A polar coordinate transformation for estimating bivariate survival functions with randomly censored and truncated data

This paper proposes a new estimator for bivariate distribution functions under random truncation and random censoring. The new method is based on a polar coordinate transformation, which enables us to transform a bivariate survival function to a univariate survival function. A consistent estimator for the transformed univariate function is proposed. Then the univariate estimator is transformed back to a bivariate estimator. The estimator converges weakly to a zero-mean Gaussian process with an easily estimated covariance function. Consistent truncation probability estimate is also provided. Numerical studies show that the distribution estimator and truncation probability estimator perform remarkably well.     

A semi-parametric estimation of a survival function from incomplete and doubly censored data

The purpose of this paper is to present a semi-parametric estimation of a survival function when analyzing incomplete and doubly censored data. Under the assumption that the chance of censoring is not related to the individual's survivorship, we propose a consistent estimation of survival. The derived estimator treats the uncensored observations nonparametrically and uses parametric models for both right and left censored data. Some asymptotic properties and simulation studies are also presented in order to analyze the behavior of the proposed estimator.     

A class of weighted dependence measures for bivariate failure time data

This paper considers a class of summary measures of the dependence between a pair of failure time variables over a finite follow-up region. The class consists of measures that are weighted averages of local dependence measures, and includes the cross-ratio-measure and finite region version of Kendall's τ; recently proposed by the authors. Two new special cases are identified that can avoid the need to estimate the bivariate survivor function and that admit explicit variance estimators. Nonparametric estimators of such dependence measures are proposed and are shown to be consistent and asymptotically normal with variances that can be consistently estimated. Properties of selected estimators are evaluated in a simulation study, and the method is illustrated through an analysis of Australian Twin Study data.     

Tests of independence for censored bivariate failure time data

Bivariate failure time data is widely used in survival analysis, for example, in twins study. This article presents a class of chi2-type tests for independence between pairs of failure times after adjusting for covariates. A bivariate accelerated failure time model is proposed for the joint distribution of bivariate failure times while leaving the dependence structures for related failure times completely unspecified. Theoretical properties of the proposed tests are derived and variance estimates of the test statistics are obtained using a resampling technique. Simulation studies show that the proposed tests are appropriate for practical use. Two examples including the study of infection in catheters for patients on dialysis and the diabetic retinopathy study are also given to illustrate the methodology.     

Nonparametric analysis of interval censored and doubly truncated data

Doubly truncated data appear in a number of applications, including astronomy and survival analysis. For double-truncated data, the lifetime T is observable only when U≤T≤V, where U and V are the left-truncated and right-truncated time, respectively. In some situations, the lifetime T also suffers interval censoring. Using the EM algorithm of Turnbull [The empirical distribution function with arbitrarily grouped censored and truncated data, J. R. Stat. Soc. Ser. B 38 (1976), pp. 290–295] and iterative convex minorant algorithm [P. Groeneboom and J.A. Wellner, Information Bounds and Nonparametric Maximum Likelihood Estimation, Birkhäuser, Basel, 1992], we study the performance of the nonparametric maximum-likelihood estimates (NPMLEs) of the distribution function of T. Simulation results indicate that the NPMLE performs adequately for the finite sample.     

Continuous covariate frailty models for censored and truncated clustered data

Using some logarithmic and integral transformation we transform a continuous covariate frailty model into a polynomial regression model with a random effect. The responses of this mixed model can be ‘estimated’ via conditional hazard function estimation. The random error in this model does not have zero mean and its variance is not constant along the covariate and, consequently, these two quantities have to be estimated. Since the asymptotic expression for the bias is complicated, the two-large-bandwidth trick is proposed to estimate the bias. The proposed transformation is very useful for clustered incomplete data subject to left truncation and right censoring (and for complex clustered data in general). Indeed, in this case no standard software is available to fit the frailty model, whereas for the transformed model standard software for mixed models can be used for estimating the unknown parameters in the original frailty model. A small simulation study illustrates the good behavior of the proposed method. This method is applied to a bladder cancer data set.     

