Model Identifiability for Bivariate Failure Time Data with Competing Risks: Parametric Baseline Cause-specific Hazards and Non-parametric Frailty

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.

A marginal hazards model of multivariate failure times has been developed based on the ‘working independence’ assumption [L.J. Wei, D.Y. Lin, and L. Wessfeld, Regression analysis of multivariate incomplete failure time data by modeling marginal distributions, J. Amer. Statist. Assoc. 84 (1989), pp. 1065–1073.]. In this article, we study the marginal hazards model of multivariate failure times with continuous auxiliary covariates. We consider the case of common baseline hazards for subjects from the same clusters. We extend the kernel smoothing procedure of Zhou and Wang [H. Zhou and C.Y. Wang, Failure time regression with continuous covariates measured with error, J. Roy. Statist. Soc. B 62 (2000), pp. 657–665.] to correlated failure time data. Through semiparametric estimation of the marginal partial likelihood function, we obtain the estimated partial likelihood based estimator of the regression coefficients. We present asymptotic properties of the induced estimator and demonstrate the performance of the proposed estimator through a finite sample simulation study. Finally, a real data application is conducted to elucidate the use of the method.

In this paper, a class of tests is developed for comparing the cause-specific hazard rates of m competing risks simultaneously in K (> or = 2) groups. The data available for a unit are the failure time of the unit along with the identifier of the risk claiming the failure. In practice, the failure time data are generally right censored. The tests are based on the difference between the weighted averages of the cause-specific hazard rates corresponding to each risk. No assumption regarding the dependence of the competing risks is made. It is shown that the proposed test statistic has asymptotically chi-squared distribution. The proposed test is shown to be optimal for a specific type of local alternatives. The choice of weight function is also discussed. A simulation study is carried out using multivariate Gumbel distribution to compare the optimal weight function with a proposed weight function which is to be used in practice. Also, the proposed test is applied to real data on the termination of an intrauterine device.

In randomized clinical trials or observational studies, subjects are recruited at multiple treating sites. Factors that vary across sites may have some influence on outcomes; therefore, they need to be taken into account to get better results. We apply the accelerated failure time (AFT) model with linear mixed effects to analyze failure time data, accounting for correlations between outcomes. Specifically, we use Bayesian approach to fit the data, computing the regression parameters by Gibbs sampler combined with Buckley-James method. This approach is compared with the marginal independence approach and other methods through simulations and an application to a real example.

Reliability analysis of repairable systems has been widely conducted based on collected information such as the systems’ failure counting processes. In many real-life situations, the failure counting processes may not be completely observed when some failure times are not recorded or missing due to various practical reasons. In the literature, only a limited number of reliability studies focus on missing failure time data for repairable systems with an assumption that the system repair is either perfect or minimal. When the repair effect is general as in many real-world situations, challenges arise for reliability analysis in that the observed and unobserved failure times are statistically dependent in a complex manner, and the time censoring problem becomes unconventional. In this paper, to overcome these challenges, we propose a reliability model for repairable systems with incomplete failure time data under general repair. The proposed model is suitable for modeling both a single system and multiple systems subject to failure heterogeneity. A maximum likelihood estimation method is developed to estimate the model parameters. Based on the proposed model, statistical inference of the missing times is developed. Simulation and real-life case studies are conducted to verify the developed methods.

An alternative estimation method for the accelerated failure time frailty model

In the competing risks analysis, most inferences have been developed based on continuous failure time data. However, failure times are sometimes observed as being discrete. We propose nonparametric inferences for the cumulative incidence function for pure discrete data with competing risks. When covariate information is available, we propose semiparametric inferences for direct regression modelling of the cumulative incidence function for grouped discrete failure time data with competing risks. Simulation studies show that the procedures perform well. The proposed methods are illustrated with a study of contraceptive use in Indonesia.

Interval-censored failure time data commonly occur in many periodic follow-up studies such as epidemiological experiments, medical studies and clinical trials. By intervalcensored data, we usually mean that one cannot observe the failure time of interest and instead we know that it belongs to a time interval. Correlated failure time data commonly occur when there are multiple events on one individual or when the study subjects are clustered into some small groups. In this situation, study subjects from same subgroup or the failure events from same individuals are usually regarded as dependent, but the subjects in different clusters or failure events from different individuals are assumed to be independent. Besides the correlation between the cluster, sometimes the cluster size may be informative or carry some information about the failure time of interest. Cured subgroup is another interesting topic that has been discussed by many authors. For this situation, unlike the assumptions in traditional survival model that all study subjects would experience the failure event of interest eventually if the follow-up time is long enough, some subjects may never experience or not be susceptible to the event. Such subjects are treated as cured and assumed to belong to a cured subgroup in a study population. The research in this dissertation focuses on regression analysis of correlated intervalcensored data with a cured subgroup via different approaches based on different data structures. In the first part of this dissertation, we discuss clustered interval-censored data with a cured subgroup and informative cluster size. To address this, we present a within-cluster resampling method and in the approach, the multiple imputation procedure is applied for estimation of unknown parameters. To assess the performance of the proposed method, a simulation study is conducted and suggests that it works well in practical situations. Also, the method is applied to a set of real data that motivated this study. In the second part of this dissertation, we consider the clustered interval-censored data with a cured subgroup via a non-mixture cure model. We present a maximum likelihood estimation procedure under the semiparametric transformation nonmixture cure model. To estimate the unknown parameters, an expectation maximization (EM) algorithm based on an augmentation of Poisson variable is developed. To assess the performance of the proposed method, a simulation study is conducted and suggests that it works well in practical situations. An application to a study conducted by the National Aeronautics and Space Administration that motivated this study is also provided. In the third part of this dissertation, we investigate the bivariate interval-censored data with a cured subgroup. A sieve maximum likelihood estimation procedure under the semiparametric transformation non-mixture cure model based on Bernstein polynomials is presented. A simulation study is conducted to assess the finite sample performance of the proposed method and suggests that the proposed model works well. Also, a real data application from the study of AIDS Clinical Trial Group 181 is provided.

