Single-index Varying-coefficient Model Research Articles

Longitudinal varying coefficient single-index model with censored covariates.

It is of interest to health policy research to estimate the population-averaged longitudinal medical cost trajectory from initial cancer diagnosis to death, and understand how the trajectory curve is affected by patient characteristics. This research question leads to a number of statistical challenges because the longitudinal cost data are often non-normally distributed with skewness, zero-inflation, and heteroscedasticity. The trajectory is nonlinear, and its length and shape depend on survival, which are subject to censoring. Modeling the association between multiple patient characteristics and nonlinear cost trajectory curves of varying lengths should take into consideration parsimony, flexibility, and interpretation. We propose a novel longitudinal varying coefficient single-index model. Multiple patient characteristics are summarized in a single-index, representing a patient's overall propensity for healthcare use. The effects of this index on various segments of the cost trajectory depend on both time and survival, which is flexibly modeled by a bivariate varying coefficient function. The model is estimated by generalized estimating equationswith an extended marginal mean structure to accommodate censored survival time as a covariate. We established the pointwise confidence interval of the varying coefficient and a test for the covariate effect. The numerical performance was extensively studied in simulations. We applied the proposed methodology to medical cost data of prostate cancer patients from the Surveillance, Epidemiology, and End Results-Medicare-Linked Database.

Empirical likelihood and estimation in single-index varying-coefficient models with censored data

Empirical likelihood and estimation in single-index varying-coefficient models with censored data

Two-stage estimation and bias-corrected empirical likelihood in a partially linear single-index varying-coefficient model

Abstract The estimation and empirical likelihood (EL) of the parameters of interest in a partially linear single-index varying-coefficient model are studied. A two-stage method is presented to estimate the regression parameters and the coefficient functions. The asymptotic distributions of the proposed estimators are obtained. Meanwhile, a bias-corrected EL ratio for the regression parameters is proposed. It is shown that the ratio is asymptotically standard chi-squared. The result can be directly used to construct the EL confidence regions of the regression parameters. Simulation studies are carried out to evaluate the finite sample behaviour of the proposed method. An application example of a real data set is given.

Robust estimation for dynamic single index varying coefficient models

In this article, we propose a new estimation procedure based on modal regression for dynamic single index varying coefficient models, where the index coefficient functions and the link functions are approximated by B-spline basis, respectively. Under some mild regularity conditions, we establish the asymptotic normalities of the obtained spline-based estimators. By introducing an additional tuning parameter (e.g. h), the proposed method can produce robust estimates when the data are not well behaved (e.g. in the presence of outliers and/or heavy-tailed error distributions), and we further calculate the robust estimates by using a modified expectation-maximization-type iteration algorithm, and develop a data driven procedure to select the tuning parameters for the proposed approaches. Finally, some simulations and a real data application are conducted to illustrate the utility of the proposed methodology.

Robust Variable Selection with Exponential Squared Loss for the Spatial Single-Index Varying-Coefficient Model

As spatial correlation and heterogeneity often coincide in the data, we propose a spatial single-index varying-coefficient model. For the model, in this paper, a robust variable selection method based on spline estimation and exponential squared loss is offered to estimate parameters and identify significant variables. We establish the theoretical properties under some regularity conditions. A block coordinate descent (BCD) algorithm with the concave-convex process (CCCP) is composed uniquely for solving algorithms. Simulations show that our methods perform well even though observations are noisy or the estimated spatial mass matrix is inaccurate.

Correction: Robust MAVE for single-index varying-coefficient models

Correction: Robust MAVE for single-index varying-coefficient models

Penalized estimation of a class of single-index varying-coefficient models for integrative genomic analysis.

Recent technological advances have made it possible to collect high-dimensional genomic data along with clinical data on a large number of subjects. In the studies of chronic diseases such as cancer, it is of great interest to integrate clinical and genomic data to build a comprehensive understanding of the disease mechanisms. Despite extensive studies on integrative analysis, it remains an ongoing challenge to model the interaction effects between clinical and genomic variables, due to high dimensionality of the data and heterogeneity across data types. In this paper, we propose an integrative approach that models interaction effects using a single-index varying-coefficient model, where the effects of genomic features can be modified by clinical variables. We propose a penalized approach for separate selection of main and interaction effects. Notably, the proposed methods can be applied to right-censored survival outcomes based on a Cox proportional hazards model. We demonstrate the advantages of the proposed methods through extensive simulation studies and provide applications to a motivating cancer genomicstudy.

