Left-truncated And Right-censored Research Articles

Inference of dependent left-truncated and right-censored competing risks data from a general bivariate class of inverse exponentiated distributions

In this paper, inference is discussed for left-truncated and right-censored (LTRC) competing risks data. For modelling the dependent competing risks variables, a flexible bivariate lifetime model is constructed with a class of inverse exponentiated distribution, and associated distribution properties are provided correspondingly. When the LTRC competing risks data is available with partially observed failure causes, a dependent competing risks model is further established. Maximum likelihood estimators of the unknown parameters and the reliability indices are established, and the approximate confidence intervals are also constructed based on asymptotic theory. In addition, Bayesian estimators are obtained based on flexible priors, Markov Chain Monte Carlo sampling approach is also proposed for computing the Bayesian estimates as well as the highest posterior density credible intervals. Finally, simulation studies are carried out to investigate the performances of our results, and a real-life data set is analysed to illustrate the applicability of the proposed methods.

Prediction accuracy measures for time-to-event models with left-truncated and right-censored data

ABSTRACT Many time-to-event models have been developed for left-truncated and right-censored (LTRC) data, which arise in many applications involving follow-up studies. However, there is no work on evaluating the prediction accuracy of the time-to-event models for LTRC data. This paper develops two novel weighted prediction summary measures for a nonlinear prediction function with LTRC data. They are based on a weighted variance decomposition and a weighted prediction error decomposition, by the inverse probability weighting technique. The resulting measures are shown to be consistent and asymptotically normal. Simulation studies are conducted to evaluate their good finite sample performance. An empirical application to the Channing House data set illustrates the methodology.

Model selection and model averaging for analysis of truncated and censored data with measurement error

Model selection plays a critical role in statistical inference and a large literature has been devoted to this topic. Despite extensive research attention on model selection, research gaps still remain. An important but relatively unexplored problem concerns truncated and censored data with measurement error. Although analysis of left-truncated and right-censored (LTRC) data has received extensive interests in survival analysis, there has been no research on model selection for LTRC data with measurement error. In this paper, we take up this important problem and develop inferential procedures to handle model selection for LTRC data with measurement error in covariates. Our development employs the local model misspecification framework ([6]; [10]) and emphasizes the use of the focus information criterion (FIC). We develop valid estimators using the model averaging scheme and establish theoretical results to justify the validity of our methods. Numerical studies are conducted to assess the performance of the proposed methods.

Empirical likelihood ratio confidence intervals in terms of cumulative hazard function for left-truncated and right-censored data

Based on the approach of Pan and Zhou, we demonstrate that empirical likelihood ratios in terms of cumulative hazard function for left-truncated and right-censored (LTRC) data can be used to form confidence intervals for the parameters that are linear functionals of the cumulative hazard function. Simulation studies indicate that the empirical likelihood ratio based confidence intervals work well in finite samples.

On semiparametric transformation model with LTRC data: pseudo likelihood approach

When the distribution of the truncation time is known up to a finite-dimensional parameter vector, many researches have been conducted with the objective to improve the efficiency of estimation for nonparametric or semiparametric model with left-truncated and right-censored (LTRC) data. When the distribution of truncation times is unspecified, one approach is to use the conditional maximum likelihood estimators (cMLE) (Chen and Shen in Lifetime Data Anal https://doi.org/10.1007/s10985-016-9385-9, 2017). Although the cMLE has nice asymptotic properties, it is not efficient since the conditional likelihood function does not incorporate information on the distribution of truncation time. In this article, we aim to develop a more efficient estimator by considering the full likelihood function. Following Turnbull (J R Stat Soc B 38:290–295, 1976) and Qin et al. (J Am Stat Assoc 106:1434–1449, 2011), we treat the unobserved (left-truncated) subpopulation as missing data and propose a two-stage approach for obtaining the pseudo maximum likelihood estimators (PMLE) of regression parameters. In the first stage, the distribution of left truncation time is estimated by the inverse-probability-weighted (IPW) estimator (Wang in J Am Stat Assoc 86:130–143, 1991). In the second stage, we obtain the pseudo complete-data likelihood function by replacing the distribution of truncation time with the IPW estimator in the full likelihood. We propose an expectation–maximization algorithm for obtaining the PMLE and establish the consistency of the PMLE. Simulation results show that the PMLE outperforms the cMLE in terms of mean squared error. The PMLE can also be used to analyze the length-biased data, where the truncation time is uniformly distributed. We demonstrate that the PMLE works more robust against the support assumption of truncation time for length-biased data compared with the MLE proposed by Qin et al. (2011). We apply our proposed method to the channing house data. While the PMLE is quite appealing under specific cases with independent censoring and time-invariant covariates, its applicability, as shown in simulation study, can be rather restricted for more general settings.

