Biometrika Advance Access published online on October 29, 2008
Biometrika, doi:10.1093/biomet/asn041
Article |
On the asymptotics of marginal regression splines with longitudinal data
Department of Statistics, Fudan University, Shanghai 200433, China zhuzy{at}fudan.edu.cn
Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, China wingfung{at}hku.hk
Department of Statistics, University of Illinois at Urbana-Champaign, 725 South Wright Street, Champaign, Illinois 61820, U.S.A. x-he{at}uiuc.edu
Received for publication 1 March 2007. Revision received 1 April 2008.
There have been studies on how the asymptotic efficiency of a nonparametric function estimator depends on the handling of the within-cluster correlation when nonparametric regression models are used on longitudinal or cluster data. In particular, methods based on smoothing splines and local polynomial kernels exhibit different behaviour. We show that the generalized estimation equations based on weighted least squares regression splines for the nonparametric function have an interesting property: the asymptotic bias of the estimator does not depend on the working correlation matrix, but the asymptotic variance, and therefore the mean squared error, is minimized when the true correlation structure is specified. This property of the asymptotic bias distinguishes regression splines from smoothing splines.
Key Words: Asymptotic bias • B-spline • Generalized estimating equation • Generalized linear model • Least squares • Longitudinal data.
References
-
Agarwal G. G., Studden W. J. Asymptotic integrated mean square error using least squares and bias minimizing splines. Ann. Statist. (1980) 8:1307–25.[CrossRef]
Barrow L., Smith P. W. Asymptotic properties of best L2[0, 1] approximation by splines with variable knots. Quart. Appl. Math. (1978) 36:293–304.
Chen K., Jin Z. H. Local polynomial regression analysis of cluster data. Biometrika (2005) 92:59–74.
De Boor C. A bound on the L
-norm of L2-approximation by splines in terms of a global mesh ratio. Math. Comput. (1976) 30:765–71.[CrossRef]
He X., Zhu Z. Y., Fung W. K. Estimation in a semiparametric model for longitudinal data with unspecified dependence structure. Biometrika (2002) 89:579–90.
He X., Fung W. K., Zhu Z. Y. Robust estimation in generalized partial linear models for clustered data. J. Am. Statist. Assoc. (2005) 100:1176–84.[CrossRef][Web of Science]
Huang J. Z. Local asymptotics for polynomial spline regression. Ann. Statist. (2003) 31:1600–35.[CrossRef]
Huang J. Z., Zhang L., Zhou L. Efficient estimation in marginal partially linear models for longitudinal/clustered data using splines. Scand. J. Statist. (2007) 34:451–77.[CrossRef]
Huggins R. Understanding nonparametric estimation for clustered data. Biometrika (2006) 93:486–9.
Li Y., Ruppert D. On the asymptotics of penalized splines. Biometrika (2008) 95. to appear.
Liang K. Y., Zeger S. L. Longitudinal data analysis using generalized linear models. Biometrika (1986) 73:13–22.
Lin X., Carroll R. J. Nonparametric function estimation for clustered data when the predictor is measured without/with error. J. Am. Statist. Assoc. (2000) 95:520–34.[CrossRef][Web of Science]
Lin X., Wang N., Welsh A. H., Carroll R. J. Equivalent kernels of smoothing splines in nonparametric regression for clustered/longitudinal data. Biometrika (2004) 91:177–93.
Nychka D. Splines as local smoothers. Ann. Statist. (1995) 22:1175–97.
Schumaker L. L. Spline Functions: Basic Theory (1981) New York: Wiley.
Severini T. A., Staniswalis J. G. Quasi-likelihood estimation in semiparametric models. J. Am. Statist. Assoc. (1994) 89:501–11.[CrossRef][Web of Science]
Silverman B. W. Spline smoothing: the equivalent variable kernel method. Ann. Statist. (1984) 12:898–916.[CrossRef]
Wang N. Marginal nonparametric kernel regression accounting for within-subject correlation. Biometrika (2003) 90:43–52.
Wang N., Carroll R. J., Lin X. Efficient semiparametric marginal estimation for longitudinal/clustered data. J. Am. Statist. Assoc. (2005) 100:147–57.[CrossRef][Web of Science]
Welsh A. H., Lin X., Carroll R. J. Marginal longitudinal nonparametric regression: locality and efficiency of spline and kernel methods. J. Am. Statist. Assoc. (2002) 97:482–93.[CrossRef][Web of Science]
Zeger S. L., Diggle P. J. Semiparametric models for longitudinal data with application to CD4 cell numbers in HIV seroconverters. Biometrics (1994) 50:689–99.[CrossRef][Web of Science][Medline]
Zhang D., Lin X., Raz J., Sower M. F. Semiparametric stochastic mixed models for longitudinal data. J. Am. Statist. Assoc. (1998) 93:710–9.[CrossRef][Web of Science]
Zhou S., Shen X., Wolfe D. A. Local asymptotics for regression splines and confidence regions. Ann. Statist. (1998) 26:1760–82.[CrossRef]
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