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Nonparametric mixed effects models for unequally sampled noisy curves. (English) Zbl 1209.62061
Summary: We propose a method of analyzing collections of related curves in which the individual curves are modeled as spline functions with random coefficients. The method is applicable when the individual curves are sampled at variable and irregularly spaced points. This produces a low-rank, low-frequency approximation to the covariance structure, which can be estimated naturally by the EM algorithm. Smooth curves for individual trajectories are constructed as best linear unbiased predictor (BLUP) estimates, combining data from that individual and the entire collection. This framework leads naturally to methods for examining the effects of covariates on the shapes of the curves. We use model selection techniques — Akaike information criterion (AIC), Bayesian information criterion (BIC), and cross-validation — to select the number of breakpoints for the spline approximation. We believe that the methodology we propose provides a simple, flexible, and computationally efficient means of functional data analysis.

62G07 Density estimation
65C60 Computational problems in statistics (MSC2010)
fda (R)
Full Text: DOI
[1] Besse, Simultaneous non-parametric regressions of unbalanced longitudinal data, Computational Statistics and Data Analysis 24 pp 255– (1997) · Zbl 0900.62199
[2] Brumback, Smoothing spline models for the analysis of nested and crossed samples of curves, Journal of the American Statistical Association 93 pp 944– (1998) · Zbl 1064.62515
[3] Diggle, Nonparametric estimation of covariance structure in longitudinal data, Biometrics 54 pp 401– (1998) · Zbl 1058.62600
[4] Grenander, Abstract Inference (1981)
[5] Hoover, Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data, Biometrika 85 pp 809– (1998) · Zbl 0921.62045
[6] Kaslow, The Multicenter AIDS Cohort Study: Rationale, organization, and selected characteristics of participants, American Journal of Epidemiology 126 pp 310– (1987)
[7] Laird, Random-effects models for longitudinal data, Biometrics 38 pp 963– (1982) · Zbl 0512.62107
[8] Lin, Variance component testing in generalized linear models with random effects, Biometrika 84 pp 317– (1997) · Zbl 0881.62074
[9] Olshen, Gait analysis and the bootstrap, Annals of Statistics 17 pp 1419– (1989) · Zbl 0695.62244
[10] Ramsay, Functional Data Analysis (1997) · Zbl 0882.62002
[11] Rice, Estimating the mean and covariance structure nonparametrically when the data are curves, Journal of the Royal Statistical Society, Series B 53 pp 233– (1991) · Zbl 0800.62214
[12] Robinson, That BLUP is a good thing: The estimation of random effects, Statistical Science 6 pp 15– (1991) · Zbl 0955.62500
[13] Smith, Nonparametric regression using Bayesian variable selection, Journal of Econometrics 75 pp 317– (1996) · Zbl 0864.62025
[14] Staniswalis, Nonparametric regression analysis of longitudinal data, Journal of the American Statistical Association 93 pp 1403– (1998) · Zbl 1064.62522
[15] Stone, Polynomial splines and their tensor products in extended linear modeling, Annals of Statistics 25 pp 1371– (1997) · Zbl 0924.62036
[16] Zeger, Semi-parametric models for longitudinal data with applications to CD4 cell numbers in HIV seroconverters, Biometrics 50 pp 689– (1994) · Zbl 0821.62093
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