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.     

Influence Function Based Variance Estimation and Missing Data Issues in Case-Cohort Studies

Recognizing that the efficiency in relative risk estimation for the Cox proportional hazards model is largely constrained by the total number of cases, Prentice (1986) proposed the case-cohort design in which covariates are measured on all cases and on a random sample of the cohort. Subsequent to Prentice, other methods of estimation and sampling have been proposed for these designs. We formalize an approach to variance estimation suggested by Barlow (1994), and derive a robust variance estimator based on the influence function. We consider the applicability of the variance estimator to all the proposed case-cohort estimators, and derive the influence function when known sampling probabilities in the estimators are replaced by observed sampling fractions. We discuss the modifications required when cases are missing covariate information. The missingness may occur by chance, and be completely at random; or may occur as part of the sampling design, and depend upon other observed covariates. We provide an adaptation of S-plus code that allows estimating influence function variances in the presence of such missing covariates. Using examples from our current case-cohort studies on esophageal and gastric cancer, we illustrate how our results our useful in solving design and analytic issues that arise in practice.     

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.     

Aalen's Linear Model for Sampled Risk Set Data: A Large Sample Study

Borgan and Langholz (1997) describe a method for estimating the parameter functions in Aalen's linear hazard regression model from sampled risk set data. Using a counting process formulation and the martingale central limit theorem, we provide a study of the asymptotic distributional properties of the estimator. The results are applied to study the efficiencies of the nested case-control and counter-matched designs relative to a full cohort analysis.     

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.     

Quantile regression for competing risks analysis under case-cohort design

The case-cohort design brings cost reduction in large cohort studies. In this paper, we consider a nonlinear quantile regression model for censored competing risks under the case-cohort design. Two different estimation equations are constructed with or without the covariates information of other risks included, respectively. The large sample properties of the estimators are obtained. The asymptotic covariances are estimated by using a fast resampling method, which is useful to consider further inferences. The finite sample performance of the proposed estimators is assessed by simulation studies. Also a real example is used to demonstrate the application of the proposed methods.     

Statistical inference for generalized case-cohort design under the proportional hazards model with parameter constraints

ABSTRACT

The generalized case-cohort design is widely used in large cohort studies to reduce the cost and improve the efficiency. Taking prior information of parameters into consideration in modeling process can further raise the inference efficiency. In this paper, we consider fitting proportional hazards model with constraints for generalized case-cohort studies. We establish a working likelihood function for the estimation of model parameters. The asymptotic properties of the proposed estimator are derived via the Karush-Kuhn-Tucker conditions, and their finite properties are assessed by simulation studies. A modified minorization-maximization algorithm is developed for the numerical calculation of the constrained estimator. An application to a Wilms tumor study demonstrates the utility of the proposed method in practice.     

Case-cohort analysis of clusters of recurrent events

The case-cohort sampling, first proposed in Prentice (Biometrika 73:1–11, 1986), is one of the most effective cohort designs for analysis of event occurrence, with the regression model being the typical Cox proportional hazards model. This paper extends to consider the case-cohort design for recurrent events with certain specific clustering feature, which is captured by a properly modified Cox-type self-exciting intensity model. We discuss the advantage of using this model and validate the pseudo-likelihood method. Simulation studies are presented in support of the theory. Application is illustrated with analysis of a bladder cancer data.     

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.     

Computing the Cox Model for Case Cohort Designs

Prentice (1986) proposed a case-cohort design as an efficient subsampling mechanism for survival studies. Several other authors have expanded on these ideas to create a family of related sampling plans, along with estimators for the covariate effects. We describe how to obtain the proposed parameter estimates and their variance estimates using standard software packages, with SAS and SPLUS as particular examples.     

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.     

Hypothesis Testing in Multivariate Linear Models with Randomly Missing Data

Tests for mean equality proposed by Weerahandi (1995) and Chen and Chen (1998), tests that do not require equality of population variances, were examined when data were not only heterogeneous but, as well, nonnormal in unbalanced completely randomized designs. Furthermore, these tests were compared to a test examined by Lix and Keselman (1998), a test that uses a heteroscedastic statistic (i.e., Welch, 1951) with robust estimators (20% trimmed means and Winsorized variances). Our findings confirmed previously published data that the tests are indeed robust to variance heterogeneity when the data are obtained from normal populations. However, the Weerahandi (1995) and Chen and Chen (1998) tests were not found to be robust when data were obtained from nonnormal populations. Indeed, rates of Type I error were typically in excess of 10% and, at times, exceeded 50%. On the other hand, the statistic presented by Lix and Keselman (1998) was generally robust to variance heterogeneity and nonnormality.     

A class of goodness-of-fit test for the additive hazards model with case-cohort data

The case-cohort design is commonly used in epidemiological studies due to its cost-effectiveness. The additive hazards model is widely used in survival analysis when the hazards difference is constant. In this article, we propose a class of goodness-of-fit test statistics for the assumption of the additive hazards model with case-cohort data through a class of asymptotically mean-zero multiparameter stochastic processes. We also establish the asymptotic theory of the proposed test statistics and a resampling scheme is adopted to approximate its asymptotic distribution. The performance of the proposed test statistics is evaluated through simulation studies and a real dataset is analyzed to illustrate the proposed method.     

