Cox Regression in Nested Case–Control Studies with Auxiliary Covariates

Summary Nested case–control (NCC) design is a popular sampling method in large epidemiological studies for its cost effectiveness to investigate the temporal relationship of diseases with environmental exposures or biological precursors. Thomas' maximum partial likelihood estimator is commonly used to estimate the regression parameters in Cox's model for NCC data. In this article, we consider a situation in which failure/censoring information and some crude covariates are available for the entire cohort in addition to NCC data and propose an improved estimator that is asymptotically more efficient than Thomas' estimator. We adopt a projection approach that, heretofore, has only been employed in situations of random validation sampling and show that it can be well adapted to NCC designs where the sampling scheme is a dynamic process and is not independent for controls. Under certain conditions, consistency and asymptotic normality of the proposed estimator are established and a consistent variance estimator is also developed. Furthermore, a simplified approximate estimator is proposed when the disease is rare. Extensive simulations are conducted to evaluate the finite sample performance of our proposed estimators and to compare the efficiency with Thomas' estimator and other competing estimators. Moreover, sensitivity analyses are conducted to demonstrate the behavior of the proposed estimator when model assumptions are violated, and we find that the biases are reasonably small in realistic situations. We further demonstrate the proposed method with data from studies on Wilms' tumor.     

Sex‐ and age‐biased mortality in wild Grey Partridge Perdix perdix populations

Differing reproductive effort, individual qualities and local environmental conditions can lead to uneven mortality risk among individuals within populations and may result in survival differences according to age and sex. Identification of factors contributing to unequal operational sex ratios has been important for understanding population dynamics and conservation management. In this study, sex‐ and age‐specific mortality was estimated in three wild Grey Partridge populations from analysis of year‐round radiotracking data from 168 individuals. Survival days were counted in three periods defined individually for each bird: the pairing period (covey break‐up to laying of the first egg); the nesting period (between clutch initiation date and failure of the last nesting attempt, or the date when chicks were 14 days old); and the covey period (the end of the nesting period or joining a group until covey break‐up). Predation was the main cause of mortality. A significant effect of age on survival was found during the pairing period, when older individuals paired off faster and survived better. The highest mortality risk overall was found during the nesting period. Furthermore, significantly higher mortality of females was recorded during the nesting period, suggesting that greater investments in reproduction, behaviour at the nest or the quality of nesting habitats can decrease survival of females and cause a male‐skewed sex ratio. No significant effect of age or sex was found during the covey period, or for the year as a whole, but there was a significant difference in annual mortality rates between the three study populations. Our results confirm age‐ and sex‐specific variation of adult mortality in a ground‐nesting bird with biparental care during the annual cycle, documenting differing sensitivities of various population cohorts to predation.     

Double‐Robust Semiparametric Estimator for Differences in Restricted Mean Lifetimes in Observational Studies

Summary Restricted mean lifetime is often of direct interest in epidemiologic studies involving censored survival times. Differences in this quantity can be used as a basis for comparing several groups. For example, transplant surgeons, nephrologists, and of course patients are interested in comparing posttransplant lifetimes among various types of kidney transplants to assist in clinical decision making. As the factor of interest is not randomized, covariate adjustment is needed to account for imbalances in confounding factors. In this report, we use semiparametric theory to develop an estimator for differences in restricted mean lifetimes although accounting for confounding factors. The proposed method involves building working models for the time‐to‐event and coarsening mechanism (i.e., group assignment and censoring). We show that the proposed estimator possesses the double robust property; i.e., when either the time‐to‐event or coarsening process is modeled correctly, the estimator is consistent and asymptotically normal. Simulation studies are conducted to assess its finite‐sample performance and the method is applied to national kidney transplant data.     

Dynamic inference in general nested case‐control designs

Nested case‐control designs are attractive in studies with a time‐to‐event endpoint if the outcome is rare or if interest lies in evaluating expensive covariates. The appeal is that these designs restrict to small subsets of all patients at risk just prior to the observed event times. Only these small subsets need to be evaluated. Typically, the controls are selected at random and methods for time‐simultaneous inference have been proposed in the literature. However, the martingale structure behind nested case‐control designs allows for more powerful and flexible non‐standard sampling designs. We exploit that structure to find simultaneous confidence bands based on wild bootstrap resampling procedures within this general class of designs. We show in a simulation study that the intended coverage probability is obtained for confidence bands for cumulative baseline hazard functions. We apply our methods to observational data about hospital‐acquired infections.     

