Class Of Semiparametric Models Research Articles

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.

A class of proportional win-fractions regression models for composite outcomes.

The win ratio is gaining traction as a simple and intuitive approach to analysis of prioritized composite endpoints in clinical trials. To extend it from two-sample comparison to regression, we propose a novel class of semiparametric models that includes as special cases both the two-sample win ratio and the traditional Cox proportional hazards model on time to the first event. Under the assumption that the covariate-specific win and loss fractions are proportional over time, the regression coefficient is unrelated to the censoring distribution and can be interpreted as the log win ratio resulting from one-unit increase in the covariate. U-statistic estimating functions, in the form of an arbitrary covariate-specific weight process integrated by a pairwise residual process, are constructed to obtain consistent estimators for the regression parameter. The asymptotic properties of the estimators are derived using uniform weak convergence theory for U-processes. Visual inspection of a "score" process provides useful clues as to the plausibility of the proportionality assumption. Extensive numerical studies using both simulated and real data from a major cardiovascular trial show that the regression methods provide valid inference on covariate effects and outperform the two-sample win ratio in both efficiency and robustness. The proposed methodology is implemented in the R-package WR, publicly available from the Comprehensive R Archive Network (CRAN).

Jump-preserving varying-coefficient models for nonlinear time series

An important and widely used class of semiparametric models is formed by the varying-coefficient models. Although the varying coefficients are traditionally assumed to be smooth functions, the varying-coefficient model is considered here with the coefficient functions containing a finite set of discontinuities. Contrary to the existing nonparametric and varying-coefficient estimation of piecewise smooth functions, the varying-coefficient models are considered here under dependence and are applicable in time series with heteroskedastic and serially correlated errors. Additionally, the conditional error variance is allowed to exhibit discontinuities at a finite set of points too. The (uniform) consistency and asymptotic normality of the proposed estimators are established and the finite-sample performance is tested via a simulation study and in a real-data example.

Tree-based modeling of time-varying coefficients in discrete time-to-event models.

Hazard models are popular tools for the modeling of discrete time-to-event data. In particular two approaches for modeling time dependent effects are in common use. The more traditional one assumes a linear predictor with effects of explanatory variables being constant over time. The more flexible approach uses the class of semiparametric models that allow the effects of the explanatory variables to vary smoothly over time. The approach considered here is in between these modeling strategies. It assumes that the effects of the explanatory variables are piecewise constant. It allows, in particular, to evaluate at which time points the effect strength changes and is able to approximate quite complex variations of the change of effects in a simple way. A tree-based method is proposed for modeling the piecewise constant time-varying coefficients, which is embedded into the framework of varying-coefficient models. One important feature of the approach is that it automatically selects the relevant explanatory variables and no separate variable selection procedure is needed. The properties of the method are investigated in several simulation studies and its usefulness is demonstrated by considering two real-world applications.

Jump-Preserving Varying-Coefficient Models for Nonlinear Time Series

An important and widely used class of semiparametric models is formed by the varying-coefficient models. Although the varying coefficients are traditionally assumed to be smooth functions, the varying-coefficient model is considered here with the coefficient functions containing a finite set of discontinuities. Contrary to the existing nonparametric and varying-coefficient estimation of piecewise smooth functions, the varying-coefficient models are considered here under dependence and are applicable in time series with heteroscedastic and serially correlated errors. Additionally, the conditional error variance is allowed to exhibit discontinuities at a finite set of points too. The (uniform) consistency and asymptotic normality of the proposed estimators are established and the finite-sample performance is tested via a simulation study.