Nonparametric maximum likelihood estimation of the probability of failing from a particular cause by time t in the presence of other acting causes (i.e., the cause-specific failure probability) is discussed. A commonly used incorrect approach is to take 1 minus the Kaplan-Meier (KM) estimator (1 – KM), whereby patients who fail of extraneous causes are treated as censored observations. Examples showing the extent of bias in using the 1-KM approach are presented using clinical oncology data. This bias can be quite large if the data are uncensored or if a large percentage of patients fail from extraneous causes prior to the occurrence of failures from the cause of interest. Each cause-specific failure probability is mathematically defined as a function of all of the cause-specific hazards. Therefore, nonparametric estimates of the cause-specific failure probabilities may not be able to identify categorized covariate effects on the cause-specific hazards. These effects would be correctly identified by cause-specific cumulative hazard or KM plots in which the extraneous causes of failure are treated as censored observations. Examples are provided. Finally, nonparametric graphical representation of the two distinct cause-specific failure components of the mixture model (i.e., the probability of ever failing from a particular cause and the time-to-failure distribution given that a patient will fail of that cause) are presented. The difficulty in extrapolating the nonparametric estimates beyond the range of observed failure times is highlighted. In addition, the mathematical relationship of a single covariate z and the two cause-specific failure components is shown for the case where z acts multiplicatively on the cause-specific hazards. Examples are considered using the important prognostic factors in adult soft tissue sarcoma (STS) of the extremity and adult acute lymphoblastic leukemia (ALL). In each of the mathematical and actual examples considered, the characteristic that was associated with a higher hazard rate of failure from the disease was also associated with a higher probability of ever failing from the disease as well as a shorter time-to-failure distribution given that failure due to the disease will occur. Thus it may be quite common in chronic disease situations to find prognosticators that are associated with both cause-specific failure components.

In clinical trials and engineering studies that are followed by periodic follow‐ups, it is predominantly to have partly interval‐censored failure time data. Partly interval‐censored failure time data is composed of exact observations and interval‐censored observations. This paper discusses two‐sample parametric comparison of reliability function in the existence of partly interval‐censored failure time data. We have constructed a score test and likelihood ratio test for this kind of failure time data under piecewise exponential distribution by using multiple imputation technique. Simulation study is established to assess the proposed test, which indicates that the presented procedure works well. Finally, an example is given for illustration purposes.

Clustered interval-censored failure time data occur when the failure times of interest are clustered into small groups and known only to lie in certain intervals. A number of methods have been proposed for regression analysis of clustered failure time data, but most of them apply only to clustered right-censored data. In this paper, a sieve estimation procedure is proposed for fitting a Cox frailty model to clustered interval-censored failure time data. In particular, a two-step algorithm for parameter estimation is developed and the asymptotic properties of the resulting sieve maximum likelihood estimators are established. The finite sample properties of the proposed estimators are investigated through a simulation study and the method is illustrated by the data arising from a lymphatic filariasis study.

Interval censored (IC) failure time data are often observed in medical follow-up studies and clinical trials where subjects can only be followed periodically, and the failure time can only be known to lie in an interval. In this paper, we propose a weighted Wilcoxon-type rank test for the problem of comparing two IC samples. Under a very general sampling technique developed by Fay (1999), the mean and variance of the test statistics under the null hypothesis can be derived. Through simulation studies, we find that the performance of the proposed test is better than that of the two existing Wilcoxon-type rank tests proposed by Mantel (1967) and R. Peto and J. Peto (1972). The proposed test is illustrated by means of an example involving patients in AIDS cohort studies.

In this work, we introduce and study a new alternative Lomax model. The maximum likelihood method is used to estimate the unknown model parameters. We show empirically the importance and wide flexibility of the new model in modeling two types of failure times data sets. The new model is much better than the gamma Lomax, exponentiated Lomax, beta Lomax and Lomax models so the new model is a good alternative to these models.

Non-parametric estimation from simultaneous degradation and failure time data

A decision dependent stochastic process model for repairable systems with applications