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.     

A Bayesian semiparametric method for analyzing length-biased data

Survival data obtained from prevalent cohort study designs are often subject to length-biased sampling. Frequentist methods including estimating equation approaches, as well as full likelihood methods, are available for assessing covariate effects on survival from such data. Bayesian methods allow a perspective of probability interpretation for the parameters of interest, and may easily provide the predictive distribution for future observations while incorporating weak prior knowledge on the baseline hazard function. There is lack of Bayesian methods for analyzing length-biased data. In this paper, we propose Bayesian methods for analyzing length-biased data under a proportional hazards model. The prior distribution for the cumulative hazard function is specified semiparametrically using I-Splines. Bayesian conditional and full likelihood approaches are developed for analyzing simulated and real data.     

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.     

Generalized case–cohort sampling

A class of cohort sampling designs, including nested case–control, case–cohort and classical case–control designs involving survival data, is studied through a unified approach using Cox's proportional hazards model. By finding an optimal sample reuse method via local averaging, a closed form estimating function is obtained, leading directly to the estimators of the regression parameters that are relatively easy to compute and are more efficient than some commonly used estimators in case–cohort and nested case–control studies. A semiparametric efficient estimator can also be found with some further computation. In addition, the class of sampling designs in this study provides a variety of sampling options and relaxes the restrictions of sampling schemes that are currently available.     

Bayesian Methods for Missing Covariates in Cure Rate Models

We propose methods for Bayesian inference for missing covariate data with a novel class of semi-parametric survival models with a cure fraction. We allow the missing covariates to be either categorical or continuous and specify a parametric distribution for the covariates that is written as a sequence of one dimensional conditional distributions. We assume that the missing covariates are missing at random (MAR) throughout. We propose an informative class of joint prior distributions for the regression coefficients and the parameters arising from the covariate distributions. The proposed class of priors are shown to be useful in recovering information on the missing covariates especially in situations where the missing data fraction is large. Properties of the proposed prior and resulting posterior distributions are examined. Also, model checking techniques are proposed for sensitivity analyses and for checking the goodness of fit of a particular model. Specifically, we extend the Conditional Predictive Ordinate (CPO) statistic to assess goodness of fit in the presence of missing covariate data. Computational techniques using the Gibbs sampler are implemented. A real data set involving a melanoma cancer clinical trial is examined to demonstrate the methodology.     

Optimal generalized case–cohort sampling design under the additive hazard model

Generalized case–cohort designs have been proved to be a cost-effective way to enhance effectiveness in large epidemiological cohort. In generalized case–cohort design, we first select a subcohort from the underlying cohort by simple random sampling, and then sample a subset of the failures in the remaining subjects. In this article, we propose the inference procedure for the unknown regression parameters in the additive hazards model and develop an optimal sample size allocations to achieve maximum power at a given budget in generalized case–cohort design. The finite sample performance of the proposed method is evaluated through simulation studies. The proposed method is applied to a real data set from the National Wilm's Tumor Study Group.     

Case-cohort analysis with semiparametric transformation models

Semiparametric transformation models provide flexible regression models for survival analysis, including the Cox proportional hazards and the proportional odds models as special cases. We consider the application of semiparametric transformation models in case-cohort studies, where the covariate data are observed only on cases and on a subcohort randomly sampled from the full cohort. We first propose an approximate profile likelihood approach with full-cohort data, which amounts to the pseudo-partial likelihood approach of Zucker [2005. A pseudo-partial likelihood method for semiparametric survival regression with covariate errors. J. Amer. Statist. Assoc. 100, 1264–1277]. Simulation results show that our proposal is almost as efficient as the nonparametric maximum likelihood estimator. We then extend this approach to the case-cohort design, applying the Horvitz–Thompson weighting method to the estimating equations from the approximated profile likelihood. Two levels of weights can be utilized to achieve unbiasedness and to gain efficiency. The resulting estimator has a closed-form asymptotic covariance matrix, and is found in simulations to be substantially more efficient than the estimator based on martingale estimating equations. The extension to left-truncated data will be discussed. We illustrate the proposed method on data from a cardiovascular risk factor study conducted in Taiwan.     

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.     

Rank-order choice-based conjoint experiments: Efficiency and design

In a rank-order choice-based conjoint experiment, the respondent is asked to rank a number of alternatives of a number of choice sets. In this paper, we study the efficiency of those experiments and propose a D-optimality criterion for rank-order experiments to find designs yielding the most precise parameter estimators. For that purpose, an expression of the Fisher information matrix for the rank-ordered conditional logit model is derived which clearly shows how much additional information is provided by each extra ranking step. A simulation study shows that, besides the Bayesian D-optimal ranking design, the Bayesian D-optimal choice design is also an appropriate design for this type of experiments. Finally, it is shown that considerable improvements in estimation and prediction accuracy are obtained by including extra ranking steps in an experiment.     

