Left Truncation Research Articles

A pairwise likelihood augmented Cox estimator for left-truncated data.

Survival data collected from a prevalent cohort are subject to left truncation and the analysis is challenging. Conditional approaches for left-truncated data could be inefficient as they ignore the information in the marginal likelihood of the truncation times. Length-biased sampling methods may improve the estimation efficiency but only when the underlying truncation time is uniform; otherwise, they may generate biased estimates. We propose a semiparametric method for left-truncated data under the Cox model with no parametric distributional assumption about the truncation times. Our approach is to make inference based on the conditional likelihood augmented with a pairwise likelihood, which eliminates the truncation distribution, yet retains the information about the regression coefficients and the baseline hazard function in the marginal likelihood. An iterative algorithm is provided to solve for the regression coefficients and the baseline hazard function simultaneously. By empirical process and U-process theories, it has been shown that the proposed estimator is consistent and asymptotically normal with a closed-form consistent variance estimator. Simulation studies show substantial efficiency gain of our estimator in both the regression coefficients and the cumulative baseline hazard function over the conditional approach estimator. When the uniform truncation assumption holds, our estimator enjoys smaller biases and efficiency comparable to that of the full maximum likelihood estimator. An application to the analysis of a chronic kidney disease cohort study illustrates the utility of the method.

Analysis of repairable systems with severe left censoring or truncation

ABSTRACTLeft censoring or left truncation occurs when specific failure information on machines is not available before a certain age. If only the number of failures but not the actual failure times before a certain age is known, we have left censoring. If neither the number of failures nor the times of failure are known, we have left truncation. A datacenter will typically include servers and storage equipment installed on different dates. However, data collection on failures and repairs may not begin on the installation date. Often, the capture of reliability data starts only after the initiation of a service contract on a particular date. Thus, such data may exhibit severe left censoring or truncation, since machines may have operated for considerable time periods without any reliability history being recorded. This situation is quite different from the notion of left censoring in non-repairable systems, which has been dealt with extensively in the literature. Parametric modeling methods are less intuitive when the data has severe left censoring. In contrast, non-parametric methods based on the Mean Cumulative Function (MCF), recurrence rate plots, and calendar time analysis are simple to use and can provide valuable insights into the reliability of repairable systems, even under severe left censoring or truncation. The techniques shown have been successfully applied at a large server manufacturer to quantify the reliability of computer servers at customer sites. In this discussion, the techniques will be illustrated with actual field examples.

Abstract 2780: High nuclear FTH1 protein expression predicts early disease recurrence for BRCA1/2 mutation carriers

Abstract FTH1 is a 21kD subunit of the ferritin complex, which is involved in intracellular iron storage. In a previous study we found that FTH1 proteins can be located either in the cytoplasm or nucleus and, depending on its cellular localization, is associated with good or poor prognosis in triple negative breast cancer (TNBC). Furthermore, cytoplasmic localization of cFTH1 was associated with an increased anti-tumor (CD8+) T-cell / tumor-promoting (CD4+) T-cell ratio suggesting an immune modulatory role for FTH1. In the current study we have assessed the prognostic value of FTH1 expression for BRCA1/2 mutation carriers. FTH1 proteins were visualized by immunohistochemistry on a tissue microarray containing primary breast cancer specimens from 222 BRCA1/2 mutation carriers that were diagnosed between 1982 and 2008. Primary breast cancers from BRCA1 carriers were predominantly of the TNBC subtype, whereas primary breast cancers from BRCA2 mutation carriers were mainly ER-positive. The association between FTH1 expression and clinical outcome was evaluated by a Cox proportional hazards regression model using left truncation to adjust for study inclusion bias. The median follow-up time of the study was 9.2 years. High nuclear FTH1 (nFTH1), but not cytoplasmic FTH1 (cFTH1), was associated with decreased metastasis-free and disease-free survival as compared with low nFTH1 (multivariable HR=3.54, 95% CI=1.45-8.66, P=0.0056 adjusted for tumor size and nodal status; multivariable HR=2.71, 95% CI=1.49-4.92, P=0.0011 adjusted for tumor size). Neither nFTH1 nor cFTH1 were associated with breast cancer-specific or overall survival. This suggests that high nFTH1 expression in primary breast cancer predicts earlier disease recurrence for BRCA1/2 mutation carriers. Since we had observed that CD4+:CD8+ T-cell ratios were increased among TNBCs with high nFTH1, we hypothesize that nFTH1 promotes recruitment of immune suppressive CD4+/FOXP3+ regulatory T-cells in TNBC and BRCA1/2-associated tumors, suppressing CD4+/FOXP3- and CD8+ T-cell anti-tumor immunity. Citation Format: Antoinette Hollestelle, Mieke Timmermans, Renée Broeren-Foekens, Anita Trapman - Jansen, Carolien H. van Deurzen, Arzu Umar, John W. Martens, Maartje Hooning. High nuclear FTH1 protein expression predicts early disease recurrence for BRCA1/2 mutation carriers [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2017; 2017 Apr 1-5; Washington, DC. Philadelphia (PA): AACR; Cancer Res 2017;77(13 Suppl):Abstract nr 2780. doi:10.1158/1538-7445.AM2017-2780

