Concordance indices with left-truncated and right-censored data

In the context of time-to-event analysis, a primary objective is to model the risk of experiencing a particular event in relation to a set of observed predictors. The Concordance Index (C-Index) is a statistic frequently used in practice to assess how well such models discriminate between various risk levels in a population. However, the properties of conventional C-Index estimators when applied to left-truncated time-to-event data have not been well studied, despite the fact that left-truncation is commonly encountered in observational studies. We show that the limiting values of the conventional C-Index estimators depend on the underlying distribution of truncation times, which is similar to the situation with right-censoring as discussed in Uno et al. (2011) [On the C-statistics for evaluating overall adequacy of risk prediction procedures with censored survival data. Statistics in Medicine 30(10), 1105–1117]. We develop a new C-Index estimator based on inverse probability weighting (IPW) that corrects for this limitation, and we generalize this estimator to settings with left-truncated and right-censored data. The proposed IPW estimators are highly robust to the underlying truncation distribution and often outperform the conventional methods in terms of bias, mean squared error, and coverage probability. We apply these estimators to evaluate a predictive survival model for mortality among patients with end-stage renal disease.     

A semiparametric mixture cure survival model for left‐truncated and right‐censored data

In follow‐up studies, the disease event time can be subject to left truncation and right censoring. Furthermore, medical advancements have made it possible for patients to be cured of certain types of diseases. In this article, we consider a semiparametric mixture cure model for the regression analysis of left‐truncated and right‐censored data. The model combines a logistic regression for the probability of event occurrence with the class of transformation models for the time of occurrence. We investigate two techniques for estimating model parameters. The first approach is based on martingale estimating equations (EEs). The second approach is based on the conditional likelihood function given truncation variables. The asymptotic properties of both proposed estimators are established. Simulation studies indicate that the conditional maximum‐likelihood estimator (cMLE) performs well while the estimator based on EEs is very unstable even though it is shown to be consistent. This is a special and intriguing phenomenon for the EE approach under cure model. We provide insights into this issue and find that the EE approach can be improved significantly by assigning appropriate weights to the censored observations in the EEs. This finding is useful in overcoming the instability of the EE approach in some more complicated situations, where the likelihood approach is not feasible. We illustrate the proposed estimation procedures by analyzing the age at onset of the occiput‐wall distance event for patients with ankylosing spondylitis.     

Semiparametric additive time-varying coefficients model for longitudinal data with censored time origin

Statistical analysis of longitudinal data often involves modeling treatment effects on clinically relevant longitudinal biomarkers since an initial event (the time origin). In some studies including preventive HIV vaccine efficacy trials, some participants have biomarkers measured starting at the time origin, whereas others have biomarkers measured starting later with the time origin unknown. The semiparametric additive time-varying coefficient model is investigated where the effects of some covariates vary nonparametrically with time while the effects of others remain constant. Weighted profile least squares estimators coupled with kernel smoothing are developed. The method uses the expectation maximization approach to deal with the censored time origin. The Kaplan–Meier estimator and other failure time regression models such as the Cox model can be utilized to estimate the distribution and the conditional distribution of left censored event time related to the censored time origin. Asymptotic properties of the parametric and nonparametric estimators and consistent asymptotic variance estimators are derived. A two-stage estimation procedure for choosing weight is proposed to improve estimation efficiency. Numerical simulations are conducted to examine finite sample properties of the proposed estimators. The simulation results show that the theory and methods work well. The efficiency gain of the two-stage estimation procedure depends on the distribution of the longitudinal error processes. The method is applied to analyze data from the Merck 023/HVTN 502 Step HIV vaccine study.     

Semiparametric analysis of survival data with left truncation and dependent right censoring

Studies of chronic life-threatening diseases often involve both mortality and morbidity. In observational studies, the data may also be subject to administrative left truncation and right censoring. Because mortality and morbidity may be correlated and mortality may censor morbidity, the Lynden-Bell estimator for left-truncated and right-censored data may be biased for estimating the marginal survival function of the non-terminal event. We propose a semiparametric estimator for this survival function based on a joint model for the two time-to-event variables, which utilizes the gamma frailty specification in the region of the observable data. First, we develop a novel estimator for the gamma frailty parameter under left truncation. Using this estimator, we then derive a closed-form estimator for the marginal distribution of the non-terminal event. The large sample properties of the estimators are established via asymptotic theory. The methodology performs well with moderate sample sizes, both in simulations and in an analysis of data from a diabetes registry.     

