A flexible model based on piecewise linear approximation for the analysis of left truncated right censored data with covariates, and applications to Worcester Heart Attack Study data and Channing House data.

Left truncated right censored (LTRC) data arise quite commonly from survival studies. In this article, a model based on piecewise linear approximation is proposed for the analysis of LTRC data with covariates. Specifically, the model involves a piecewise linear approximation for the cumulative baseline hazard function of the proportional hazards model. The principal advantage of the proposed model is that it does not depend on restrictive parametric assumptions while being flexible and data-driven. Likelihood inference for the model is developed. Through detailed simulation studies, the robustness property of the model is studied by fitting it to LTRC data generated from different processes covering a wide range of lifetime distributions. A sensitivity analysis is also carried out by fitting the model to LTRC data generated from a process with a piecewise constant baseline hazard. It is observed that the performance of the model is quite satisfactory in all those cases. Analyses of two real LTRC datasets by using the model are provided as illustrative examples. Applications of the model in some practical prediction issues are discussed. In summary, the proposed model provides a comprehensive and flexible approach to model a general structure for LTRC lifetime data.

On the role of Volterra integral equations in self-consistent, product-limit, inverse probability of censoring weighted, and redistribution-to-the-right estimators for the survival function.

This paper reconsiders several results of historical and current importance to nonparametric estimation of the survival distribution for failure in the presence of right-censored observation times, demonstrating in particular how Volterra integral equations help inter-connect the resulting estimators. The paper begins by considering Efron's self-consistency equation, introduced in a seminal 1967 Berkeley symposium paper. Novel insights provided in the current work include the observations that (i) the self-consistency equation leads directly to an anticipating Volterra integral equation whose solution is given by a product-limit estimator for the censoring survival function; (ii) a definition used in this argument immediately establishes the familiar product-limit estimator for the failure survival function; (iii) the usual Volterra integral equation for the product-limit estimator of the failure survival function leads to an immediate and simple proof that it can be represented as an inverse probability of censoring weighted estimator; (iv) a simple identity characterizes the relationship between natural inverse probability of censoring weighted estimators for the survival and distribution functions of failure; (v) the resulting inverse probability of censoring weighted estimators, attributed to a highly influential 1992 paper of Robins and Rotnitzky, were implicitly introduced in Efron's 1967 paper in its development of the redistribution-to-the-right algorithm. All results developed herein allow for ties between failure and/or censored observations.

Assessing model prediction performance for the expected cumulative number of recurrent events.

In a recurrent event setting, we introduce a new score designed to evaluate the prediction ability, for a given model, of the expected cumulative number of recurrent events. This score can be seen as an extension of the Brier Score for single time to event data but works for recurrent events with or without a terminal event. Theoretical results are provided that show that under standard assumptions in a recurrent event context, our score can be asymptotically decomposed as the sum of the theoretical mean squared error between the model and the true expected cumulative number of recurrent events and an inseparability term that does not depend on the model. This decomposition is further illustrated on simulations studies. It is also shown that this score should be used in comparison with a reference model, such as a nonparametric estimator that does not include the covariates. Finally, the score is applied for the prediction of hospitalisations on a dataset of patients suffering from atrial fibrillation and a comparison of the prediction performances of different models, such as the Cox model, the Aalen Model or the Ghosh and Lin model, is investigated.

Model averaging for right censored data with measurement error.

This paper studies a novel model averaging estimation issue for linear regression models when the responses are right censored and the covariates are measured with error. A novel weighted Mallows-type criterion is proposed for the considered issue by introducing multiple candidate models. The weight vector for model averaging is selected by minimizing the proposed criterion. Under some regularity conditions, the asymptotic optimality of the selected weight vector is established in terms of its ability to achieve the lowest squared loss asymptotically. Simulation results show that the proposed method is superior to the other existing related methods. A real data example is provided to supplement the actual performance.

CASANOVA: Permutation inference in factorial survival designs.

