Jane-Ling Wang

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.

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.

Density and hazard rate estimation for censored data via strong representation of the Kaplan-Meier estimator

SummaryWe study the estimation of a density and a hazard rate function based on censored data by the kernel smoothing method. Our technique is facilitated by a recent result of Lo and Singh (1986) which establishes a strong uniform approximation of the Kaplan-Meier estimator by an average of independent random variables. (Note that the approximation is carried out on the original probability space, which should be distinguished from the Hungarian embedding approach.) Pointwise strong consistency and a law of iterated logarithm are derived, as well as the mean squared error expression and asymptotic normality, which is obtain using a more traditional method, as compared with the Hajek projection employed by Tanner and Wong (1983).

Estimation for a partial-linear single-index model

In this paper, we study the estimation for a partial-linear single-index model. A two-stage estimation procedure is proposed to estimate the link function for the single index and the parameters in the single index, as well as the parameters in the linear component of the model. Asymptotic normality is established for both parametric components. For the index, a constrained estimating equation leads to an asymptotically more efficient estimator than existing estimators in the sense that it is of a smaller limiting variance. The estimator of the nonparametric link function achieves optimal convergence rates, and the structural error variance is obtained. In addition, the results facilitate the construction of confidence regions and hypothesis testing for the unknown parameters. A simulation study is performed and an application to a real dataset is illustrated. The extension to multiple indices is briefly sketched.

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.

Longevity-fertility trade-offs in the tephritid fruit fly, Anastrepha ludens, across dietary-restriction gradients.

Although it is widely known that dietary restriction (DR) not only extends the longevity of a wide range of species but also reduces their reproductive output, the interrelationship of DR, longevity extension and reproduction is not well understood in any organism. Here we address the question: ‘Under what nutritional conditions do the longevity‐enhancing effects resulting from food restriction either counteract, complement or reinforce the mortality costs of reproduction? To answer this question we designed a fine‐grained DR study involving 4800 individuals of the tephritid fruit fly, Anastrepha ludens, in which we measured sex‐specific survival and daily reproduction in females in each of 20 different treatments (sugar : yeast ratios) plus 4 starvation controls. The database generated from this 3‐year study consisted of approximately 100 000 life‐days for each sex and 750 000 eggs distributed over the reproductive lives of 2400 females. The fertility and longevity‐extending responses were used to create contour maps (X‐Y grid) that show the demographic responses (Z‐axis) across dietary gradients that range from complete starvation to both ad libitum sugar‐only and ad libitum standard diet (3 : 1 sugar : yeast). The topographic perspectives reveal demographic equivalencies along nutritional gradients, differences in the graded responses of males and females, egg production costs that are sensitive to the interaction of food amounts and constituents, and orthogonal contours (equivalencies in longevity or reproduction) representing demographic thresholds related to both caloric content and sugar : yeast ratios. In general, the finding that lifespan and reproductive maxima occur at much different nutritional coordinates poses a major challenge for the use of food restriction (or a mimetic) in humans to improve health and extend longevity in humans.

Functional canonical analysis for square integrable stochastic processes

We study the extension of canonical correlation from pairs of random vectors to the case where a data sample consists of pairs of square integrable stochastic processes. Basic questions concerning the definition and existence of functional canonical correlation are addressed and sufficient criteria for the existence of functional canonical correlation are presented. Various properties of functional canonical analysis are discussed. We consider a canonical decomposition, in which the original processes are approximated by means of their canonical components.

Risk of dementia after anaesthesia and surgery

BACKGROUND The potential relationship between anaesthesia, surgery and onset of dementia remains elusive. AIMS To determine whether the risk of dementia increases after surgery with anaesthesia, and to evaluate possible associations among age, mode of anaesthesia, type of surgery and risk of dementia. METHOD The study cohort comprised patients aged 50 years and older who were anaesthetised for the first time since 1995 between 1 January 2004 and 31 December 2007, and a control group of randomly selected patients matched for age and gender. Patients were followed until 31 December 2010 to identify the emergence of dementia. RESULTS Relative to the control group, patients who underwent anaesthesia and surgery exhibited an increased risk of dementia (hazard ratio = 1.99) and a reduced mean interval to dementia diagnosis. The risk of dementia increased in patients who received intravenous or intramuscular anaesthesia, regional anaesthesia and general anaesthesia. CONCLUSIONS The results of our nationwide, population-based study suggest that patients who undergo anaesthesia and surgery may be at increased risk of dementia.

Locally adaptive hazard smoothing

SummaryWe consider nonparametric estimation of hazard functions and their derivatives under random censorship, based on kernel smoothing of the Nelson (1972) estimator. One critically important ingredient for smoothing methods is the choice of an appropriate bandwidth. Since local variance of these estimates depends on the point where the hazard function is estimated and the bandwidth determines the trade-off between local variance and local bias, data-based local bandwidth choice is proposed. A general principle for obtaining asymptotically efficient data-based local bandwiths, is obtained by means of weak convergence of a local bandwidth process to a Gaussian limit process. Several specific asymptotically efficient bandwidth estimators are discussed. We propose in particular an, asymptotically efficient method derived from direct pilot estimators of the hazard function and of the local mean squared error. This bandwidth choice method has practical advantages and is also of interest in the uncensored case as well as for density estimation.