Semiparametric estimation of treatment effect with time-lagged response in the presence of informative censoring

In many randomized clinical trials, the primary response variable, for example, the survival time, is not observed directly after the patients enroll in the study but rather observed after some period of time (lag time). It is often the case that such a response variable is missing for some patients due to censoring that occurs when the study ends before the patient’s response is observed or when the patients drop out of the study. It is often assumed that censoring occurs at random which is referred to as noninformative censoring; however, in many cases such an assumption may not be reasonable. If the missing data are not analyzed properly, the estimator or test for the treatment effect may be biased. In this paper, we use semiparametric theory to derive a class of consistent and asymptotically normal estimators for the treatment effect parameter which are applicable when the response variable is right censored. The baseline auxiliary covariates and post-treatment auxiliary covariates, which may be time-dependent, are also considered in our semiparametric model. These auxiliary covariates are used to derive estimators that both account for informative censoring and are more efficient then the estimators which do not consider the auxiliary covariates.     

A Profile Conditional Likelihood Approach for the Semiparametric Transformation Regression Model with Missing Covariates

We propose a profile conditional likelihood approach to handle missing covariates in the general semiparametric transformation regression model. The method estimates the marginal survival function by the Kaplan-Meier estimator, and then estimates the parameters of the survival model and the covariate distribution from a conditional likelihood, substituting the Kaplan-Meier estimator for the marginal survival function in the conditional likelihood. This method is simpler than full maximum likelihood approaches, and yields consistent and asymptotically normally distributed estimator of the regression parameter when censoring is independent of the covariates. The estimator demonstrates very high relative efficiency in simulations. When compared with complete-case analysis, the proposed estimator can be more efficient when the missing data are missing completely at random and can correct bias when the missing data are missing at random. The potential application of the proposed method to the generalized probit model with missing continuous covariates is also outlined.     

Buckley–James Type Estimator for Censored Data with Covariates Missing by Design

Abstract. The Buckley–James estimator (BJE) is a well‐known estimator for linear regression models with censored data. Ritov has generalized the BJE to a semiparametric setting and demonstrated that his class of Buckley–James type estimators is asymptotically equivalent to the class of rank‐based estimators proposed by Tsiatis. In this article, we revisit such relationship in censored data with covariates missing by design. By exploring a similar relationship between our proposed class of Buckley–James type estimating functions to the class of rank‐based estimating functions recently generalized by Nan, Kalbfleisch and Yu, we establish asymptotic properties of our proposed estimators. We also conduct numerical studies to compare asymptotic efficiencies from various estimators.     

Investigation of the performance of trimmed estimators of life time distributions with censoring

For the lifetime (or negative) exponential distribution, the trimmed likelihood estimator has been shown to be explicit in the form of a β‐trimmed mean which is representable as an estimating functional that is both weakly continuous and Fréchet differentiable and hence qualitatively robust at the parametric model. It also has high efficiency at the model. The robustness is in contrast to the maximum likelihood estimator (MLE) involving the usual mean which is not robust to contamination in the upper tail of the distribution. When there is known right censoring, it may be perceived that the MLE which is the most asymptotically efficient estimator may be protected from the effects of ‘outliers’ due to censoring. We demonstrate that this is not the case generally, and in fact, based on the functional form of the estimators, suggest a hybrid defined estimator that incorporates the best features of both the MLE and the β‐trimmed mean. Additionally, we study the pure trimmed likelihood estimator for censored data and show that it can be easily calculated and that the censored observations are not always trimmed. The different trimmed estimators are compared by a modest simulation study.     

Exact inference for the two-parameter exponential distribution under Type-II hybrid censoring

Epstein (1954) introduced the Type-I hybrid censoring scheme as a mixture of Type-I and Type-II censoring schemes. Childs et al. (2003) introduced the Type-II hybrid censoring scheme as an alternative to Type-I hybrid censoring scheme, and provided the exact distribution of the maximum likelihood estimator of the mean of a one-parameter exponential distribution based on Type-II hybrid censored samples. The associated confidence interval also has been provided. The main aim of this paper is to consider a two-parameter exponential distribution, and to derive the exact distribution of the maximum likelihood estimators of the unknown parameters based on Type-II hybrid censored samples. The marginal distributions and the exact confidence intervals are also provided. The results can be used to derive the exact distribution of the maximum likelihood estimator of the percentile point, and to construct the associated confidence interval. Different methods are compared using extensive simulations and one data analysis has been performed for illustrative purposes.     

