this article we develop new methods for analyzing the data from an experiment using rodent models to investigate the effect of type of dietary fat on O -methylguanine-DNA-methyltransferase (MGMT), an important biomarker in early colon carcinogenesis. The data consist of observed pro# les over a spatial variable contained within a two-stage hierarchy, a structure that we dub hierarchical functional data. We present a new method providing a uni# ed framework for modeling these data, simultaneously yielding estimates and posterior samples for mean, individual, and subsample-level pro# les, as well as covariance parameters at the various hierarchical levels. Our method is nonparametric in that it does not require the prespeci# cation of parametric forms for the functions and involves modeling in the wavelet space, which is especially effective for spatially heterogeneous functions as encountered in the MGMT data. Our approach is Bayesian; the only informative hyperparameters in our model are effectively smoothing parameters. Analysis of this dataset yields interesting new insights into how MGMT operates in early colon carcinogenesis, and how this may depend on diet. Our method is general, so it can be applied to other settings where hierarchical functional data are encountered

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

Abstract: The method of covariate adjusted regression was recently proposed for situations where both predictors and response in a regression model are not directly observed, but are observed after being contaminated by unknown functions of a common observable confounder in a multiplicative fashion. One example is data collected for a study on diabetes, where the variables of interest, systolic and diastolic blood pressures and glycosolated hemoglobin levels are known to be influenced by an observable confounder, body mass index. An estimation procedure based on equidistant binning (EB), currently available, gives consistent estimators for the regression coefficients adjusted for the confounder. In this paper, we propose two new estimation procedures based on nearest neighbor binning (NB) and local polynomial modeling (LP). Even though, the three methods perform similarly in terms of their bias, it is shown through simulation studies that NB has smaller variance compared to EB, and LP yields substantially lower variance relative to the two binning methods for small to moderate sample sizes. The consistency and convergence rates of the proposed estimators of LP, with the smallest MSE, are also established. We illustrate the proposed method of LP with the above mentioned diabetes data, where the goal is to uncover the regression relation between the response, glycosolated hemoglobin levels, and the predictors, systolic and diastolic blood pressures, adjusted for body mass index.

In some applications, quality of a process is characterized by the functional relationship between a response variable and one or more explanatory variables. Profile monitoring is for checking the stability of this relationship over time. Control charts for monitoring nonparametric profiles are useful when the relationship is too complicated to be described parametrically. Most existing control charts in the literature are for monitoring parametric profiles. They require the assumption that within-profile measurements are independent of each other, which is often invalid in practice. This paper focuses on nonparametric profile monitoring when within-profile data are correlated. A novel control chart is suggested, which incorporates local linear kernel smoothing into the exponentially weighted moving average (EWMA) control scheme. In this method, within-profile correlation is described by a nonparametric mixed-effects model. Our proposed control chart is fast to compute and convenient to use. Numerical examples show that it works well in various cases. Some technical details are provided in an appendix available online as supplemental materials.

er: problems with one may be solved by tinkering with one of the other aspects, and model misspecification in one respect may lead to consequences in other respects. E.g., unrecognized level-one heteroscedasticity may lead to a significant random slope variance, which then disappears if the heteroscedasticity is taken into account; non-linear effects of some variables in X , when unrecognized, may show up as heteroscedasticity at level 1 or as a random slope; and non-zero expected residuals sometimes can be dealt with by transformations of variables in X . This presentation of diagnostic techniques starts with techniques that can be represented as model checks remaining within the framework of the HLM. This is followed by a section on model checking based on various types of residuals. An important type of misspecification can reside in non-linearity of the effects of explanatory variables. The last part of the chapter presents methods to identify such misspecifications and estimate t

In this article we study the relationship between virologic and immunologic responses in AIDS clinical trials. Since plasma HIV RNA copies (viral load) and CD4+ cell counts are crucial virologic and immunologic markers for HIV infection, it is important to study their relationship during HIV/AIDS treatment. We propose a mixed-effects varyingcoefficient model based on an exploratory analysis of a clinical trial data. Since both viral load and CD4+ cell counts are subject to measurement error, we also consider the measurement error problem in covariates in our model. The regression spline method is proposed for inference for parameters in the proposed model. The regression spline method transforms the unknown nonparametric components into parametric functions.

