Statistical inference under imputation for proportional hazard model with missing covariates

Missing covariate data are common in biomedical studies. In this article, by using the non parametric kernel regression technique, a new imputation approach is developed for the Cox-proportional hazard regression model with missing covariates. This method achieves the same efficiency as the fully augmented weighted estimators (Qi et al. 2005. Journal of the American Statistical Association, 100:1250) and has a simpler form. The asymptotic properties of the proposed estimator are derived and analyzed. The comparisons between the proposed imputation method and several other existing methods are conducted via a number of simulation studies and a mouse leukemia data.     

Asymptotic results for fitting marginal hazards models from stratified case-cohort studies with multiple disease outcomes

In stratified case-cohort designs, samplings of case-cohort samples are conducted via a stratified random sampling based on covariate information available on the entire cohort members. In this paper, we extended the work of Kang & Cai (2009) to a generalized stratified case-cohort study design for failure time data with multiple disease outcomes. Under this study design, we developed weighted estimating procedures for model parameters in marginal multiplicative intensity models and for the cumulative baseline hazard function. The asymptotic properties of the estimators are studied using martingales, modern empirical process theory, and results for finite population sampling.     

Maximum Likelihood Estimations and EM Algorithms with Length-biased Data

Length-biased sampling has been well recognized in economics, industrial reliability, etiology applications, epidemiological, genetic and cancer screening studies. Length-biased right-censored data have a unique data structure different from traditional survival data. The nonparametric and semiparametric estimations and inference methods for traditional survival data are not directly applicable for length-biased right-censored data. We propose new expectation-maximization algorithms for estimations based on full likelihoods involving infinite dimensional parameters under three settings for length-biased data: estimating nonparametric distribution function, estimating nonparametric hazard function under an increasing failure rate constraint, and jointly estimating baseline hazards function and the covariate coefficients under the Cox proportional hazards model. Extensive empirical simulation studies show that the maximum likelihood estimators perform well with moderate sample sizes and lead to more efficient estimators compared to the estimating equation approaches. The proposed estimates are also more robust to various right-censoring mechanisms. We prove the strong consistency properties of the estimators, and establish the asymptotic normality of the semi-parametric maximum likelihood estimators under the Cox model using modern empirical processes theory. We apply the proposed methods to a prevalent cohort medical study. Supplemental materials are available online.     

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.     

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.     

Regression analysis of case-cohort studies in the presence of dependent interval censoring

The case-cohort design is widely used as a means of reducing the cost in large cohort studies, especially when the disease rate is low and covariate measurements may be expensive, and has been discussed by many authors. In this paper, we discuss regression analysis of case-cohort studies that produce interval-censored failure time with dependent censoring, a situation for which there does not seem to exist an established approach. For inference, a sieve inverse probability weighting estimation procedure is developed with the use of Bernstein polynomials to approximate the unknown baseline cumulative hazard functions. The proposed estimators are shown to be consistent and the asymptotic normality of the resulting regression parameter estimators is established. A simulation study is conducted to assess the finite sample properties of the proposed approach and indicates that it works well in practical situations. The proposed method is applied to an HIV/AIDS case-cohort study that motivated this investigation.     

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.     

Cox regression in cohort studies with validation sampling

An estimation procedure is proposed for the Cox model in cohort studies with validation sampling, where crude covariate information is observed for the full cohort and true covariate information is collected on a validation set sampled randomly from the full cohort. The method proposed makes use of the partial information from data that are available on the entire cohort by fitting a working Cox model relating crude covariates to the failure time. The resulting estimator is consistent regardless of the specification of the working model and is asymptotically more efficient than the validation-set-only estimator. Approximate asymptotic relative efficiencies with respect to some alternative methods are derived under a simple scenario and further studied numerically. The finite sample performance is investigated and compared with alternative methods via simulation studies. A similar procedure also works for the case where the validation set is a stratified random sample from the cohort.     

