Focused information criterion and model averaging for generalized additive partial linear models

We study model selection and model averaging in generalized additive partial linear models (GAPLMs). Polynomial spline is used to approximate nonparametric functions. The corresponding estimators of the linear parameters are shown to be asymptotically normal. We then develop a focused information criterion (FIC) and a frequentist model average (FMA) estimator on the basis of the quasi-likelihood principle and examine theoretical properties of the FIC and FMA. The major advantages of the proposed procedures over the existing ones are their computational expediency and theoretical reliability. Simulation experiments have provided evidence of the superiority of the proposed procedures. The approach is further applied to a real-world data example.Comment: Published in at http://dx.doi.org/10.1214/10-AOS832 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Flexible Partially Linear Single Index Regression Models for Multivariate Survival Data

Survival regression models usually assume that covariate effects have a linear form. In many circumstances, however, the assumption of linearity may be violated. The present work addresses this limitation by adding nonlinear covariate effects to survival models. Nonlinear covariates are handled using a single index structure, which allows high-dimensional nonlinear effects to be reduced to a scalar term. The nonlinear single index approach is applied to modeling of survival data with multivariate responses, in three popular models: the proportional hazards (PH) model, the proportional odds (PO) model, and the generalized transformation model. Another extension of the PH and PO model is the handling of the baseline function. Instead of modeling it in a parametric way, which is fairly restrictive, or leaving it unspecified, which makes it impossible to calculate the survival and hazard functions, a weakly parametric approach is used here. As a result, the full likelihood can be applied for inference. The new developments are realized by adding a number of weakly parametric elements to the standard parametric regression models. The marginal baseline hazard functions are modeled using piecewise constants. Marginal survival functions are combined in using copula models, such as the Clayton model, to incorporate association among the multivariate responses. The nonlinear covariate effect is brought into the model through a smooth function with the single-index structure as the input. The smooth function is modeled using a spline. The performance of the PH, PO, and transformation models with the proposed extensions is evaluated through extensive simulation studies. The PH and PO models are also applied to a real-world data set. The results suggest that the proposed methods can capture the nonlinear covariate effects well, and that there is benefit to modeling the association between the correlated responses. Individual-level survival or hazard function estimates also provide information of interest to researchers. The proposed transformation model in particular is very promising. Some discussion of how this model may be further developed is provided

Semiparametric Regression During 2003–2007

Semiparametric regression is a fusion between parametric regression and nonparametric regression and the title of a book that we published on the topic in early 2003. We review developments in the field during the five year period since the book was written. We find semiparametric regression to be a vibrant field with substantial involvement and activity, continual enhancement and widespread application

The single-index hazards model

We first propose the single-index hazards model for right censored survival data. As an extension of the Cox model, this model allows nonparametric modeling of covariate effects in a parsimonious way via a single-index. In addition, the relative importance of covariates can be assessed via this model. We consider the conventional profile-kernel method based on the local likelihood for model estimation. It is shown that this method may give consistent estimation under certain restrictive conditions, but in general it can yield biased estimation. Simulation studies are conducted to demonstrate the bias phenomena. The existence and nature of the failure of this commonly used approach is somewhat surprising. The interpretation of covariate effects in the aforementioned single-index hazards model is difficult. Thus, we further propose the partly proportional single-index hazards model in which the effect of covariates of primary interest is represented by the regression parameter while "nuisance" covariates can have nonparametric effect on the survival time. We again consider the conventional profile-kernel method and it leads to biased estimation as well. A bias correction method is then proposed and the corrected profile local likelihood estimators are shown to be consistent, asymptotically normal and semiparametrically efficient. We evaluate the finite-sample properties of our estimators through simulation studies and illustrate the proposed model and method with an application to a dataset from the Multicenter AIDS Cohort Study (MACS). Besides the profile-kernel method, we also study the profile stratified likelihood method based on stratification of the single-index. In the single-index hazards model, this method may give consistent estimation under the restrictive "independent censoring" condition, but in general it can yield biased estimation. Simulation studies are conducted to demonstrate the situations in which the bias phenomena do (or do not) exist; In the partly proportional single-index hazards model, we demonstrate numerically the existence of the bias and then propose a bias correction method. The estimators from the corrected profile stratified likelihood method are shown to be consistent. Their finite-sample properties are evaluated through simulation studies. The corrected profile stratified method is applied to the aforementioned MACS study for illustration

A generalized Fellner-Schall method for smoothing parameter estimation with application to Tweedie location, scale and shape models

We consider the estimation of smoothing parameters and variance components in models with a regular log likelihood subject to quadratic penalization of the model coefficients, via a generalization of the method of Fellner (1986) and Schall (1991). In particular: (i) we generalize the original method to the case of penalties that are linear in several smoothing parameters, thereby covering the important cases of tensor product and adaptive smoothers; (ii) we show why the method's steps increase the restricted marginal likelihood of the model, that it tends to converge faster than the EM algorithm, or obvious accelerations of this, and investigate its relation to Newton optimization; (iii) we generalize the method to any Fisher regular likelihood. The method represents a considerable simplification over existing methods of estimating smoothing parameters in the context of regular likelihoods, without sacrificing generality: for example, it is only necessary to compute with the same first and second derivatives of the log-likelihood required for coefficient estimation, and not with the third or fourth order derivatives required by alternative approaches. Examples are provided which would have been impossible or impractical with pre-existing Fellner-Schall methods, along with an example of a Tweedie location, scale and shape model which would be a challenge for alternative methods

Estimating and Interpreting Effects from Nonlinear Exposure-Response Curves in Occupational Cohorts Using Truncated Power Basis Expansions and Penalized Splines

Truncated power basis expansions and penalized spline methods are demonstrated for estimating nonlinear exposure-response relationships in the Cox proportional hazards model. R code is provided for fitting models to get point and interval estimates. The method is illustrated using a simulated data set under a known exposure-response relationship and in a data application examining risk of carpal tunnel syndrome in an occupational cohort