Rank tests for independence based on partially right-censored pairs

Linear rank procedures are developed for testing independence with right-censored matched pairs. It is assumed that censoring Is Independent of the random variables under study. The test statistics are derived as score statistics (Hajek and Sidak, 1967) based on the probability of the generalised rank vectors (Prentice, 1978). Applications to survival data analysis are also discussed.     

On Consistency of the Self-Consistent Estimator of Survival Functions with Interval-Censored Data

The self-consistent estimator is commonly used for estimating a survival function with interval-censored data. Recent studies on interval censoring have focused on case 2 interval censoring, which does not involve exact observations, and double censoring, which involves only exact, right-censored or left-censored observations. In this paper, we consider an interval censoring scheme that involves exact, left-censored, right-censored and strictly interval-censored observations. Under this censoring scheme, we prove that the self-consistent estimator is strongly consistent under certain regularity conditions.     

Nonparametric and semiparametric regression estimation for length-biased survival data

For the past several decades, nonparametric and semiparametric modeling for conventional right-censored survival data has been investigated intensively under a noninformative censoring mechanism. However, these methods may not be applicable for analyzing right-censored survival data that arise from prevalent cohorts when the failure times are subject to length-biased sampling. This review article is intended to provide a summary of some newly developed methods as well as established methods for analyzing length-biased data.     

Nonparametric tests for left-truncated and interval-censored data

This paper considers two-sample nonparametric comparison of survival function when data are subject to left truncation and interval censoring. We propose a class of rank-based tests, which are generalization of weighted log-rank tests for right-censored data. Simulation studies indicate that the proposed tests are appropriate for practical use.     

Gini index estimation for lifetime data

Lifetime data is often right-censored. Recent literature on the Gini index estimation with censored data focuses on independent censoring. However, the censoring mechanism is likely to be dependent censoring in practice. This paper proposes two estimators of the Gini index under independent censoring and covariate-dependent censoring, respectively. The proposed estimators are consistent and asymptotically normal. We also evaluate the performance of our estimators in finite samples through Monte Carlo simulations. Finally, the proposed methods are applied to real data.     

Identification of the minimum effective dose based on weighted Kaplan–Meier statistics

Testing procedures are considered for identifying the minimum effective dose (MED) in a dose–response study with randomly right-censored survival data, where the MED is defined to be the smallest dose level under study that has survival advantage over the zero dose control. The proposed testing procedures are implemented in a step-down manner together with three different types of weighted Kaplan–Meier statistics. Comparative results of a Monte Carlo error rate and power/bias study for a variety of survival and censoring distributions are then presented and discussed. The application of the proposed procedures is finally illustrated for identifying the MED of the diethylstilbestrol in the treatment of prostate cancer.     

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.     

Variable screening for survival data in the presence of heterogeneous censoring

Variable screening for censored survival data is most challenging when both survival and censoring times are correlated with an ultrahigh-dimensional vector of covariates. Existing approaches to handling censoring often make use of inverse probability weighting by assuming independent censoring with both survival time and covariates. This is a convenient but rather restrictive assumption which may be unmet in real applications, especially when the censoring mechanism is complex and the number of covariates is large. To accommodate heterogeneous (covariate-dependent) censoring that is often present in high-dimensional survival data, we propose a Gehan-type rank screening method to select features that are relevant to the survival time. The method is invariant to monotone transformations of the response and of the predictors, and works robustly for a general class of survival models. We establish the sure screening property of the proposed methodology. Simulation studies and a lymphoma data analysis demonstrate its favorable performance and practical utility.     

Nonparametric estimation in the illness-death model using prevalent data

We study nonparametric estimation of the illness-death model using left-truncated and right-censored data. The general aim is to estimate the multivariate distribution of a progressive multi-state process. Maximum likelihood estimation under censoring suffers from problems of uniqueness and consistency, so instead we review and extend methods that are based on inverse probability weighting. For univariate left-truncated and right-censored data, nonparametric maximum likelihood estimation can be considerably improved when exploiting knowledge on the truncation distribution. We aim to examine the gain in using such knowledge for inverse probability weighting estimators in the illness-death framework. Additionally, we compare the weights that use truncation variables with the weights that integrate them out, showing, by simulation, that the latter performs more stably and efficiently. We apply the methods to intensive care units data collected in a cross-sectional design, and discuss how the estimators can be easily modified to more general multi-state models.     