We propose inference procedures for general factorial designs with time-to-event endpoints. Similar to additive Aalen models, null hypotheses are formulated in terms of cumulative hazards. Deviations are measured in terms of quadratic forms in Nelson-Aalen-type integrals. Different from existing approaches, this allows to work without restrictive model assumptions as proportional hazards. In particular, crossing survival or hazard curves can be detected without a significant loss of power. For a distribution-free application of the method, a permutation strategy is suggested. The resulting procedures' asymptotic validity is proven and small sample performances are analyzed in extensive simulations. The analysis of a data set on asthma illustrates the applicability.

An inverse probability weighted regression method that accounts for right-censoring for causal inference with multiple treatments and a binary outcome.

Comparative effectiveness research often involves evaluating the differences in the risks of an event of interest between two or more treatments using observational data. Often, the post-treatment outcome of interest is whether the event happens within a pre-specified time window, which leads to a binary outcome. One source of bias for estimating the causal treatment effect is the presence of confounders, which are usually controlled using propensity score-based methods. An additional source of bias is right-censoring, which occurs when the information on the outcome of interest is not completely available due to dropout, study termination, or treatment switch before the event of interest. We propose an inverse probability weighted regression-based estimator that can simultaneously handle both confounding and right-censoring, calling the method CIPWR, with the letter C highlighting the censoring component. CIPWR estimates the average treatment effects by averaging the predicted outcomes obtained from a logistic regression model that is fitted using a weighted score function. The CIPWR estimator has a double robustness property such that estimation consistency can be achieved when either the model for the outcome or the models for both treatment and censoring are correctly specified. We establish the asymptotic properties of the CIPWR estimator for conducting inference, and compare its finite sample performance with that of several alternatives through simulation studies. The methods under comparison are applied to a cohort of prostate cancer patients from an insurance claims database for comparing the adverse effects of four candidate drugs for advanced stage prostate cancer.

Performance of Cox regression models for composite time-to-event endpoints with component-wise censoring in randomized trials.

BACKGROUND: Composite time-to-event endpoints are beneficial for assessing related outcomes jointly in clinical trials, but components of the endpoint may have different censoring mechanisms. For example, in the PRagmatic EValuation of evENTs And Benefits of Lipid-lowering in oldEr adults (PREVENTABLE) trial, the composite outcome contains one endpoint that is right censored (all-cause mortality) and two endpoints that are interval censored (dementia and persistent disability). Although Cox regression is an established method for time-to-event outcomes, it is unclear how models perform under differing component-wise censoring schemes for large clinical trial data. The goal of this article is to conduct a simulation study to investigate the performance of Cox models under different scenarios for composite endpoints with component-wise censoring. METHODS: We simulated data by varying the strength and direction of the association between treatment and outcome for the two component types, the proportion of events arising from the components of the outcome (right censored and interval censored), and the method for including the interval-censored component in the Cox model (upper value and midpoint of the interval). Under these scenarios, we compared the treatment effect estimate bias, confidence interval coverage, and power. RESULTS: Based on the simulation study, Cox models generally have adequate power to achieve statistical significance for comparing treatments for composite outcomes with component-wise censoring. In our simulation study, we did not observe substantive bias for scenarios under the null hypothesis or when the treatment has a similar relative effect on each component outcome. Performance was similar regardless of if the upper value or midpoint of the interval-censored part of the composite outcome was used. CONCLUSION: Cox regression is a suitable method for analysis of clinical trial data with composite time-to-event endpoints subject to different component-wise censoring mechanisms.

Weighted log-rank tests for censored data under Wei's urn design: saddlepoint approximation and confidence intervals.

The randomization designs in clinical trials provide probabilistic basis for the statistical inference of the permutation tests. One of the widely used designs to avoid the problems of imbalance and selection bias for one of the treatments is Wei's urn design. In this article, the saddlepoint approximation is suggested to approximate the p-values of the weighted log-rank class of two-sample tests under Wei's urn design. To show the accuracy of the proposed method and to clarify its procedure, two sets of real data are analyzed, and a simulation study is conducted using different sample sizes and three different life time distributions. Through the illustrative examples and simulation study, a comparison is made between the proposed method and the traditional method, which is the normal approximation method. All of these procedures confirmed that the proposed method is more accurate and efficient than the normal approximation method in approximating the exact p-value of the considered class of tests. As a result, the nominal 95% confidence intervals for the treatment effect are determined.

