Iterated Reweighted Rank-Based Estimates for GEE Models

Repeated measurement designs occur in many areas of statistical research. In 1986, Liang and Zeger offered an elegant analysis of these problems based on a set of generalized estimating equations (GEEs) for regression parameters, that specify only the relationship between the marginal mean of the response variable and covariates. Their solution is based on iterated reweighted least squares fitting. In this paper, we propose a rank-based fitting procedure that only involves substituting a norm based on a score function for the Euclidean norm used by Liang and Zeger. Our subsequent fitting, while also an iterated reweighted least squares solution to GEEs, is robust to outliers in response space and the weights can easily be adapted for robustness in factor space. As with the fitting of Liang and Zeger, our rank-based fitting utilizes a working covariance matrix. We prove that our estimators of the regression coefficients are asymptotically normal. The results of a simulation study show that the our proposed estimators are empirically efficient and valid. We illustrate our analysis on a real data set drawn from a hierarchical (three-way nested) design.