Robust estimation for varying index coefficient models

Varying index coefficient models (VICMs) proposed by Ma and Song (J Am Stat Assoc, 2014. doi:10.1080/01621459.2014.903185) are a new class of semiparametric models, which encompass most of the existing semiparametric models. So far, only the profile least squares method and local linear fitting were developed for the VICM, which are very sensitive to the outliers and will lose efficiency for the heavy tailed error distributions. In this paper, we propose an efficient and robust estimation procedure for the VICM based on modal regression which depends on a bandwidth. We establish the consistency and asymptotic normality of proposed estimators for index coefficients by utilizing profile spline modal regression method. The oracle property of estimators for the nonparametric functions is also established by utilizing a two-step spline backfitted local linear modal regression approach. In addition, we discuss the bandwidth selection for achieving better robustness and efficiency and propose a modified expectation---maximization-type algorithm for the proposed estimation procedure. Finally, simulation studies and a real data analysis are carried out to assess the finite sample performance of the proposed method.

Varying Index Coefficient Models

It has been a long history of using interactions in regression analysis to investigate alterations in covariate-effects on response variables. In this article, we aim to address two kinds of new challenges arising from the inclusion of such high-order effects in the regression model for complex data. The first kind concerns a situation where interaction effects of individual covariates are weak but those of combined covariates are strong, and the other kind pertains to the presence of nonlinear interactive effects directed by low-effect covariates. We propose a new class of semiparametric models with varying index coefficients, which enables us to model and assess nonlinear interaction effects between grouped covariates on the response variable. As a result, most of the existing semiparametric regression models are special cases of our proposed models. We develop a numerically stable and computationally fast estimation procedure using both profile least squares method and local fitting. We establish both estimation consistency and asymptotic normality for the proposed estimators of index coefficients as well as the oracle property for the nonparametric function estimator. In addition, a generalized likelihood ratio test is provided to test for the existence of interaction effects or the existence of nonlinear interaction effects. Our models and estimation methods are illustrated by simulation studies, and by an analysis of child growth data to evaluate alterations in growth rates incurred by mother’s exposures to endocrine disrupting compounds during pregnancy. Supplementary materials for this article are available online.

Maximum likelihood estimation of semiparametric mixture component models for competing risks data.

In the analysis of competing risks data, the cumulative incidence function is a useful quantity to characterize the crude risk of failure from a specific event type. In this article, we consider an efficient semiparametric analysis of mixture component models on cumulative incidence functions. Under the proposed mixture model, latency survival regressions given the event type are performed through a class of semiparametric models that encompasses the proportional hazards model and the proportional odds model, allowing for time-dependent covariates. The marginal proportions of the occurrences of cause-specific events are assessed by a multinomial logistic model. Our mixture modeling approach is advantageous in that it makes a joint estimation of model parameters associated with all competing risks under consideration, satisfying the constraint that the cumulative probability of failing from any cause adds up to one given any covariates. We develop a novel maximum likelihood scheme based on semiparametric regression analysis that facilitates efficient and reliable estimation. Statistical inferences can be conveniently made from the inverse of the observed information matrix. We establish the consistency and asymptotic normality of the proposed estimators. We validate small sample properties with simulations and demonstrate the methodology with a data set from a study of follicular lymphoma.

Polynomial spline estimation for generalized varying coefficient partially linear models with a diverging number of components

Generalized varying coefficient partially linear models are a flexible class of semiparametric models that deal with data with different types of responses. In this paper, we focus on polynomial spline estimator as a computationally easier alternative to the more commonly used local polynomial regression approach, since one can directly take advantage of many existing implementations for generalized linear models. Furthermore, motivated by the high dimensionality characteristics that accompany many modern data sets nowadays, we investigate its asymptotic properties when both the number of nonparametric and the number of parametric components grows with, but is still smaller than, the sample size. Simulations and a real data example are used to illustrate our proposal.

Are generalized additive models for location, scale, and shape an improvement on existing models for estimating skewed and heteroskedastic cost data?