Nonparametric estimation in an illness-death model with component-wise censoring.

In disease settings where study participants are at risk for death and a serious nonfatal event, composite endpoints defined as the time until the earliest of death or the nonfatal event are often used as the primary endpoint in clinical trials. In practice, if the nonfatal event can only be detected at clinic visits and the death time is known exactly, the resulting composite endpoint exhibits "component-wise censoring." The standard method used to estimate event-free survival in this setting fails to account for component-wise censoring. We apply a kernel smoothing method previously proposed for a marker process in a novel way to produce a nonparametric estimator for event-free survival that accounts for component-wise censoring. The key insight that allows us to apply this kernel method is thinking of nonfatal event status as an intermittently observed binary time-dependent variable rather than thinking of time to the nonfatal event as interval-censored. We also propose estimators for the probability in state and restricted mean time in state for reversible or irreversible illness-death models, under component-wise censoring, and derive their large-sample properties. We perform a simulation study to compare our method to existing multistate survival methods and apply the methods on data from a large randomized trial studying a multifactor intervention for reducing morbidity and mortality among men at above average risk of coronary heart disease.

Inverse probability weighted estimators of vaccine effects accommodating partial interference and censoring.

Estimating population-level effects of a vaccine is challenging because there may be interference, that is, the outcome of one individual may depend on the vaccination status of another individual. Partial interference occurs when individuals can be partitioned into groups such that interference occurs only within groups. In the absence of interference, inverse probability weighted (IPW) estimators are commonly used to draw inference about causal effects of an exposure or treatment. Tchetgen Tchetgen and VanderWeele proposed a modified IPW estimator for causal effects in the presence of partial interference. Motivated by a cholera vaccine study in Bangladesh, this paper considers an extension of the Tchetgen Tchetgen and VanderWeele IPW estimator to the setting where the outcome is subject to right censoring using inverse probability of censoring weights (IPCW). Censoring weights are estimated using proportional hazards frailty models. The large sample properties of the IPCW estimators are derived, and simulation studies are presented demonstrating the estimators' performance in finite samples. The methods are then used to analyze data from the cholera vaccine study.

Self-reporting and screening: Data with right-censored, left-censored, and complete observations.

We consider survival data that combine three types of observations: uncensored, right-censored, and left-censored. Such data arises from screening a medical condition, in situations where self-detection arises naturally. Our goal is to estimate the failure-time distribution, based on these three observation types. We propose a novel methodology for distribution estimation using both semiparametric and nonparametric techniques. We then evaluate the performance of these estimators via simulated data. Finally, as a case study, we estimate the patience of patients who arrive at an emergency department and wait for treatment. Three categories of patients are observed: those who leave the system and announce it, and thus their patience time is observed; those who get service and thus their patience time is right-censored by the waiting time; and those who leave the system without announcing it. For this third category, the patients' absence is revealed only when they are called to service, which is after they have already left; formally, their patience time is left-censored. Other applications of our proposed methodology are discussed.

Analysis of composite endpoints with component-wise censoring in the presence of differential visit schedules.

Composite endpoints are very common in clinical research, such as recurrence-free survival in oncology research, defined as the earliest of either death or disease recurrence. Because of the way data are collected in such studies, component-wise censoring is common, where, for example, recurrence is an interval-censored event and death is a right-censored event. However, a common way to analyze such component-wise censored composite endpoints is to treat them as right-censored, with the date at which the non-fatal event was detected serving as the date the event occurred. This approach is known to introduce upward bias when the Kaplan-Meier estimator is applied, but has more complex impact on semi-parametric regression approaches. In this article we compare the performance of the Cox model estimators for right-censored data and the Cox model estimators for interval-censored data in the context of component-wise censored data where the visit process differs across levels of a covariate of interest, a common scenario in observational data. We additionally examine estimators of the cause-specific hazard when applied to the individual components of such component-wise censored composite endpoints. We found that when visit schedules differed according to levels of a covariate of interest, the Cox model estimators for right-censored data and the estimators for cause-specific hazards were increasingly biased as the frequency of visits decreased. The Cox model estimator for interval-censored data with censoring at the last disease-free date is recommended for use in the presence of differential visit schedules.