Weighted regression analysis to correct for informative monitoring times and confounders in longitudinal studies

We address estimation of the marginal effect of a time‐varying binary treatment on a continuous longitudinal outcome in the context of observational studies using electronic health records, when the relationship of interest is confounded, mediated, and further distorted by an informative visit process. We allow the longitudinal outcome to be recorded only sporadically and assume that its monitoring timing is informed by patients' characteristics. We propose two novel estimators based on linear models for the mean outcome that incorporate an adjustment for confounding and informative monitoring process through generalized inverse probability of treatment weights and a proportional intensity model, respectively. We allow for a flexible modeling of the intercept function as a function of time. Our estimators have closed‐form solutions, and their asymptotic distributions can be derived. Extensive simulation studies show that both estimators outperform standard methods such as the ordinary least squares estimator or estimators that only account for informative monitoring or confounders. We illustrate our methods using data from the Add Health study, assessing the effect of depressive mood on weight in adolescents.     

Estimation of separable direct and indirect effects in continuous time

Many research questions involve time-to-event outcomes that can be prevented from occurring due to competing events. In these settings, we must be careful about the causal interpretation of classical statistical estimands. In particular, estimands on the hazard scale, such as ratios of cause-specific or subdistribution hazards, are fundamentally hard to interpret causally. Estimands on the risk scale, such as contrasts of cumulative incidence functions, do have a clear causal interpretation, but they only capture the total effect of the treatment on the event of interest; that is, effects both through and outside of the competing event. To disentangle causal treatment effects on the event of interest and competing events, the separable direct and indirect effects were recently introduced. Here we provide new results on the estimation of direct and indirect separable effects in continuous time. In particular, we derive the nonparametric influence function in continuous time and use it to construct an estimator that has certain robustness properties. We also propose a simple estimator based on semiparametric models for the two cause-specific hazard functions. We describe the asymptotic properties of these estimators and present results from simulation studies, suggesting that the estimators behave satisfactorily in finite samples. Finally, we reanalyze the prostate cancer trial from Stensrud et al. (2020).     

On the validity of time‐dependent AUC estimation in the presence of cure fraction

During the last decades, several approaches have been proposed to estimate the time‐dependent area under the receiver operating characteristic curve (AUC) of risk tools derived from survival data. The validity of these estimators relies on some regularity assumptions among which a survival function being proper. In practice, this assumption is not always satisfied because a fraction of the population may not be susceptible to experience the event of interest even for long follow‐up. Studying the sensitivity of the proposed estimators to the violation of this assumption is of substantial interest. In this paper, we investigate the performance of a nonparametric simple estimator, developed for classical survival data, in the case when the population exhibits a cure fraction. Motivated from the current practice of deriving risk tools in oncology and cardiovascular disease prevention, we also assess the loss, in terms of predictive performance, when deriving risk tools from survival models that do not acknowledge the presence of cure. The simulation results show that the investigated method is valid even under the presence of cure. They also show that risk tools derived from survival models that ignore the presence of cure have smaller AUC compared to those derived from survival models that acknowledge the presence of cure. This was also attested with a real data analysis from a breast cancer study.     

Case-cohort analysis with accelerated failure time model

Summary .  In a case–cohort design, covariates are assembled only for a subcohort that is randomly selected from the entire cohort and any additional cases outside the subcohort. This design is appealing for large cohort studies of rare disease, especially when the exposures of interest are expensive to ascertain for all the subjects. We propose statistical methods for analyzing the case–cohort data with a semiparametric accelerated failure time model that interprets the covariates effects as to accelerate or decelerate the time to failure. Asymptotic properties of the proposed estimators are developed. The finite sample properties of case–cohort estimator and its relative efficiency to full cohort estimator are assessed via simulation studies. A real example from a study of cardiovascular disease is provided to illustrate the estimating procedure.     

