Regression Analysis of Current Status Data Under the Additive Hazards Model with Auxiliary Covariates

This paper discusses regression analysis of current status or case I interval‐censored failure time data arising from the additive hazards model. In this situation, some covariates could be missing because of various reasons, but there may exist some auxiliary information about the missing covariates. To address the problem, we propose an estimated partial likelihood approach for estimation of regression parameters, which makes use of the available auxiliary information. The method can be easily implemented, and the asymptotic properties of the resulting estimates are established. To assess the finite sample performance of the proposed method, an extensive simulation study is conducted and indicates that the method works well.     

Additive hazards regression of current status data with auxiliary covariates

This paper discusses the regression analysis of current status failure time data arising from the additive hazards model with auxiliary covariates. As often occurs in practice, it is impossible or impractical to measure the exact magnitude of covariates for all subjects in a study. To compensate the missing information, some auxiliary covariates are utilized instead. We propose two easy-to-implement procedures for estimation of regression parameters by making use of auxiliary information. The asymptotic properties of the resulting estimators are established and extensive numerical studies indicate that both procedures work well in practice.     

Marginal hazard regression for correlated failure time data with auxiliary covariates

In many biomedical studies, it is common that due to budget constraints, the primary covariate is only collected in a randomly selected subset from the full study cohort. Often, there is an inexpensive auxiliary covariate for the primary exposure variable that is readily available for all the cohort subjects. Valid statistical methods that make use of the auxiliary information to improve study efficiency need to be developed. To this end, we develop an estimated partial likelihood approach for correlated failure time data with auxiliary information. We assume a marginal hazard model with common baseline hazard function. The asymptotic properties for the proposed estimators are developed. The proof of the asymptotic results for the proposed estimators is nontrivial since the moments used in estimating equation are not martingale-based and the classical martingale theory is not sufficient. Instead, our proofs rely on modern empirical process theory. The proposed estimator is evaluated through simulation studies and is shown to have increased efficiency compared to existing methods. The proposed method is illustrated with a data set from the Framingham study.     

Penalized empirical likelihood for quantile regression with missing covariates and auxiliary information

Based on the inverse probability weight method, we, in this article, construct the empirical likelihood (EL) and penalized empirical likelihood (PEL) ratios of the parameter in the linear quantile regression model when the covariates are missing at random, in the presence and absence of auxiliary information, respectively. It is proved that the EL ratio admits a limiting Chi-square distribution. At the same time, the asymptotic normality of the maximum EL and PEL estimators of the parameter is established. Also, the variable selection of the model in the presence and absence of auxiliary information, respectively, is discussed. Simulation study and a real data analysis are done to evaluate the performance of the proposed methods.     

Auxiliary covariate in additive hazards regression for survival data

We consider the additive hazards regression analysis by utilising auxiliary covariate information to improve the efficiency of the statistical inference when the primary covariate is ascertained only for a randomly selected subsample. We construct a martingale-based estimating equation for the regression parameter and establish the asymptotic consistency and normality of the resultant estimator. Simulation study shows that our proposed method can improve the efficiency compared with the estimator which discards the auxiliary covariate information. A real example is also analysed as an illustration.     

Using Auxiliary Data for Binomial Parameter Estimation with Nonignorable Nonresponse

Nonignorable nonresponse is a nonresponse mechanism that depends on the values of the variable having nonresponse. When an observed data of a binomial distribution suffer missing values from a nonignorable nonresponse mechanism, the binomial distribution parameters become unidentifiable without any other auxiliary information or assumption. To address the problems of non identifiability, existing methods mostly based on the log-linear regression model. In this article, we focus on the model when the nonresponse is nonignorable and we consider to use the auxiliary data to improve identifiability; furthermore, we derive the maximum likelihood estimator (MLE) for the binomial proportion and its associated variance. We present results for an analysis of real-life data from the SARS study in China. Finally, the simulation study shows that the proposed method gives promising results.     

基于半参数方法的模型辅助抽样估计研究

在总结现有模型辅助估计方法的基础上,本文通过构造一种半参数超总体模型,同时结合广义差分估计思想提出一种新型的模型辅助估计量。该估计量比传统的非参数和半参数回归估计利用更少、更易得到的辅助信息,即只需利用和广义回归估计相同的辅助信息,但一般会比广义回归估计拥有更高的估计精度。理论证明了该估计量是渐近设计无偏和设计一致的,其渐近设计均方误差为广义差分估计量的方差。模拟结果显示：其至少与广义回归估计一样好;对于线性程度越低的超总体模型,其估计精度比广义回归估计有越明显的提高;就本文模拟而言,光滑参数在0.04～0.12间适当取值时其会取到相对较好的估计效果。     