Generalized additive models for location, scale, and shape (GAMLSS) are a class of semi-parametric models with potential applicability to health care cost data. We compared the bias, accuracy, and coverage of GAMLSS estimators with two distributions [gamma and generalized inverse gaussian (GIG)] using a log link to the generalized linear model (GLM) with log link and gamma family and the log-transformed OLS. The evaluation using simulated gamma data showed that the GAMLSS and GLM gamma model had similar bias, accuracy, and coverage and outperformed the GAMLSS GIG. When applied to simulated GIG data, the GLM gamma was similar or improved in bias, accuracy, and coverage compared to the GAMLSS GIG and gamma; furthermore, the GAMLSS estimators produced wildly inaccurate or overly-precise results in certain circumstances. Applying all models to empirical data on health care costs after a fall-related injury, all estimators produced similar coefficient estimates, but GAMLSS estimators produced spuriously smaller standard errors. Although no single alternative was best for all simulations, the GLM gamma was the most consistent, so we recommend against using GAMLSS estimators using GIG or gamma to test for differences in mean health care costs. Since GAMLSS offers many other flexible distributions, future work should evaluate whether GAMLSS is useful when predicting health care costs.

Estimation and model selection in a class of semiparametric models for cluster data

Stimulated by a study in Bangladesh about the first birth interval, we propose a semivarying-coefficient model for cluster data analysis. We consider the estimation procedure for the proposed model and establish the asymptotic results of the proposed estimators. Furthermore, we employ the cross-validation (CV) to identify the constant coefficients. The associated asymptotic properties are rigorously examined. Simulation studies are conducted to investigate the performance of the proposed estimation and the CV-based model selection procedure for finite sample size. Finally, our methods are used to analyse the aforementioned data set to explore how several factors affect the first birth interval in Bangladesh.

Relative risk regression for current status data in case-cohort studies

We propose using the weighted likelihood method to fit a general relative risk regression model for the current status data with missing data as arise, for example, in case-cohort studies. The missingness probability is either known or can be reasonably estimated. Asymptotic properties of the weighted likelihood estimators are established. For the case of using estimated weights, we construct a general theorem that guarantees the asymptotic normality of the M-estimator of a finite dimensional parameter in a class of semiparametric models, where the infinite dimensional parameter is allowed to converge at a slower than parametric rate, and some other parameters in the objective function are estimated a priori. The weighted bootstrap method is employed to estimate the variances. Simulations show that the proposed method works well for finite sample sizes. A motivating example of the case-cohort study from an HIV vaccine trial is used to demonstrate the proposed method. The Canadian Journal of Statistics 39: 557-577; 2011 © 2011 Statistical Society of Canada R´

Bandwidth selection through cross-validation for semi-parametric varying-coefficient partially linear models

We study bandwidth selection for a class of semi-parametric models. The proper choice of optimal bandwidth minimizes the prediction errors of the model. We provide detailed derivation of our procedure and the corresponding computation algorithms. Our proposed method simplifies the computation of the cross-validation criteria and facilitates more complicated inference and analysis in practice. A data set from Wisconsin Diabetes Registry has been analysed as an illustration.

Proportional functional coefficient time series models

In this paper, we study a new class of semiparametric models, termed as the proportional functional-coefficient linear regression models for time series data. The model can be viewed as a generalization of the functional-coefficient regression models but it has different proportional functions of parameter and different smoothing variables in the same coefficient function in different position. When the parameter is known, the local linear technique is employed to give the initial estimator of the coefficient function in the model, which does not share the optimal rate of convergence. To improve its convergent rate, a one-step backfitting technique is used to obtain the optimal estimator of the coefficient function. The asymptotic properties of the proposed estimators are investigated. When the parameter is unknown, the method of estimating parameter is given. It can be shown that the estimator of the parameter is n -consistent. The bandwidths and the smoothing variables are selected by a data-driven method. A simulated example with two cases and two real data examples are used to illustrate the applications of the model.