Robust Prediction of t‐Year Survival with Data from Multiple Studies

Summary Recently meta‐analysis has been widely utilized to combine information across multiple studies to evaluate a common effect. Integrating data from similar studies is particularly useful in genomic studies where the individual study sample sizes are not large relative to the number of parameters of interest. In this article, we are interested in developing robust prognostic rules for the prediction of t ‐year survival based on multiple studies. We propose to construct a composite score for prediction by fitting a stratified semiparametric transformation model that allows the studies to have related but not identical outcomes. To evaluate the accuracy of the resulting score, we provide point and interval estimators for the commonly used accuracy measures including the time‐specific receiver operating characteristic curves, and positive and negative predictive values. We apply the proposed procedures to develop prognostic rules for the 5‐year survival of breast cancer patients based on five breast cancer genomic studies.     

Proportional Hazards Regression for the Analysis of Clustered Survival Data from Case–Cohort Studies

Summary Case–cohort sampling is a commonly used and efficient method for studying large cohorts. Most existing methods of analysis for case–cohort data have concerned the analysis of univariate failure time data. However, clustered failure time data are commonly encountered in public health studies. For example, patients treated at the same center are unlikely to be independent. In this article, we consider methods based on estimating equations for case–cohort designs for clustered failure time data. We assume a marginal hazards model, with a common baseline hazard and common regression coefficient across clusters. The proposed estimators of the regression parameter and cumulative baseline hazard are shown to be consistent and asymptotically normal, and consistent estimators of the asymptotic covariance matrices are derived. The regression parameter estimator is easily computed using any standard Cox regression software that allows for offset terms. The proposed estimators are investigated in simulation studies, and demonstrated empirically to have increased efficiency relative to some existing methods. The proposed methods are applied to a study of mortality among Canadian dialysis patients.     

Using Penalized Splines to Model Age‐ and Season‐of‐Birth‐Dependent Effects of Childhood Mortality Risk Factors in Rural Burkina Faso

Several previous studies have identified risk factors for childhood mortality in high risk areas, such as Sub‐Saharan Africa. Among these are lifestyle factors related for example to nutrition or sanitation. Other factors are related to social class, ethnicity and poverty in general. Few studies have investigated a dependence of these factors by age and season of birth which is the focus in this study. We perform a survival analysis of 9121 children born between 1998 and 2001 in a rural area of western Burkina Faso. The whole population is under demographic surveillance since 1993. All cause mortality is used as the endpoint and follow‐up information until the age of five years is available. Recently developed spline regression methods are used for the analysis. Ethnic group, religion, age of mother, twin status, sex, and distance to next health center are used as covariates all of which having a clear effect on survival in standard Cox regression analysis. With penalized spline regression, a more detailed risk pattern is observed. Ethnicity is more related to death at early age, as well as age of mother. The effect of the risk factors considered also appear to be related with season of birth (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Survival Analysis with Error‐Prone Time‐Varying Covariates: A Risk Set Calibration Approach

Summary Occupational, environmental, and nutritional epidemiologists are often interested in estimating the prospective effect of time‐varying exposure variables such as cumulative exposure or cumulative updated average exposure, in relation to chronic disease endpoints such as cancer incidence and mortality. From exposure validation studies, it is apparent that many of the variables of interest are measured with moderate to substantial error. Although the ordinary regression calibration (ORC) approach is approximately valid and efficient for measurement error correction of relative risk estimates from the Cox model with time‐independent point exposures when the disease is rare, it is not adaptable for use with time‐varying exposures. By recalibrating the measurement error model within each risk set, a risk set regression calibration (RRC) method is proposed for this setting. An algorithm for a bias‐corrected point estimate of the relative risk using an RRC approach is presented, followed by the derivation of an estimate of its variance, resulting in a sandwich estimator. Emphasis is on methods applicable to the main study/external validation study design, which arises in important applications. Simulation studies under several assumptions about the error model were carried out, which demonstrated the validity and efficiency of the method in finite samples. The method was applied to a study of diet and cancer from Harvard's Health Professionals Follow‐up Study (HPFS).     