Robust Variable Selection for Single-Index Varying-Coefficient Model with Missing Data in Covariates

As applied sciences grow by leaps and bounds, semiparametric regression analyses have broad applications in various fields, such as engineering, finance, medicine, and public health. Single-index varying-coefficient model is a common class of semiparametric models due to its flexibility and ease of interpretation. The standard single-index varying-coefficient regression models consist mainly of parametric regression and semiparametric regression, which assume that all covariates can be observed. The assumptions are relaxed by taking the models with missing covariates into consideration. To eliminate the possibility of bias due to missing data, we propose a probability weighted objective function. In this paper, we investigate the robust variable selection for a single-index varying-coefficient model with missing covariates. Using parametric and nonparametric estimates of the likelihood of observations with fully observed covariates, we examine the estimators for estimating the likelihood of observations. For variable selection, we use a weighted objective function penalized by a non-convex SCAD. Theoretical challenges include the treatment of missing data and a single-index varying-coefficient model that uses both the non-smooth loss function and the non-convex penalty function. We provide Monte Carlo simulations to evaluate the performance of our approach.

Local Walsh-average regression for single index varying coefficient models

The local Walsh-average regression has been shown to be a comparable method to the others in nonparametric regression. In this paper, we develop a new estimation procedure based on local Walsh-average regression for single index varying coefficient models. Under some general assumptions, we establish the asymptotic properties of the proposed estimators for both the parametric and nonparametric parts. We further demonstrate that the proposed estimates have great efficiency gains across a wide spectrum of non-normal error distributions and almost not lose any efficiency for the normal error compared with that of local polynomial regression estimates under the least squares loss. Even in the worst case scenarios, the asymptotic relative efficiency owns a lower bound equaling to 0.864 for estimating the single-index parameter and a lower bound being 0.8896 for estimating the nonparametric link functions respectively, versus the least squares estimators. A simulation study is conducted to assess the finite sample properties of the proposed procedure, and a real data analysis is followed to further illustrate the application of the proposed methodology.

Statistical inference for a single-index varying coefficient model with measurement errors in all covariates

In this paper, a single-index varying coefficient model with measurement errors in all covariates is investigated. We propose a new approach to estimate the link function and the index parameter β, which combines the simulation-extrapolation (SIMEX) technique with local linear regression and estimating equation. Under some regularity conditions, large sample properties of the proposed estimators are established. Simulation studies and an application are presented to evaluate the proposed method.

Extreme quantile regression for tail single-index varying-coefficient models

This paper studies the estimation of extreme conditional quantile for single-index varying-coefficient models (SIVCM). Our contribution is a three-step extreme quantile regression procedure for SIVCM. In the first step, we estimate the intermediate conditional quantile in a quantile regression framework. In the second step, we estimate the extreme value index (EVI) based on the estimated intermediate quantile. In the third step, we extrapolate the intermediate conditional quantile to the extreme tails by adapting the extreme value theory (EVT). We obtain the asymptotation of tail single-index varying-coefficient estimators and demonstrate through simulation study and real data example that the proposed methods enjoy higher accuracy than the conventional quantile regression estimates.

Statistical estimation for single-index varying-coefficient models with multiplicative distortion measurement errors

During the observation, various factors can cause variables to be contaminated by measurement errors. In this paper, we consider the estimation for single-index varying-coefficient models where both the response variable and covariates are measured with multiplicative distortion measurement errors. First, we use the conditional absolute mean calibration to calibrate the distorted variables. Based on the calibrated variables, we obtain the estimators of the parameters and coefficient functions by local linear smoothing and estimating equations. Furthermore, we prove that the obtained estimators have asymptotic normality and the estimator of parametric component is still root-n consistent. Finally, two simulation examples and the analysis of body fat data are given to illustrate that the proposed estimators are unbiased, robust and practical.

