Hans-Georg Müller

Functional Data Analysis for Sparse Longitudinal Data

We propose a nonparametric method to perform functional principal components analysis for the case of sparse longitudinal data. The method aims at irregularly spaced longitudinal data, where the number of repeated measurements available per subject is small. In contrast, classical functional data analysis requires a large number of regularly spaced measurements per subject. We assume that the repeated measurements are located randomly with a random number of repetitions for each subject and are determined by an underlying smooth random (subject-specific) trajectory plus measurement errors. Basic elements of our approach are the parsimonious estimation of the covariance structure and mean function of the trajectories, and the estimation of the variance of the measurement errors. The eigenfunction basis is estimated from the data, and functional principal components score estimates are obtained by a conditioning step. This conditional estimation method is conceptually simple and straightforward to implement. A key step is the derivation of asymptotic consistency and distribution results under mild conditions, using tools from functional analysis. Functional data analysis for sparse longitudinal data enables prediction of individual smooth trajectories even if only one or few measurements are available for a subject. Asymptotic pointwise and simultaneous confidence bands are obtained for predicted individual trajectories, based on asymptotic distributions, for simultaneous bands under the assumption of a finite number of components. Model selection techniques, such as the Akaike information criterion, are used to choose the model dimension corresponding to the number of eigenfunctions in the model. The methods are illustrated with a simulation study, longitudinal CD4 data for a sample of AIDS patients, and time-course gene expression data for the yeast cell cycle.

Generalized functional linear models

We propose a generalized functional linear regression model for a regression situation where the response variable is a scalar and the predictor is a random function. A linear predictor is obtained by forming the scalar product of the predictor function with a smooth parameter function, and the expected value of the response is related to this linear predictor via a link function. If, in addition, a variance function is specified, this leads to a functional estimating equation which corresponds to maximizing a functional quasi-likelihood. This general approach includes the special cases of the functional linear model, as well as functional Poisson regression and functional binomial regression. The latter leads to procedures for classification and discrimination of stochastic processes and functional data. We also consider the situation where the link and variance functions are unknown and are estimated nonparametrically from the data, using a semiparametric quasi-likelihood procedure. An essential step in our proposal is dimension reduction by approximating the predictor processes with a truncated Karhunen-Loeve expansion. We develop asymptotic inference for the proposed class of generalized regression models. In the proposed asymptotic approach, the truncation parameter increases with sample size, and a martingale central limit theorem is applied to establish the resulting increasing dimension asymptotics. We establish asymptotic normality for a properly scaled distance between estimated and true functions that corresponds to a suitable L 2 metric and is defined through a generalized covariance operator. As a consequence, we obtain asymptotic tests and simultaneous confidence bands for the parameter function that determines the model. The proposed estimation, inference and classification procedures and variants with unknown link and variance functions are investigated in a simulation study. We find that the practical selection of the number of components works well with the AIC criterion, and this finding is supported by theoretical considerations. We include an application to the classification of medflies regarding their remaining longevity status, based on the observed initial egg-laying curve for each of 534 female medflies.We propose a generalized functional linear regression model for a regression situation where the response variable is a scalar and the predictor is a random function. A linear predictor is obtained by forming the scalar product of the predictor function with a smooth parameter function, and the expected value of the response is related to this linear predictor via a link function. If in addition a variance function is specified, this leads to a functional estimating equation which corresponds to maximizing a functional quasi-likelihood. This general approach includes the special cases of the functional linear model, as well as functional Poisson regression and functional binomial regression. The latter leads to procedures for classification and discrimination of stochastic processes and functional data. We also consider the situation where the link and variance functions are unknown and are estimated nonparametrically from the data, using a semiparametric quasi-likelihood procedure. An essential step in our proposal is dimension reduction by approximating the predictor processes with a truncated Karhunen-Loève expansion. We develop asymptotic inference for the proposed class of generalized regression models. In the proposed asymptotic approach, the truncation parameter increases with sample size, and a martingale central limit theorem is applied to establish the resulting increasing dimension asymptotics. We establish asymptotic normality for a properly scaled distance between estimated and true functions that corresponds to a suitable L2 metric and is defined through a generalized covariance operator. As a consequence, we obtain asymptotic tests and simultaneous confidence bands for the parameter function that determines the model. The proposed estimation, inference and classification procedures and variants with unknown link and variance functions are investigated in a simulation study. We find that the practical selection of the number of components works well with the AIC criterion, and this finding is supported by theoretical considerations. We include an application to the classification of medflies regarding their remaining longevity status, based on the observed initial egg-laying curve for each of 534 female medflies.

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 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.

Hazard rate estimation under random censoring with varying kernels and bandwidths.

We discuss the estimation of hazard rates under random censoring with the kernel method. Two practically relevant problems that occur when applying unmodified kernel estimators are boundary effects near the endpoints of the support of the hazard rate, and a substantial increase in the variance from left to right over the range of abscissae where the hazard rate is estimated. A new class of boundary kernels is proposed for the first problem. Explicit formulas for these kernels are developed, and it is shown that this boundary correction works well in practice. A data-adaptive varying bandwidth selection procedure is proposed for the second problem. This procedure generally will lead to increasing bandwidths near the left endpoint and toward the right endpoint, and will lead to smaller integrated mean squared error of the hazard rate estimator as compared to a fixed bandwidth method. A practically feasible method incorporating the new boundary kernels and local bandwidth choices is implemented and illustrated with survival data from a leukemia study.