Causal inference on the difference of the restricted mean lifetime between two groups

When comparing survival times between two treatment groups, it may be more appropriate to compare the restricted mean lifetime, i.e., the expectation of lifetime restricted to a time L, rather than mean lifetime in order to accommodate censoring. When the treatments are not assigned to patients randomly, as in observational studies, we also need to account for treatment imbalances in confounding factors. In this article, we propose estimators for the difference of the restricted mean lifetime between two groups that account for treatment imbalances in prognostic factors assuming a proportional hazards relationship. Large-sample properties of our estimators based on martingale theory for counting processes are also derived. Simulation studies were conducted to compare these estimators and to assess the adequacy of the large-sample approximations. Our methods are also applied to an observational database of acute coronary syndrome patients from Duke University Medical Center to estimate the treatment effect on the restricted mean lifetime over 5 years.     

Evaluating the Predictive Value of Biomarkers with Stratified Case‐Cohort Design

Summary Identification of novel biomarkers for risk assessment is important for both effective disease prevention and optimal treatment recommendation. Discovery relies on the precious yet limited resource of stored biological samples from large prospective cohort studies. Case‐cohort sampling design provides a cost‐effective tool in the context of biomarker evaluation, especially when the clinical condition of interest is rare. Existing statistical methods focus on making efficient inference on relative hazard parameters from the Cox regression model. Drawing on recent theoretical development on the weighted likelihood for semiparametric models under two‐phase studies ( Breslow and Wellner, 2007 ), we propose statistical methods to evaluate accuracy and predictiveness of a risk prediction biomarker, with censored time‐to‐event outcome under stratified case‐cohort sampling. We consider nonparametric methods and a semiparametric method. We derive large sample properties of proposed estimators and evaluate their finite sample performance using numerical studies. We illustrate new procedures using data from Framingham Offspring Study to evaluate the accuracy of a recently developed risk score incorporating biomarker information for predicting cardiovascular disease.     

Heterogeneous treatment effects in stratified clinical trials with time‐to‐event endpoints

When analyzing clinical trials with a stratified population, homogeneity of treatment effects is a common assumption in survival analysis. However, in the context of recent developments in clinical trial design, which aim to test multiple targeted therapies in corresponding subpopulations simultaneously, the assumption that there is no treatment‐by‐stratum interaction seems inappropriate. It becomes an issue if the expected sample size of the strata makes it unfeasible to analyze the trial arms individually. Alternatively, one might choose as primary aim to prove efficacy of the overall (targeted) treatment strategy. When testing for the overall treatment effect, a violation of the no‐interaction assumption renders it necessary to deviate from standard methods that rely on this assumption. We investigate the performance of different methods for sample size calculation and data analysis under heterogeneous treatment effects. The commonly used sample size formula by Schoenfeld is compared to another formula by Lachin and Foulkes, and to an extension of Schoenfeld's formula allowing for stratification. Beyond the widely used (stratified) Cox model, we explore the lognormal shared frailty model, and a two‐step analysis approach as potential alternatives that attempt to adjust for interstrata heterogeneity. We carry out a simulation study for a trial with three strata and violations of the no‐interaction assumption. The extension of Schoenfeld's formula to heterogeneous strata effects provides the most reliable sample size with respect to desired versus actual power. The two‐step analysis and frailty model prove to be more robust against loss of power caused by heterogeneous treatment effects than the stratified Cox model and should be preferred in such situations.     

Addressing an Idiosyncrasy in Estimating Survival Curves Using Double Sampling in the Presence of Self-Selected Right Censoring

We investigate the use of follow-up samples of individuals to estimate survival curves from studies that are subject to right censoring from two sources: (i) early termination of the study, namely, administrative censoring, or (ii) censoring due to lost data prior to administrative censoring, so-called dropout. We assume that, for the full cohort of individuals, administrative censoring times are independent of the subjects' inherent characteristics, including survival time. To address the loss to censoring due to dropout, which we allow to be possibly selective, we consider an intensive second phase of the study where a representative sample of the originally lost subjects is subsequently followed and their data recorded. As with double-sampling designs in survey methodology, the objective is to provide data on a representative subset of the dropouts. Despite assumed full response from the follow-up sample, we show that, in general in our setting, administrative censoring times are not independent of survival times within the two subgroups, nondropouts and sampled dropouts. As a result, the stratified Kaplan-Meier estimator is not appropriate for the cohort survival curve. Moreover, using the concept of potential outcomes, as opposed to observed outcomes, and thereby explicitly formulating the problem as a missing data problem, reveals and addresses these complications. We present an estimation method based on the likelihood of an easily observed subset of the data and study its properties analytically for large samples. We evaluate our method in a realistic situation by simulating data that match published margins on survival and dropout from an actual hip-replacement study. Limitations and extensions of our design and analytic method are discussed.     

