Pang Du

Penalized variable selection procedure for Cox models with semiparametric relative risk

We study the Cox models with semiparametric relative risk, which can be partially linear with one nonparametric component, or multiple additive or nonadditive nonparametric components. A penalized partial likelihood procedure is proposed to simultaneously estimate the parameters and select variables for both the parametric and the nonparametric parts. Two penalties are applied sequentially. The first penalty, governing the smoothness of the multivariate nonlinear covariate effect function, provides a smoothing spline ANOVA framework that is exploited to derive an empirical model selection tool for the nonparametric part. The second penalty, either the smoothly-clipped-absolute-deviation (SCAD) penalty or the adaptive LASSO penalty, achieves variable selection in the parametric part. We show that the resulting estimator of the parametric part possesses the oracle property, and that the estimator of the nonparametric part achieves the optimal rate of convergence. The proposed procedures are shown to work well in simulation experiments, and then applied to a real data example on sexually transmitted diseases.

Corticosterone Level Changes throughout Larval Development in the Amphibians Rana sylvatica and Ambystoma jeffersonianum Reared under Laboratory, Mesocosm, or Free-living Conditions

Abstract Studies of a few “model” amphibians continue to advance our mechanistic understanding of the endocrine control of larval amphibian development and metamorphosis, but there are few studies examining steroid profiles across species during larval amphibian development. We used censored regression analysis to address our primary objective, which was to examine baseline corticosterone level changes and responses to a standardized stressor throughout larval development in two amphibian species: one anuran (Wood Frogs, Rana sylvatica) and one caudate (Jefferson Salamanders, Ambystoma jeffersonianum). In addition, we looked at two additional factors that could influence the study of corticosterone during larval development, namely the rearing location of the animals (free-living, mesocosm-held, or laboratory-held) and for A. jeffersonianum, the method of induction of the stress response (ACTH injection or a confinement-agitation [CA] protocol). As has been documented for other anurans, baseline corticosterone content of R. sylvatica increased close to metamorphic climax in all rearing locations, although the absolute level varied with rearing location. Baseline corticosterone content of A. jeffersonianum increased gradually over development, and the increase in corticosterone content following CA mirrored the increase in baseline levels, although the absolute magnitude of the increase with CA varied based on rearing location. In larvae of A. jeffersonianum, both the CA method and ACTH injection significantly increased corticosterone content, with 30 min eliciting the maximum hormonal response level. Our results suggest that rearing location can influence corticosterone levels and the response to a standardized CA protocol, and that care should be taken in extrapolating results from laboratory studies to free-living amphibian populations.

Two-Component Mixture Cure Rate Model with Spline Estimated Nonparametric Components

In some survival analysis of medical studies, there are often long-term survivors who can be considered as permanently cured. The goals in these studies are to estimate the noncured probability of the whole population and the hazard rate of the susceptible subpopulation. When covariates are present as often happens in practice, to understand covariate effects on the noncured probability and hazard rate is of equal importance. The existing methods are limited to parametric and semiparametric models. We propose a two-component mixture cure rate model with nonparametric forms for both the cure probability and the hazard rate function. Identifiability of the model is guaranteed by an additive assumption that allows no time-covariate interactions in the logarithm of hazard rate. Estimation is carried out by an expectation-maximization algorithm on maximizing a penalized likelihood. For inferential purpose, we apply the Louis formula to obtain point-wise confidence intervals for noncured probability and hazard rate. Asymptotic convergence rates of our function estimates are established. We then evaluate the proposed method by extensive simulations. We analyze the survival data from a melanoma study and find interesting patterns for this study.

Nonparametric modeling of the gap time in recurrent event data

Recurrent event data arise in many biomedical and engineering studies when failure events can occur repeatedly over time for each study subject. In this article, we are interested in nonparametric estimation of the hazard function for gap time. A penalized likelihood model is proposed to estimate the hazard as a function of both gap time and covariate. Method for smoothing parameter selection is developed from subject-wise cross-validation. Confidence intervals for the hazard function are derived using the Bayes model of the penalized likelihood. An eigenvalue analysis establishes the asymptotic convergence rates of the relevant estimates. Empirical studies are performed to evaluate various aspects of the method. The proposed technique is demonstrated through an application to the well-known bladder tumor cancer data.

Pond acidification may explain differences in corticosterone among salamander populations.

Physiological tolerances play a key role in determining species distributions and abundance across a landscape, and understanding these tolerances can therefore be useful in predicting future changes in species distributions that might occur. Vertebrates possess several highly conserved physiological mechanisms for coping with environmental stressors, including the hormonal stress response that involves an endocrine cascade resulting in the increased production of glucocorticoids. We examined the function of this endocrine axis by assessing both baseline and acute stress–induced concentrations of corticosterone in larvae from eight natural breeding populations of Jefferson’s salamander Ambystoma jeffersonianum. We surveyed individuals from each pond and also examined a variety of environmental pond parameters. We found that baseline and stress-induced corticosterone concentrations differed significantly among ponds. Population-level baseline corticosterone concentrations were negatively related to pH and positively related to nitrate, and stress-induced concentrations were again negatively related to pH, positively related to nitrate, and positively related to temperature. We followed the field survey with an outdoor mesocosm experiment in which we manipulated pH and again examined baseline and acute stress–induced corticosterone in A. jeffersonianum larvae. As in the field survey, we observed an increase in the baseline corticosterone concentration of individuals exposed to the lowest pH treatment (pH 5–5.8). Examining physiological indices using a combined approach of field surveys and experiments can be a powerful tool for trying to unravel the complexities of environmental impacts on species distributions.

