Estimation from Censored Data with Incomplete Information

Phillips and Sweeting [J. R. Statist. Soc. B 58 (1996) 775–783.] considered estimation of the parameter of the exponential distribution with censored failure time data when there is incomplete knowledge of the censoring times. It was shown that, under particular models for the censoring mechanism and censoring errors, it will usually be safe to ignore such errors provided they are not expected to be too large. A flexible model is introduced which includes the extreme cases of no censoring errors and no information on the censoring values. The effect of alternative assumptions about knowledge of the censoring values on the estimation of failure rate is investigated.     

Dealing with Ties in Failure Time Data

In dealing with ties in failure time data the mechanism by which the data are observed should be considered. If the data are discrete, the process is relatively simple and is determined by what is actually observed. With continuous data, ties are not supposed to occur, but they do because the data are grouped into intervals (even if only rounding intervals). In this case there is actually a non–identifiability problem which can only be resolved by modelling the process. Various reasonable modelling assumptions are investigated in this paper. They lead to better ways of dealing with ties between observed failure times and censoring times of different individuals. The current practice is to assume that the censoring times occur after all the failures with which they are tied.     

Estimation with Interval Censored Data and Covariates

In biostatistical applications interest often focuses on the estimation of the distribution of time T between two consecutive events. If the initial event time is observed and the subsequent event time is only known to be larger or smaller than an observed monitoring time C, then the data conforms to the well understood singly-censored current status model, also known as interval censored data, case I. Additional covariates can be used to allow for dependent censoring and to improve estimation of the marginal distribution of T. Assuming a wrong model for the conditional distribution of T, given the covariates, will lead to an inconsistent estimator of the marginal distribution. On the other hand, the nonparametric maximum likelihood estimator of FT requires splitting up the sample in several subsamples corresponding with a particular value of the covariates, computing the NPMLE for every subsample and then taking an average. With a few continuous covariates the performance of the resulting estimator is typically miserable. In van der Laan, Robins (1996) a locally efficient one-step estimator is proposed for smooth functionals of the distribution of T, assuming nothing about the conditional distribution of T, given the covariates, but assuming a model for censoring, given the covariates. The estimators are asymptotically linear if the censoring mechanism is estimated correctly. The estimator also uses an estimator of the conditional distribution of T, given the covariates. If this estimate is consistent, then the estimator is efficient and if it is inconsistent, then the estimator is still consistent and asymptotically normal. In this paper we show that the estimators can also be used to estimate the distribution function in a locally optimal way. Moreover, we show that the proposed estimator can be used to estimate the distribution based on interval censored data (T is now known to lie between two observed points) in the presence of covariates. The resulting estimator also has a known influence curve so that asymptotic confidence intervals are directly available. In particular, one can apply our proposal to the interval censored data without covariates. In Geskus (1992) the information bound for interval censored data with two uniformly distributed monitoring times at the uniform distribution (for T has been computed. We show that the relative efficiency of our proposal w.r.t. this optimal bound equals 0.994, which is also reflected in finite sample simulations. Finally, the good practical performance of the estimator is shown in a simulation study. This revised version was published online in July 2006 with corrections to the Cover Date.     

Failure Inference From a Marker Process Based on a Bivariate Wiener Model

Many models have been proposed that relate failure times and stochastic time-varying covariates. In some of these models, failure occurs when a particular observable marker crosses a threshold level. We are interested in the more difficult, and often more realistic, situation where failure is not related deterministically to an observable marker. In this case, joint models for marker evolution and failure tend to lead to complicated calculations for characteristics such as the marginal distribution of failure time or the joint distribution of failure time and marker value at failure. This paper presents a model based on a bivariate Wiener process in which one component represents the marker and the second, which is latent (unobservable), determines the failure time. In particular, failure occurs when the latent component crosses a threshold level. The model yields reasonably simple expressions for the characteristics mentioned above and is easy to fit to commonly occurring data that involve the marker value at the censoring time for surviving cases and the marker value and failure time for failing cases. Parametric and predictive inference are discussed, as well as model checking. An extension of the model permits the construction of a composite marker from several candidate markers that may be available. The methodology is demonstrated by a simulated example and a case application.     

A relation between product-limit estimate and joint-risk estimate

The joint-risk estimate of the survival function, used for censored survival data grouped into fixed intervals, is shown to be the geometric mean of all the product-limit estimates that correspond to all the possible orderings of all the failure times and censoring times in the group. The joint-risk estimate is proposed as a more appropriate and better means of dealing with ties for data containing tied failure times and censoring times. It is also applicable to competing risk problems with tied failure times involving different causes. It could be used as a substitute for the product-limit estimate in discrete failure time analysis.     