this paper explores a class of varying-coefficient linear models in which the index is unknown and is estimated as a linear combination of regressors and/or other variables. We search for the index such that the derived varying-coefficient model provides the least squares approximation to the underlying unknown multidimensional regression function. The search is implemented through a newly proposed hybrid backfitting algorithm.The core of the algorithm is the alternating iteration between estimating the index through a one-step scheme and estimating coefficient functions through one-dimensional local linear smoothing. The locally significant variables are selected in terms of a combined use of the t -statistic and the Akaike information criterion. We further extend the algorithm for models with two indices. Simulation shows that the methodology proposed has appreciable flexibility to model complex multivariate nonlinear structure and is practically feasible with average modern computers. The methods are further illustrated through the Canadian mink--muskrat data in 1925--1994 and the pound--dollar exchange rates in 1974--1983

this paper, we use \kernel methods" to describe the previously proposed methods which consist essentially of the method introduced by Severini and Staniswalis (1994) and some natural alternatives. \Kernel methods" should not be read as \all possible kernel-type methods". Rather than annoying the reader by continually making this distinction, except in the discussion we will adopt the convention that \kernel methods" means \previously proposed kernel methods." The purpose of this paper is to investigate whether the kernel result holds for marginal spline methods, especially smoothing splines (see Wahba, 1990; Green and Silverman, 1994 among many others) and penalized regression splines (P{splines, see Eilers and Marx, 1996; Ruppert and Carroll, 2000). One might reasonably suppose that the kernel result will hold for splines because, in the case of independent data, Silverman (1984) showed that smoothing splines are asymptotically equivalent to kernel regression with a particular higher{order kernel. It would not appear to be too great a leap to conclude that if smoothing splines and kernels are equivalent in the usual independence case, then they ought also to be equivalent in marginal longitudinal nonparametric models, and hence with splines it is also better to ignore the correlation structure entirely. For simplicity, we assume in this paper that the true covariance matrix V is known and compare the nite sample and asymptotic performance of spline and kernel methods when using either working independence or the true covariance

In this chapter we examine two different settings in which sparseness can be important in a functional data analysis (FDA). The first setting involves sparseness in the functions. The classical assumption of FDA is that each function has been measured at all time points. However, in practice it is often the case that the functions have only been observed at a relatively small number of points. Here we discuss different general approaches that can be applied in this setting, such as basis functions, mixed effects models and local smoothing, and examine the relative merits of each. Then we briefly outline several specific methodologies that have been developed for dealing with sparse functional data in the principal components, clustering, classification and regression paradigms. The second setting involves using sparsity ideas from high dimensional regression problems, where most of the regression coefficients are assumed to be zero, to perform a dimension reduction in the functional space. We discuss two specific approaches that have been suggested in the literature.

Abstract: We propose a general class of quantile residual life models, where a specific quantile of the residual life time, conditional on an individual has survived up to time t, is a function of certain covariates with their coefficients varying over time. The varying coefficients are assumed to be smooth unspecified functions of t. We propose to estimate the coefficient functions using spline approximation. Incorporating the spline representation directly into a set of unbiased estimating equations, we obtain a one-step estimation procedure, and we show that this leads to a uniformly consistent estimator. To obtain further computational simplification, we propose a two-step estimation approach in which we estimate the coefficients on a series of time points first, and follow this with spline smoothing. We compare the two methods in terms of their asymptotic efficiency and computational complexity. We further develop inference tools to test the significance of the covariate effect on residual life. The finite sample performance of the estimation and testing procedures are further illustrated through numerical experiments. We also apply the methods to a data set from a neurological study. Key words and phrases: Censored data, nonparametric regression, quantile regression, residual life, spline. 1.