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Biometrika 2005 92(1):59-74; doi:10.1093/biomet/92.1.59
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© 2004 Biometrika Trust

Local polynomial regression analysis of clustered data

Kani Chen1 and Zhezhen Jin2

1 Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong makchen{at}ust.hk, 2 Department of Biostatistics, Mailman School of Public Health, Columbia University, 722 West 168th Street, New York, NY 10032, U.S.A. zjin{at}biostat.columbia.edu

This paper proposes a classical weighted least squares type of local polynomial smoothing for the analysis of clustered data, with the key idea of using generalised inverses of correlation matrices. The estimator has a simple closed-form expression. Simplicity is achieved also for nonparametric generalised linear models with arbitrary link function via a transformation. Our approach can be characterised by ‘local observations with local variances’, which yields intuitively correct results in the sense that correct/incorrect specification of within-cluster correlation has respective positive/negative effects. The approach is a natural extension of classical local polynomial smoothing. Consequently, existing theory can be largely carried over and important issues such as bandwidth selection can be tackled in the classical fashion. Moreover, the approach can handle various types of covariate, such as cluster-level, subject-level or partially cluster-level. Numerical studies support the theoretical results. The method is illustrated with a real example on luteinising hormone levels in cows.

Key Words: Asymptotic bias; Bandwidth selection; Generalised estimating equation; Kernel function; Mean squared error; Nonparametric curve estimation

Received July 2003. Revised April 2004.


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