Estimation of parameters in logistic and log-logistic distribution with grouped data

In many situations, instead of a complete sample, data are available only in grouped form. For example, grouped failure time data occur in studies in which subjects are monitored periodically to determine whether failure has occurred in the predetermined intervals. Here the model under consideration is the log-logistic distribution. This paper demonstrates the existence and uniqueness of the MLEs of the parameters of the logistic distribution under mild conditions with grouped data. The times with the maximum failure rate and the mode of the p.d.f. of the log-logistic distribution are also estimated based on the MLEs. The methodology is further studied with simulations and exemplified with a data set with artificially introduced grouping from a locomotive life test study.     

Planning Life Tests Based on Progressively Type-I Grouped Censored Data from the Weibull Distribution

In this article, we apply the simulated annealing algorithm to determine optimally spaced inspection times for the two-parameter Weibull distribution for any given progressive Type-I grouped censoring plan. We examine how the asymptotic relative efficiencies of the estimates are affected by the position of the monitoring points and the number of monitoring points used. A comparison of different inspection plans is made that will enable the user to select a plan for a specified quality goal. Using the same algorithm, we can also determine an optimal progressive Type-I grouped censoring plan when the inspection times and the expected proportions of total failures in the experiment are pre-fixed. Finally, we discuss the sample size and the acceptance constant of the progressively Type-I grouped censored reliability sampling plan when the optimal inspection times are used.     

Deviance residuals in generalised log-gamma regression models with censored observations

In this article, we compare three residuals based on the deviance component in generalised log-gamma regression models with censored observations. For different parameter settings, sample sizes and censoring percentages, various simulation studies are performed and the empirical distribution of each residual is displayed and compared with the standard normal distribution. For all cases studied, the empirical distributions of the proposed residuals are in general symmetric around zero, but only a martingale-type residual presented negligible kurtosis for the majority of the cases studied. These studies suggest that the residual analysis usually performed in normal linear regression models can be straightforwardly extended for the martingale-type residual in generalised log-gamma regression models with censored data. A lifetime data set is analysed under log-gamma regression models and a model checking based on the martingale-type residual is performed.     

Semiparametric analysis of recurrent events: artificial censoring,truncation, pairwise estimation and inference

The analysis of recurrent failure time data from longitudinal studies can be complicated by the presence of dependent censoring. There has been a substantive literature that has developed based on an artificial censoring device. We explore in this article the connection between this class of methods with truncated data structures. In addition, a new procedure is developed for estimation and inference in a joint model for recurrent events and dependent censoring. Estimation proceeds using a mixed U-statistic based estimating function approach. New resampling-based methods for variance estimation and model checking are also described. The methods are illustrated by application to data from an HIV clinical trial as with a limited simulation study.     

The log-odd log-logistic Weibull regression model: modelling,estimation, influence diagnostics and residual analysis

In applications of survival analysis, the failure rate function may frequently present a unimodal shape. In such cases, the log-normal and log-logistic distributions are used. In this paper, we shall be concerned only with parametric forms, so a location-scale regression model based on the odd log-logistic Weibull distribution is proposed for modelling data with a decreasing, increasing, unimodal and bathtub failure rate function as an alternative to the log-Weibull regression model. For censored data, we consider a classic method to estimate the parameters of the proposed model. We derive the appropriate matrices for assessing local influences on the parameter estimates under different perturbation schemes and present some ways to assess global influences. Further, for different parameter settings, sample sizes and censoring percentages, various simulations are performed. In addition, the empirical distribution of some modified residuals is determined and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be extended to a modified deviance residual in the new regression model applied to censored data. We analyse a real data set using the log-odd log-logistic Weibull regression model.     

On Nonparametric Maximum Likelihood Estimations of Multivariate Distribution Function Based on Interval-Censored Data

This article considers the nonparametric maximum likelihood estimator (NPMLE) of a joint distribution function when the multivariate failure times of interest are interval-censored. With different types of interval censoring mechanism, the NPMLE's of the multivariate distribution function are studied and the strong consistency for the NPMLEs is obtained in terms of a self-consistency equation. Furthermore, the convergence rate of the estimator is given, which depends on the types of interval censoring mechanism.     

Residuals for log-Burr XII regression models in survival analysis

In this paper, we compare three residuals to assess departures from the error assumptions as well as to detect outlying observations in log-Burr XII regression models with censored observations. These residuals can also be used for the log-logistic regression model, which is a special case of the log-Burr XII regression model. For different parameter settings, sample sizes and censoring percentages, various simulation studies are performed and the empirical distribution of each residual is displayed and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be straightforwardly extended to the modified martingale-type residual in log-Burr XII regression models with censored data.     

