More efficient estimators for case-cohort studies

Improving estimation efficiency of case-cohort studies with interval-censored failure time data

The case-cohort design is a commonly used cost-effective sampling strategy for large cohort studies, where some covariates are expensive to measure or obtain. In this paper, we consider regression analysis under a case-cohort study with interval-censored failure time data, where the failure time is only known to fall within an interval instead of being exactly observed. A common approach to analyzing data from a case-cohort study is the inverse probability weighting approach, where only subjects in the case-cohort sample are used in estimation, and the subjects are weighted based on the probability of inclusion into the case-cohort sample. This approach, though consistent, is generally inefficient as it does not incorporate information outside the case-cohort sample. To improve efficiency, we first develop a sieve maximum weighted likelihood estimator under the Cox model based on the case-cohort sample and then propose a procedure to update this estimator by using information in the full cohort. We show that the update estimator is consistent, asymptotically normal, and at least as efficient as the original estimator. The proposed method can flexibly incorporate auxiliary variables to improve estimation efficiency. A weighted bootstrap procedure is employed for variance estimation. Simulation results indicate that the proposed method works well in practical situations. An application to a Phase 3 HIV vaccine efficacy trial is provided for illustration.

Efficient estimation for left-truncated competing risks regression for case-cohort studies

The case-cohort study design provides a cost-effective study design for a large cohort study with competing risk outcomes. The proportional subdistribution hazards model is widely used to estimate direct covariate effects on the cumulative incidence function for competing risk data. In biomedical studies, left truncation often occurs and brings extra challenges to the analysis. Existing inverse probability weighting methods for case-cohort studies with competing risk data not only have not addressed left truncation, but also are inefficient in regression parameter estimation for fully observed covariates. We propose an augmented inverse probability-weighted estimating equation for left-truncated competing risk data to address these limitations of the current literature. We further propose a more efficient estimator when extra information from the other causes is available. The proposed estimators are consistent and asymptotically normally distributed. Simulation studies show that the proposed estimator is unbiased and leads to estimation efficiency gain in the regression parameter estimation. We analyze the Atherosclerosis Risk in Communities study data using the proposed methods.

A semi-parametric weighted likelihood approach for regression analysis of bivariate interval-censored outcomes from case-cohort studies

The case-cohort design was developed to reduce costs when disease incidence is low and covariates are difficult to obtain. However, most of the existing methods are for right-censored data and there exists only limited research on interval-censored data, especially on regression analysis of bivariate interval-censored data. Interval-censored failure time data frequently occur in many areas and a large literature on their analyses has been established. In this paper, we discuss the situation of bivariate interval-censored data arising from case-cohort studies. For the problem, a class of semiparametric transformation frailty models is presented and for inference, a sieve weighted likelihood approach is developed. The large sample properties, including the consistency of the proposed estimators and the asymptotic normality of the regression parameter estimators, are established. Moreover, a simulation is conducted to assess the finite sample performance of the proposed method and suggests that it performs well in practice.

The Case for Case-Cohort: An Applied Epidemiologist's Guide to Reframing Case-Cohort Studies to Improve Usability and Flexibility

When research questions require the use of precious samples, expensive assays or equipment, or labor-intensive data collection or analysis, nested case-control or case-cohort sampling of observational cohort study participants can often reduce costs. These study designs have similar statistical precision for addressing a singular research question, but case-cohort studies have broader efficiency and superior flexibility. Despite this, case-cohort designs are comparatively underutilized in the epidemiologic literature. Recent advances in statistical methods and software have made analyses of case-cohort data easier to implement, and advances from casual inference, such as inverse probability of sampling weights, have allowed the case-cohort design to be used with a variety of target parameters and populations. To provide an accessible link to this technical literature, we give a conceptual overview of case-cohort study analysis with inverse probability of sampling weights. We show how this general analytic approach can be leveraged to more efficiently study subgroups of interest or disease subtypes or to examine associations independent of case status. A brief discussion of how this framework could be extended to incorporate other related methodologic applications further demonstrates the broad cost-effectiveness and adaptability of case-cohort methods for a variety of modern epidemiologic applications in resource-limited settings.