The Cox–Aalen model for left-truncated and right-censored data

ABSTRACTWe analyze left-truncated and right-censored (LTRC) data using an additive-multiplicative Cox–Aalen model proposed by Scheike and Zhang (2002), which extends the Cox regression model as well as the additive Aalen model. Based on the conditional likelihood function, we derive the weighted least-squared (WLS) estimators for the regression parameters and cumulative intensity functions of the model. The estimators are shown to be consistent and asymptotically normal. A simulation study is conducted to investigate the performance of the proposed estimators.

Survival trees for left-truncated and right-censored data, with application to time-varying covariate data.

Tree methods (recursive partitioning) are a popular class of nonparametric methods for analyzing data. One extension of the basic tree methodology is the survival tree, which applies recursive partitioning to censored survival data. There are several existing survival tree methods in the literature, which are mainly designed for right-censored data. We propose two new survival trees for left-truncated and right-censored (LTRC) data, which can be seen as a generalization of the traditional survival tree for right-censored data. Further, we show that such trees can be used to analyze survival data with time-varying covariates, essentially building a time-varying covariates survival tree. Implementation of the methods is easy, and simulations and real data analysis results show that the proposed methods work well for LTRC data and survival data with time-varying covariates, respectively.

Estimating equation methods for quantile regression with length-biased and right-censored data

In this paper, we first develop the quantile regression approach for survival data arising from lengthbiased sampling, where the survival times are left-truncated by uniformly distributed random truncated times. As a special case of the left-truncated and right-censored (LTRC) data, length-biased right-censored (LBRC) data supports more information than LTRC data. When we apply the general methods to LBRC data, the resulting estimators may be inefficient because they do not exploit the special structure in LBRC data. In order to improve efficiency, we then develop a composite quantile regression approach for LBRC data, which do not need to estimate the distribution of censoring time. The proposed estimating equations lead to a simple algorithm that involves minimizations of L 1 type convex functions. We establish uniform consistency and weak convergence of the resultant estimators using empirical process and stochastic integral techniques. Simulation studies are conducted to evaluate the finite sample performance of the proposed methods. A real data example is also given.

Quantile regression methods for left-truncated and right-censored data

Left-truncated and right-censored (LTRC) data are encountered frequently due to a prevalent cohort sampling in follow-up studies. Because of the skewness of the distribution of survival time, quantile regression is a useful alternative to the Cox's proportional hazards model and the accelerated failure time model for survival analysis. In this paper, we apply the quantile regression model to LTRC data and develops an unbiased estimating equation for regression coefficients. The proposed estimation methods use the inverse probabilities of truncation and censoring weighting technique. The resulting estimator is uniformly consistent and asymptotically normal. The finite-sample performance of the proposed estimation methods is also evaluated using extensive simulation studies. Finally, analysis of real data is presented to illustrate our proposed estimation methods.

Aalen’s Additive Risk Model for Left-Truncated and Right-Censored Data

We analyze left-truncated and right-censored (LTRC) data using Aalen’s linear models. The integrated square error (ISE) is used to select an optimal bandwidth of the weighted least-squared estimator. We also consider a semiparametric approach for the case when the distribution of the left-truncated variable is parameterized. A simulation study is conducted to investigate the performance of the proposed estimators. The approaches are illustrated with the data of Stanford heart transplant.

Relative deficiency of quantile estimators for left truncated and right censored data

The quantity deficiency which was proposed by Hodges and Lehmann (1970) is used to compare different statistical procedures. In this article, the deficiency of the sample quantile estimator with respect to the kernel quantile estimator for left truncated and right censored (LTRC) data in the sense of Hodges and Lehmann is considered. We also give the optimal bandwidth for the kernel quantile estimator. Monte Carlo studies are conducted to illustrate our results.

Semiparametric analysis of transformation models with left-truncated and right-censored data

We analyze left-truncated and right-censored (LTRC) data using semiparametric transformation models. It is demonstrated that the approach of Chen et al. (Biometrika 89: 659–668, 2002) can be extended to LTRC data. Furthermore, when covariates are discrete, similar to the approach of Cai and Cheng (Biometrika 91: 277–290, 2004), we propose an alternative estimator. A simulation study is conducted to investigate the performance of the proposed estimators.

Semiparametric analysis of survival data with left truncation and right censoring

Let T , C and V denote the lifetime, censoring and truncation variables, respectively. Assume that ( C , V ) is independent of T and P ( C ≥ V ) = 1 . Let F , Q and G denote the common distribution functions of T , C and V , respectively. For left-truncated and right-censored (LTRC) data, one can observe nothing if T < V and observe ( X , δ , V ) , with X = min ( T , C ) and δ = I [ T ≤ C ] , if T ≥ V . For LTRC data, the truncation product-limit estimate F ˆ n is the maximum likelihood estimate (MLE) for nonparametric models. If the distribution of V is parameterized as G ( x ; θ ) and the distributions of T and C are left unspecified, the product-limit estimate F ˆ n is not the MLE for this semiparametric model. In this article, for LTRC data, two semiparametric estimates are proposed for the semiparametric model. A simulation study is conducted to compare the performances of the two semiparametric estimators against that of F ˆ n . The proposed semiparametric method is applied to a Channing House data.

