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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. [ABSTRACT FROM AUTHOR]

Background Shorter average lifespans for minoritized populations are hypothesized to stem from 'weathering' or accelerated health declines among minoritized individuals due to systemic marginalization. However, evidence is mixed on whether racial/ethnic differences exist in reproductive ageing, potentially due to selection biases in cohort studies that may systematically exclude 'weathered' participants. This study examines racial/ethnic disparities in the age of menopause after accounting for differential selection 'into' (left truncation) and 'out of' (right censoring) a cohort of midlife women. Methods Using data from the Study of Women's Health Across the Nation (SWAN) cross-sectional screener (N  = 15 695) and accompanying ∼20-year longitudinal cohort (N  = 3302) (1995–2016), we adjusted for potential selection bias using inverse probability weighting (left truncation) to account for socio-demographic/health differences between the screening and cohort study, and multiple imputation (right censoring) to estimate racial/ethnic differences in age at menopause (natural and surgical). Results Unadjusted for selection, no Black/White differences in menopausal timing [hazard ratio (HR)=0.98 (0.86, 1.11)] were observed. After adjustment, Black women had an earlier natural [HR = 1.13 (1.00, 1.26)] and surgical [HR= 3.21 (2.80, 3.62)] menopause than White women with natural menopause—corresponding to a 1.2-year Black/White difference in menopause timing overall. Conclusions Failure to account for multiple forms of selection bias masked racial/ethnic disparities in the timing of menopause in SWAN. Results suggest that there may be racial differences in age at menopause and that selection particularly affected the estimated menopausal age for women who experienced earlier menopause. Cohorts should consider incorporating methods to account for all selection biases, including left truncation, as they impact our understanding of health in 'weathered' populations. [ABSTRACT FROM AUTHOR]

The analysis of left truncated and right censored data is very common in survival and reliability analysis. In lifetime studies patients are often subject to left truncation in addition to right censoring. For example, in bone marrow transplant studies based on International Bone Marrow Transplant Registry (IBMTR), the patients who die while waiting for the transplants will not be reported to the IBMTR. In this paper, we develop novel U-statistics under left truncation and right censoring. We prove the n -consistency of the proposed U-statistics. We derive the asymptotic distribution of the U-statistics using the counting process technique. As an application of the U-statistics, we develop a simple non-parametric test for testing the independence between time to failure and cause of failure in competing risks when the observations are subject to left truncation and right censoring. The finite sample performance of the proposed test is evaluated through a Monte Carlo simulation study. Finally, we illustrate our test procedure using the lifetime data of transformers. [ABSTRACT FROM AUTHOR]

A platform trial is a trial involving an innovative adaptive design with a single master protocol to efficiently evaluate multiple interventions. It offers flexible features such as dropping interventions for futility and adding new interventions to be evaluated during the course of a trial. Although there is a consensus that platform trials can identify beneficial interventions with fewer patients, less time, and a higher probability of success than traditional trials, there remains debate on certain issues, one of which is whether (and how) the non-concurrent control (NCC) (i.e., patients in the control group recruited prior to the new interventions) can be combined with the current control (CC) in the analysis, especially if there is a change of standard of care during the trial. Methods: In this paper, considering time-to-event endpoints under the proportional hazard model assumption, we introduce a new concept of NCC concurrent observation time (NCC COT), and propose to borrow NCC COT through left truncation. This assumes that the NCC COT and CC are comparable. If the protocol does not prohibit NCC patients to change the standard of care while on study, NCC COT and CC likely will share the same standard of care. A simulated example is provided to demonstrate the approach. Results: Using exponential distributions, the simulated example assumes that NCC COT and CC have the same hazard, and the treatment group has a lower hazard. The estimated HR comparing treatment to the pooled control group is 0.744 (95% CI 0.575, 0.962), whereas the comparison to the CC group alone is 0.755 (95% CI 0.566, 1.008), with corresponding p-values of 0.024 versus 0.057, respectively. This suggests that borrowing NCC COT can improve statistical efficiency when the exchangeability assumption holds. Conclusion: This article proposes an innovative approach of borrowing NCC COT to enhance statistical inference in platform trials under appropriate scenarios.