The Efficiency of Simple and Counter-matched Nested Case-control Sampling

This paper presents a study of the performance of simple and counter-matched nested case-control sampling relative to a full cohort study. First we review methods for estimating the regression parameters and the integrated baseline hazard for Cox's proportional hazards model from cohort and case-control data. Then the asymptotic distributional properties of these estimators are recapitulated, and relative efficiency results are presented both for regression and baseline hazard estimation.     

Analysis of ordered covariate effects among groups with repeated measurements

Testing procedures for ordered covariate effects are developed in the repeated measures experiment. The maximum likelihood estimators of covariate effects under the ordered hypothesis are approximated by the isotonic regression of their unconstrained estimators. The asymptotic null distributions of the test statistics are chi-bar-square distributions which are mixtures of chi-square distributions. A Monte-Carlo simulation reveals that the proposed test for ordered covariate effects is seriously more powerful than the usual chi-square test that neglects the information on the order restriction. These testing methods are applied for analyzing the effect of vitamin E diet supplement on growth rate of animals.     

Marginal Regression Model with Time-Varying Coefficients for Panel Data

In this article, we formulate a class of semiparametric marginal means models with a mixture of time-varying and time-independent parameters for analyzing panel data. For inference about the regression parameters, an estimation procedure is developed and asymptotic properties of the proposed estimators are established. In addition, some tests are presented for investigating whether or not covariate effects vary with time. The finite-sample behavior of the proposed methods is examined in simulation studies, and the data from an AIDS clinical trial study are used to illustrate the methodology.     

A comparison of regression calibration approaches for designs with internal validation data

We compare the asymptotic relative efficiency of several regression calibration methods of correcting for measurement error in studies with internal validation data, when a single covariate is measured with error. The estimators we consider are appropriate in main study/hybrid validation study designs, where the latter study includes internal validation and may include external validation data. Although all of the methods we consider produce consistent estimates, the method proposed by Spiegelman et al. (Statistics in Medicine, 20 (2001) 139) has an asymptotically smaller variance than the other methods. The methods for measurement error correction are illustrated using a study of the effect of in utero lead exposure on infant birth weight.     

A class of partially linear transformation models for recurrent gap times

In this article, we propose a general class of partially linear transformation models for recurrent gap time data, which extends the linear transformation models by incorporating non linear covariate effects and includes the partially linear proportional hazards and the partially linear proportional odds models as special cases. Both global and local estimating equations are developed to estimate the parametric and non parametric covariate effects, and the asymptotic properties of the resulting estimators are established. The finite-sample behavior of the proposed estimators is evaluated through simulation studies, and an application to a clinic study on chronic granulomatous disease is provided.     

A class of chain ratio–product type estimators with two auxiliary variables under double sampling scheme

In this paper, a chain ratio–product type estimators has been developed for estimating population mean of the study variable using two auxiliary variables under double sampling scheme, when the information on another additional auxiliary character is available along with the main auxiliary character. The optimum property of the proposed strategy has been identified in two different cases with their variances. Theoretical and empirical studies have been done to demonstrate the efficiency of the proposed estimator over other estimators, which utilized the information on two auxiliary characteristics.     

Stratified proportional subdistribution hazards model with covariate-adjusted censoring weight for case-cohort studies

The case-cohort study design is widely used to reduce cost when collecting expensive covariates in large cohort studies with survival or competing risks outcomes. A case-cohort study dataset consists of two parts: (a) a random sample and (b) all cases or failures from a specific cause of interest. Clinicians often assess covariate effects on competing risks outcomes. The proportional subdistribution hazards model directly evaluates the effect of a covariate on the cumulative incidence function under the non-covariate-dependent censoring assumption for the full cohort study. However, the non-covariate-dependent censoring assumption is often violated in many biomedical studies. In this article, we propose a proportional subdistribution hazards model for case-cohort studies with stratified data with covariate-adjusted censoring weight. We further propose an efficient estimator when extra information from the other causes is available under case-cohort studies. The proposed estimators are shown to be consistent and asymptotically normal. Simulation studies show (a) the proposed estimator is unbiased when the censoring distribution depends on covariates and (b) the proposed efficient estimator gains estimation efficiency when using extra information from the other causes. We analyze a bone marrow transplant dataset and a coronary heart disease dataset using the proposed method.     