Semiparametric Models for Missing Covariate and Response Data in Regression Models

We consider a class of semiparametric models for the covariate distribution and missing data mechanism for missing covariate and/or response data for general classes of regression models including generalized linear models and generalized linear mixed models. Ignorable and nonignorable missing covariate and/or response data are considered. The proposed semiparametric model can be viewed as a sensitivity analysis for model misspecification of the missing covariate distribution and/or missing data mechanism. The semiparametric model consists of a generalized additive model (GAM) for the covariate distribution and/or missing data mechanism. Penalized regression splines are used to express the GAMs as a generalized linear mixed effects model, in which the variance of the corresponding random effects provides an intuitive index for choosing between the semiparametric and parametric model. Maximum likelihood estimates are then obtained via the EM algorithm. Simulations are given to demonstrate the methodology, and a real data set from a melanoma cancer clinical trial is analyzed using the proposed methods.

Tests against inequality constraints in semiparametric models

We consider the problem of testing that a subvector of a finite dimensional parameter is zero against that it satisfies some inequality constraints in a semiparametric model. To our knowledge, this problem has not been studied in the literature in a general context. Huang (Ann. Statist. 24 (1996) 540) developed maximum likelihood estimation in a class of semiparametric models, and Choi et al. (Ann. Statist. 24 (1996) 84) developed efficient tests against unrestricted alternatives in a large class of semiparametric models which includes the adaptable ones. In this paper, we introduce local tests as a general approach to developing tests against inequality constraints, and apply them in different semiparametric models. In particular, we apply our local tests to develop adaptive tests and efficient tests in the classes of semiparametric models studied by Huang (1996) and Choi et al. (1996), thus extending their results to inequality constrained framework.

Asymptotical Analysis of Semiparametric Models in Survival Analysis and Accelerated Life Testing

Classes of semiparametric models, generalizing the proportional and additive hazards, proportional odds and other models are considered. Asymptotical properties of estimators of regression parameters and baseline survival function arc investigated in the case of one of considered classes.

Efficient estimation for the proportional hazards model with interval censoring

The maximum likelihood estimator (MLE) for the proportional hazards model with case 1 interval censored data is studied. It is shown that the MLE for the regression parameter is asymptotically normal with $\sqrt{n}$ convergence rate and achieves the information bound, even though the MLE for the baseline cumulative hazard function only converges at $n^{1/3}$ rate. Estimation of the asymptotic variance matrix for the MLE of the regression parameter is also considered. To prove our main results, we also establish a general theorem showing that the MLE of the finite-dimensional parameter in a class of semiparametric models is asymptotically efficient even though the MLE of the infinite-dimensional parameter converges at a rate slower than $\sqrt{n}$. The results are illustrated by applying them to a data set from a tumorigenicity study.

Nonparametric Methods for Stratified Two-Sample Designs with Application to Multiclinic Trials

Abstract Motivated by some problems arising from multiclinic trials, we consider stratified two-sample designs. Nonparametric effects are defined and nonparametric hypotheses are formulated in a design where treatment, centers (strata), and interactions are assumed to be fixed factors. The interpretation of the nonparametric effects and hypotheses is analyzed in two classes of semiparametric models: the linear models and models with Lehmann alternatives. The case where centers and interactions are assumed to be random factors, the so-called mixed model, is also considered. Nonparametric effects and hypotheses are defined for general models, and their properties are analyzed in corresponding linear models and in models with Lehmann alternatives. The nonparametric effects are estimated by linear rank statistics where the ranks over all centers are used. The mixed model for repeated (baseline and endpoint) observations is briefly considered, and rank procedures are also proposed for this model. All procedure...

Profile Likelihood and Conditionally Parametric Models

In this paper, we outline a general approach to estimating the parametric component of a semiparametric model. For the case of a scalar parametric component, the method is based on the idea of first estimating a one-dimensional subproblem of the original problem that is least favorable in the sense of Stein. The likelihood function for the scalar parameter along this estimated subproblem may be viewed as a generalization of the profile likelihood for the problem. The scalar parameter is then estimated by maximizing this generalized profile likelihood. This method of estimation is applied to a particular class of semiparametric models, where it is shown that the resulting estimator is asymptotically efficient.