A Multiple‐Absorbent Markov Process in Survival Studies: Application to Breast Cancer

A Markov process with several absorbent states is applied for analyzing a breast cancer dataset. The study examines the evolution of patients until death, and shows that two well‐differentiated ways can be considered in the evolution of patients towards the death state: those who relapse and those who not. The risk groups we have considered are determined by the application of treatments radiotherapy and chemotherapy, which are introduced as covariates. Four states are distinguished: no relapse, relapse, death after metastasis, and death without metastasis, the last two absorbent. We apply a methodology that uses algorithmic procedures, avoiding differential equations. The transition probability functions and the likelihood function in the model are calculated. For the dataset, the survival functions and the mean times in states for the different group of risks are determined. We show that the metastasis is the main cause of death in this cohort, but the number of deaths by relapse is not negligible.     

Correcting for noncompliance and dependent censoring in an AIDS Clinical Trial with inverse probability of censoring weighted (IPCW) log-rank tests

AIDS Clinical Trial Group (ACTG) randomized trial 021 compared the effect of bactrim versus aerosolized pentamidine (AP) as prophylaxis therapy for pneumocystis pneumonia (PCP) in AIDS patients. Although patients randomized to the bactrim arm experienced a significant delay in time to PCP, the survival experience in the two arms was not significantly different (p = .32). In this paper, we present evidence that bactrim therapy improves survival but that the standard intent-to-treat comparison failed to detect this survival advantage because a large fraction of the subjects either crossed over to the other therapy or stopped therapy altogether. We obtain our evidence of a beneficial bactrim effect on survival by artificially regarding the subjects as dependently censored at the first time the subject either stops or switches therapy; we then analyze the data with the inverse probability of censoring weighted Kaplan-Meier and Cox partial likelihood estimators of Robins (1993, Proceedings of the Biopharmaceutical Section, American Statistical Association, pp. 24-33) that adjust for dependent censoring by utilizing data collected on time-dependent prognostic factors.     

A Robust Test for Two‐Stage Design in Genome‐Wide Association Studies

Summary A two‐stage design is cost‐effective for genome‐wide association studies (GWAS) testing hundreds of thousands of single nucleotide polymorphisms (SNPs). In this design, each SNP is genotyped in stage 1 using a fraction of case–control samples. Top‐ranked SNPs are selected and genotyped in stage 2 using additional samples. A joint analysis, combining statistics from both stages, is applied in the second stage. Follow‐up studies can be regarded as a two‐stage design. Once some potential SNPs are identified, independent samples are further genotyped and analyzed separately or jointly with previous data to confirm the findings. When the underlying genetic model is known, an asymptotically optimal trend test (TT) can be used at each analysis. In practice, however, genetic models for SNPs with true associations are usually unknown. In this case, the existing methods for analysis of the two‐stage design and follow‐up studies are not robust across different genetic models. We propose a simple robust procedure with genetic model selection to the two‐stage GWAS. Our results show that, if the optimal TT has about 80% power when the genetic model is known, then the existing methods for analysis of the two‐stage design have minimum powers about 20% across the four common genetic models (when the true model is unknown), while our robust procedure has minimum powers about 70% across the same genetic models. The results can be also applied to follow‐up and replication studies with a joint analysis.     

Reduced rank hazard regression with fixed and time‐varying effects of the covariates

Modelling survival data from long‐term follow‐up studies presents challenges. The commonly used proportional hazards model should be extended to account for dynamic behaviour of the effects of fixed covariates. This work illustrates the use of reduced rank models in survival data, where some of the covariate effects are allowed to behave dynamically in time and some as fixed. Time‐varying effects of the covariates can be fitted by using interactions of the fixed covariates with flexible transformations of time based on b‐splines. To avoid overfitting, a reduced rank model will restrict the number of parameters, resulting in a more sensible fit to the data. This work presents the basic theory and the algorithm to fit such models. An application to breast cancer data is used for illustration of the suggested methods.