The negative binomial–beta Weibull regression model to predict the cure of prostate cancer

In this article, for the first time, we propose the negative binomial–beta Weibull (BW) regression model for studying the recurrence of prostate cancer and to predict the cure fraction for patients with clinically localized prostate cancer treated by open radical prostatectomy. The cure model considers that a fraction of the survivors are cured of the disease. The survival function for the population of patients can be modeled by a cure parametric model using the BW distribution. We derive an explicit expansion for the moments of the recurrence time distribution for the uncured individuals. The proposed distribution can be used to model survival data when the hazard rate function is increasing, decreasing, unimodal and bathtub shaped. Another advantage is that the proposed model includes as special sub-models some of the well-known cure rate models discussed in the literature. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes. We analyze a real data set for localized prostate cancer patients after open radical prostatectomy.     

Regression analysis of competing risks data via semi-parametric additive hazard model

When the subjects in a study possess different demographic and disease characteristics and are exposed to more than one types of failure, a practical problem is to assess the covariate effects on each type of failure as well as on all-cause failure. The most widely used method is to employ the Cox models on each cause-specific hazard and the all-cause hazard. It has been pointed out that this method causes the problem of internal inconsistency. To solve such a problem, the additive hazard models have been advocated. In this paper, we model each cause-specific hazard with the additive hazard model that includes both constant and time-varying covariate effects. We illustrate that the covariate effect on all-cause failure can be estimated by the sum of the effects on all competing risks. Using data from a longitudinal study on breast cancer patients, we show that the proposed method gives simple interpretation of the final results, when the primary covariate effect is constant in the additive manner on each cause-specific hazard. Based on the given additive models on the cause-specific hazards, we derive the inferences for the adjusted survival and cumulative incidence functions.     

Modelling population-based cancer survival trends using join point models for grouped survival data

Summary.  In the USA cancer as a whole is the second leading cause of death and a major burden to health care; thus medical progress against cancer is a major public health goal. There are many individual studies to suggest that cancer treatment breakthroughs and early diagnosis have significantly improved the prognosis of cancer patients. To understand better the relationship between medical improvements and the survival experience for the patient population at large, it is useful to evaluate cancer survival trends on the population level, e.g. to find out when and how much the cancer survival rates changed. We analyse population-based grouped cancer survival data by incorporating join points into the survival models. A join point survival model facilitates the identification of trends with significant change-points in cancer survival, when related to cancer treatments or interventions. The Bayesian information criterion is used to select the number of join points. The performance of the join point survival models is evaluated with respect to cancer prognosis, join point locations, annual percentage changes in death rates by year of diagnosis and sample sizes through intensive simulation studies. The model is then applied to grouped relative survival data for several major cancer sites from the 'Surveillance, epidemiology and end results' programme of the National Cancer Institute. The change-points in the survival trends for several major cancer sites are identified and the potential driving forces behind such change-points are discussed.     

Nonparametric change point estimation for survival distributions with a partially constant hazard rate

Lifetime Data Analysis - We present a new method for estimating a change point in the hazard function of a survival distribution assuming a constant hazard rate after the change point and a...     

Restricted tests for testing independence of time to failure and cause of failure in a competing-risks model

We consider the competing-risks problem without making any assumption concerning the independence of the risks. Maximum-likelihood estimates of the cause-specific hazard rates are obtained under the condition that their ratio is monotone. We also consider the likelihood-ratio test for testing the proportionality of two cause-specific hazard rates against the alternative that the ratio of these two hazard rates is monotonic. This testing problem is equivalent to testing independence against likelihood-ratio dependence of the time to failure and the cause of failure in the competing-risks setup. We allow for random censoring on the right. The asymptotic null distribution of the test statistic is obtained and is found to be of the chi-bar-square type. The problem is extended to the case of more than two risks. A numerical example is given to illustrate the procedure.     

Fitting a Semi-Parametric Mixture Model for Competing Risks in Survival Data

A model for survival analysis is studied that is relevant for samples which are subject to multiple types of failure. In comparison with a more standard approach, through the appropriate use of hazard functions and transition probabilities, the model allows for a more accurate study of cause-specific failure with regard to both the timing and type of failure. A semiparametric specification of a mixture model is employed that is able to adjust for concomitant variables and allows for the assessment of their effects on the probabilities of eventual causes of failure through a generalized logistic model, and their effects on the corresponding conditional hazard functions by employing the Cox proportional hazards model. A carefully formulated estimation procedure is presented that uses an EM algorithm based on a profile likelihood construction. The methods discussed, which could also be used for reliability analysis, are applied to a prostate cancer data set.     