Competing risks regression models with covariates-adjusted censoring weight under the generalized case-cohort design

A generalized case-cohort design has been used when measuring exposures is expensive and events are not rare in the full cohort. This design collects expensive exposure information from a (stratified) randomly selected subset from the full cohort, called the subcohort, and a fraction of cases outside the subcohort. For the full cohort study with competing risks, He et al. (Scand J Stat 43:103-122, 2016) studied the non-stratified proportional subdistribution hazards model with covariate-dependent censoring to directly evaluate covariate effects on the cumulative incidence function. In this paper, we propose a stratified proportional subdistribution hazards model with covariate-adjusted censoring weights for competing risks data under the generalized case-cohort design. We consider a general class of weight functions to account for the generalized case-cohort design. Then, we derive the optimal weight function which minimizes the asymptotic variance of parameter estimates within the general class of weight functions. The proposed estimator is shown to be consistent and asymptotically normally distributed. The simulation studies show (i) the proposed estimator with covariate-adjusted weight is unbiased when the censoring distribution depends on covariates; and (ii) the proposed estimator with the optimal weight function gains parameter estimation efficiency. We apply the proposed method to stem cell transplantation and diabetes data sets.

Case-cohort design in hematopoietic cell transplant studies

SERIES EDITORS- NOTE

Imagine you and your colleagues have done 1000 transplants in persons with acute myeloid leukaemia (AML) in 1st remission. 5 percent of the 20 percent of recipients relapsing posttransplant have an isolated central nervous system relapse. You are curious and want to know whether there is anything special about this 5 percent, specifically whether this risk corelates with any pretransplant clinical and laboratory co-variates. You have extensive clinical data and some typical laboratory data on all 1000 but you suspect the culprit is mutation topography. What to do? Fortunately you have bio-banked DNA from the 1000. If resources and monies are not limiting you can do targeted or next generation sequencing on all 1000 DNA samples and off you go. However, most of us lack unlimited resources and monies. How can you sensibly and efficiently tackle this research problem? The answer is a case-cohort design study. In the typescript which follows Profs. Cai and Kim explain how to accomplish this. If you follow their advice you may need only to analyze samples from < 300 recipients rather than 1000 to test your hypothesis. They explain how to design such a study and provide references to estimate sample size.Sadly, their typescript will not tell you how to get funding for the study, whish poor devil who will have to write the protocol, worse, who will shepherd it though endless committees for approval and the like. Help on these issues is outside the scope of our statistics series. In this context we suggest advice from Woody Allen's article in the New Yorker: The Kugelmass Episode (April 24, 1977). When Prof. Kugelmass (English, City College) tells his analyst Dr. Mandel he has fallen in love with Emma Bovary who died of arsenic poisoning near Rouen, France 120 years earlier the analyst says: After all, I'm an analyst, not a magician. Kugelmass' reply: Then perhaps what I need is a magician and is off to Coney Island to find one. Good luck, the magician may still be there! (Note: This typescript is R-rated. It contains an equation.)Robert Peter Gale, Imperial College London, and Mei-Jie Zhang, Medical College of Wisconsin and CIBMTR.

Conditional screening for ultrahigh-dimensional survival data in case-cohort studies

The case-cohort design has been widely used to reduce the cost of covariate measurements in large cohort studies. In many such studies, the number of covariates is very large, and the goal of the research is to identify active covariates which have great influence on response. Since the introduction of sure independence screening, screening procedures have achieved great success in terms of effectively reducing the dimensionality and identifying active covariates. However, commonly used screening methods are based on marginal correlation or its variants, they may fail to identify hidden active variables which are jointly important but are weakly correlated with the response. Moreover, these screening methods are mainly proposed for data under the simple random sampling and can not be directly applied to case-cohort data. In this paper, we consider the ultrahigh-dimensional survival data under the case-cohort design, and propose a conditional screening method by incorporating some important prior known information of active variables. This method can effectively detect hidden active variables. Furthermore, it possesses the sure screening property under some mild regularity conditions and does not require any complicated numerical optimization. We evaluate the finite sample performance of the proposed method via extensive simulation studies and further illustrate the new approach through a real data set from patients with breast cancer.