Nonparametric inference strategies for the quantile functions under left truncation and right censoring

This paper addresses the problem of estimating the quantile function in a multiple-sample set up when the data are left-truncated and right-censored (LTRC). Assuming an uncertain prior non-sample information on the value of the quantile, we propose improved estimators based on Stein-type shrinkage estimators. A test statistic is also proposed to define improved estimators for the quantile function. The asymptotic bias and risk of the estimators are derived and compared with the benchmark estimator analytically. For several choices of parameters, a Monte Carlo simulation experiment is conducted to appraise the risk reduction of the proposed estimators at different levels of censoring and truncation. We demonstrate that the proposed estimators have superior performance in terms of risk reduction over the benchmark estimator.

A general semiparametric model for left-truncated and right-censored data

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*, and denote the lifetime, left truncation, type A and type B censoring variables, respectively. Assume that T*, and are independent of one another but V* and are dependent with . Let F, G and Q denote the common distribution functions of T*, V* and , respectively. Let . For left-truncated and right-censored (LTRC) data, one can observe nothing if Z*<V*, and observe (X*, δ*), if Z*≥V*, where , and δ* is equal to 1 if X*=T*, equal to 2 if and zero otherwise. For LTRC data, the truncation product-limit estimate [Fcirc] n is the maximum likelihood estimate (MLE) for nonparametric models. If the distribution of V* is parameterized as G (x; θ) and the distributions of T* and are left unspecified, the product-limit estimate [Fcirc] n is not the MLE for this semiparametric model. When (i.e., left-truncated data), Wang [Wang M.-C. 1989, A semiparametric model for randomly truncated data. Journal of the American Statistical Association, 86, 130–143.] derived the MLE of F for the semiparametric model and established its weak convergence properties. When G(x; θ)=x/θ and (the so-called stationarity assumption), Asgharian et al. [Asgharian, M., M’Lan, C.E. and Wolfson, D.B., 2002, Length-biased sampling with right censoring: an unconditional approach. Journal of the American Statistical Association, 97(457), 201–209; Asgharian, M. and Wolfson, D.B., 2005, Asymptotic behavior of the unconditional NPMLE of the length-biased survivor function from right censored prevalent cohort data. Annals of Statistics, 33, 2109–2131.] derived an unconditional MLE of F and established its asymptotic properties. In this paper, we extend previous models by distinguishing two types of censoring. Iterative algorithms are proposed to obtain a semiparametric estimate, [Fcirc] n (x; ˆθ n ). The asymptotic properties of [Fcirc] n (x; ˆθ n ) are discussed. A simulation study is conducted to compare the performance of [Fcirc] n (x; ˆθ n ) against that of [Fcirc] n (x).

Genetic Mixed Linear Models for Twin Survival Data

Twin studies are useful for assessing the relative importance of genetic or heritable component from the environmental component. In this paper we develop a methodology to study the heritability of age-at-onset or lifespan traits, with application to analysis of twin survival data. Due to limited period of observation, the data can be left truncated and right censored (LTRC). Under the LTRC setting we propose a genetic mixed linear model, which allows general fixed predictors and random components to capture genetic and environmental effects. Inferences are based upon the hierarchical-likelihood (h-likelihood), which provides a statistically efficient and unified framework for various mixed-effect models. We also propose a simple and fast computation method for dealing with large data sets. The method is illustrated by the survival data from the Swedish Twin Registry. Finally, a simulation study is carried out to evaluate its performance.

Inference concerning quantile for left truncated and right censored data

The problem of both testing and estimating the quantile function when the data are left truncated and right censored (LTRC) is considered. The aim of this communication is two-fold. First, a large sample test statistic to test for the quantile function under the LTRC model is defined and its null and non-null distributions are derived. A Monte Carlo simulation study is conducted to assess the power of the proposed test statistic that is used to define the estimators. Secondly, an improved estimation of the quantile function is investigated. In the spirit of the shrinkage principle in parameter estimation, three estimators assuming an uncertain prior non-sample information on the value of the quantile are proposed. The asymptotic bias and mean square error of the estimators are derived and compared with the usual estimator. The method is illustrated with hypothetical data as well as real data.

Variance of the bivariate density estimator for left truncated right censored data

In this study the variance of the bivariate kernel density estimators for the left truncated and right censored (LTRC) observations are considered. In LTRC models, the complete observation of the variable Y is prevented by the truncating variable T and the censoring variable C. Consequently, one observes the i.i.d. samples from the triplets ( T, Z, δ) only if T⩽ Z, Z=min( Y, C) and δ is one if Z= Y and zero otherwise. Gürler and Prewitt (1997, submitted for publication) consider the estimation of the bivariate density function via nonparametric kernel methods and establish an i.i.d. representation of their estimators. Asymptotic variance of the i.i.d. part of their representation is developed in this paper. Application of the results are also discussed for the data-driven and the least-squares cross validation bandwidth choice procedures.