Analysis of two-phase sampling data with semiparametric additive hazards models

Under the case-cohort design introduced by Prentice (Biometrica 73:1–11, 1986), the covariate histories are ascertained only for the subjects who experience the event of interest (i.e., the cases) during the follow-up period and for a relatively small random sample from the original cohort (i.e., the subcohort). The case-cohort design has been widely used in clinical and epidemiological studies to assess the effects of covariates on failure times. Most statistical methods developed for the case-cohort design use the proportional hazards model, and few methods allow for time-varying regression coefficients. In addition, most methods disregard data from subjects outside of the subcohort, which can result in inefficient inference. Addressing these issues, this paper proposes an estimation procedure for the semiparametric additive hazards model with case-cohort/two-phase sampling data, allowing the covariates of interest to be missing for cases as well as for non-cases. A more flexible form of the additive model is considered that allows the effects of some covariates to be time varying while specifying the effects of others to be constant. An augmented inverse probability weighted estimation procedure is proposed. The proposed method allows utilizing the auxiliary information that correlates with the phase-two covariates to improve efficiency. The asymptotic properties of the proposed estimators are established. An extensive simulation study shows that the augmented inverse probability weighted estimation is more efficient than the widely adopted inverse probability weighted complete-case estimation method. The method is applied to analyze data from a preventive HIV vaccine efficacy trial.     

Inference on finite population categorical response: nonparametric regression-based predictive approach

Suppose that a finite population consists of N distinct units. Associated with the ith unit is a polychotomous response vector, d _i , and a vector of auxiliary variable x _i . The values x _i ’s are known for the entire population but d _i ’s are known only for the units selected in the sample. The problem is to estimate the finite population proportion vector P. One of the fundamental questions in finite population sampling is how to make use of the complete auxiliary information effectively at the estimation stage. In this article a predictive estimator is proposed which incorporates the auxiliary information at the estimation stage by invoking a superpopulation model. However, the use of such estimators is often criticized since the working superpopulation model may not be correct. To protect the predictive estimator from the possible model failure, a nonparametric regression model is considered in the superpopulation. The asymptotic properties of the proposed estimator are derived and also a bootstrap-based hybrid re-sampling method for estimating the variance of the proposed estimator is developed. Results of a simulation study are reported on the performances of the predictive estimator and its re-sampling-based variance estimator from the model-based viewpoint. Finally, a data survey related to the opinions of 686 individuals on the cause of addiction is used for an empirical study to investigate the performance of the nonparametric predictive estimator from the design-based viewpoint.     

Some comments on efficiency gains from auxiliary information for right-censored data

Heavily right-censored time to event, or survival, data arise frequently in research areas such as medicine and industrial reliability. Recently, there have been suggestions that auxiliary outcomes which are more fully observed may be used to “enhance” or increase the efficiency of inferences for a primary survival time variable. However, efficiency gains from this approach have mostly been very small. Most of the situations considered have involved semiparametric models, so in this note we consider two very simple fully parametric models. In the one case involving a correlated auxiliary variable that is always observed, we find that efficiency gains are small unless the response and auxiliary variable are very highly correlated and the response is heavily censored. In the second case, which involves an intermediate stage in a three-stage model of failure, the efficiency gains can be more substantial. We suggest that careful study of specific situations is needed to identify opportunities for “enhanced” inferences, but that substantial gains seem more likely when auxiliary information involves structural information about the failure process.     

Proportional Rate Model with Incomplete Covariate and Complete Auxiliary Information

This paper deals with the analysis of proportional rate model for recurrent event data when covariates are subject to missing. The true covariate is measured only on a randomly chosen validation set, whereas auxiliary information is available for all cohort subjects. To further utilize the auxiliary information to improve study efficiency, we propose an estimated estimating equation for the regression parameters. The resulting estimators are shown to be consistent and asymptotically normal. Both graphical and numerical techniques for checking the adequacy of the model are presented. Simulations are conducted to evaluate the finite sample performance of the proposed estimators. Illustration with a real medical study is provided.     

Failure time regression with continuous covariates measured with error

We consider failure time regression analysis with an auxiliary variable in the presence of a validation sample. We extend the nonparametric inference procedure of Zhou and Pepe to handle a continuous auxiliary or proxy covariate. We estimate the induced relative risk function with a kernel smoother and allow the selection probability of the validation set to depend on the observed covariates. We present some asymptotic properties for the kernel estimator and provide some simulation results. The method proposed is illustrated with a data set from an on-going epidemiologic study.     

Estimation of population mean based on dual use of auxiliary information in non response

Whenever there is auxiliary information available in any form, the researchers want to utilize it in the method of estimation to obtain the most efficient estimator. When there exists enough amount of correlation between the study and the auxiliary variables, and parallel to these associations, the ranks of the auxiliary variables are also correlated with the study variable, which can be used a valuable device for enhancing the precision of an estimator accordingly. This article addresses the problem of estimating the finite population mean that utilizes the complementary information in the presence of (i) the auxiliary variable and (ii) the ranks of the auxiliary variable for non response. We suggest an improved estimator for estimating the finite population mean using the auxiliary information in the presence of non response. Expressions for bias and mean squared error of considered estimators are derived up to the first order of approximation. The performance of estimators is compared theoretically and numerically. A numerical study is carried out to evaluate the performances of estimators. It is observed that the proposed estimator is more efficient than the usual sample mean and the regression estimators, and some other families of ratio and exponential type of estimators.     