Smoothing Spline ANOVA Frailty Model for Recurrent Event Data

Gap time hazard estimation is of particular interest in recurrent event data. This article proposes a fully nonparametric approach for estimating the gap time hazard. Smoothing spline analysis of variance (ANOVA) decompositions are used to model the log gap time hazard as a joint function of gap time and covariates, and general frailty is introduced to account for between-subject heterogeneity and within-subject correlation. We estimate the nonparametric gap time hazard function and parameters in the frailty distribution using a combination of the Newton-Raphson procedure, the stochastic approximation algorithm (SAA), and the Markov chain Monte Carlo (MCMC) method. The convergence of the algorithm is guaranteed by decreasing the step size of parameter update and/or increasing the MCMC sample size along iterations. Model selection procedure is also developed to identify negligible components in a functional ANOVA decomposition of the log gap time hazard. We evaluate the proposed methods with simulation studies and illustrate its use through the analysis of bladder tumor data.

Partially linear structure identification in generalized additive models with NP-dimensionality

Separation of the linear and nonlinear components in additive models based on penalized likelihood has received attention recently. However, it remains unknown whether consistent separation is possible in generalized additive models, and how high dimensionality is allowed. In this article, we study the doubly SCAD-penalized approach for partial linear structure identification problems of non-polynomial (NP) dimensionality and demonstrate its oracle property. In particular, if the number of nonzero components is fixed, the dimensionality of the total number of components can be of order exp{n^d^/^(^2^d^+^1^)} where d is the smoothness of the component functions. Under such dimensionality assumptions, we show that with probability approaching one, the proposed procedure can correctly identify the zero, linear, and nonlinear components in the model. We further establish the convergence rate of the estimator for the nonlinear component and the asymptotic normality of the estimator for the linear component. Performance of the proposed method is evaluated by simulation studies. The methods are demonstrated by analyzing a gene data set.

Optimal Penalized Function-on-Function Regression Under a Reproducing Kernel Hilbert Space Framework

ABSTRACT Many scientific studies collect data where the response and predictor variables are both functions of time, location, or some other covariate. Understanding the relationship between these functional variables is a common goal in these studies. Motivated from two real-life examples, we present in this article a function-on-function regression model that can be used to analyze such kind of functional data. Our estimator of the 2D coefficient function is the optimizer of a form of penalized least squares where the penalty enforces a certain level of smoothness on the estimator. Our first result is the representer theorem which states that the exact optimizer of the penalized least squares actually resides in a data-adaptive finite-dimensional subspace although the optimization problem is defined on a function space of infinite dimensions. This theorem then allows us an easy incorporation of the Gaussian quadrature into the optimization of the penalized least squares, which can be carried out through standard numerical procedures. We also show that our estimator achieves the minimax convergence rate in mean prediction under the framework of function-on-function regression. Extensive simulation studies demonstrate the numerical advantages of our method over the existing ones, where a sparse functional data extension is also introduced. The proposed method is then applied to our motivating examples of the benchmark Canadian weather data and a histone regulation study. Supplementary materials for this article are available online.

Promotion time cure rate model with nonparametric form of covariate effects

Survival data with a cured portion are commonly seen in clinical trials. Motivated from a biological interpretation of cancer metastasis, promotion time cure model is a popular alternative to the mixture cure rate model for analyzing such data. The existing promotion cure models all assume a restrictive parametric form of covariate effects, which can be incorrectly specified especially at the exploratory stage. In this paper, we propose a nonparametric approach to modeling the covariate effects under the framework of promotion time cure model. The covariate effect function is estimated by smoothing splines via the optimization of a penalized profile likelihood. Point-wise interval estimates are also derived from the Bayesian interpretation of the penalized profile likelihood. Asymptotic convergence rates are established for the proposed estimates. Simulations show excellent performance of the proposed nonparametric method, which is then applied to a melanoma study.

Variance Change Point Detection Under a Smoothly-Changing Mean Trend with Application to Liver Procurement

Abstract Literature on change point analysis mostly requires a sudden change in the data distribution, either in a few parameters or the distribution as a whole. We are interested in the scenario, where the variance of data may make a significant jump while the mean changes in a smooth fashion. The motivation is a liver procurement experiment monitoring organ surface temperature. Blindly applying the existing methods to the example can yield erroneous change point estimates since the smoothly changing mean violates the sudden-change assumption. We propose a penalized weighted least-squares approach with an iterative estimation procedure that integrates variance change point detection and smooth mean function estimation. The procedure starts with a consistent initial mean estimate ignoring the variance heterogeneity. Given the variance components the mean function is estimated by smoothing splines as the minimizer of the penalized weighted least squares. Given the mean function, we propose a likelihood ratio test statistic for identifying the variance change point. The null distribution of the test statistic is derived together with the rates of convergence of all the parameter estimates. Simulations show excellent performance of the proposed method. Application analysis offers numerical support to non invasive organ viability assessment by surface temperature monitoring. Supplementary materials for this article are available online.