A platform trial is a trial involving an innovative adaptive design with a single master protocol to efficiently evaluate multiple interventions. It offers flexible features such as dropping interventions for futility and adding new interventions to be evaluated during the course of a trial. Although there is a consensus that platform trials can identify beneficial interventions with fewer patients, less time, and a higher probability of success than traditional trials, there remains debate on certain issues, one of which is whether (and how) the non-concurrent control (NCC) (i.e., patients in the control group recruited prior to the new interventions) can be combined with the current control (CC) in the analysis, especially if there is a change of standard of care during the trial. Methods: In this paper, considering time-to-event endpoints under the proportional hazard model assumption, we introduce a new concept of NCC concurrent observation time (NCC COT), and propose to borrow NCC COT through left truncation. This assumes that the NCC COT and CC are comparable. If the protocol does not prohibit NCC patients to change the standard of care while on study, NCC COT and CC likely will share the same standard of care. A simulated example is provided to demonstrate the approach. Results: Using exponential distributions, the simulated example assumes that NCC COT and CC have the same hazard, and the treatment group has a lower hazard. The estimated HR comparing treatment to the pooled control group is 0.744 (95% CI 0.575, 0.962), whereas the comparison to the CC group alone is 0.755 (95% CI 0.566, 1.008), with corresponding p-values of 0.024 versus 0.057, respectively. This suggests that borrowing NCC COT can improve statistical efficiency when the exchangeability assumption holds. Conclusion: This article proposes an innovative approach of borrowing NCC COT to enhance statistical inference in platform trials under appropriate scenarios. [ABSTRACT FROM AUTHOR]

Cross-sectional sampling is often used when investigating inter-event times, resulting in left-truncated and right-censored data. In this paper, we consider a semiparametric truncation model in which the truncating variable is assumed to belong to a certain parametric family. We examine two methods of estimating both the truncation and the lifetime distributions. We obtain asymptotic representations of the estimators for the lifetime distribution and establish their weak convergence. Both of the proposed estimators perform better than Wang 's (1991) nonparametric maximum likelihood estimator in terms of the integrated mean squared error, when the parametric family for the truncation is sufficiently close to its true distribution. The full likelihood approach is preferable to the conditional likelihood approach in estimating the lifetime distribution, though not necessarily the truncation distribution. In an application to Alzheimer's disease data, hypothesis tests reject the uniform truncation distribution, but several other parametric models lead to similar behaviour of the truncation and lifetime distributions after disease onset. [ABSTRACT FROM AUTHOR]

Left-truncation and right-censoring arise frequently when considering lifetime data. When both incompleteness conditions occur, a product-limit estimator was proposed and investigated in the independent case by Tsai et al. (Biometrika74, 883–886, 1987). In the presence of covariates, the conditional version was studied in the α-mixing setting by Liang et al. (Test21, 790–810, 2012). Our objective in the present paper is to derive strong uniform consistency rates for the cumulative hazard and the product-limit estimates when the lifetime observations form an associated sequence. Then, as an application we derive a strong uniform consistency rate for the kernel estimator of the hazard rate function considered by Uzunoḡullari and Wang (Biometrika79, 297–310, 1992) in the iid case. [ABSTRACT FROM AUTHOR]

A time to event, |$X$|⁠ , is left-truncated by |$T$| if |$X$| can be observed only if |$T