Local composite partial likelihood estimation for length-biased and right-censored data

Length-biased data, which are often encountered in engineering, economics and epidemiology studies, are generally subject to right censoring caused by the research ending or the follow-up loss. The structure of length-biased data is distinct from conventional survival data, since the independent censoring assumption is often violated due to the biased sampling. In this paper, a proportional hazard model with varying coefficients is considered for the length-biased and right-censored data. A local composite likelihood procedure is put forward for the estimation of unknown coefficient functions in the model, and large sample properties of the proposed estimators are also obtained. Additionally, an extensive simulation studies are conducted to assess the finite sample performance of the proposed method and a data set from the Academy Awards is analyzed.     

Comparing waiting times in a multi-stage model: A log-rank approach

A proper log-rank test for comparing two waiting (i.e. sojourn, gap) times under right censored data has been absent in the survival literature. The classical log-rank test provides a biased comparison even under independent right censoring since the censoring induced on the time since state entry depends on the entry time unless the hazards are semi-Markov. We develop test statistics for comparing K waiting time distributions from a multi-stage model in which censoring and waiting times may be dependent upon the transition history in the multi-stage model. To account for such dependent censoring, the proposed test statistics utilize an inverse probability of censoring weighted (IPCW) approach previously employed to define estimators for the cumulative hazard and survival function for waiting times in multi-stage models. We develop the test statistics as analogues to K-sample log-rank statistics for failure time data, and weak convergence to a Gaussian limit is demonstrated. A simulation study demonstrates the appropriateness of the test statistics in designs that violate typical independence assumptions for multi-stage models, under which naive test statistics for failure time data perform poorly, and illustrates the superiority of the test under proportional hazards alternatives to a Mann–Whitney type test. We apply the test statistics to an existing data set of burn patients.     

Nonparametric Estimation of Sojourn Time Distributions for Truncated Serial Event Data—a Weight-adjusted Approach

In follow-up studies, survival data often include subjects who have had a certain event at recruitment and may potentially experience a series of subsequent events during the follow-up period. This kind of survival data collected under a cross-sectional sampling criterion is called truncated serial event data. The outcome variables of interest in this paper are serial sojourn times between successive events. To analyze the sojourn times in truncated serial event data, we need to confront two potential sampling biases arising simultaneously from a sampling criterion and induced informative censoring. In this study, nonparametric estimation of the joint probability function of serial sojourn times is developed by using inverse probabilities of the truncation and censoring times as weight functions to accommodate these two sampling biases under various situations of truncation and censoring. Relevant statistical properties of the proposed estimators are also discussed. Simulation studies and two real data are presented to illustrate the proposed methods.     

Tobit regression for modeling mean survival time using data subject to multiple sources of censoring

Mean survival time is often of inherent interest in medical and epidemiologic studies. In the presence of censoring and when covariate effects are of interest, Cox regression is the strong default, but mostly due to convenience and familiarity. When survival times are uncensored, covariate effects can be estimated as differences in mean survival through linear regression. Tobit regression can validly be performed through maximum likelihood when the censoring times are fixed (ie, known for each subject, even in cases where the outcome is observed). However, Tobit regression is generally inapplicable when the response is subject to random right censoring. We propose Tobit regression methods based on weighted maximum likelihood which are applicable to survival times subject to both fixed and random censoring times. Under the proposed approach, known right censoring is handled naturally through the Tobit model, with inverse probability of censoring weighting used to overcome random censoring. Essentially, the re‐weighting data are intended to represent those that would have been observed in the absence of random censoring. We develop methods for estimating the Tobit regression parameter, then the population mean survival time. A closed form large‐sample variance estimator is proposed for the regression parameter estimator, with a semiparametric bootstrap standard error estimator derived for the population mean. The proposed methods are easily implementable using standard software. Finite‐sample properties are assessed through simulation. The methods are applied to a large cohort of patients wait‐listed for kidney transplantation.     

Improved precision in the analysis of randomized trials with survival outcomes,without assuming proportional hazards

