Article |
Nonparametric additive regression for repeatedly measured data
Department of Statistics, Texas A&M University, College Station, Texas 77843, U.S.A. carroll{at}stat.tamu.edu
Department of Biostatistics, Harvard School of Public Health, 655 Huntington Avenue, Boston, Massachusetts 02115, U.S.A. amaity{at}hsph.harvard.edu
Department of Economics, University of Mannheim, L 7, 3-5, 68131 Mannheim, Germany emammen{at}rumms.uni-mannheim.de kyusangu{at}yahoo.co.kr
Received for publication 1 October 2007. Revision received 1 October 2008.
We develop an easily computed smooth backfitting algorithm for additive model fitting in repeated measures problems. Our methodology easily copes with various settings, such as when some covariates are the same over repeated response measurements. We allow for a working covariance matrix for the regression errors, showing that our method is most efficient when the correct covariance matrix is used. The component functions achieve the known asymptotic variance lower bound for the scalar argument case. Smooth backfitting also leads directly to design-independent biases in the local linear case. Simulations show our estimator has smaller variance than the usual kernel estimator. This is also illustrated by an example from nutritional epidemiology.
Key Words: Additive model • Generalized least square • Nonparametric regression • Repeated measure • Smooth backfitting
References
-
Bickel P., Klaassen A., Ritov Y., Wellner J. Efficient and Adaptive Estimation for Semiparametric Models. (1993) Baltimore, MD: Johns Hopkins University Press.
Chen K., Jin Z. Local polynomial regression analysis of clustered data. (2005) 92:59–74.
Fan J., Gijbels I. Local Polynomial Modeling and Its Applications (1996) London: Chapman and Hall.
Hoover D. R., Rice J. A., Wu C. O., Yang Y. Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data. (1998) 85:809–22.
Huggins R. Understanding nonparametric estimation for clustered data. (2006) 93:486–89.
Kipnis V., Subar A. F., Midthune D., Freedman L. S., Ballard-Barbash R., Troiano R., Bingham S., Schoeller D. A., Schatzkin A., Carroll R. J. The structure of dietary measurement error: results of the OPEN biomarker study. (2003) 158:14–21.
Lin X., Carroll R. J. Nonparametric function estimation for clustered data when the predictor is measured without/with error. (2000) 95:520–34.
Lin X., Carroll R. J. Semiparametric regression for clustered data using generalized estimating equations. (2001) 96:1045–56.
Lin X., Carroll R. J. Semiparametric estimation in general repeated measures problems. J. R. Statist. Soc. (2006) B 68:68–88.
Lin D. Y., Ying Z. Semiparametric and nonparametric regression analysis of longitudinal data (with discussion). (2001) 96:103–26.
Lin X., Wang N., Welsh A. H., Carroll R. J. Equivalent kernels of smoothing splines in nonparametric regression for longitudinal/clustered data. (2004) 91:177–94.
Linton O., Mammen E. Estimating semiparametric ARCH(
) models by kernel smoothing methods. Econometrica (2005) 73:771–836.[CrossRef][Web of Science]
Linton O., Mammen E. Nonparametric transformation to white noise. J. Economet. (2008) 142:241–64.[CrossRef]
Linton O. B., Mammen E., Lin X., Carroll R. J. Correlation and marginal longitudinal kernel nonparametric regression. In: Proceedings of the Second Seattle Symposium in Biostatistics: Analysis of Correlated Data—Lin D. Y., Heagerty P. J., eds. (2004) New York: Springer. 23–33.
Nielsen J., Linton O. An optimization interpretation of integration and back-fitting estimators for separable nonparametric models. J. R. Statist. Soc. (1998) B 60:217–22.[CrossRef]
Mammen E., Linton O., Nielsen J. The existence and asymptotic properties of a backfitting projection algorithm under weak conditions. (1999) 27:1443–90.
Mammen E., Nielsen J. P. Generalised structured models. (2003) 90:551–66.
Mammen E., Park B. U. Bandwidth selection for smooth backfitting in additive models. (2005) 33:1260–94.
Mammen E., Park B. U. A simple smooth backfitting method for additive models. (2006) 34:2252–71.
Nielsen J., Sperlich S. Smooth backfitting in practice. J. R. Statist. Soc. (2005) B 67:43–61.[CrossRef]
Ruckstuhl A., Welsh A. H., Carroll R. J. Nonparametric function estimation of the relationship between two repeatedly measured variables. Statist. Sinica (2000) 10:51–71.
Wang N. Marginal nonparametric kernel regression accounting for within-subject correlation. (2003) 90:43–52.
Willett W. C. Nutritional Epidemiology (1990) New York: Oxford University Press.
Yu K., Mammen E., Park B. Smooth backfitting in generalized additive models. (2008) 36:228–60.
CiteULike 
Connotea 
Del.icio.us What's this?
| ||||||||||||||||||||||||||||||||||||||||||||||||