Weighted Local Regression and Kernel Methods for Nonparametric Curve Fitting

Abstract Weighted local regression, a popular technique for smoothing scatterplots, is shown to be asymptotically equivalent to certain kernel smoothers. Since both methods are local weighted averages of the data, it is proved that in the fixed design regression model, given a weighted local regression procedure with any weight function, there is a corresponding kernel method such that the quotients of weights distributed by both methods tend uniformly to 1 as the number of observations increases to infinity. It is demonstrated by examples that in some instances the weights are nearly the same for both methods, even for small samples. The asymptotic equivalence allows the derivation of the leading terms of the mean squared error and of the local limit distribution for weighted local regression. Further, a close correspondence is found between the orders of the polynomial to be locally fitted in weighted local regression and the order (number of vanishing moments) of the kernel employed in the kernel smoot...

Functional Additive Models

In commonly used functional regression models, the regression of a scalar or functional response on the functional predictor is assumed to be linear. This means that the response is a linear function of the functional principal component scores of the predictor process. We relax the linearity assumption and propose to replace it by an additive structure, leading to a more widely applicable and much more flexible framework for functional regression models. The proposed functional additive regression models are suitable for both scalar and functional responses. The regularization needed for effective estimation of the regression parameter function is implemented through a projection on the eigenbasis of the covariance operator of the functional components in the model. The use of functional principal components in an additive rather than linear way leads to substantial broadening of the scope of functional regression models and emerges as a natural approach, because the uncorrelatedness of the functional principal components is shown to lead to a straightforward implementation of the functional additive model, based solely on a sequence of one-dimensional smoothing steps and without the need for backfitting. This facilitates the theoretical analysis, and we establish the asymptotic consistency of the estimates of the components of the functional additive model. We illustrate the empirical performance of the proposed modeling framework and estimation methods through simulation studies and in applications to gene expression time course data.

Classification using functional data analysis for temporal gene expression data

MOTIVATION Temporal gene expression profiles provide an important characterization of gene function, as biological systems are predominantly developmental and dynamic. We propose a method of classifying collections of temporal gene expression curves in which individual expression profiles are modeled as independent realizations of a stochastic process. The method uses a recently developed functional logistic regression tool based on functional principal components, aimed at classifying gene expression curves into known gene groups. The number of eigenfunctions in the classifier can be chosen by leave-one-out cross-validation with the aim of minimizing the classification error. RESULTS We demonstrate that this methodology provides low-error-rate classification for both yeast cell-cycle gene expression profiles and Dictyostelium cell-type specific gene expression patterns. It also works well in simulations. We compare our functional principal components approach with a B-spline implementation of functional discriminant analysis for the yeast cell-cycle data and simulations. This indicates comparative advantages of our approach which uses fewer eigenfunctions/base functions. The proposed methodology is promising for the analysis of temporal gene expression data and beyond. AVAILABILITY MATLAB programs are available upon request.

Age of ovary determines remaining life expectancy in old ovariectomized mice.

We investigated the capacity of young ovaries, transplanted into old ovariectomized CBA mice, to improve remaining life expectancy of the hosts. Donor females were sexually mature 2‐month‐olds; recipients were prepubertally ovariectomized at 3 weeks and received transplants at 5, 8 or 11 months of age. Relative to ovariectomized control females, life expectancy at 11 months was increased by 60% in 11‐month recipient females and by 40% relative to intact control females. Only 20% of the 11‐month transplant females died in the 300‐day period following ovarian transplantation, whereas nearly 65% of the ovariectomized control females died during this same period. The 11‐month‐old recipient females resumed oestrus and continued to cycle up to several months beyond the age of control female reproductive senescence. Across the three recipient age groups, transplantation of young ovaries increased life expectancy in proportion to the relative youth of the ovary. Our results relate to recent findings on the gonadal input upon aging in Caenorhabditis elegans and may suggest how the mammalian gonad, including that of humans, could regulate aging and determine longevity.

Velocity and acceleration of height growth using kernel estimation

A method is introduced for estimating acceleration, velocity and distance of longitudinal growth curves and it is illustrated by analysing human height growth. This approach, called kernel estimation, belongs to the class of smoothing methods and does not assume an a priori fixed functional model, and not even that one and the same model is applicable for all children. The examples presented show that acceleration curves might allow a better quantification of the mid-growth spurt (MS) and a more differentiated analysis of the pubertal spurt (PS). Accelerations are prone to follow random variations present in the data, and parameters defined in terms of acceleration are, therefore, validated by a comparison with parameters defined in terms of velocity. Our non-parametric-curve-fitting approach is also compared with parametric fitting via a model suggested by Preece and Baines (1978).

Life history response of Mediterranean fruit flies to dietary restriction.

The purpose of this study was to investigate medfly longevity and reproduction across a broad spectrum of diet restriction using a protocol similar to those applied in most rodent studies. Age‐specific reproduction and age of death were monitored for 1200 adult males and 1200 females, each individually maintained on one of 12 diets from ad libitum to 30% of ad libitum. Diet was provided in a fixed volume of solution that was fully consumed each day, ensuring control of total nutrient consumption for every fly. Contrary to expectation and precedence, increased longevity was not observed at any level of diet restriction. Among females, reproduction continued across all diet levels despite the cost in terms of increased mortality. Among males, life expectancy exceeded that of females at most diet levels. However, in both sexes, mortality increased more sharply and the pattern of survival changed abruptly once the diet level fell to 50% of ad libitum or below, even though the energetic demands of egg production has no obvious counterpart in males. We believe that a more complete picture of the life table response to dietary restriction will emerge when studies are conducted on a wider range of species and include both sexes, more levels of diet, and the opportunity for mating and reproduction.