Regression on Quantile Residual Life

Summary A time‐specific log‐linear regression method on quantile residual lifetime is proposed. Under the proposed regression model, any quantile of a time‐to‐event distribution among survivors beyond a certain time point is associated with selected covariates under right censoring. Consistency and asymptotic normality of the regression estimator are established. An asymptotic test statistic is proposed to evaluate the covariate effects on the quantile residual lifetimes at a specific time point. Evaluation of the test statistic does not require estimation of the variance–covariance matrix of the regression estimators, which involves the probability density function of the survival distribution with censoring. Simulation studies are performed to assess finite sample properties of the regression parameter estimator and test statistic. The new regression method is applied to a breast cancer data set with long‐term follow‐up to estimate the patients' median residual lifetimes, adjusting for important prognostic factors.     

Semiparametric frailty models for zero‐inflated event count data in the presence of informative dropout

Recurrent events data are commonly encountered in medical studies. In many applications, only the number of events during the follow‐up period rather than the recurrent event times is available. Two important challenges arise in such studies: (a) a substantial portion of subjects may not experience the event, and (b) we may not observe the event count for the entire study period due to informative dropout. To address the first challenge, we assume that underlying population consists of two subpopulations: a subpopulation nonsusceptible to the event of interest and a subpopulation susceptible to the event of interest. In the susceptible subpopulation, the event count is assumed to follow a Poisson distribution given the follow‐up time and the subject‐specific characteristics. We then introduce a frailty to account for informative dropout. The proposed semiparametric frailty models consist of three submodels: (a) a logistic regression model for the probability such that a subject belongs to the nonsusceptible subpopulation; (b) a nonhomogeneous Poisson process model with an unspecified baseline rate function; and (c) a Cox model for the informative dropout time. We develop likelihood‐based estimation and inference procedures. The maximum likelihood estimators are shown to be consistent. Additionally, the proposed estimators of the finite‐dimensional parameters are asymptotically normal and the covariance matrix attains the semiparametric efficiency bound. Simulation studies demonstrate that the proposed methodologies perform well in practical situations. We apply the proposed methods to a clinical trial on patients with myelodysplastic syndromes.     

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.     

Testing goodness of fit of a uniform truncation model

Several goodness-of-fit tests of a lifetime distribution have been suggested in the literature; many take into account censoring and/or truncation of event times. In some contexts, a goodness-of-fit test for the truncation distribution is of interest. In particular, better estimates of the lifetime distribution can be obtained when knowledge of the truncation law is exploited. In cross-sectional sampling, for example, there are theoretical justifications for the assumption of a uniform truncation distribution, and several studies have used it to improve the efficiency of their survival estimates. The duality of lifetime and truncation in the absence of censoring enables methods for testing goodness of fit of the lifetime distribution to be used for testing goodness of fit of the truncation distribution. However, under random censoring, this duality does not hold and different tests are required. In this article, we introduce several goodness-of-fit tests for the truncation distribution and investigate their performance in the presence of censored event times using simulation. We demonstrate the use of our tests on two data sets.     

Methods for Estimating Center Effects on Recurrent Events

In this article, we develop methods for quantifying center effects with respect to recurrent event data. In the models of interest, center effects are assumed to act multiplicatively on the recurrent event rate function. When the number of centers is large, traditional estimation methods that treat centers as categorical variables have many parameters and are sometimes not feasible to implement, especially with large numbers of distinct recurrent event times. We propose a new estimation method for center effects which avoids including indicator variables for centers. We then show that center effects can be consistently estimated by the center-specific ratio of observed to expected cumulative numbers of events. We also consider the case where the recurrent event sequence can be stopped permanently by a terminating event. Large-sample results are developed for the proposed estimators. We assess the finite-sample properties of the proposed estimators through simulation studies. The methods are then applied to national hospital admissions data for end stage renal disease patients.     

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.     

Additive partial linear models with measurement errors

We consider statistical inference for additive partial linearmodels when the linear covariate is measured with error. Wepropose attenuation-to-correction and simulation-extrapolation,simex, estimators of the parameter of interest. It is shownthat the first resulting estimator is asymptotically normaland requires no undersmoothing. This is an advantage of ourestimator over existing backfitting-based estimators for semiparametricadditive models which require undersmoothing of the nonparametriccomponent in order for the estimator of the parametric componentto be root-n consistent. This feature stems from a decreaseof the bias of the resulting estimator, which is appropriatelyderived using a profile procedure. A similar characteristicin semiparametric partially linear models was obtained by Wanget al. (2005). We also discuss the asymptotics of the proposedsimex approach. Finite-sample performance of the proposed estimatorsis assessed by simulation experiments. The proposed methodsare applied to a dataset from a semen study.     