Estimation for Location-Scale Models with Censored Data

This article considers a class of estimators for the location and scale parameters in the location-scale model based on ‘synthetic data’ when the observations are randomly censored on the right. The asymptotic normality of the estimators is established using counting process and martingale techniques when the censoring distribution is known and unknown, respectively. In the case when the censoring distribution is known, we show that the asymptotic variances of this class of estimators depend on the data transformation and have a lower bound which is not achievable by this class of estimators. However, in the case that the censoring distribution is unknown and estimated by the Kaplan–Meier estimator, this class of estimators has the same asymptotic variance and attains the lower bound for variance for the case of known censoring distribution. This is different from censored regression analysis, where asymptotic variances depend on the data transformation. Our method has three valuable advantages over the method of maximum likelihood estimation. First, our estimators are available in a closed form and do not require an iterative algorithm. Second, simulation studies show that our estimators being moment-based are comparable to maximum likelihood estimators and outperform them when sample size is small and censoring rate is high. Third, our estimators are more robust to model misspecification than maximum likelihood estimators. Therefore, our method can serve as a competitive alternative to the method of maximum likelihood in estimation for location-scale models with censored data. A numerical example is presented to illustrate the proposed method.     

Nonparametric Survival Estimation when Death is Reported with Delay

In disease registries there can be a delay between death of asubject and the reporting of this death to the data analyst.If researchers use the Kaplan-Meier estimator and implicitlyassumed that subjects who have yet to have death reported arestill alive, i.e. are censored at the time of analysis, the Kaplan-Meierestimator is typically inconsistent. Assuming censoring is independentof failure, we provide a simple estimator that is consistentand asymptotically efficient. We also provide estimates of theasymptotic variance of our estimator and simulations that demonstratethe favorable performance of these estimators. Finally, we demonstrateour methods by analyzing AIDS survival data. This analysis underscoresthe pitfalls of not accounting for delay when estimating thesurvival distribution and suggests a significant reduction inbias by using our estimator.     

Linear censored quantile regression: A novel minimum-distance approach

In this article, we investigate a new procedure for the estimation of a linear quantile regression with possibly right-censored responses. Contrary to the main literature on the subject, we propose in this context to circumvent the formulation of conditional quantiles through the so-called “check” loss function that stems from the influential work of Koenker and Bassett (1978). Instead, our suggestion is here to estimate the quantile coefficients by minimizing an alternative measure of distance. In fact, our approach could be qualified as a generalization in a parametric regression framework of the technique consisting in inverting the conditional distribution of the response given the covariates. This is motivated by the knowledge that the main literature for censored data already relies on some nonparametric conditional distribution estimation as well. The ideas of effective dimension reduction are then exploited in order to accommodate for higher dimensional settings as well in this context. Extensive numerical results then suggest that such an approach provides a strongly competitive procedure to the classical approaches based on the check function, in fact both for complete and censored observations. From a theoretical prospect, both consistency and asymptotic normality of the proposed estimator for linear regression are obtained under classical regularity conditions. As a by-product, several asymptotic results on some “double-kernel” version of the conditional Kaplan–Meier distribution estimator based on effective dimension reduction, and its corresponding density estimator, are also obtained and may be of interest on their own. A brief application of our procedure to quasar data then serves to further highlight the relevance of the latter for quantile regression estimation with censored data.     

Estimation for two-parameter weibull distribution and extreme-value distribution under multiply type-II censoring

Compared to Type-II censoring, multiply Type-II censoring is a more general, yet mathematically and numerically much more complicated censoring scheme. For multiply Type II censored data from a two-parameter Weibull distribution, we propose several estimators, including MLE, approximate MLE, and estimators corresponding to the BLUE and BLIE from estimating parameters in extreme-value distribution. An approximately unbiased estimator for the shape parameter is also proposed which has the smallest MSE. Numerical examples show that this estimator is the best in terms of bias and MSE. Numerical examples also show that the approximate MLE which admits a closed form is better for estimating the scale parameter.     