A conditional perspective of weighted variance estimation of the optimal regression estimator

The estimation of the variance for the GREG (general regression) estimator by weighted residuals is widely accepted as a method which yields estimators with good conditional properties. Since the optimal (regression) estimator shares the properties of GREG estimators which are used in the construction of weighted variance estimators, we introduce the weighting procedure also for estimating the variance of the optimal estimator. This method of variance estimation was originally presented in a seemingly ad hoc manner, and we shall discuss it from a conditional point of view and also look at an alternative way of utilizing the weights. Examples that stress conditional behaviour of estimators are then given for elementary sampling designs such as simple random sampling, stratified simple random sampling and Poisson sampling, where for the latter design we have conducted a small simulation study.     

Fast accelerated failure time modeling for case-cohort data

Semiparametric accelerated failure time (AFT) models directly relate the expected failure times to covariates and are a useful alternative to models that work on the hazard function or the survival function. For case-cohort data, much less development has been done with AFT models. In addition to the missing covariates outside of the sub-cohort in controls, challenges from AFT model inferences with full cohort are retained. The regression parameter estimator is hard to compute because the most widely used rank-based estimating equations are not smooth. Further, its variance depends on the unspecified error distribution, and most methods rely on computationally intensive bootstrap to estimate it. We propose fast rank-based inference procedures for AFT models, applying recent methodological advances to the context of case-cohort data. Parameters are estimated with an induced smoothing approach that smooths the estimating functions and facilitates the numerical solution. Variance estimators are obtained through efficient resampling methods for nonsmooth estimating functions that avoids full blown bootstrap. Simulation studies suggest that the recommended procedure provides fast and valid inferences among several competing procedures. Application to a tumor study demonstrates the utility of the proposed method in routine data analysis.     

Weighted Likelihood for Semiparametric Models and Two-phase Stratified Samples, with Application to Cox Regression

Abstract.  We consider semiparametric models for which solution of Horvitz–Thompson or inverse probability weighted (IPW) likelihood equations with two-phase stratified samples leads to consistent and asymptotically Gaussian estimators of both Euclidean and non-parametric parameters. For Bernoulli (independent and identically distributed) sampling, standard theory shows that the Euclidean parameter estimator is asymptotically linear in the IPW influence function. By proving weak convergence of the IPW empirical process, and borrowing results on weighted bootstrap empirical processes, we derive a parallel asymptotic expansion for finite population stratified sampling. Several of our key results have been derived already for Cox regression with stratified case–cohort and more general survey designs. This paper is intended to help interpret this previous work and to pave the way towards a general Horvitz–Thompson approach to semiparametric inference with data from complex probability samples.     

Strategies for Cohort Sampling Under the Cox Proportional Hazards Model,Application to an AIDS Clinical Trial

In some studies that relate covariates to times of failure it is not feasible to observe all covariates for all subjects. For example, some covariates may be too costly in terms of time, money, or effect on the subject to record for all subjects. This paper considers the relative efficiencies of several designs for sampling a portion of the cohort on which the costly covariates will be observed. Such designs typically measure all covariates for each failure and control for covariates of lesser interest. Control subjects are sampled either from risk sets at times of observed failures or from the entire cohort. A new design in which the sampling probability for each individual depends on the amount of information that the individual can contribute to estimated coefficients is shown to be superior to other sampling designs under certain conditions. Primary focus of our designs is on time-invariant covariates, but some methods easily generalize to the time-varying setting. Data from a study conducted by the AIDS Clinical Trials Group are used to illustrate the new sampling procedure and to explore the relative efficiency of several sampling schemes.     

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.     

Catalogue of Group Structures for Three-level Fractional Factorial Designs

Taguchi (1959) introduced the concept of split-unit design to sort the factors into different groups depending upon the difficulties involved in changing the levels of factors. Li et al. (1991) renamed it as split-plot design. Chen et al. (1993) have given a catalogue of small designs for two- and three-level fractional factorial designs pertaining to a single type of factors. Aggarwal et al. (1997) have given a catalogue of group structure for two-level fractional factorial designs developed under the concept of split-plot design. In this paper, an algorithm has been developed for generating group structure and possible allocations for various 3^n-k fractional factorial designs.     

Variance estimation of DEE and regression estimator when a first-phase cluster sample is restratified

We consider a variance estimation when a stratified single stage cluster sample is selected in the first phase and a stratified simple random element sample is selected in the second phase. We propose explicit formulas of (asymptotically), we propose explicit formulas of (asymptotically) unbiased variance estimators for the double expansion estimator and regression estimator. We perform a small simulation study to investigate the performance of the proposed variance estimators. In our simulation study, the proposed variance estimator showed better or comparable performance to the Jackknife variance estimator. We also extend the results to a two-phase sampling design in which a stratified pps with replacement cluster sample is selected in the first phase.