Robust modified profile estimating function with application to the generalized estimating equation

We consider methods for reducing the effects of fitting nuisance parameters on a general estimating function, when the estimating function depends on not only a vector of parameters of interest, θ, but also a vector of nuisance parameters, λ. We propose a class of modified profile estimating functions with plug-in bias reduced by two orders. A robust version of the adjustment term does not require any information about the probability mechanism beyond that required by the original estimating function. An important application of this method is bias correction for the generalized estimating equation in analyzing stratified longitudinal data, where the stratum-specific intercepts are considered as fixed nuisance parameters, the dependence of the expected outcome on the covariates is of interest, and the intracluster correlation structure is unspecified. Furthermore, when the quasi-scores for θ and λ are available, we propose an additional multiplicative adjustment term such that the modified profile estimating function is approximately information unbiased. This multiplicative adjustment term can serve as an optimal weight in the analysis of stratified studies. A brief simulation study shows that the proposed method considerably reduces the impact of the nuisance parameters.