1,302 research outputs found
Functional single index models for longitudinal data
A new single-index model that reflects the time-dynamic effects of the single
index is proposed for longitudinal and functional response data, possibly
measured with errors, for both longitudinal and time-invariant covariates. With
appropriate initial estimates of the parametric index, the proposed estimator
is shown to be -consistent and asymptotically normally distributed.
We also address the nonparametric estimation of regression functions and
provide estimates with optimal convergence rates. One advantage of the new
approach is that the same bandwidth is used to estimate both the nonparametric
mean function and the parameter in the index. The finite-sample performance for
the proposed procedure is studied numerically.Comment: Published in at http://dx.doi.org/10.1214/10-AOS845 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Modeling left-truncated and right-censored survival data with longitudinal covariates
There is a surge in medical follow-up studies that include longitudinal
covariates in the modeling of survival data. So far, the focus has been largely
on right-censored survival data. We consider survival data that are subject to
both left truncation and right censoring. Left truncation is well known to
produce biased sample. The sampling bias issue has been resolved in the
literature for the case which involves baseline or time-varying covariates that
are observable. The problem remains open, however, for the important case where
longitudinal covariates are present in survival models. A joint likelihood
approach has been shown in the literature to provide an effective way to
overcome those difficulties for right-censored data, but this approach faces
substantial additional challenges in the presence of left truncation. Here we
thus propose an alternative likelihood to overcome these difficulties and show
that the regression coefficient in the survival component can be estimated
unbiasedly and efficiently. Issues about the bias for the longitudinal
component are discussed. The new approach is illustrated numerically through
simulations and data from a multi-center AIDS cohort study.Comment: Published in at http://dx.doi.org/10.1214/12-AOS996 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Covariate adjusted functional principal components analysis for longitudinal data
Classical multivariate principal component analysis has been extended to
functional data and termed functional principal component analysis (FPCA). Most
existing FPCA approaches do not accommodate covariate information, and it is
the goal of this paper to develop two methods that do. In the first approach,
both the mean and covariance functions depend on the covariate and time
scale while in the second approach only the mean function depends on the
covariate . Both new approaches accommodate additional measurement errors
and functional data sampled at regular time grids as well as sparse
longitudinal data sampled at irregular time grids. The first approach to fully
adjust both the mean and covariance functions adapts more to the data but is
computationally more intensive than the approach to adjust the covariate
effects on the mean function only. We develop general asymptotic theory for
both approaches and compare their performance numerically through simulation
studies and a data set.Comment: Published in at http://dx.doi.org/10.1214/09-AOS742 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Basis Expansions for Functional Snippets
Estimation of mean and covariance functions is fundamental for functional
data analysis. While this topic has been studied extensively in the literature,
a key assumption is that there are enough data in the domain of interest to
estimate both the mean and covariance functions. In this paper, we investigate
mean and covariance estimation for functional snippets in which observations
from a subject are available only in an interval of length strictly (and often
much) shorter than the length of the whole interval of interest. For such a
sampling plan, no data is available for direct estimation of the off-diagonal
region of the covariance function. We tackle this challenge via a basis
representation of the covariance function. The proposed approach allows one to
consistently estimate an infinite-rank covariance function from functional
snippets. We establish the convergence rates for the proposed estimators and
illustrate their finite-sample performance via simulation studies and two data
applications.Comment: 51 pages, 10 figure
Functional linear regression analysis for longitudinal data
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 subjects are investigated under mild conditions,
as , 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 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.Comment: Published at http://dx.doi.org/10.1214/009053605000000660 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Properties of principal component methods for functional and longitudinal data analysis
The use of principal component methods to analyze functional data is
appropriate in a wide range of different settings. In studies of ``functional
data analysis,'' it has often been assumed that a sample of random functions is
observed precisely, in the continuum and without noise. While this has been the
traditional setting for functional data analysis, in the context of
longitudinal data analysis a random function typically represents a patient, or
subject, who is observed at only a small number of randomly distributed points,
with nonnegligible measurement error. Nevertheless, essentially the same
methods can be used in both these cases, as well as in the vast number of
settings that lie between them. How is performance affected by the sampling
plan? In this paper we answer that question. We show that if there is a sample
of functions, or subjects, then estimation of eigenvalues is a
semiparametric problem, with root- consistent estimators, even if only a few
observations are made of each function, and if each observation is encumbered
by noise. However, estimation of eigenfunctions becomes a nonparametric problem
when observations are sparse. The optimal convergence rates in this case are
those which pertain to more familiar function-estimation settings. We also
describe the effects of sampling at regularly spaced points, as opposed to
random points. In particular, it is shown that there are often advantages in
sampling randomly. However, even in the case of noisy data there is a threshold
sampling rate (depending on the number of functions treated) above which the
rate of sampling (either randomly or regularly) has negligible impact on
estimator performance, no matter whether eigenfunctions or eigenvectors are
being estimated.Comment: Published at http://dx.doi.org/10.1214/009053606000000272 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Inverse regression for longitudinal data
Sliced inverse regression (Duan and Li [Ann. Statist. 19 (1991) 505-530], Li
[J. Amer. Statist. Assoc. 86 (1991) 316-342]) is an appealing dimension
reduction method for regression models with multivariate covariates. It has
been extended by Ferr\'{e} and Yao [Statistics 37 (2003) 475-488, Statist.
Sinica 15 (2005) 665-683] and Hsing and Ren [Ann. Statist. 37 (2009) 726-755]
to functional covariates where the whole trajectories of random functional
covariates are completely observed. The focus of this paper is to develop
sliced inverse regression for intermittently and sparsely measured longitudinal
covariates. We develop asymptotic theory for the new procedure and show, under
some regularity conditions, that the estimated directions attain the optimal
rate of convergence. Simulation studies and data analysis are also provided to
demonstrate the performance of our method.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1193 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org). With Correction
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