1. Estimation in Marginal Models (GEE and Robust Estimation)

2. 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

3. 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

4. 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

5. Working covariance matrix
V is called the working covariance matrix to distinguish for the true underlying covariance of Y

6. GEE
minimize
GEE equations
Solution of the GEE equation

7. 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

8. Asymptotic distribution of GEE estimator
In large sample, the GEE estimator is multivariate normal
True covariance matrix

9. Sandwich estimate of bread meat Consistent estimate of the true
Covariance matrix of Y

10. Link to stata command xtgee for continuous data
substitute into GEE equations, got
xtgee,identity link, corr(exch)
Use Weighted Least Square for

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

12. 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