Joint models for longitudinal counts and left-truncated time-to event data with applications to health insurance

Aging societies have given rise to important challenges in the field of health insurance. Elderly policyholders need to be provided with fair premiums based on their individual health status, whereas insurance companies want to plan for the potential costs of tackling lifetimes above mean expectations. In this article, we focus on a large cohort of policyholders in Barcelona (Spain), aged 65 years and over. A shared-parameter joint model is proposed to analyse the relationship between annual demand for emergency claims and time until death outcomes, which are subject to left truncation. We compare different functional forms of the association between both processes, and, furthermore, we illustrate how the fitted model provides time-dynamic predictions of survival probabilities. The parameter estimation is performed under the Bayesian framework using Markov chain Monte Carlo methods.

Left Truncation Bias to Explain the Protective Effect of Smoking on Preeclampsia: Potential, But How Plausible?

An inverse association between maternal smoking and preeclampsia has been frequently observed in epidemiologic studies for several decades. In the May 2015 issue of this journal, Lisonkova and Joseph described a simulation study suggesting that bias from left truncation might explain the inverse association. The simulations were based on strong assumptions regarding the underlying mechanisms through which bias might occur. To examine the sensitivity of the previous authors' conclusions to these assumptions, we constructed a new Monte Carlo simulation using published estimates to frame our data-generating parameters. We estimated the association between smoking and preeclampsia across a range of scenarios that incorporated abnormal placentation and early pregnancy loss. Our results confirmed that the previous authors' findings are highly dependent on assumptions regarding the strength of association between abnormal placentation and preeclampsia. Thus, the bias they described may be less pronounced than was suggested. Under empirically derived constraints of these critical assumptions, left truncation does not appear to fully explain the inverse association between smoking and preeclampsia. Furthermore, when considering processes in which left truncation may result from the exposure, it is important to precisely describe the target population and parameter of interest before assessing potential bias. We comment on the specification of a meaningful target population when assessing maternal smoking and preeclampsia as a public health issue. We describe considerations for defining a target population in studies of perinatal exposures when those exposures cause competing events (e.g., early pregnancy loss) for primary outcomes of interest.

Drospirenone-Containing Oral Contraceptive Pills and the Risk of Venous Thromboembolism: An Assessment of Risk in First-Time Users and Restarters.

The effects of drospirenone-containing combined oral contraceptives (COCs) on the risk of venous thromboembolism (VTE) remain controversial due to the challenge in distinguishing between first-time users and restarters, and their different underlying VTE risks, in healthcare databases. The aim of this study was to describe the challenge of studying the risk of VTE among first-time users of drospirenone-containing COCs in a healthcare database and assess the risk among first-time users and restarters. We used data from the Clinical Practice Research Datalink to construct two cohorts. The first-time user cohort included all women aged 16-45years who received a first ever prescription of drospirenone- or levonorgestrel-containing COCs between May 2002 and March 2015. The restarter cohort included those who were restarting a COC after a period of non-use of ≥6months. Cox proportional hazards models adjusted for high dimensional propensity scores were used to estimate hazard ratios (HRs) and 95% confidence intervals (CIs). The final cohorts included 55,139 first-time users (3582 drospirenone and 51,557 levonorgestrel) and 162,959 restarters (23,191 drospirenone and 139,768 levonorgestrel). The adjusted HR of VTE associated with drospirenone versus levonorgestrel was 3.19 (95% CI 1.12-9.08) for first-time users and 1.96 (95% CI 1.12-3.41) for restarters. We found an elevated risk of VTE associated with drospirenone-containing COCs in comparison with levonorgestrel-containing COCs in both cohorts. While left truncation of healthcare databases is a concern for the identification of first-time users, the use of a more explicit cohort of restarters suggests a doubling of VTE risk with drospirenone-containing COCs.