Improved Doubly Robust Estimation When Data Are Monotonely Coarsened,with Application to Longitudinal Studies with Dropout

Summary A routine challenge is that of making inference on parameters in a statistical model of interest from longitudinal data subject to dropout, which are a special case of the more general setting of monotonely coarsened data. Considerable recent attention has focused on doubly robust (DR) estimators, which in this context involve positing models for both the missingness (more generally, coarsening) mechanism and aspects of the distribution of the full data, that have the appealing property of yielding consistent inferences if only one of these models is correctly specified. DR estimators have been criticized for potentially disastrous performance when both of these models are even only mildly misspecified. We propose a DR estimator applicable in general monotone coarsening problems that achieves comparable or improved performance relative to existing DR methods, which we demonstrate via simulation studies and by application to data from an AIDS clinical trial.     

Nonparametric estimation of the survival function for ordered multivariate failure time data: A comparative study

In longitudinal studies of disease, patients may experience several events through a follow‐up period. In these studies, the sequentially ordered events are often of interest and lead to problems that have received much attention recently. Issues of interest include the estimation of bivariate survival, marginal distributions, and the conditional distribution of gap times. In this work, we consider the estimation of the survival function conditional to a previous event. Different nonparametric approaches will be considered for estimating these quantities, all based on the Kaplan–Meier estimator of the survival function. We explore the finite sample behavior of the estimators through simulations. The different methods proposed in this article are applied to a dataset from a German Breast Cancer Study. The methods are used to obtain predictors for the conditional survival probabilities as well as to study the influence of recurrence in overall survival.     

Nonparametric inference for competing risks current status data with continuous, discrete or grouped observation times

New methods and theory have recently been developed to nonparametrically estimate cumulative incidence functions for competing risks survival data subject to current status censoring. In particular, the limiting distribution of the nonparametric maximum likelihood estimator and a simplified naive estimator have been established under certain smoothness conditions. In this paper, we establish the large-sample behaviour of these estimators in two additional models, namely when the observation time distribution has discrete support and when the observation times are grouped. These asymptotic results are applied to the construction of confidence intervals in the three different models. The methods are illustrated on two datasets regarding the cumulative incidence of different types of menopause from a cross-sectional sample of women in the United States and subtype-specific HIV infection from a sero-prevalence study in injecting drug users in Thailand.     

A Note on Variance Estimation of the Aalen–Johansen Estimator of the Cumulative Incidence Function in Competing Risks,with a View towards Left‐Truncated Data

The Aalen–Johansen estimator is the standard nonparametric estimator of the cumulative incidence function in competing risks. Estimating its variance in small samples has attracted some interest recently, together with a critique of the usual martingale‐based estimators. We show that the preferred estimator equals a Greenwood‐type estimator that has been derived as a recursion formula using counting processes and martingales in a more general multistate framework. We also extend previous simulation studies on estimating the variance of the Aalen–Johansen estimator in small samples to left‐truncated observation schemes, which may conveniently be handled within the counting processes framework. This investigation is motivated by a real data example on spontaneous abortion in pregnancies exposed to coumarin derivatives, where both competing risks and left‐truncation have recently been shown to be crucial methodological issues (Meister and Schaefer (2008), Reproductive Toxicology 26 , 31–35). Multistate‐type software and data are available online to perform the analyses. The Greenwood‐type estimator is recommended for use in practice.     

Analysis of longitudinal marginal structural models

In this article we construct and study estimators of the causal effect of a time-dependent treatment on survival in longitudinal studies. We employ a particular marginal structural model (MSM), proposed by Robins (2000), and follow a general methodology for constructing estimating functions in censored data models. The inverse probability of treatment weighted (IPTW) estimator of Robins et al. (2000) is used as an initial estimator and forms the basis for an improved, one-step estimator that is consistent and asymptotically linear when the treatment mechanism is consistently estimated. We extend these methods to handle informative censoring. The proposed methodology is employed to estimate the causal effect of exercise on mortality in a longitudinal study of seniors in Sonoma County. A simulation study demonstrates the bias of naive estimators in the presence of time-dependent confounders and also shows the efficiency gain of the IPTW estimator, even in the absence such confounding. The efficiency gain of the improved, one-step estimator is demonstrated through simulation.