Location and scale parameter estimation from randomly censored data

The problem of location and scale parameter estimation from randomly censored data is analyzed through use of a regression model for the Kaplan-Meier quantlle process. Continuous time regression techniques are employed to construct estimators that are both asymptotically normal and efficient. Estimators with a particularly simple form are obtained for the Koziol-Green model for random censorship. In the event of no censoring the regression model, and resulting estimators, reduce to those proposed by Parzen (1979 a, b).     

Estimation of bivariate and marginal distributions with censored data

Summary. Consider a pair of random variables, both subject to random right censoring. New estimators for the bivariate and marginal distributions of these variables are proposed. The estimators of the marginal distributions are not the marginals of the corresponding estimator of the bivariate distribution. Both estimators require estimation of the conditional distribution when the conditioning variable is subject to censoring. Such a method of estimation is proposed. The weak convergence of the estimators proposed is obtained. A small simulation study suggests that the estimators of the marginal and bivariate distributions perform well relatively to respectively the Kaplan–Meier estimator for the marginal distribution and the estimators of Pruitt and van der Laan for the bivariate distribution. The use of the estimators in practice is illustrated by the analysis of a data set.     

Empirical likelihood inference for the bivariate survival function under univariate censoring

In this article, we apply the empirical likelihood method to make inference on the bivariate survival function of paired failure times by estimating the survival function of censored time with the Kaplan–Meier estimator. Adjusted empirical likelihood (AEL) confidence intervals for the bivariate survival function are developed. We conduct a simulation study to compare the proposed AEL method with other methods. The simulation study shows the proposed AEL method has better performance than other existing methods. We illustrate the proposed method by analyzing the skin graft data.     

A new class of bivariate lifetime distributions

This paper introduces a new class of bivariate lifetime distributions. Let {X_i}_{i ? 1} and {Y_i}_{i ? 1} be two independent sequences of independent and identically distributed positive valued random variables. Define T₁ = min?(X₁, …, X_M) and T₂ = min?(Y₁, …, Y_N), where (M, N) has a discrete bivariate phase-type distribution, independent of {X_i}_{i ? 1} and {Y_i}_{i ? 1}. The joint survival function of (T₁, T₂) is studied.     

Simple nonparametric tests for a known standard survival based on censored data

It is often of interest in survival analysis to test whether the distribution of lifetimes from which the sample under study was derived is the same as a reference distribution. The latter can be specified on the basis of previous studies or on subject matter considerations. In this paper several tests are developed for the above hypothesis, suitable for right-censored observations. The tests are based on modifications of Moses' one-sample limits of some classical two-sample rank tests. The asymptotic distributions of the test statistics are derived, consistency is established for alternatives which are stochastically ordered with respect to the null, and Pitman asymptotic efficiencies are calculated relative to competing tests. Simulated power comparisons are reported. An example is given with data on the survival times of lung cancer patients.     

Monte carlo results for conditional survival tests under randomly censored data

Within a Monte Carlo study finite sample results are obtained for different generalized rank tests based on randomly censored life time data. It is pointed out that conditional tests should be applied in practice whenever drastic differences between the censoring distributions for the underlying groups do not appear. The tests are slight modifications of known permutation tests for censored data.     

Maximum likelihood estimation of bivariate survival probabilities and its existence and uniqueness

Nonparametric maximum likelihood estimation of bivariate survival probabilities is developed for interval censored survival data. We restrict our attention to the situation where response times within pairs are not distinguishable, and the univariate survival distribution is the same for any individual within any pair. Campbell's (1981) model is modified to incorporate this restriction. Existence and uniqueness of maximum likelihood estimators are discussed. This methodology is illustrated with a bivariate life table analysis of an angioplasty study where each patient undergoes two procedures.     

Nonparametric estimation of the bivariate cdf for arbitrarily censored data

Right, left or interval censored multivariate data can be represented by an intersection graph. Focussing on the bivariate case, the authors relate the structure of such an intersection graph to the support of the nonparametric maximum likelihood estimate (NPMLE) of the cumulative distribution function (CDF) for such data. They distinguish two types of non‐uniqueness of the NPMLE: representational, arising when the likelihood is unaffected by the distribution of the estimated probability mass within regions, and mixture, arising when the masses themselves are not unique. The authors provide a brief overview of estimation techniques and examine three data sets.     

