Functional linear regression analysis for longitudinal data

By Fang Yao, H. Muller, Jane-ling Wang. 2005

769 citations recorded by the source corpus.

We propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data, where both the predictor and response are functions of a covariate such as time. Predictor and response processes have smooth random trajectories, and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects. In longitudinal studies, the number of repeated measurements per subject is often small and may be modeled as a discrete random number and, accordingly, only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit. We propose a functional regression approach for this situation, using functional principal component analysis, where we estimate the functional principal component scores through conditional expectations. This allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory. The resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories. Asymptotic properties for a sample of n subjects are investigated under mild conditions, as n → oo, and we obtain consistent estimation for the regression function. Besides convergence results for the components of functional linear regression, such as the regression parameter function, we construct asymptotic pointwise confidence bands for the predicted trajectories. A functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced, extending the standard R 2 to the functional case. The proposed methods are illustrated with a simulation study, longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index.

Subfields: Artificial Intelligence, Statistics and Probability

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Citations in the corpus

Local Linear Forests (2018, methodology): “Local linear regression has been particularly well-studied for longitudinal data, as in Li and Hsing (2010) and Yao, Muller, and Wang (2005).”
Shrinkage estimation and selection for multiple functional regression (2011, background): “nt the smoothness of the high dimensional covariates, and often suggests new approaches to the problems that have not been discovered before. Some recent developments in functional regression include Yao et al. (2005); Cai and Hall (2006); Crambes et al. (2009); Yuan and Cai (2010+). The literature ”
A Geometric Approach to Maximum Likelihood Estimation of the Functional Principal Components From Sparse Longitudinal Data (2007, methodology): “This assumption has been used in various studies, including Boente and Fraiman (2000), Cardot (2000), James, Hastie, and Sugar (2000), Yao, Müller, and Wang (2005, 2006), and Hall, Müller, and Wang (2006).”
Second-Order Comparison of Gaussian Random Functions and the Geometry of DNA Minicircles (2010, methodology): “(4) When the null hypothesis is true, we expect to have ϕ̂j,NX ≈ ϕ̂ j,N XY ; this essentially reduces PFCX(K) to the Gaussian pseudolikelihood-based Akaike information criterion (AIC) employed by Yao, Müller, and Wang (2005a) (see also Yao, Müller, and Wang 2005b).”
Varying-coefficient functional linear regression (2010, methodology): “Adopting the general framework of functional data analysis, we assume, for each z, that there exist orthogonal expansions of the covariance functions GX,z(·, ·) (resp.”
Functional Varying Coefficient Models for Longitudinal Data (2010, methodology): “While in functional linear models with both predictors and responses as random functions (Ramsay and Dalzell 1991; Yao, Müller, and Wang 2005b), it is assumed that past, present, and future values of the predictor process influence current response, only the current value of the predictor process…”
Detecting changes in the mean of functional observations (2009, methodology): “To name a few recent applications of FDA which illustrate its advantages alluded to above, we cite Antoniadis and Sapatinas (2003), Fernández de Castro et al. (2005), Müller and Stadtmüller (2005), Yao et al. (2005) and Glendinning and Fleet (2007).”
Determining the order of the functional autoregressive model (2013, background): “The last assumption is made in most contributions studying the functional linear model, see Yao et al. (2005a, 2005b), Müller and Stadtmüller (2005), Cai and Hall (2006), Li and Hsing (2007), Horvath et al. (2009), Mckeague and Sen (2010) and Li et al. (2010), to name only a handful of recent…”
A survey of functional principal component analysis (2014, neutral): “( 2006 ) , Yao et al .”