Cox proportional hazards models with left truncation and time-varying coefficient: Application of age at event as outcome in cohort studies.

When analyzing time-to-event cohort data, two different ways of choosing a time scale have been discussed in the literature: time-on-study or age at onset of disease. One advantage of choosing the latter is interpretability of the hazard ratio as a function of age. To handle the analysis of age at onset in a principled manner, we present an analysis of the Cox Proportional Hazards model with time-varying coefficient for left-truncated and right-censored data. In the analysis of Northern Manhattan Study (NOMAS) with age at onset of stroke as outcome, we demonstrate that well-established risk factors may be important only around a certain age span and less established risk factors can have a strong effect in a certain age span.

Estimation of parametric failure time distributions based on interval-censored data with irregular dependent follow-up.

Event history studies based on disease clinic data often face several complications. Specifically, patients may visit the clinic irregularly, and the intermittent observation times could depend on disease-related variables; this can cause a failure time outcome to be dependently interval-censored. We propose a weighted estimating function approach so that dependently interval-censored failure times can be analysed consistently. A so-called inverse-intensity-of-visit weight is employed to adjust for the informative inspection times. Left truncation of failure times can also be easily handled. Additionally, in observational studies, treatment assignments are typically non-randomized and may depend on disease-related variables. An inverse-probability-of-treatment weight is applied to estimating functions to further adjust for measured confounders. Simulation studies are conducted to examine the finite sample performances of the proposed estimators. Finally, the Toronto Psoriatic Arthritis Cohort Study is used for illustration. Copyright © 2017 John Wiley & Sons, Ltd.

On Truncation Invariant Copulas and their Estimation

Abstract The paper deals with the family of irreducible left truncation invariant bivariate copulas, which admit a nontrivial lower tail dependence function. Such copulas, similarly as the Archimedean ones, are characterized by a functional parameter, a generator being an increasing convex function.We provide a nonparametric, piece-wise linear estimator of such generators.

Regression models for the restricted residual mean life for right‐censored and left‐truncated data

The hazard ratios resulting from a Cox's regression hazards model are hard to interpret and to be converted into prolonged survival time. As the main goal is often to study survival functions, there is increasing interest in summary measures based on the survival function that are easier to interpret than the hazard ratio; the residual mean time is an important example of those measures. However, because of the presence of right censoring, the tail of the survival distribution is often difficult to estimate correctly. Therefore, we consider the restricted residual mean time, which represents a partial area under the survival function, given any time horizon τ, and is interpreted as the residual life expectancy up to τ of a subject surviving up to time t. We present a class of regression models for this measure, based on weighted estimating equations and inverse probability of censoring weighted estimators to model potential right censoring. Furthermore, we show how to extend the models and the estimators to deal with delayed entries. We demonstrate that the restricted residual mean life estimator is equivalent to integrals of Kaplan-Meier estimates in the case of simple factor variables. Estimation performance is investigated by simulation studies. Using real data from Danish Monitoring Cardiovascular Risk Factor Surveys, we illustrate an application to additive regression models and discuss the general assumption of right censoring and left truncation being dependent on covariates. Copyright © 2017 John Wiley & Sons, Ltd.

Quantile regression and variable selection for partially linear model with randomly truncated data

This paper focuses on the problem of estimation and variable selection for quantile regression (QR) of partially linear model (PLM) where the response is subject to random left truncation. We propose a three-stage estimation procedure for parametric and nonparametric parts based on the weights which are random quantities and determined by the product-limit estimates of the distribution function of truncated variable. The estimators obtained in the second and third stages are more efficient than the initial estimators in the first stage. Furthermore, we propose a variable selection procedure for the QR of PLM by combining the estimation method with the smoothly clipped absolute deviation penalty to get sparse estimation of the regression parameter. The oracle properties of the variable selection approach are established. Simulation studies are conducted to examine the performance of our estimators and variable selection method.

