEE equations
Solution of the GEE equation
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Properties of GEE estimates

• The GEE estimator is consistent whether or not
  the within subject associations/correlations have
  been correctly modelled
• That is, for GEE estimator to provide a valid
  estimate of the true beta, we only require that
  the model for the mean response has been
  correctly specified

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Asymptotic distribution of GEE estimator

• In large sample, the GEE estimator is
  multivariate normal

True covariance matrix
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Sandwich estimate of
bread
meat
Consistent estimate of the true Covariance
matrix of Y
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Link to stata command xtgee for continuous data

• substitute into GEE equations, got
• xtgee,identity link, corr(exch)
• Use Weighted Least Square for

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• xtgee, identity link, corr(exch), robust
• Use Sandwich Estimator for

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Link to stata commands xtgee for binary data

• Substitute into GEE equation, but no closed-form
  solution, need iteration.
• Difference between using robust or not analogous
  to continuous data
• xtgee,logit link, corr(exch)
• xtgee, logit link, corr(exch), robust