Estimation and goodness-of-fit for the Cox model with various types of censored data

The currently existing estimation methods and goodness-of-fit tests for the Cox model mainly deal with right censored data, but they do not have direct extension to other complicated types of censored data, such as doubly censored data, interval censored data, partly interval-censored data, bivariate right censored data, etc. In this article, we apply the empirical likelihood approach to the Cox model with complete sample, derive the semiparametric maximum likelihood estimators (SPMLE) for the Cox regression parameter and the baseline distribution function, and establish the asymptotic consistency of the SPMLE. Via the functional plug-in method, these results are extended in a unified approach to doubly censored data, partly interval-censored data, and bivariate data under univariate or bivariate right censoring. For these types of censored data mentioned, the estimation procedures developed here naturally lead to Kolmogorov-Smirnov goodness-of-fit tests for the Cox model. Some simulation results are presented.     

Small sample study on estimators of life distributions and of mean survival times using randomly censored data

Whereas large-sample properties of the estimators of survival distributions using censored data have been studied by many authors, exact results for small samples have been difficult to obtain. In this paper we obtain the exact expression for the ath moment (a > 0) of the Bayes estimator of survival distribution using the censored data under proportional hazard model. Using the exact expression we compute the exact mean, variance and MSE of the Bayes estimator. Also two estimators ofthe mean survival time based on the Kaplan-Meier estimator and the Bayes estimator are compared for small samples under proportional hazards.     

A computationally simple bivariate survival estimator for efficacy and safety

Both treatment efficacy and safety are typically the primary endpoints in Phase II, and even in some Phase III, clinical trials. Efficacy is frequently measured by time to response, death, or some other milestone event and thus is a continuous, possibly censored, outcome. Safety, however, is frequently measured on a discrete scale; in Eastern Cooperative Oncology Group clinical trial E2290, it was measured as the number of weekly rounds of chemotherapy that were tolerable to colorectal cancer patients. For the joint analysis of efficacy and safety, we propose a non-parametric, computationally simple estimator for the bivariate survival function when one time-to-event is continuous, one is discrete, and both are subject to right-censoring. The bivariate censoring times may depend on each other, but they are assumed to be independent of both event times. We derive a closed-form covariance estimator for the survivor function which allows for inference to be based on any of several possible statistics of interest. In addition, we derive its covariance with respect to calendar time of analysis, allowing for its use in sequential studies.     

A semiparametric shock model for a pair of event-related dependent censored failure times

In this paper, we propose a semiparametric shock model for two dependent failure times where the risk indicator of one failure time plays the part of a time-varying covariate for the other one. According to Hougaard [2000. Analysis of Multivariate Survival Data. Springer, New York], the dependence between the two failure times is therefore of event-related type.     

Flexible modelling of simultaneously interval censored and truncated time‐to‐event data

This paper deals with the analysis of data from a HET‐CAM^VT experiment. From a statistical perspective, such data yield many challenges. First of all, the data are typically time‐to‐event like data, which are at the same time interval censored and right truncated. In addition, one has to cope with overdispersion as well as clustering. Traditional analysis approaches ignore overdispersion and clustering and summarize the data into a continuous score that can be analysed using simple linear models. In this paper, a novel combined frailty model is developed that simultaneously captures all of the aforementioned statistical challenges posed by the data. Copyright © 2015 John Wiley & Sons, Ltd.     

Nonparametric estimation of the bivariate survival function with left-truncated and right-censored data

In this note, we consider estimating the bivariate survival function when both components are subject to left truncation and right censoring. We propose two types of estimators as generalizations of the Dabrowska and Campbell and Földes estimators. The consistency of the proposed estimators is established. A simple bootstrap method is used for obtaining precision estimation of the proposed estimators. A simulation study is conducted to investigate the performance of the proposed estimators.