Estimation in Marginal Models (GEE and Robust Estimation)

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

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

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

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

GEE minimize GEE equations Solution of the GEE equation

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

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

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

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

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

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