Statistical inference for the accelerated failure time model under two-stage generalized case–cohort design

Abstract

In this article, we propose a two-stage generalized case–cohort design and develop an efficient inference procedure for the data collected with this design. In the first-stage, we observe the failure time, censoring indicator and covariates which are easy or cheap to measure, and in the second-stage, select a subcohort by simple random sampling and a subset of failures in remaining subjects from the first-stage subjects to observe their exposures which are different or expensive to measure. We derive estimators for regression parameters in the accelerated failure time model under the two-stage generalized case–cohort design through the estimated augmented estimating equation and the kernel function method. The resulting estimators are shown to be consistent and asymptotically normal. The finite sample performance of the proposed method is evaluated through the simulation studies. The proposed method is applied to a real data set from the National Wilm’s Tumor Study Group.     

Bayesian methods for generalized linear models with covariates missing at random

The authors propose methods for Bayesian inference for generalized linear models with missing covariate data. They specify a parametric distribution for the covariates that is written as a sequence of one‐dimensional conditional distributions. They propose an informative class of joint prior distributions for the regression coefficients and the parameters arising from the covariate distributions. They examine the properties of the proposed prior and resulting posterior distributions. They also present a Bayesian criterion for comparing various models, and a calibration is derived for it. A detailed simulation is conducted and two real data sets are examined to demonstrate the methodology.     

Model Averaging for Prediction With Fragmentary Data

ABSTRACT

One main challenge for statistical prediction with data from multiple sources is that not all the associated covariate data are available for many sampled subjects. Consequently, we need new statistical methodology to handle this type of “fragmentary data” that has become more and more popular in recent years. In this article, we propose a novel method based on the frequentist model averaging that fits some candidate models using all available covariate data. The weights in model averaging are selected by delete-one cross-validation based on the data from complete cases. The optimality of the selected weights is rigorously proved under some conditions. The finite sample performance of the proposed method is confirmed by simulation studies. An example for personal income prediction based on real data from a leading e-community of wealth management in China is also presented for illustration.     

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.     

Feature screening for case‐cohort studies with failure time outcome

Case‐cohort design has been demonstrated to be an economical and efficient approach in large cohort studies when the measurement of some covariates on all individuals is expensive. Various methods have been proposed for case‐cohort data when the dimension of covariates is smaller than sample size. However, limited work has been done for high‐dimensional case‐cohort data which are frequently collected in large epidemiological studies. In this paper, we propose a variable screening method for ultrahigh‐dimensional case‐cohort data under the framework of proportional model, which allows the covariate dimension increases with sample size at exponential rate. Our procedure enjoys the sure screening property and the ranking consistency under some mild regularity conditions. We further extend this method to an iterative version to handle the scenarios where some covariates are jointly important but are marginally unrelated or weakly correlated to the response. The finite sample performance of the proposed procedure is evaluated via both simulation studies and an application to a real data from the breast cancer study.     

Bivariate versus univariate ordinal categorical data with reference to an ophthalmologic study

The Wisconsin Epidemiologic Study of Diabetic Retinopathy is a population-based epidemiological study carried out in Southern Wisconsin during the 1980s. The resulting data were analysed by different statisticians and ophthalmologists during the last two decades. Most of the analyses were carried out on the baseline data, although there were two follow-up studies on the same population. A Bayesian analysis of the first follow-up data, taken four years after the baseline study, was carried out by Angers and Biswas [Angers, J.-F. and Biswas, A., 2004, A Bayesian analysis of the four-year follow-up data of theWisconsin epidemiologic study of diabetic retinopathy. Statistics in Medicine, 23, 601–615.], where the choice of the best model in terms of the covariate inclusion is done, and estimates of the associated covariate effects were obtained using the baseline data to set the prior for the parameters. In the present article we consider an univariate transformation of the bivariate ordinal data, and a parallel analysis with the much simpler univariate data is carried out. The results are then compared with the results of Angers and Biswas (2004). In conclusion, our analyses suggest that the univariate analysis fails to detect features of the data found by the bivariate analysis. Even an univariate transformation of our data with quite high correlation with both left and right eyes is inadequate.     

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.     

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.     

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.     

Exposure Stratified Case-Cohort Designs

A variant of the case-cohort design is proposed for the situation in which a correlate of the exposure (or prognostic factor) of interest is available for all cohort members, and exposure information is to be collected for a case-cohort sample. The cohort is stratified according to the correlate, and the subcohort is selected by stratified random sampling. A number of possible methods for the analysis of such exposure stratified case-cohort samples are presented, some of their statistical properties developed, and approximate relative efficiency and optimal allocation to the strata discussed. The methods are compared to each other, and to randomly sampled case-cohort studies, in a limited computer simulation study. We found that all of the proposed analysis methods performed well and were more efficient than a randomly sampled case-cohort study.     

Use of multiple imputation in supersampled nested case-control and case-cohort studies

Nested case-control and case-cohort studies are useful for studying associations between covariates and time-to-event when some covariates are expensive to measure. Full covariate information is collected in the nested case-control or case-cohort sample only, while cheaply measured covariates are often observed for the full cohort. Standard analysis of such case-control samples ignores any full cohort data. Previous work has shown how data for the full cohort can be used efficiently by multiple imputation of the expensive covariate(s), followed by a full-cohort analysis. For large cohorts this is computationally expensive or even infeasible. An alternative is to supplement the case-control samples with additional controls on which cheaply measured covariates are observed. We show how multiple imputation can be used for analysis of such supersampled data. Simulations show that this brings efficiency gains relative to a traditional analysis and that the efficiency loss relative to using the full cohort data is not substantial.