Semiparametric regression for clustered data

We consider estimation in a semiparametric partially generalised linear model for clustered data using estimating equations. A marginal model is assumed where the mean of the outcome variable depends on some covariates parametrically and a cluster-level covariate nonparametrically. A profile-kernel method allowing for working correlation matrices is developed. We show that the nonparametric part of the model can be estimated using standard nonparametric methods, including smoothing-parameter estimation, and the parametric part of the model can be estimated in a profile fashion. The asymptotic distributions of the parameter estimators are derived, and the optimal estimators of both the nonparametric and parametric parts are shown to be obtained when the working correlation matrix equals the actual correlation matrix. The asymptotic covariance matrix of the parameter estimator is consistently estimated by the sandwich estimator. We show that the semiparametric efficient score takes on a simple form and our profile-kernel method is semiparametric efficient. The results for the case where the nonparametric part of the model is an observation-level covariate are noted to be dramatically different.