We present a new estimator of the restricted mean survival time in randomized trials where there is right censoring that may depend on treatment and baseline variables. The proposed estimator leverages prognostic baseline variables to obtain equal or better asymptotic precision compared to traditional estimators. Under regularity conditions and random censoring within strata of treatment and baseline variables, the proposed estimator has the following features: (i) it is interpretable under violations of the proportional hazards assumption; (ii) it is consistent and at least as precise as the Kaplan–Meier and inverse probability weighted estimators, under identifiability conditions; (iii) it remains consistent under violations of independent censoring (unlike the Kaplan–Meier estimator) when either the censoring or survival distributions, conditional on covariates, are estimated consistently; and (iv) it achieves the nonparametric efficiency bound when both of these distributions are consistently estimated. We illustrate the performance of our method using simulations based on resampling data from a completed, phase 3 randomized clinical trial of a new surgical treatment for stroke; the proposed estimator achieves a 12% gain in relative efficiency compared to the Kaplan–Meier estimator. The proposed estimator has potential advantages over existing approaches for randomized trials with time-to-event outcomes, since existing methods either rely on model assumptions that are untenable in many applications, or lack some of the efficiency and consistency properties (i)–(iv). We focus on estimation of the restricted mean survival time, but our methods may be adapted to estimate any treatment effect measure defined as a smooth contrast between the survival curves for each study arm. We provide R code to implement the estimator.

     

Testing for Linear Regression Relationships in Randomly Right-Censored Varying-Coefficient Models

A test statistic is constructed to test linear relationships in randomly right-censored varying-coefficient models. A residual-based bootstrap procedure is employed to derive the p-value of the test. The performance of the test is examined by extensive simulations. The simulation results show that the bootstrap estimate of the null distribution of the test statistic is approximately valid and the test method with the residual-based bootstrap works satisfactorily for at least moderate censoring rates of the response. Furthermore, the proposed test is applied to the Stanford heart transplant data for exploring a linear regression relationship between the logrithm of the survival time and the age of the patients.     

A Mass Redistribution Algorithm for Right Censored and Left Truncated Time to Event Data

Failure times are often right-censored and left-truncated. In this paper we give a mass redistribution algorithm for right-censored and/or left-truncated failure time data. We show that this algorithm yields the Kaplan-Meier estimator of the survival probability. One application of this algorithm in modeling the subdistribution hazard for competing risks data is studied. We give a product-limit estimator of the cumulative incidence function via modeling the subdistribution hazard. We show by induction that this product-limit estimator is identical to the left-truncated version of Aalen-Johansen (1978) estimator for the cumulative incidence function.     

Imputation for semiparametric transformation models with biased-sampling data

Widely recognized in many fields including economics, engineering, epidemiology, health sciences, technology and wildlife management, length-biased sampling generates biased and right-censored data but often provide the best information available for statistical inference. Different from traditional right-censored data, length-biased data have unique aspects resulting from their sampling procedures. We exploit these unique aspects and propose a general imputation-based estimation method for analyzing length-biased data under a class of flexible semiparametric transformation models. We present new computational algorithms that can jointly estimate the regression coefficients and the baseline function semiparametrically. The imputation-based method under the transformation model provides an unbiased estimator regardless whether the censoring is independent or not on the covariates. We establish large-sample properties using the empirical processes method. Simulation studies show that under small to moderate sample sizes, the proposed procedure has smaller mean square errors than two existing estimation procedures. Finally, we demonstrate the estimation procedure by a real data example.     

Likelihood-based confidence sets for partially identified parameters

There has been growing interest in partial identification of probability distributions and parameters. This paper considers statistical inference on parameters that are partially identified because data are incompletely observed, due to nonresponse or censoring, for instance. A method based on likelihood ratios is proposed for constructing confidence sets for partially identified parameters. The method can be used to estimate a proportion or a mean in the presence of missing data, without assuming missing-at-random or modeling the missing-data mechanism. It can also be used to estimate a survival probability with censored data without assuming independent censoring or modeling the censoring mechanism. A version of the verification bias problem is studied as well.     

Estimation of association parameters in copula models for bivariate left-truncated and right-censored data

We investigate the problem of estimating the association between two related survival variables when they follow a copula model and bivariate left-truncated and right-censored data are available. By expressing truncation probability as the functional of marginal survival functions, we propose a two-stage estimation procedure for estimating the parameters of Archimedean copulas. The asymptotic properties of the proposed estimators are established. Simulation studies are conducted to investigate the finite sample properties of the proposed estimators. The proposed method is applied to a bivariate RNA data.     

Presmoothed Estimation with Left-Truncated and Right-Censored Data

We propose a new method to estimate the cumulative hazard function and the corresponding distribution function of survival times under randomly left-truncated and right-censored observations (LTRC). The new estimators are based on presmoothing ideas, the estimation of the conditional expectation m of the censoring indicator. An almost sure representation for both estimators is established, from which a strong consistency rate and asymptotic normality are derived. It is shown that the presmoothed modification leads to a gain in terms of asymptotic mean squared error. This efficiency with respect to the classical estimators is also shown in a simulation study. Finally, an application to a real data set is provided.