To shed light on the etiology of metabolic syndrome development, it is important to understand whether its 5 component disorders follow certain onset sequences. To explore disease progression of the syndrome, we studied the ages at onset of 5 cardiometabolic diseases: abdominal obesity, diabetes, hypertension, hypertriglyceridemia, and hypo-α-lipoproteinemia. In analyzing longitudinal data from the Cardiovascular Disease Risk Factors Two-Township Study (1989-2002) in Taiwan, we adjusted for nonsusceptibility, utilizing the logistic-accelerated failure time location-scale mixture regression models for left-truncated and interval-censored data to simultaneously estimate the associations of township and sex with the susceptibility probability and the age-at-onset distribution of susceptible individuals for each disease. We then validated the onset sequences of 5 cardiometabolic diseases by comparing the overall probability density curves across township-sex strata. Visualization of these curves indicates that women tended to have onsets of abdominal obesity and hypo-α-lipoproteinemia in young adulthood, hypertension and hypertriglyceridemia in middle age, and diabetes later; men tended to have onsets of abdominal obesity, hypo-α-lipoproteinemia, and hypertriglyceridemia in young adulthood, hypertension in middle age, and diabetes later. Different onset patterns of abdominal obesity, hypo-α-lipoproteinemia, and male hypertension were identified between townships. Our proposed method provides a novel strategy for investigating both pathogenesis and preventive measures of complex syndromes. [ABSTRACT FROM AUTHOR]

The idea that epidemiologic studies should start from first exposure onward has been advocated in the past few years. The study of incident exposures is contrasted with studies of prevalent exposures in which follow-up may commence after first exposure. The former approach is seen as a hallmark of a good study and necessary for causal inference. We argue that studying incident exposures may be necessary in some situations, but it is not always necessary and is not the preferred option in many instances. Conducting a study involves decisions as to which person-time experience should be included. Although studies of prevalent exposures involve left truncation (missingness on the left), studies of incident exposures may involve right censoring (missingness on the right) and therefore may not be able to assess the long-term effects of exposure. These considerations have consequences for studies of dynamic (open) populations that involve a mixture of prevalent and incident exposures. We argue that studies with prevalent exposures will remain a necessity for epidemiology. The purpose of this paper is to restore the balance between the emphasis on first exposure cohorts and the richness of epidemiologic information obtained when studying prevalent exposures. [ABSTRACT FROM AUTHOR]

For a linear regression model subject to left-truncation and right-censoring where the truncation and censoring points are known constants (or always observed if random), Karlsson and Laitila (Stat Probab Lett 78:2567-2571, ) proposed a semiparametric estimator which deals with left-truncation by trimming and right-censoring by 'winsorizing'. The estimator was motivated by a zero moment condition where a transformed error term appears with trimmed and winsorized tails. This paper takes the semiparametric estimator further by deriving the asymptotic distribution that was not shown in Karlsson and Laitila (Stat Probab Lett 78:2567-2571, ) and discusses its implementation aspects in practice, albeit brief. [ABSTRACT FROM AUTHOR]

Quantile regression methods have been used to estimate upper and lower quantile reference curves as the function of several covariates. In this article, it is demonstrated that the estimating equation of Zhou [A weighted quantile regression for randomly truncated data, Comput. Stat. Data Anal. 55 (2011), pp. 554–566.] can be extended to analyse left-truncated and right-censored data. We evaluate the finite sample performance of the proposed estimators through simulation studies. The proposed estimator βˆ(q) is applied to the Veteran's Administration lung cancer data reported by Prentice [Exponential survival with censoring and explanatory variables, Biometrika 60 (1973), pp. 279–288]. [ABSTRACT FROM AUTHOR]

The maximum likelihood estimates (MLEs) of the parameters of a two-parameter lognormal distribution with left truncation and right censoring are developed through the Expectation Maximization (EM) algorithm. For comparative purpose, the MLEs are also obtained by the Newton–Raphson method. The asymptotic variance-covariance matrix of the MLEs is obtained by using the missing information principle, under the EM framework. Then, using asymptotic normality of the MLEs, asymptotic confidence intervals for the parameters are constructed. Asymptotic confidence intervals are also obtained using the estimated variance of the MLEs by the observed information matrix, and by using parametric bootstrap technique. Different confidence intervals are then compared in terms of coverage probabilities, through a Monte Carlo simulation study. A prediction problem concerning the future lifetime of a right censored unit is also considered. A numerical example is given to illustrate all the inferential methods developed here. [ABSTRACT FROM PUBLISHER]