A Berry-Esseen type bound for the kernel density estimator based on a weakly dependent and randomly left truncated data

In many applications, the available data come from a sampling scheme that causes loss of information in terms of left truncation. In some cases, in addition to left truncation, the data are weakly dependent. In this paper we are interested in deriving the asymptotic normality as well as a Berry-Esseen type bound for the kernel density estimator of left truncated and weakly dependent data.

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.

Age-at-onset subsets of bipolar I disorders: A critical insight into admixture analyses.

Gaussian mixture analysis is frequently used to model the age-at-onset (AAO) in bipolar I disorder and identify homogeneous subsets of patients. This study aimed to examine whether, using admixture analysis of AAO, cross-sectional designs (which cause right truncation), unreliable diagnosis for individuals younger than 10years old (which causes left truncation) and the selection criterion used for admixture analysis impact the number of identified subsets. A simulation study was performed. Different criteria - the likelihood ratio test (LRT), the Akaike information criterion (AIC), and the Bayesian information criterion (BIC) - were compared using no, left and/or right truncation simulated data. The error rate of each criterion (percentage of erroneous number of detected subsets) was estimated. An application to two real databases, including 2,876 and 1,393 patients, is provided. Without data truncation and regardless of the distribution of AAO, the LRT and the AIC had much higher error rates (12% and 33%, respectively) than the BIC (1%). For a homogeneous population, the error rate increased with the introduction of left truncation. This study shows that the number of subsets identified using admixture analysis may depend on the sample size, the selection criterion, and the study design.

Periconceptional exposure to bisphenol-A and fecundability among couples trying to conceive in a prospective, preconception cohort

Background: Bisphenol-A (BPA) is an endocrine disrupting chemical with nearly ubiquitous exposure. BPA caused reproductive dysfunction in animal models, but data in humans are limited. Objective: Examine couple urinary BPA concentration and fecundability. Methods: A prospective, pre-conception cohort of heterosexual couples (women 18–35, men 18–40) without known infertility or subfertility were recruited, and women observed cervical mucus to identify an estimated day of ovulation and fertile window. Couples collected daily first-morning urine samples during the fertile window and women continued daily collection until the onset of menses or estimated day of ovulation +18 days. Cox proportional hazards models for discrete survival time, adjusting for left truncation for time trying before enrollment, were performed to assess time-varying BPA and fecundability. Models were adjusted for woman’s age and education level, household income, women and men’s BMI, and parity. Fecundability odds ratios (FOR) <1.0 indicate reduced fecundity and longer time to pregnancy, whereas FORs>1.0 indicate enhanced fecundity and shorter time to pregnancy. Results: Women collected mean 16.7±5.2 and men collected mean 5.1±2.1 urine samples per cycle. Among 183 couples, 126 conceived during the study, and 16 early pregnancy losses were observed. There was no association between periconceptional BPA exposure and fecundability either among all pregnancies (FOR 0.995, 95% CI: 0.962, 1.029) or when restricting to live births (FOR 1.002, 95% CI: 0.969, 1.036). Discussion: In this sample of 183 couples, no association between BPA and fecundability was observed. Strengths of this study were the prospective biospecimen collection during the fertile window and serial, daily urine collection to support robust BPA exposure assessment for both men and women.

Left truncation criteria for survival analysis of white‐tailed deer

ABSTRACTSurvival estimates are commonly obtained by physically capturing wildlife and marking or affixing a transmitter to a representative sample of the population. Bias induced by capture stress can occur for white‐tailed deer (Odocoileus virginianus) if capture influences the probability of mortality. To mitigate this bias, researchers often left truncate data for a threshold number of days (14–28 days for deer) after capture. Potential costs of left truncation include reduced sample size and reduced inference. Costs associated with capture and monitoring of deer are substantial, and defining a truncation period is usually arbitrary or ad hoc. Hence, researchers need to evaluate objectively the effects of left truncation. We analyzed time‐to‐event data from 1,001 radio‐collared white‐tailed deer from northern forests and eastern farmlands of Wisconsin, USA in 2011–2014 to evaluate justification for using a 2‐week truncation period by comparing the probability of mortality for deer <2 weeks post‐capture and deer ≥2 weeks post‐capture. We found little support for a difference in mortality between the groups. Results accounting for time of year, study area, and age suggested a 0.69 probability that deer ≥2 weeks post‐ capture had a lower mortality rate than deer <2 weeks post‐capture. Using a reference group of 6–10 ‐month‐old deer in the eastern farmlands in 2011, cumulative capture season mortality was 0.102 (50% CI = 0.075–0.125; SE = 0.037) for deer <2 weeks post‐capture and 0.114 (50% CI = 0.087–0.138; SE = 0.037) for deer ≥2 weeks post‐capture. Our results suggested that when a careful review of cause‐specific mortality is considered for each deer, and presumed capture‐related mortalities are removed from the sample, a 2‐week truncation period is unnecessary. © 2016 The Wildlife Society.