Optimal tests for carcinogenicity in a model with fatal and incidental tumours

In a recent paper Heimann and Neuhaus (Biometrika 88 (2001) 435) studied the combined test of Peto et al. (Long Term and Short Term Screening Assays for Carcinogens: A Critical Appraisal, IARC Monograph on the evaluation of the carcinogenic risk of chemicals to humans, Annex to supplement 2, WHO, Geneva, pp. 311–426) for the two sample testing problem. This test is an ad hoc test adding the log rank test statistic for the random censorship model and the Mantel Haenszel test statistic for the interval censored data model. Heimann and Neuhaus (Biometrika 88 (2001) 435) also introduced an appropriate statistical model combining randomly censored and interval censored data, to handle lethal and incidental tumours jointly.In the present paper we will derive asymptotically optimal tests for this model in a systematic way. The optimal test statistic is decomposed into two orthogonal parts, representing the fatal and incidental components. This will allow to use well known results from the literature for the fatal part, to estimate unknown quantities for the incidental part, and last but not least to compare with Peto's combined test.Moreover, we introduce conditional permutation versions of our tests which are finite sample distribution free under the null hypothesis with equal censoring and are asymptotically equivalent to their unconditional counterparts even under locally unequal censoring. Crucial for these results is a conditional limit theorem for the test statistics under local alternatives which follows from results of Strasser and Weber (Math. Methods Statist. 2 (1999) 220).     

Double robust estimation in longitudinal marginal structural models

In this article we consider estimation of causal parameters in a marginal structural model for the discrete intensity of the treatment specific counting process (e.g. hazard of a treatment specific survival time) based on longitudinal observational data on treatment, covariates and survival. We define three estimators: the inverse probability of treatment weighted (IPTW) estimator, the maximum likelihood estimator (MLE), and a double robust (DR) estimator. The DR estimator is obtained by following a general methodology for constructing double robust estimating functions in censored data models as described in van der Laan and Robins (Unified Methods for Censored Longitudinal Data and Causality, 2002). The double-robust estimator is consistent and asymptotically linear when either the treatment mechanism or the partial likelihood of the observed data is consistently estimated. We illustrate the superiority of the DR estimator relative to the IPTW and ML estimators in a simulation study. The proposed methodology is also applied to estimate the causal effect of exercise on physical functioning in a longitudinal study of seniors in Sonoma County.     

Survival analysis for the missing censoring indicator model using kernel density estimation techniques

This article concerns asymptotic theory for a new estimator of a survival function in the missing censoring indicator model of random censorship. Specifically, the large sample results for an inverse probability-of-non-missingness weighted estimator of the cumulative hazard function, so far not available, are derived, including an almost sure representation with rate for a remainder term, and uniform strong consistency with rate of convergence. The estimator is based on a kernel estimate for the conditional probability of non-missingness of the censoring indicator. Expressions for its bias and variance, in turn leading to an expression for the mean squared error as a function of the bandwidth, are also obtained. The corresponding estimator of the survival function, whose weak convergence is derived, is asymptotically efficient. A numerical study, comparing the performances of the proposed and two other currently existing efficient estimators, is presented.     

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.     

A Bayesian nonparametric estimator based on left censored data

Summary This paper introduces a Bayesian nonparametric estimator for an unknown distribution function based on left censored observations. Hjort (1990)/Lo (1993) introduced Bayesian nonparametric estimators derived from beta/beta-neutral processes which allow for right censoring. These processes are taken as priors from the class ofneutral to the right processes (Doksum, 1974). The Kaplan-Meier nonparametric product limit estimator can be obtained from these Bayesian nonparametric estimators in the limiting case of a vague prior. The present paper introduces what can be seen as the correspondingleft beta/beta-neutral process prior which allow for left censoring. The Bayesian nonparametyric estimator is obtained as in the corresponding product limit estimator based on left censored data.     