Estimated Generalized Estimating Equation for Correlated Failure Time Data with Auxiliary Covariates

In this article, we study a marginal hazard model with common baseline hazard for correlated failure time data. We assume that the true covariate is measured precisely in a subset of the whole study cohort, whereas an auxiliary information for the true covariate is available for the whole cohort. We first estimate the relative risk function empirically. Then we obtain the estimator for the regression parameter by replacing the relative risk function with its estimator in a generalized estimating equation (GEE) proposed by Cai (1992 Cai , J. ( 1992 ). Generalized estimation equations for censored multivariate failure time data. Ph.D. thesis, University of Washington, Seattle, Washington . [Google Scholar]). A key feature of this method is that it is nonparametric with respect to the association between the missing covariate and the observed auxiliary covariate. The proposed estimator is shown to be consistent and asymptotically normal. Furthermore, we present a corrected Breslow-type estimator for the cumulative hazard function. Simulation studies are conducted to evaluate the proposed method.     

Fitting semiparametric accelerated failure time models for nested case–control data

A nested case–control (NCC) study is an efficient cohort-sampling design in which a subset of controls are sampled from the risk set at each event time. Since covariate measurements are taken only for the sampled subjects, time and efforts of conducting a full scale cohort study can be saved. In this paper, we consider fitting a semiparametric accelerated failure time model to failure time data from a NCC study. We propose to employ an efficient induced smoothing procedure for rank-based estimating method for regression parameters estimation. For variance estimation, we propose to use an efficient resampling method that utilizes the robust sandwich form. We extend our proposed methods to a generalized NCC study that allows a sampling of cases. Finite sample properties of the proposed estimators are investigated via an extensive stimulation study. An application to a tumor study illustrates the utility of the proposed method in routine data analysis.     

Generalized M-estimation for the accelerated failure time model

The accelerated failure time (AFT) model is an important regression tool to study the association between failure time and covariates. In this paper, we propose a robust weighted generalized M (GM) estimation for the AFT model with right-censored data by appropriately using the Kaplan–Meier weights in the GM–type objective function to estimate the regression coefficients and scale parameter simultaneously. This estimation method is computationally simple and can be implemented with existing software. Asymptotic properties including the root-n consistency and asymptotic normality are established for the resulting estimator under suitable conditions. We further show that the method can be readily extended to handle a class of nonlinear AFT models. Simulation results demonstrate satisfactory finite sample performance of the proposed estimator. The practical utility of the method is illustrated by a real data example.     

MODEL-ASSISTED HIGHER-ORDER CALIBRATION OF ESTIMATORS OF VARIANCE

In survey sampling, interest often centres on inference for the population total using information about an auxiliary variable. The variance of the estimator used plays a key role in such inference. This study develops a new set of higher‐order constraints for the calibration of estimators of variance for various estimators of the population total. The proposed strategy requires an appropriate model for describing the relationship between the response and auxiliary variable, and the variance of the auxiliary variable. It is therefore referred to as a model‐assisted approach. Several new estimators of variance, including the higher‐order calibration estimators of the variance of the ratio and regression estimators suggested by Singh, Horn & Yu and Sitter & Wu are special cases of the proposed technique. The paper presents and discusses the results of an empirical study to compare the performance of the proposed estimators and existing counterparts.     

Bootstrapping with auxiliary information

In the nonparametric setting, the standard bootstrap method is based on the empirical distribution function of a random sample. The author proposes, by means of the empirical likelihood technique, an alternative bootstrap procedure under a nonparametric model in which one has some auxiliary information about the population distribution. By proving the almost sure weak convergence of the modified bootstrapped empirical process, the validity of the proposed bootstrap procedure is established. This new result is used to obtain bootstrap confidence bands for the population distribution function and to perform the bootstrap Kolmogorov test in the presence of auxiliary information. Other applications include bootstrapping means and variances with auxiliary information. Three simulation studies are presented to demonstrate the performance of the proposed bootstrap procedure for small samples.     

Covariate selection for accelerated failure time data

Selection of appropriate predictors for right censored time to event data is very often encountered by the practitioners. We consider the ?₁ penalized regression or “least absolute shrinkage and selection operator” as a tool for predictor selection in association with accelerated failure time model. The choice of the penalizing parameter λ is crucial to identify the correct set of covariates. In this paper, we propose an information theory-based method to choose λ under log-normal distribution. Furthermore, an efficient algorithm is discussed in the same context. The performance of the proposed λ and the algorithm is illustrated through simulation studies and a real data analysis. The convergence of the algorithm is also discussed.