Cause-specific hazard regression for competing risks data under interval censoring and left truncation

Inference for cause-specific hazards from competing risks data under interval censoring and possible left truncation has been understudied. Aiming at this target, a penalized likelihood approach for a Cox-type proportional cause-specific hazards model is developed, and the associated asymptotic theory is discussed. Monte Carlo simulations show that the approach performs very well for moderate sample sizes. An application to a longitudinal study of dementia illustrates the practical utility of the method. In the application, the age-specific hazards of AD, other dementia and death without dementia are estimated, and risk factors of all competing risks are studied.

The analysis of terminal endpoint events in stepped wedge designs.

The stepped wedge design is a unique clinical trial design that allows for a sequential introduction of an intervention. However, the statistical analysis is unclear when this design is applied in survival data. The time-dependent introduction of the intervention in combination with terminal endpoints and interval censoring makes the analysis more complicated. In this paper, a time-on-study scale discrete survival model was constructed. Simulations were conducted primarily to study the performance of our model for different settings of the stepped wedge design. Secondary, we compared our approach to continuous Cox proportional hazard model. The results show that the discrete survival model estimates the intervention effects unbiasedly. If the length of the censoring interval is increased, the precision of the estimates is decreased. Without left truncation and late entry, the number of steps improves the precision of the estimates, whereas in combination of left truncation and late entry, the number of steps decreases the precision. Given the same number of participants and clusters, a parallel group design has higher precision than a stepped wedge design. Copyright © 2016 John Wiley & Sons, Ltd.

Survival analysis with delayed entry in selected families with application to human longevity.

In the field of aging research, family-based sampling study designs are commonly used to study the lifespans of long-lived family members. However, the specific sampling procedure should be carefully taken into account in order to avoid biases. This work is motivated by the Leiden Longevity Study, a family-based cohort of long-lived siblings. Families were invited to participate in the study if at least two siblings were 'long-lived', where 'long-lived' meant being older than 89 years for men or older than 91 years for women. As a result, more than 400 families were included in the study and followed for around 10 years. For estimation of marker-specific survival probabilities and correlations among life times of family members, delayed entry due to outcome-dependent sampling mechanisms has to be taken into account. We consider shared frailty models to model left-truncated correlated survival data. The treatment of left truncation in shared frailty models is still an open issue and the literature on this topic is scarce. We show that the current approaches provide, in general, biased estimates and we propose a new method to tackle this selection problem by applying a correction on the likelihood estimation by means of inverse probability weighting at the family level.

Inverted-U relationship between R&D intensity and survival: Evidence on scale and complementarity effects in UK data

Existing evidence on the relationship between R&D intensity and firm survival is varied and often conflicting. We argue that this may be due to overlooking R&D scale effects and complementarity between R&D intensity and market concentration. Drawing on Schumpeterian models of competition and innovation, we address these issues by developing a formal model of firm survival and using a panel dataset of 37,930 of R&D-active UK firms over 1998–2012. We report the following findings: (i) the relationship between R&D intensity and firm survival follows an inverted-U pattern that reflects diminishing scale effects; (ii) R&D intensity and market concentration are complements in that R&D-active firms have longer survival time if they are in more concentrated industries; and (iii) creative destruction as proxied by median R&D intensity in the industry and the premium on business lending have negative effects on firm survival. Other findings concerning age, size, productivity, relative growth, Pavitt technology classes and the macroeconomic environment are in line with the existing literature. The results are strongly or moderately robust to different samples, stepwise estimations, and controls for frailty and left truncation.