The cross-ratio is an important local measure that characterizes the dependence between bivariate failure times. To estimate the cross-ratio in follow-up studies where delayed entry is present, estimation procedures need to account for left truncation. Ignoring left truncation yields biased estimates of the cross-ratio. We extend the method of Hu et al., Biometrika 98:341-354 () by modifying the risk sets and relevant indicators to handle left-truncated bivariate failure times, which yields the cross-ratio estimate with desirable asymptotic properties that can be shown by the same techniques used in Hu et al., Biometrika 98:341-354 (). Numerical studies are conducted. [ABSTRACT FROM AUTHOR]

The gamma distribution is used as a lifetime distribution widely in reliability analysis. Lifetime data are often left truncated, and right censored. The EM algorithm is developed here for the estimation of the scale and shape parameters of the gamma distribution based on left truncated and right censored data. The Newton-Raphson method is also used for the same purpose, and then these two methods of estimation are compared through an extensive Monte Carlo simulation study. The asymptotic variance-covariance matrix of the MLEs under the EM framework is obtained by using the missing information principle (Louis, 1982). Then, the asymptotic confidence intervals for the parameters are constructed. The confidence intervals based on the EM algorithm and the Newton-Raphson method are then compared empirically in terms of coverage probabilities. Finally, all the methods of inference discussed here are illustrated through a numerical example. [ABSTRACT FROM AUTHOR]

Abstract: Spontaneous abortion (SAB) is an important endpoint in studying the effects of drugs or vaccines on pregnancy outcomes. Such studies are typically observational in nature. Evaluation of the risk for SAB is often conducted as part of a prospective exposure cohort study, or pregnancy registry, where women are recruited into the study at some point after recognition of pregnancy. Subjects accrued in this way are left truncated, in the sense that the time to SAB is not followed from the start of pregnancy, and the observed data is conditional on the fact that a woman has not had SAB by enrollment. Due to left truncation the estimation of SAB rates needs to use survival analysis methods. A further complication occurs when there is loss to follow-up, so that for those individuals it is not known whether they have SAB. These factors necessarily lead to complex modeling of the SAB data, and the calculation of sample size is not straightforward. In this communication we consider a simple sample size calculation that only requires minimum assumptions on the population SAB rates and the anticipated observed SAB rates. The method is exemplified on a hypothetical prospective study of risk for SAB. [Copyright &y& Elsevier]

Abstract: The lognormal distribution is quite commonly used as a lifetime distribution. Data arising from life-testing and reliability studies are often left truncated and right censored. Here, the EM algorithm is used to estimate the parameters of the lognormal model based on left truncated and right censored data. The maximization step of the algorithm is carried out by two alternative methods, with one involving approximation using Taylor series expansion (leading to approximate maximum likelihood estimate) and the other based on the EM gradient algorithm (). These two methods are compared based on Monte Carlo simulations. The Fisher scoring method for obtaining the maximum likelihood estimates shows a problem of convergence under this setup, except when the truncation percentage is small. The asymptotic variance–covariance matrix of the MLEs is derived by using the missing information principle (), and then the asymptotic confidence intervals for scale and shape parameters are obtained and compared with corresponding bootstrap confidence intervals. Finally, some numerical examples are given to illustrate all the methods of inference developed here. [Copyright &y& Elsevier]