Estimation for partially varying-coefficient single-index models with distorted measurement errors

Estimation for partially varying-coefficient single-index models with distorted measurement errors

Semiparametric model selection for identification of environmental covariates related to adult groundfish catches and weights

Ecologists and fisheries managers are interested in monitoring economically important marine fish species and using this data to inform management strategies. Determining environmental factors that best predict changes in these populations, particularly under rapid climate change, are a priority. I illustrate the application of the least squares-based spline estimation and group LASSO (LSSGLASSO) procedure for selection of coefficient functions in single index varying coefficient models (SIVCMs) on an ecological data set that includes spatiotemporal environmental covariates suspected to play a role in the catches and weights of six groundfish species. Temporal trends in variable selection were apparent, though the selection of variables was largely unrelated to common North Pacific climate indices. These results indicate that the strength of an environmental variable’s effect on a groundfish population may change over time, and not necessarily in-step with known low-frequency patterns of ocean-climate variability commonly attributable to large-scale regime shifts in the North Pacific. My application of the LSSGLASSO procedure for SIVCMs to deep water species using environmental data from various sources illustrates how variable selection with a flexible model structure can produce informative inference for remote and hard-to-reach animal populations.

Model Averaging Estimation for Varying-Coefficient Single-Index Models

The varying-coefficient single-index model (VCSIM) is widely used in economics, statistics and biology. A model averaging method for VCSIM based on a Mallows-type criterion is proposed to improve prodictive capacity, which allows the number of candidate models to diverge with sample size. Under model misspecification, the asymptotic optimality is derived in the sense of achieving the lowest possible squared errors. The authors compare the proposed model averaging method with several other classical model selection methods by simulations and the corresponding results show that the model averaging estimation has a outstanding performance. The authors also apply the method to a real dataset.

Single-index varying-coefficient models with missing covariates at random

In this article, we consider the problem of estimation of the single-index varying-coefficient model when covariates are not fully observed. By using the bias-correction and inverse selection probability methods, a weighted estimating equations estimator for the index parameters with missing covariates is constructed, and its asymptotic properties has been established. The local linear estimator for the coefficient functions is proved to converge at an optimal rate. Numerical studies based on simulation and application suggest that the proposed estimation procedure is powerful and easy to implement.

Model identification and selection for single-index varying-coefficient models

Single-index varying-coefficient models include many types of popular semiparametric models, i.e., single-index models, partially linear models, varying coefficient models, and so on. In this paper, a two-stage efficient variable selection procedure is proposed to select important nonparametric and parametric components and obtain estimators simultaneously. We also find that the proposed procedure can separate predictors into varying-coefficient and constant-coefficient predictors automatically. Theoretically, it has the selection and estimation consistency properties. Simulation studies and a real data application are conducted to evaluate and illustrate the proposed methods.

Principal single-index varying-coefficient models for dimension reduction in quantile regression

ABSTRACTWe propose a principal single-index varying-coefficient model focusing on conditional quantiles. In this general and flexible class of models, dimension reduction is achieved in three aspects: first, standard varying-coefficient models can partially avoid curse of dimensionality of large dimensional nonparametric regression; second, a one-dimensional adaptive index is constructed from multiple index variables; finally, the number of independent functions is further reduced by using principal functions. We derive the convergence rate of the estimates and asymptotic normality of the index parameter and the coefficient functions. Penalization can be added straightforwardly to obtain joint variable selection and dimension reduction. Simulations are used to demonstrate the performances and an empirical application is presented.

Estimation for a partially linear single-index varying-coefficient model

In this paper, we study the estimation for the partial linear single-index varying-coefficient model, which is a natural extension of the partially linear varying-coefficient model. A stepwise estimation procedure is developed to obtain asymptotic normality estimators of the index parameter vector, the coefficient parameter vector, and the coefficient function vector. The asymptotic properties of the resulting estimators are established under some conditions. A simulation study is conducted to assess the performance of the stepwise estimation procedure and the results show that our proposed procedure performs well in finite samples. Furthermore, a real data example is also used to illustrate our proposed method.

Feature Screening for Ultrahigh-dimensional Censored Data with Varying Coefficient Single-index Model

In this paper, we study the sure independence screening of ultrahigh-dimensional censored data with varying coefficient single-index model. This general model framework covers a large number of commonly used survival models. The property that the proposed method is not derived for a specific model is appealing in ultrahigh dimensional regressions, as it is difficult to specify a correct model for ultrahigh dimensional predictors. Once the assuming data generating process does not meet the actual one, the screening method based on the model will be problematic. We establish the sure screening property and consistency in ranking property of the proposed method. Simulations are conducted to study the finite sample performances, and the results demonstrate that the proposed method is competitive compared with the existing methods. We also illustrate the results via the analysis of data from The National Alzheimers Coordinating Center (NACC).