Ancillarity properties of generalized residuals with applications in failure time models

This paper addresses the issue of when residuals from failure time models, which are useful in model validation and diagnostics, possess a conditional ancillarity property. This property states that the distribution of the residuals depends on the model parameters only through a many-to-one function of these parameters, which in certain models turn out to be the censoring proportion. Concrete results are obtained for models which possess an invariance structure, and these results are applied to commonly used failure time models. Aside from furthering our understanding of the distributional structure of residuals, this conditional ancillarity property can be exploited to study in a more efficient manner the distributional properties of residuals either analytically and/or through numerical methods.     

Bayesian statistical inference of the loglogistic model with interval-censored lifetime data

Interval-censored data arise when a failure time say, T cannot be observed directly but can only be determined to lie in an interval obtained from a series of inspection times. The frequentist approach for analysing interval-censored data has been developed for some time now. It is very common due to unavailability of software in the field of biological, medical and reliability studies to simplify the interval censoring structure of the data into that of a more standard right censoring situation by imputing the midpoints of the censoring intervals. In this research paper, we apply the Bayesian approach by employing Lindley's 1980, and Tierney and Kadane 1986 numerical approximation procedures when the survival data under consideration are interval-censored. The Bayesian approach to interval-censored data has barely been discussed in literature. The essence of this study is to explore and promote the Bayesian methods when the survival data been analysed are is interval-censored. We have considered only a parametric approach by assuming that the survival data follow a loglogistic distribution model. We illustrate the proposed methods with two real data sets. A simulation study is also carried out to compare the performances of the methods.     

A class of residuals for outlier identification in zero adjusted regression models

Zero adjusted regression models are used to fit variables that are discrete at zero and continuous at some interval of the positive real numbers. Diagnostic analysis in these models is usually performed using the randomized quantile residual, which is useful for checking the overall adequacy of a zero adjusted regression model. However, it may fail to identify some outliers. In this work, we introduce a class of residuals for outlier identification in zero adjusted regression models. Monte Carlo simulation studies and two applications suggest that one of the residuals of the class introduced here has good properties and detects outliers that are not identified by the randomized quantile residual.     

The Robustness of Several Estimators of the Survivorship Function with Randomly Censored Data

The problem of estimating the survivorship function, R(t) = P(T > t), arises frequently in both engineering and biomedical sciences. In many applications the data one sees are censored due to the occurrence of some competing cause of failure such as withdrawal from the study, failure from some cause not under study, etc. In the biomedical sciences the distribution free estimator suggested by Kaplan and Meier (JASA 1958) is routinely used, while in the engineering sciences a parametric approach is more commonly used. In this report we study the efficiency of these two techniques when a particular parametric model such as the exponential, Weibull, normal, log normal, exponential power, Pareto, Gompertz, gamma, or bathtub shaped hazard distribution is assumed under a variety of censoring schemes and underlying failure models. We conclude that in most cases the parametric estimators outperform the distribution free estimator. The results are particularly striking if the Weibull forms of these estimators are used routinely.     

Bayesian analysis of competing risks with partially masked cause of failure

Summary. Bayesian analysis of system failure data from engineering applications under a competing risks framework is considered when the cause of failure may not have been exactly identified but has only been narrowed down to a subset of all potential risks. In statistical literature, such data are termed masked failure data. In addition to masking, failure times could be right censored owing to the removal of prototypes at a prespecified time or could be interval censored in the case of periodically acquired readings. In this setting, a general Bayesian formulation is investigated that includes most commonly used parametric lifetime distributions and that is sufficiently flexible to handle complex forms of censoring. The methodology is illustrated in two engineering applications with a special focus on model comparison issues.     

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.     

Interval censoring: Model characterizations for the validity of the simplified likelihood

In survival data analysis, the interval censoring problem has generally been treated via likelihood methods. Because this likelihood is complex, it is often assumed that the censoring mechanisms do not affect the mortality process. The authors specify conditions that ensure the validity of such a simplified likelihood. They prove the equivalence between different characterizations of noninformative censoring and define a constant‐sum condition analogous to the one derived in the context of right censoring. They also prove that when the noninformative or constant‐sum condition holds, the simplified likelihood can be used to obtain the nonparametric maximum likelihood estimator of the death time distribution function.     

A log-linear regression model for the odd Weibull distribution with censored data

We introduce the log-odd Weibull regression model based on the odd Weibull distribution (Cooray, 2006). We derive some mathematical properties of the log-transformed distribution. The new regression model represents a parametric family of models that includes as sub-models some widely known regression models that can be applied to censored survival data. We employ a frequentist analysis and a parametric bootstrap for the parameters of the proposed model. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to assess global influence. Further, for different parameter settings, sample sizes and censoring percentages, some simulations are performed. In addition, the empirical distribution of some modified residuals are given and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be extended to a modified deviance residual in the proposed regression model applied to censored data. We define martingale and deviance residuals to check the model assumptions. The extended regression model is very useful for the analysis of real data.