Semiparametric regression analysis of case-cohort studies with multiple interval-censored disease outcomes

Interval-censored failure time data commonly arise in epidemiological and biomedical studies where the occurrence of an event or a disease is determined via periodic examinations. Subject to interval-censoring, available information on the failure time can be quite limited. Cost-effective sampling designs are desirable to enhance the study power, especially when the disease rate is low and the covariates are expensive to obtain. In this work, we formulate the case-cohort design with multiple interval-censored disease outcomes and also generalize it to nonrare diseases where only a portion of diseased subjects are sampled. We develop a marginal sieve weighted likelihood approach, which assumes that the failure times marginally follow the proportional hazards model. We consider two types of weights to account for the sampling bias, and adopt a sieve method with Bernstein polynomials to handle the unknown baseline functions. We employ a weighted bootstrap procedure to obtain a variance estimate that is robust to the dependence structure between failure times. The proposed method is examined via simulation studies and illustrated with a dataset on incident diabetes and hypertension from the Atherosclerosis Risk in Communities study.

Accelerated failure time model for data from outcome-dependent sampling

Outcome-dependent sampling designs such as the case-control or case-cohort design are widely used in epidemiological studies for their outstanding cost-effectiveness. In this article, we propose and develop a smoothed weighted Gehan estimating equation approach for inference in an accelerated failure time model under a general failure time outcome-dependent sampling scheme. The proposed estimating equation is continuously differentiable and can be solved by the standard numerical methods. In addition to developing asymptotic properties of the proposed estimator, we also propose and investigate a new optimal power-based subsamples allocation criteria in the proposed design by maximizing the power function of a significant test. Simulation results show that the proposed estimator is more efficient than other existing competing estimators and the optimal power-based subsamples allocation will provide an ODS design that yield improved power for the test of exposure effect. We illustrate the proposed method with a data set from the Norwegian Mother and Child Cohort Study to evaluate the relationship between exposure to perfluoroalkyl substances and women's subfecundity.

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 are 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.

Bi-level variable selection for case-cohort studies with group variables

The case-cohort design is an economical approach to estimate the effect of risk factors on the survival outcome when collecting exposure information or covariates on all patients is expensive in a large cohort study. Variables often have group structure such as categorical variables and highly correlated continuous variables. The existing literature for case-cohort data is limited to identifying non-zero variables at individual level only. In this article, we propose a bi-level variable selection method to select non-zero group and within-group variables for case-cohort data when variables have group structure. The proposed method allows the number of variables to diverge as the sample size increases. The asymptotic properties of the estimator including bi-level variable selection consistency and the asymptotic normality are shown. We also conduct simulations to compare our proposed method with some existing method and apply them to the Busselton Health data.

Analysis of multiple survival events in generalized case-cohort designs

Generalized case-cohort design has been proposed to assess the effects of exposures on survival outcomes when measuring exposures is expensive and events are not rare in the cohort. In such design, expensive exposure information is collected from both a (stratified) randomly selected subcohort and a subset of individuals with events. In this article, we consider extension of such design to study multiple types of survival events by selecting a proportion of cases for each type of event. We propose a general weighting scheme to analyze data. Furthermore, we examine the optimal choice of weights and show that this optimal weighting yields much improved efficiency gain both asymptotically and in simulation studies. Finally, we apply our proposed methods to data from the Atherosclerosis Risk in Communities study.

Regression analysis for secondary response variable in a case-cohort study

Case-cohort study design has been widely used for its cost-effectiveness. In any real study, there are always other important outcomes of interest beside the failure time that the original case-cohort study is based on. How to utilize the available case-cohort data to study the relationship of a secondary outcome with the primary exposure obtained through the case-cohort study is not well studied. In this article, we propose a non-parametric estimated likelihood approach for analyzing a secondary outcome in a case-cohort study. The estimation is based on maximizing a semiparametric likelihood function that is built jointly on both time-to-failure outcome and the secondary outcome. The proposed estimator is shown to be consistent, efficient, and asymptotically normal. Finite sample performance is evaluated via simulation studies. Data from the Sister Study is analyzed to illustrate our method.

Improving efficiency of parameter estimation in case-cohort studies with multivariate failure time data

The case-cohort study design is an effective way to reduce cost of assembling and measuring expensive covariates in large cohort studies. Recently, several weighted estimators were proposed for the case-cohort design when multiple diseases are of interest. However, these existing weighted estimators do not make effective use of the covariate information available in the whole cohort. Furthermore, the auxiliary information for the expensive covariates, which may be available in the studies, cannot be incorporated directly. In this article, we propose a class of updated-estimators. We show that, by making effective use of the whole cohort information, the proposed updated-estimators are guaranteed to be more efficient than the existing weighted estimators asymptotically. Furthermore, they are flexible to incorporate the auxiliary information whenever available. The advantages of the proposed updated-estimators are demonstrated in simulation studies and a real data analysis.