Model selection for survival individualized treatment rules using the jackknife estimator.

BACKGROUND: Precision medicine is an emerging field that involves the selection of treatments based on patients' individual prognostic data. It is formalized through the identification of individualized treatment rules (ITRs) that maximize a clinical outcome. When the type of outcome is time-to-event, the correct handling of censoring is crucial for estimating reliable optimal ITRs. METHODS: We propose a jackknife estimator of the value function to allow for right-censored data for a binary treatment. The jackknife estimator or leave-one-out-cross-validation approach can be used to estimate the value function and select optimal ITRs using existing machine learning methods. We address the issue of censoring in survival data by introducing an inverse probability of censoring weighted (IPCW) adjustment in the expression of the jackknife estimator of the value function. In this paper, we estimate the optimal ITR by using random survival forest (RSF) and Cox proportional hazards model (COX). We use a Z-test to compare the optimal ITRs learned by RSF and COX with the zero-order model (or one-size-fits-all). Through simulation studies, we investigate the asymptotic properties and the performance of our proposed estimator under different censoring rates. We illustrate our proposed method on a phase III clinical trial of non-small cell lung cancer data. RESULTS: Our simulations show that COX outperforms RSF for small sample sizes. As sample sizes increase, the performance of RSF improves, in particular when the expected log failure time is not linear in the covariates. The estimator is fairly normally distributed across different combinations of simulation scenarios and censoring rates. When applied to a non-small-cell lung cancer data set, our method determines the zero-order model (ZOM) as the best performing model. This finding highlights the possibility that tailoring may not be needed for this cancer data set. CONCLUSION: The jackknife approach for estimating the value function in the presence of right-censored data shows satisfactory performance when there is small to moderate censoring. Winsorizing the upper and lower percentiles of the estimated survival weights for computing the IPCWs stabilizes the estimator.

Conditional Independence Test of Failure and Truncation Times: Essential Tool for Method Selection.

Conditional independence assumption of truncation and failure times conditioning on covariates is a fundamental and common assumption in the regression analysis of left-truncated and right-censored data. Testing for this assumption is essential to ensure the correct inference on the failure time, but this has often been overlooked in the literature. With consideration of challenges caused by left truncation and right censoring, tests for this conditional independence assumption are developed in which the generalized odds ratio derived from a Cox proportional hazards model on the failure time and the concept of Kendall's tau are combined. Except for the Cox proportional hazards model, no additional model assumptions are imposed, and the distributions of the truncation time and conditioning variables are unspecified. The asymptotic properties of the test statistic are established and an easy implementation for obtaining its distribution is developed. The performance of the proposed test has been evaluated through simulation studies and two real studies.

Model selection among Dimension-Reduced generalized Cox models.

Conventional semiparametric hazards regression models rely on the specification of particular model formulations, such as proportional-hazards feature and single-index structures. Instead of checking these modeling assumptions one-by-one, we proposed a class of dimension-reduced generalized Cox models, and then a consistent model selection procedure among this class to select covariates with proportional-hazards feature and a proper model formulation for non-proportional-hazards covariates. In this class, the non-proportional-hazards covariates are treated in a nonparametric manner, and a partial sufficient dimension reduction is introduced to reduce the curse of dimensionality. A semiparametric efficient estimation is proposed to estimate these models. Based on the proposed estimation, we further constructed a cross-validation type criterion to consistently select the correct model among this class. Most importantly, this class of hazards regression models contains the fully nonparametric hazards regression model as the most saturated submodel, and hence no further model diagnosis is required. Overall speaking, this model selection approach is more effective than performing a sequence of conventional model checking. The proposed method is illustrated by simulation studies and a data example.

New classes of tests for the Weibull distribution using Stein's method in the presence of random right censoring.

We develop two new classes of tests for the Weibull distribution based on Stein's method. The proposed tests are applied in the full sample case as well as in the presence of random right censoring. We investigate the finite sample performance of the new tests using a comprehensive Monte Carlo study. In both the absence and presence of censoring, it is found that the newly proposed classes of tests outperform competing tests against the majority of the distributions considered. In the cases where censoring is present we consider various censoring distributions. Some remarks on the asymptotic properties of the proposed tests are included. We present another result of independent interest; a test initially proposed for use with full samples is amended to allow for testing for the Weibull distribution in the presence of censoring. The techniques developed in the paper are illustrated using two practical examples.