Statistical analysis of right-censored failure-time data with partially specified hazard rates

This paper discusses the analysis of right-censored failure-time data in which the failure rate may have different forms in different time intervals. Such data occur naturally, for example, in demography studies and leukemia research, and a number of methods for the analysis have been proposed in the literature. However, most methods are purely parametric or nonparametric. Matthews and Farewell (1982), for example, discussed this problem and proposed a method for testing a constant failure rate against a failure rate involving a change point. To estimate an absolute limit on the attainable human life span, Zelterman (1992) discussed a hazard function that has different parametric forms over different time intervals. We consider a different situation in which the hazard function may follow a parametric form before a change point and is completely unknown after the change point. To test the existence of the change point, a modified maximal-censored-likelihood-ratio test is proposed and its asymptotic properties are studied. A bootstrap method is described for finding critical values of the proposed test. Simulation results indicate that the test performs well.     

Identification of Longitudinal Biomarkers in Survival Analysis for Competing Risks Data

In recent years, joint analysis of longitudinal measurements and survival data has received much attention. However, previous work has primarily focused on a single failure type for the event time. In this article, we consider joint modeling of repeated measurements and competing risks failure time data to allow for more than one distinct failure type in the survival endpoint so we fit a cause-specific hazards sub-model to allow for competing risks, with a separate latent association between longitudinal measurements and each cause of failure. Besides, previous work does not focus on the hypothesis to test a separate latent association between longitudinal measurements and each cause of failure. In this article, we derive a score test to identify longitudinal biomarkers or surrogates for a time to event outcome in competing risks data. With a carefully chosen definition of complete data, the maximum likelihood estimation of the cause-specific hazard functions is performed via an EM algorithm. We extend this work and allow random effects to be present in both the longitudinal biomarker and underlying survival function. The random effects in the biomarker are introduced via an explicit term while the random effect in the underlying survival function is introduced by the inclusion of frailty into the model.

We use simulations to explore how the number of individuals, the number of time points per individual and the functional form of the random effects from the longitudinal biomarkers considering heterogeneous baseline hazards in individuals influence the power to detect the association of a longitudinal biomarker and the survival time.     

Graphical representation of survival curves in the presence of time-dependent categorical covariates with application to liver transplantation

Graphical representation of survival curves is often used to illustrate associations between exposures and time-to-event outcomes. However, when exposures are time-dependent, calculation of survival probabilities is not straightforward. Our aim was to develop a method to estimate time-dependent survival probabilities and represent them graphically. Cox models with time-dependent indicators to represent state changes were fitted, and survival probabilities were plotted using pre-specified times of state changes. Time-varying hazard ratios for the state change were also explored. The method was applied to data from the Adult-to-Adult Living Donor Liver Transplantation Cohort Study (A2ALL). Survival curves showing a ‘split’ at a pre-specified time t allow for the qualitative comparison of survival probabilities between patients with similar baseline covariates who do and do not experience a state change at time t. Time since state change interactions can be visually represented to reflect changing hazard ratios over time. A2ALL study results showed differences in survival probabilities among those who did not receive a transplant, received a living donor transplant, and received a deceased donor transplant. These graphical representations of survival curves with time-dependent indicators improve upon previous methods and allow for clinically meaningful interpretation.     

Modelling accelerated life test data by using a Bayesian approach

Summary. Because of the high reliability of many modern products, accelerated life tests are becoming widely used to obtain timely information about their time-to-failure distributions. We propose a general class of accelerated life testing models which are motivated by the actual failure process of units from a limited failure population with a positive probability of not failing during the technological lifetime. We demonstrate a Bayesian approach to this problem, using a new class of models with non-monotone hazard rates, the hazard model with potential scope for use far beyond accelerated life testing. Our methods are illustrated with the modelling and analysis of a data set on lifetimes of printed circuit boards under humidity accelerated life testing.     

Spatially explicit survival modeling for small area cancer data

In this paper we propose a novel Bayesian statistical methodology for spatial survival data. Our methodology broadens the definition of the survival, density and hazard functions by explicitly modeling the spatial dependency using direct derivations of these functions and their marginals and conditionals. We also derive spatially dependent likelihood functions. Finally we examine the applications of these derivations with geographically augmented survival distributions in the context of the Louisiana Surveillance, Epidemiology, and End Results registry prostate cancer data.     

Semiparametric Sequential Testing for Multiple Change Points in Piecewise Constant Hazard Functions with Long-Term Survivors

In this article, we consider parameter estimation in the hazard rate with multiple change points in the presence of long-term survivors. We combine two methods: maximum likelihood based and martingale based, to estimate the change points in the hazard rate for right censored survival data that accounts for long-term survivors. A simulation study is carried out to compare the performance of estimators. The method is applied to analyze two real datasets.     

