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Biometrika 2004 91(2):251-262; doi:10.1093/biomet/91.2.251
© 2004 by Biometrika Trust
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Profile-kernel versus backfitting in the partially linear models for longitudinal/clustered data

Zonghui Hu1, Naisyin Wang1 and Raymond J.Carroll1

1 Department of Statistics, Texas A&M University, College Station, Texas 77843-3143, U.S.Azhu{at}stat.tamu.edu nwang{at}stat.tamu.edu carroll{at}stat.tamu.edu

We study the profile-kernel and backfitting methods in partially linear models for clustered/longitudinal data.For independent data, despite the potential root-n inconsistency of the backfitting estimator noted by Rice (1986), the two estimators have the same asymptotic variance matrix, as shown by Opsomer & Ruppert (1999). In this paper, theoretical comparisons of the two estimators for multivariate responses are investigated. We show that, for correlated data, backfitting often produces a larger asymptotic variance than the profile-kernel method; that is, for clustered data, in addition to its bias problem, the backfitting estimator does not have the same asymptotic efficiency as the profile-kernel estimator. Consequently, the common practice of using the backfitting method to compute profile-kernel estimates is no longer advised. We illustrate this in detail by following Zeger & Diggle (1994) and Lin & Carroll (2001) with a working independence covariance structure for nonparametric estimation and a correlated covariance structure for parametric estimation. Numerical performance of the two estimators is investigated through a simulation study. Their application to an ophthalmology dataset is also described.

Key Words: Asymptotic distribution; Bandwidth; Kernel estimation; Local linear estimation; Smoother matrix; Under-smoothing

Received January 2003. Revised August 2003


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