Yehua Li

Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data

We consider nonparametric estimation of the mean and covariance functions for functional/longitudinal data. Strong uniform convergence rates are developed for estimators that are local-linear smoothers. Our results are obtained in a unified framework in which the number of observations within each curve/cluster can be of any rate relative to the sample size. We show that the convergence rates for the procedures depend on both the number of sample curves and the number of observations on each curve. For sparse functional data, these rates are equivalent to the optimal rates in nonparametric regression. For dense functional data, root-n rates of convergence can be achieved with proper choices of bandwidths. We further derive almost sure rates of convergence for principal component analysis using the estimated covariance function. The results are illustrated with simulation studies.

Generalized Functional Linear Models with Semiparametric Single-Index Interactions

We introduce a new class of functional generalized linear models, where the response is a scalar and some of the covariates are functional. We assume that the response depends on multiple covariates, a finite number of latent features in the functional predictor, and interaction between the two. To achieve parsimony, the interaction between the multiple covariates and the functional predictor is modeled semiparametrically with a single-index structure. We propose a two-step estimation procedure based on local estimating equations, and investigate two situations: (a) when the basis functions are predetermined, e.g., Fourier or wavelet basis functions and the functional features of interest are known; and (b) when the basis functions are data driven, such as with functional principal components. Asymptotic properties are developed. Notably, we show that when the functional features are data driven, the parameter estimates have an increased asymptotic variance due to the estimation error of the basis functions. Our methods are illustrated with a simulation study and applied to an empirical dataset where a previously unknown interaction is detected. Technical proofs of our theoretical results are provided in the online supplemental materials.

Selecting the Number of Principal Components in Functional Data

Functional principal component analysis (FPCA) has become the most widely used dimension reduction tool for functional data analysis. We consider functional data measured at random, subject-specific time points, contaminated with measurement error, allowing for both sparse and dense functional data, and propose novel information criteria to select the number of principal component in such data. We propose a Bayesian information criterion based on marginal modeling that can consistently select the number of principal components for both sparse and dense functional data. For dense functional data, we also develop an Akaike information criterion based on the expected Kullback–Leibler information under a Gaussian assumption. In connecting with the time series literature, we also consider a class of information criteria proposed for factor analysis of multivariate time series and show that they are still consistent for dense functional data, if a prescribed undersmoothing scheme is undertaken in the FPCA algorithm. We perform intensive simulation studies and show that the proposed information criteria vastly outperform existing methods for this type of data. Surprisingly, our empirical evidence shows that our information criteria proposed for dense functional data also perform well for sparse functional data. An empirical example using colon carcinogenesis data is also provided to illustrate the results. Supplementary materials for this article are available online.

DECIDING THE DIMENSION OF EFFECTIVE DIMENSION REDUCTION SPACE FOR FUNCTIONAL AND HIGH-DIMENSIONAL DATA

In this paper, we consider regression models with a Hilbert-space-valued predictor and a scalar response, where the response depends on the predictor only through a finite number of projections. The linear subspace spanned by these projections is called the effective dimension reduction (EDR) space. To determine the dimensionality of the EDR space, we focus on the leading principal component scores of the predictor, and propose two sequential χ 2 testing procedures under the assumption that the predictor has an elliptically contoured distribution. We further extend these procedures and introduce a test that simultaneously takes into account a large number of principal component scores. The proposed procedures are supported by theory, validated by simulation studies, and illustrated by a real-data example. Our methods and theory are applicable to functional data and high-dimensional multivariate data.

Nonparametric estimation of correlation functions in longitudinal and spatial data, with application to colon carcinogenesis experiments

In longitudinal and spatial studies, observations often demonstrate strong correlations that are stationary in time or distance lags, and the times or locations of these data being sampled may not be homogeneous. We propose a nonparametric estimator of the correlation function in such data, using kernel methods. We develop a pointwise asymptotic normal distribution for the proposed estimator, when the number of subjects is fixed and the number of vectors or functions within each subject goes to infinity. Based on the asymptotic theory, we propose a weighted block bootstrapping method for making inferences about the correlation function, where the weights account for the inhomogeneity of the distribution of the times or locations. The method is applied to a data set from a colon carcinogenesis study, in which colonic crypts were sampled from a piece of colon segment from each of the 12 rats in the experiment and the expression level of p27, an important cell cycle protein, was then measured for each cell within the sampled crypts. A simulation study is also provided to illustrate the numerical performance of the proposed method.