Case-cohort studies with interval-censored failure time data

The case-cohort design has been widely used as a means of cost reduction in assembling or measuring expensive covariates in large cohort studies. The existing literature on the case-cohort design is mainly focused on right-censored data. In practice, however, the failure time is often subject to interval-censoring; it is known only to fall within some random time interval. In this paper, we consider the case-cohort study design for interval-censored failure time and develop a sieve semiparametric likelihood approach for analyzing data from this design under the proportional hazards model. We construct the likelihood function using inverse probability weighting and build the sieves with Bernstein polynomials. The consistency and asymptotic normality of the resulting regression parameter estimator are established and a weighted bootstrap procedure is considered for variance estimation. Simulations show that the proposed method works well for practical situations, and an application to real data is provided.

Outcome-Dependent Sampling Design and Inference for Cox's Proportional Hazards Model

We propose a cost-effective outcome-dependent sampling design for the failure time data and develop an efficient inference procedure for data collected with this design. To account for the biased sampling scheme, we derive estimators from a weighted partial likelihood estimating equation. The proposed estimators for regression parameters are shown to be consistent and asymptotically normally distributed. A criteria that can be used to optimally implement the ODS design in practice is proposed and studied. The small sample performance of the proposed method is evaluated by simulation studies. The proposed design and inference procedure is shown to be statistically more powerful than existing alternative designs with the same sample sizes. We illustrate the proposed method with an existing real data from the Cancer Incidence and Mortality of Uranium Miners Study.

Recent progresses in outcome-dependent sampling with failure time data

An outcome-dependent sampling (ODS) design is a retrospective sampling scheme where one observes the primary exposure variables with a probability that depends on the observed value of the outcome variable. When the outcome of interest is failure time, the observed data are often censored. By allowing the selection of the supplemental samples depends on whether the event of interest happens or not and oversampling subjects from the most informative regions, ODS design for the time-to-event data can reduce the cost of the study and improve the efficiency. We review recent progresses and advances in research on ODS designs with failure time data. This includes researches on ODS related designs like case-cohort design, generalized case-cohort design, stratified case-cohort design, general failure-time ODS design, length-biased sampling design and interval sampling design.

Variable selection for case-cohort studies with failure time outcome

Case-cohort designs are widely used in large cohort studies to reduce the cost associated with covariate measurement. In many such studies the number of covariates is very large, so an efficient variable selection method is necessary. In this paper, we study the properties of a variable selection procedure using the smoothly clipped absolute deviation penalty in a case-cohort design with a diverging number of parameters. We establish the consistency and asymptotic normality of the maximum penalized pseudo-partial-likelihood estimator, and show that the proposed variable selection method is consistent and has an asymptotic oracle property. Simulation studies compare the finite-sample performance of the procedure with tuning parameter selection methods based on the Akaike information criterion and the Bayesian information criterion. We make recommendations for use of the proposed procedures in case-cohort studies, and apply them to the Busselton Health Study.

Improving the efficiency of estimation in the additive hazards model for stratified case-cohort design with multiple diseases

The case-cohort study design has often been used in studies of a rare disease or for a common disease with some biospecimens needing to be preserved for future studies. A case-cohort study design consists of a random sample, called the subcohort, and all or a portion of the subjects with the disease of interest. One advantage of the case-cohort design is that the same subcohort can be used for studying multiple diseases. Stratified random sampling is often used for the subcohort. Additive hazards models are often preferred in studies where the risk difference, instead of relative risk, is of main interest. Existing methods do not use the available covariate information fully. We propose a more efficient estimator by making full use of available covariate information for the additive hazards model with data from a stratified case-cohort design with rare (the traditional situation) and non-rare (the generalized situation) diseases. We propose an estimating equation approach with a new weight function. The proposed estimators are shown to be consistent and asymptotically normally distributed. Simulation studies show that the proposed method using all available information leads to efficiency gain and stratification of the subcohort improves efficiency when the strata are highly correlated with the covariates. Our proposed method is applied to data from the Atherosclerosis Risk in Communities study.