Simple non-parametric estimators for unemployment duration analysis

Summary.  We consider an extension of conventional univariate Kaplan–Meier-type estimators for the hazard rate and the survivor function to multivariate censored data with a censored random regressor. It is an Akritas-type estimator which adapts the non-parametric conditional hazard rate estimator of Beran to more typical data situations in applied analysis. We show with simulations that the estimator has nice finite sample properties and our implementation appears to be fast. As an application we estimate non-parametric conditional quantile functions with German administrative unemployment duration data.     

Estimation on dependent right censoring scheme in an ordinary bivariate geometric distribution

Discrete lifetime data are very common in engineering and medical researches. In many cases the lifetime is censored at a random or predetermined time and we do not know the complete survival time. There are many situations that the lifetime variable could be dependent on the time of censoring. In this paper we propose the dependent right censoring scheme in discrete setup when the lifetime and censoring variables have a bivariate geometric distribution. We obtain the maximum likelihood estimators of the unknown parameters with their risks in closed forms. The Bayes estimators as well as the constrained Bayes estimates of the unknown parameters under the squared error loss function are also obtained. We considered an extension to the case where covariates are present along with the data. Finally we provided a simulation study and an illustrative example with a real data.     

关于均匀分布统计分析的一些思考

对于均匀分布给出了在全样本场合下参数的区间估计,在定数截尾场合下参数的点估计和区间估计,并通过大量Monte-Carlo模拟从均方差的角度比较各种点估计的优劣,考察各种区间估计方法的精度,通过实例说明这些点估计和区间估计方法的应用。     

Nonparametric estimation of a survival function under progressive Type-I multistage censoring

Failure time data subject to three progressive Type-I multistage censoring schemes are studied. Product limit estimators are proposed for the estimation of the survival function. It is shown that the resulting estimators are asymptotically equivalent to the corresponding estimators where the data are subject to a random iid right censoring scheme. Many well-known results on confidence bands and tests for randomly right censored data hold for these progressive censoring schemes.     

Effects of model misspecification on tests of no randomized treatment effect arising from Cox's proportional hazards model

We examine the asymptotic and small sample properties of model-based and robust tests of the null hypothesis of no randomized treatment effect based on the partial likelihood arising from an arbitrarily misspecified Cox proportional hazards model. When the distribution of the censoring variable is either conditionally independent of the treatment group given covariates or conditionally independent of covariates given the treatment group, the numerators of the partial likelihood treatment score and Wald tests have asymptotic mean equal to 0 under the null hypothesis, regardless of whether or how the Cox model is misspecified. We show that the model-based variance estimators used in the calculation of the model-based tests are not, in general, consistent under model misspecification, yet using analytic considerations and simulations we show that their true sizes can be as close to the nominal value as tests calculated with robust variance estimators. As a special case, we show that the model-based log-rank test is asymptotically valid. When the Cox model is misspecified and the distribution of censoring depends on both treatment group and covariates, the asymptotic distributions of the resulting partial likelihood treatment score statistic and maximum partial likelihood estimator do not, in general, have a zero mean under the null hypothesis. Here neither the fully model-based tests, including the log-rank test, nor the robust tests will be asymptotically valid, and we show through simulations that the distortion to test size can be substantial.     

Stratified proportional subdistribution hazards model with covariate-adjusted censoring weight for case-cohort studies

The case-cohort study design is widely used to reduce cost when collecting expensive covariates in large cohort studies with survival or competing risks outcomes. A case-cohort study dataset consists of two parts: (a) a random sample and (b) all cases or failures from a specific cause of interest. Clinicians often assess covariate effects on competing risks outcomes. The proportional subdistribution hazards model directly evaluates the effect of a covariate on the cumulative incidence function under the non-covariate-dependent censoring assumption for the full cohort study. However, the non-covariate-dependent censoring assumption is often violated in many biomedical studies. In this article, we propose a proportional subdistribution hazards model for case-cohort studies with stratified data with covariate-adjusted censoring weight. We further propose an efficient estimator when extra information from the other causes is available under case-cohort studies. The proposed estimators are shown to be consistent and asymptotically normal. Simulation studies show (a) the proposed estimator is unbiased when the censoring distribution depends on covariates and (b) the proposed efficient estimator gains estimation efficiency when using extra information from the other causes. We analyze a bone marrow transplant dataset and a coronary heart disease dataset using the proposed method.