Although many authors have proposed different approaches to the analysis of length-biased survival data, a number of issues have not been fully addressed. The most important among these issues is perhaps that regarding inclusion of covariates into the analysis of length-biased lifetime data collected through cross-sectional sampling of a population. One aspect of this problem, which appears to have been neglected in the literature, concerns the effect of length bias on the sampling distribution of the covariates. In most regression analyses, it is conventional to condition on the observed covariate values; however, certain covariate values could be preferentially selected into the sample, being linked to the long-term survivors, who themselves are favored by the sampling mechanism. This observation raises two questions: (1) Does the conditional analysis of covariates lead to biased estimators of regression coefficients?; and (2) does inference through the joint I likelihood of covariates and failure times yield more efficient estimators of the regression parameters? We present a joint likelihood approach and study the large-sample behavior of the resulting maximum likelihood estimators (MLEs). We find that these MLEs are more efficient than their conditional counterparts even though the two MLEs are asymptotically equal. Our results are illustrated using data on survival with dementia, collected as part of the Canadian Study of Health and Aging. [ABSTRACT FROM AUTHOR]

In many follow-up studies, survival data are often observed according to a cross-sectional sampling scheme. Data of this type are subject to left truncation and right censoring. In many practical cases, two types of censoring may occur. The first type (type A) is due to termination of the follow-up period. The second type (type B) is a consequence of other types of failure which might occur before the cross-section time. Let T*, V*, [image omitted] and [image omitted] denote the lifetime, left truncation, type A and type B censoring variables, respectively. Assume that T*, [image omitted] and [image omitted] are independent of one another but V* and [image omitted] are dependent with [image omitted] . Let F, G and Q denote the common distribution functions of T*, V* and [image omitted] , respectively. Let [image omitted] . For left-truncated and right-censored (LTRC) data, one can observe nothing if Z*

Let (Ui*, Ci*, Vi*) be i.i.d. random vectors such that (Ci*, Vi*) is independent of Ui* and P(Ci* ≥ Vi*) = 1. Let F, Q, and G denote the common distribution function of Ui*, Ci*, and Vi*, respectively. For left-truncated and right-censored data, one can observe nothing if Ui* < Vi* and observe (Xi*, δi*), with Xi* = min (Ui*, Ci*) and δi* = I[Ui* < Ci*], if Ui* ≥ Vi*. A problem of interest is the estimation of the truncation probability α = P(Ui* ≥ Ui*). Under the constraint that P (Ci* ≥ Vi*) = 1, Wang [Wang, M.-C., 1991, Nonparametric estimation from cross-sectional survival data. Journal of Americal Statistical Association , 86, 130-143.] suggested estimating α by αn = ∫ [1-Fn (s-)]d Gn(s), where Fn and Gn are non-parametric maximum likelihood estimate of the distributions F and G , respectively. In this note, using inverse-probability-weighted estimators of F , G , and Q , we obtain two alternative representations, ...n and ...n , for αn . Simulation study shows that the estimator αn works satisfactorily for moderate sample size. [ABSTRACT FROM AUTHOR]

In this paper we consider nonparametric regression with left-truncated and right-censored data. An estimator of the regression function is developed when censoring and truncation are independent of covariates and the response. The estimation is based on the product limit estimator of the response variable. Under certain conditions, the L2 rate of convergence of the estimated regression function is obtained when tensor products of B-splines are used. [Copyright &y& Elsevier]

Retrospectively ascertained data are common in many areas, including demography, epidemiology, and actuarial science. The main objective of this article is to study the effect of retrospective ascertainment on inference regarding recurrent events of time to pregnancy (TTP) data. For the particular TTP dataset that we consider, couples are included retrospectively based on their first pregnancy and then followed prospectively to a second pregnancy or to end of study. We consider a conditional model for the recurrent events data where the second TTP is included only if it is observed and a full model where the nonobserved second TTPs are included as suitably right censored. We furthermore consider two different approaches to modeling the dependencies of the recurrent events. A traditional frailty model, where the frailty enters the model as an unobserved covariate, and a marginal frailty model are applied. We find that efficiency is gained from including the second TTPs, with the full model being the most efficient. Further, the marginal frailty model is preferred over the traditional frailty model because estimates of covariate effects are easier to interpret and are more robust to changes in the frailty distribution. [ABSTRACT FROM AUTHOR]