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Model Robust Regression Based on Generalized Estimating Equations

Many studies produce dependent data. For example, repeated measures experiments produce repeated measurements on independent subjects and longitudinal studies generate datasets with several observations collected over time. These data have a clustered correlated structure. The clusters are defined as groups of observations that are correlated, but are uncorrelated with observations in other clusters. Generalized Estimating Equations (GEEs) (Zeger and Liang, 1986) are estimating functions used to estimate mean models for cluster correlated data when only the mean model and the variance function (of the mean) have been specified. They are often appropriate for modeling the population mean of counts, binary, and continuous responses when the correlation structure is a nuisance rather than the focus. The variance function of the mean can be taken from a distribution in the exponential family, though it need not be, since GEE makes no distributional assumptions beyond the first two moments. Like many other statistical models, the GEE mean model and variance function may be misspecified. Robustification against variance function misspecification, especially correlation structure misspecification, is done through a semiparametric “sandwich” variance estimation of the standard errors. Robustification of the mean function is possible through a kernel-based localization of GEEs, which makes weak assumptions about the functional form of the mean function. The work presented here concerns a situation that lies in-between the two extremes of knowing the mean model entirely and not knowing it at all. We imagine that a model builder specifies a parametric model that he or she believes may only be partly correct. Then hybridization of global (parametric) and local (nonparametric) models is used to formulate a semiparametric mean model that draws on the strengths of global and local regression for estimation in this setting. We seek then to robustify the GEE mean model specification using local GEEs. The ideas presented here are an extension of the works on model robust regression (MRR) by Mays, Birch, and Starnes (2001) for the linear model, and model robust regression for quantal regression by Nottingham and Birch (2000). This article is organized as follows. In section 2 we review parametric, nonparametric and semiparametric GEEs and discuss bandwidth selection for local GEEs. In section 3 we review MRR, propose extensions of MRR to GEEs, and give cross-validation and optimal mixing parameter estimators. An example is presented in section 4. A discussion follows in section 5.