Log-rank tests for censored clustered data under generalized randomized block design: Saddlepoint approximation.

The weighted log-rank class is the common and widely used class of two-sample tests for clustered data. Clustered data with censored failure times often arise in tumorigenicity investigations and clinical trials. The randomized block design is a significant design that reduces both unintentional bias and selection bias. Accordingly, the p-values of the null permutation distribution of weighted log-rank class for clustered data are approximated using the double saddlepoint approximation technique. Comprehensive simulation studies are carried out to appraise the accuracy of the saddlepoint approximation. This approximation exhibits a significant improvement in precision over the asymptotic approximation. This precision motivates us to determine the approximated confidence intervals for the treatment impact.

Estimating Consignment-Level Infestation Rates from the Proportion of Consignment that Failed Border Inspection: Possibilities and Limitations in the Presence of Overdispersed Data.

Introduction of pests and diseases through trade is one of the main socioecological challenges worldwide. Targeted sampling at border security can efficiently provide information about biosecurity threats and also reduce pest entry risk. Prioritizing sampling effort requires knowing which pathways are most infested. However, border security inspection data are often right-censored, as inspection agencies often only report that a consignment has failed inspection (i.e., there was at least one unit infested), not how many infested units were found. A method has been proposed to estimate the mean infestation rate of a pathway from such right-censored data (Chen et al.). Using simulations and case study data from imported germplasm consignments inspected at the border, we show that the proposed method results in negatively biased estimates of the pathway infestation rate when the inspection data exhibit overdispersion (i.e., varying infestation rates among different consignments of the same pathway). The case study data also show that overdispersion is prevalent in real data sets. We demonstrate that the method proposed by Chen et al. recovers the median infestation rate of the pathway, rather than its mean. Therefore, it underpredicts the infestation rate when the data are overdispersed (in right-skewed distributions, the mean is above the median). To allow better monitoring and optimizing sampling effort at the border, we recommend that border protection agencies report all the data (the number of infested units found together with the sample size of the inspection) instead of only that the consignment failed inspection.

Using Cure Models to Estimate the Serial Interval of Tuberculosis With Limited Follow-up.

Serial interval (SI), defined as the time between symptom onset in an infector and infectee pair, is commonly used to understand infectious diseases transmission. Slow progression to active disease, as well as the small percentage of individuals who will eventually develop active disease, complicate the estimation of the SI for tuberculosis (TB). In this paper, we showed via simulation studies that when there is credible information on the percentage of those who will develop TB disease following infection, a cure model, first introduced by Boag in 1949, should be used to estimate the SI for TB. This model includes a parameter in the likelihood function to account for the study population being composed of those who will have the event of interest and those who will never have the event. We estimated the SI for TB to be approximately 0.5 years for the United States and Canada (January 2002 to December 2006) and approximately 2.0 years for Brazil (March 2008 to June 2012), which might imply a higher occurrence of reinfection TB in a developing country like Brazil.

Nonparametric Bayesian inference for mean residual life functions in survival analysis.

Modeling and inference for survival analysis problems typically revolves around different functions related to the survival distribution. Here, we focus on the mean residual life (MRL) function, which provides the expected remaining lifetime given that a subject has survived (i.e. is event-free) up to a particular time. This function is of direct interest in reliability, medical, and actuarial fields. In addition to its practical interpretation, the MRL function characterizes the survival distribution. We develop general Bayesian nonparametric inference for MRL functions built from a Dirichlet process mixture model for the associated survival distribution. The resulting model for the MRL function admits a representation as a mixture of the kernel MRL functions with time-dependent mixture weights. This model structure allows for a wide range of shapes for the MRL function. Particular emphasis is placed on the selection of the mixture kernel, taken to be a gamma distribution, to obtain desirable properties for the MRL function arising from the mixture model. The inference method is illustrated with a data set of two experimental groups and a data set involving right censoring. The supplementary material available at Biostatistics online provides further results on empirical performance of the model, using simulated data examples.