Efficient estimation for the conditional autoregressive model

Recently, spatial regression models have been attracting a great deal of attention in areas ranging from effect of traffic congestion on accident rates to the analysis of trends in gastric cancer mortality. In this paper, we propose efficient estimators for the regression coefficients of the spatial conditional autoregressive model, when uncertain auxiliary information is available about these coefficients. We provide efficiency comparisons of the proposed estimators based on asymptotic risk analysis and Monte Carlo simulations. We apply the proposed methods to real data on Boston housing prices and illustrate how a bootstrapping approach can be employed to compute prediction errors of the estimators.     

Reweighted estimators for additive hazard model with censoring indicators missing at random

Survival data with missing censoring indicators are frequently encountered in biomedical studies. In this paper, we consider statistical inference for this type of data under the additive hazard model. Reweighting methods based on simple and augmented inverse probability are proposed. The asymptotic properties of the proposed estimators are established. Furthermore, we provide a numerical technique for checking adequacy of the fitted model with missing censoring indicators. Our simulation results show that the proposed estimators outperform the simple and augmented inverse probability weighted estimators without reweighting. The proposed methods are illustrated by analyzing a dataset from a breast cancer study.     

Least squares estimation of varying‐coefficient hazard regression with application to breast cancer dose‐intensity data

To enhance modeling flexibility, the authors propose a nonparametric hazard regression model, for which the ordinary and weighted least squares estimation and inference procedures are studied. The proposed model does not assume any parametric specifications on the covariate effects, which is suitable for exploring the nonlinear interactions between covariates, time and some exposure variable. The authors propose the local ordinary and weighted least squares estimators for the varying‐coefficient functions and establish the corresponding asymptotic normality properties. Simulation studies are conducted to empirically examine the finite‐sample performance of the new methods, and a real data example from a recent breast cancer study is used as an illustration. The Canadian Journal of Statistics 37: 659–674; 2009 © 2009 Statistical Society of Canada     

Efficient M-estimators with auxiliary information

This paper introduces a new class of M-estimators based on generalised empirical likelihood (GEL) estimation with some auxiliary information available in the sample. The resulting class of estimators is efficient in the sense that it achieves the same asymptotic lower bound as that of the efficient generalised method of moment (GMM) estimator with the same auxiliary information. The paper also shows that in case of smooth estimating equations the proposed estimators enjoy a small second order bias property compared to both efficient GMM and full GEL estimators. Analytical formulae to obtain bias corrected estimators are also provided. Simulations show that with correctly specified auxiliary information the proposed estimators and in particular those based on empirical likelihood outperform standard M and efficient GMM estimators both in terms of finite sample bias and efficiency. On the other hand with moderately misspecified auxiliary information estimators based on the nonparametric tilting method are typically characterised by the best finite sample properties.     

On Local Polynomial Modelling of the Additive Risk Model

The additive risk model provides an alternative modelling technique for failure time data to the proportional hazards model. In this article, we consider the additive risk model with a nonparametric risk effect. We study estimation of the risk function and its derivatives with a parametric and an unspecified baseline hazard function respectively. The resulting estimators are the local likelihood and the local score estimators. We establish the asymptotic normality of the estimators and show that both methods have the same formula for asymptotic bias but different formula for variance. It is found that, in some special cases, the local score estimator is of the same efficiency as the local likelihood estimator though it does not use the information about the baseline hazard function. Another advantage of the local score estimator is that it has a closed form and is easy to implement. Some simulation studies are conducted to evaluate and compare the performance of the two estimators. A numerical example is used for illustration.