Bayesian Parametric Accelerated Failure Time Spatial Model and its Application to Prostate Cancer

Prostate cancer is the most common cancer diagnosed in American men and the second leading cause of death from malignancies. There are large geographical variation and racial disparities existing in the survival rate of prostate cancer. Much work on the spatial survival model is based on the proportional hazards model, but few focused on the accelerated failure time model. In this paper, we investigate the prostate cancer data of Louisiana from the SEER program and the violation of the proportional hazards assumption suggests the spatial survival model based on the accelerated failure time model is more appropriate for this data set. To account for the possible extra-variation, we consider spatially-referenced independent or dependent spatial structures. The deviance information criterion (DIC) is used to select a best fitting model within the Bayesian frame work. The results from our study indicate that age, race, stage and geographical distribution are significant in evaluating prostate cancer survival.     

A Test for Constant Hazard Against a Change-Point Alternative

In this article, we consider the change-point hazard rate model which arises quite commonly in mechanical or biological systems, which experience a high hazard rate early in their lifetime due to infant mortality and then a constant or steady hazard rate after the threshold time. We first derive the corresponding mean residual life function (MRLF) and observe that the MRLF is initially increasing and then constant. Here, we derive a test statistic for exponentiality against Increasing Initially then Constant Mean Residual Life (ICMRL). We also derive the asymptotic distribution of the test statistic and compare the power of the test with other existing tests such as likelihood ratio, Weibull, and Log gamma tests considered in the literature. The test performs quite well as compared to other alternatives studied.     

An extended semiparametric model for short-term and long-term hazard ratios

In survival analysis, one way to deal with non-proportional hazards is to model short-term and long-term hazard ratios. The existing model of this nature has no control over how fast the hazard ratio is changing over time. We add a parameter to the existing model to allow the hazard ratio to change over time at different speed. A nonparametric maximum likelihood approach is used to estimate the model parameters. The existing model is a special case of the extended model when the speed parameter is 0, which leads naturally to a way of testing the adequacy of the existing model. Simulation results show that there can be substantial bias in the estimation of the short-term and long-term hazard ratio if the speed parameter is fixed incorrectly at 0 rather than estimated. The extended model is fitted to three real data sets to shed new insights, including the observation that converging hazards does not necessarily imply the odds are proportional.     

Detecting change in a hazard regression model with right-censoring

The hazard function plays an important role in survival analysis and reliability, since it quantifies the instantaneous failure rate of an individual at a given time point t, given that this individual has not failed before t. In some applications, abrupt changes in the hazard function are observed, and it is of interest to detect the location of such a change. In this paper, we consider testing of existence of a change in the parameters of an exponential regression model, based on a sample of right-censored survival times and the corresponding covariates. Likelihood ratio type tests are proposed and non-asymptotic bounds for the type II error probability are obtained. When the tests lead to acceptance of a change, estimators for the location of the change are proposed. Non-asymptotic upper bounds of the underestimation and overestimation probabilities are obtained. A short simulation study illustrates these results.     

Estimation of a Change Point in a Hazard Function Based on Censored Data

The hazard function plays an important role in reliability or survival studies since it describes the instantaneous risk of failure of items at a time point, given that they have not failed before. In some real life applications, abrupt changes in the hazard function are observed due to overhauls, major operations or specific maintenance activities. In such situations it is of interest to detect the location where such a change occurs and estimate the size of the change. In this paper we consider the problem of estimating a single change point in a piecewise constant hazard function when the observed variables are subject to random censoring. We suggest an estimation procedure that is based on certain structural properties and on least squares ideas. A simulation study is carried out to compare the performance of this estimator with two estimators available in the literature: an estimator based on a functional of the Nelson-Aalen estimator and a maximum likelihood estimator. The proposed least squares estimator tums out to be less biased than the other two estimators, but has a larger variance. We illustrate the estimation method on some real data sets.     

Reconstructing the history of forest fire frequency: Identifying hazard rate change points using the bayes information criterion

A problem in model selection, namely the identification of multiple change points for a piece‐wise constant hazard rate, is discussed. A methodology using the Bayes Information Criterion is developed in an overdispersed survival model (with corresponding quasi‐likelihood function). The technique is used to identify changes in the historical frequency of forest fire. It is applied to two datasets derived from time‐since‐fire maps in the Canadian Rockies.     

Proportional Hazards Model for Multivariate Failure Time Data

Multivariate failure time data also referred to as correlated or clustered failure time data, often arise in survival studies when each study subject may experience multiple events. Statistical analysis of such data needs to account for intracluster dependence. In this article, we consider a bivariate proportional hazards model using vector hazard rate, in which the covariates under study have different effect on two components of the vector hazard rate function. Estimation of the parameters as well as base line hazard function are discussed. Properties of the estimators are investigated. We illustrated the method using two real life data. A simulation study is reported to assess the performance of the estimator.