Functional Principal Component Analysis of Spatiotemporal Point Processes With Applications in Disease Surveillance

In disease surveillance applications, the disease events are modeled by spatiotemporal point processes. We propose a new class of semiparametric generalized linear mixed model for such data, where the event rate is related to some known risk factors and some unknown latent random effects. We model the latent spatiotemporal process as spatially correlated functional data, and propose Poisson maximum likelihood and composite likelihood methods based on spline approximations to estimate the mean and covariance functions of the latent process. By performing functional principal component analysis to the latent process, we can better understand the correlation structure in the point process. We also propose an empirical Bayes method to predict the latent spatial random effects, which can help highlight hot areas with unusually high event rates. Under an increasing domain and increasing knots asymptotic framework, we establish the asymptotic distribution for the parametric components in the model and the asymptotic convergence rates for the functional principal component estimators. We illustrate the methodology through a simulation study and an application to the Connecticut Tumor Registry data. Supplementary materials for this article are available online.

Joint Modeling and Clustering Paired Generalized Longitudinal Trajectories With Application to Cocaine Abuse Treatment Data

In a cocaine dependence treatment study, we have paired binary longitudinal trajectories that record the cocaine use patterns of each patient before and after a treatment. To better understand the drug-using behaviors among the patients, we propose a general framework based on functional data analysis to jointly model and cluster these paired non-Gaussian longitudinal trajectories. Our approach assumes that the response variables follow distributions from the exponential family, with the canonical parameters determined by some latent Gaussian processes. To reduce the dimensionality of the latent processes, we express them by a truncated Karhunen-Lóeve (KL) expansion allowing the mean and covariance functions to be different across clusters. We further represent the mean and eigenfunctions functions by flexible spline bases, and determine the orders of the truncated KL expansions using data-driven methods. By treating the cluster membership as a missing value, we cluster the cocaine use trajectories by a likelihood-based approach. The cluster membership and parameter estimates are jointly estimated by a Monte Carlo EM algorithm with Gibbs sampling steps. We discover subgroups of patients with distinct behaviors in terms of overall probability to use, binge verses periodic use pattern, etc. The joint modeling approach also sheds new lights on relating relapse behavior to baseline pattern in each subgroup. Supplementary materials for this article are available online.

Generalized Quasi-Likelihood Ratio Tests for Semiparametric Analysis of Covariance Models in Longitudinal Data

ABSTRACT We model generalized longitudinal data from multiple treatment groups by a class of semiparametric analysis of covariance models, which take into account the parametric effects of time dependent covariates and the nonparametric time effects. In these models, the treatment effects are represented by nonparametric functions of time and we propose a generalized quasi-likelihood ratio test procedure to test if these functions are identical. Our estimation procedure is based on profile estimating equations combined with local linear smoothers. We find that the much celebrated Wilks phenomenon which is well established for independent data still holds for longitudinal data if a working independence correlation structure is assumed in the test statistic. However, this property does not hold in general, especially when the working variance function is misspecified. Our empirical study also shows that incorporating correlation into the test statistic does not necessarily improve the power of the test. The proposed methods are illustrated with simulation studies and a real application from opioid dependence treatments. Supplementary materials for this article are available online.

Joint modeling of longitudinal drug using pattern and time to first relapse in cocaine dependence treatment data

An important endpoint variable in a cocaine rehabilitation study is the time to first relapse of a patient after the treatment. We propose a joint modeling approach based on functional data analysis to study the relationship between the baseline longitudinal cocaine-use pattern and the interval censored time to first relapse. For the baseline cocaine-use pattern, we consider both self-reported cocaine-use amount trajectories and dichotomized use trajectories. Variations within the generalized longitudinal trajectories are modeled through a latent Gaussian process, which is characterized by a few leading functional principal components. The association between the baseline longitudinal trajectories and the time to first relapse is built upon the latent principal component scores. The mean and the eigenfunctions of the latent Gaussian process as well as the hazard function of time to first relapse are modeled nonparametrically using penalized splines, and the parameters in the joint model are estimated by a Monte Carlo EM algorithm based on Metropolis-Hastings steps. An Akaike information criterion (AIC) based on effective degrees of freedom is proposed to choose the tuning parameters, and a modified empirical information is proposed to estimate the variance-covariance matrix of the estimators.

Cocaine dependence treatment data: Methods for measurement error problems with predictors derived from stationary stochastic processes

In a cocaine dependence treatment study, we use linear and nonlinear regression models to model posttreatment cocaine craving scores and first cocaine relapse time. A subset of the covariates are summary statistics derived from baseline daily cocaine use trajectories, such as baseline cocaine use frequency and average daily use amount. These summary statistics are subject to estimation error and can therefore cause biased estimators for the regression coefficients. Unlike classical measurement error problems, the error we encounter here is heteroscedastic with an unknown distribution, and there are no replicates for the error-prone variables or instrumental variables. We propose two robust methods to correct for the bias: a computationally efficient method-of-moments-based method for linear regression models and a subsampling extrapolation method that is generally applicable to both linear and nonlinear regression models. Simulations and an application to the cocaine dependence treatment data are used to illustrate the efficacy of the proposed methods. Asymptotic theory and variance estimation for the proposed subsampling extrapolation method and some additional simulation results are described in the online supplementary material.