A Bayesian semi-parametric bivariate failure time model

In this paper we introduce a Bayesian semiparametric model for bivariate and multivariate survival data. The marginal densities are well-known nonparametric survival models and the joint density is constructed via a mixture. Our construction also defines a copula and the properties of this new copula are studied. We also consider the model in the presence of covariates and, in particular, we find a simple generalisation of the widely used frailty model, which is based on a new bivariate gamma distribution.     

A Bayesian semi-parametric bivariate failure time model

In this paper we introduce a Bayesian semiparametric model for bivariate and multivariate survival data. The marginal densities are well-known nonparametric survival models and the joint density is constructed via a mixture. Our construction also defines a copula and the properties of this new copula are studied. We also consider the model in the presence of covariates and, in particular, we find a simple generalisation of the widely used frailty model, which is based on a new bivariate gamma distribution.     

Mixture and non-mixture cure fraction models based on the generalized modified Weibull distribution with an application to gastric cancer data

The cure fraction models are usually used to model lifetime time data with long-term survivors. In the present article, we introduce a Bayesian analysis of the four-parameter generalized modified Weibull (GMW) distribution in presence of cure fraction, censored data and covariates. In order to include the proportion of “cured” patients, mixture and non-mixture formulation models are considered. To demonstrate the ability of using this model in the analysis of real data, we consider an application to data from patients with gastric adenocarcinoma. Inferences are obtained by using MCMC (Markov Chain Monte Carlo) methods.     

On the independence between risk profiles in the compound collective risk actuarial model

This paper examines a compound collective risk model in which the primary distribution comprised the Poisson–Lindley distribution with a λ parameter, and where the secondary distribution is an exponential one with a θ parameter. We consider the case of dependence between risk profiles (i.e., the parameters λ and θ), where the dependence is modelled by a Farlie–Gumbel–Morgenstern family. We analyze the consequences of the dependence on the Bayes premium. We conclude that the consequences of the dependence on the Bayes premium may vary considerably.     

Marginal analysis of multivariate failure time data with a surviving fraction based on semiparametric transformation cure models

In biomedical, genetic and social studies, there may exist a fraction of individuals not experiencing the event of interest such that the survival curves eventually level off to nonzero proportions. These people are referred to as “cured” or “nonsusceptible” individuals. Models that have been developed to address this issue are known as cured models. The mixture model, which consists of a model for the binary cure status and a survival model for the event times of the noncured individuals, is one of the widely used cure models. In this paper, we propose a class of semiparametric transformation cure models for multivariate survival data with a surviving fraction by fitting a logistic regression model to the cure status and a semiparametric transformation model to the event time of the noncured individual. Both models allow incorporating covariates and do not require any assumption of the association structure. The statistical inference is based on the marginal approach by constructing a system of estimating equations. The asymptotic properties of the proposed estimators are proved, and the performance of the estimation is demonstrated via simulations. In addition, the approach is illustrated by analyzing the smoking cessation data.     

Mixture cure models for multivariate survival data

Mixture cure models (MCMs) have been widely used to analyze survival data with a cure fraction. The MCMs postulate that a fraction of the patients are cured from the disease and that the failure time for the uncured patients follows a proper survival distribution, referred to as latency distribution. The MCMs have been extended to bivariate survival data by modeling the marginal distributions. In this paper, the marginal MCM is extended to multivariate survival data. The new model is applicable to the survival data with varied cluster size and interval censoring. The proposed model allows covariates to be incorporated into both the cure fraction and the latency distribution for the uncured patients. The primary interest is to estimate the marginal parameters in the mean structure, where the correlation structure is treated as nuisance parameters. The marginal parameters are estimated consistently by treating the observations within the cluster as independent. The variances of the parameters are estimated by the one-step jackknife method. The proposed method does not depend on the specification of correlation structure. Simulation studies show that the new method works well when the marginal model is correct. The performance of the MCM is also examined when the clustered survival times share common random effect. The MCM is applied to the data from a smoking cessation study.     

A SAS macro for parametric and semiparametric mixture cure models

Cure models have been developed to analyze failure time data with a cured fraction. For such data, standard survival models are usually not appropriate because they do not account for the possibility of cure. Mixture cure models assume that the studied population is a mixture of susceptible individuals, who may experience the event of interest, and non-susceptible individuals that will never experience it. The aim of this paper is to propose a SAS macro to estimate parametric and semiparametric mixture cure models with covariates. The cure fraction can be modelled by various binary regression models. Parametric and semiparametric models can be used to model the survival of uncured individuals. The maximization of the likelihood function is performed using SAS PROC NLMIXED for parametric models and through an EM algorithm for the Cox's proportional hazards mixture cure model. Indications and limitations of the proposed macro are discussed and an example in the field of cancer clinical trials is shown.     

Semiparametric Bayesian approaches to joinpoint regression for population-based cancer survival data

According to the American Cancer Society report (1999), cancer surpasses heart disease as the leading cause of death in the United States of America (USA) for people of age less than 85. Thus, medical research in cancer is an important public health interest. Understanding how medical improvements are affecting cancer incidence, mortality and survival is critical for effective cancer control. In this paper, we study the cancer survival trend on the population level cancer data. In particular, we develop a parametric Bayesian joinpoint regression model based on a Poisson distribution for the relative survival. To avoid identifying the cause of death, we only conduct analysis based on the relative survival. The method is further extended to the semiparametric Bayesian joinpoint regression models wherein the parametric distributional assumptions of the joinpoint regression models are relaxed by modeling the distribution of regression slopes using Dirichlet process mixtures. We also consider the effect of adding covariates of interest in the joinpoint model. Three model selection criteria, namely, the conditional predictive ordinate (CPO), the expected predictive deviance (EPD), and the deviance information criteria (DIC), are used to select the number of joinpoints. We analyze the grouped survival data for distant testicular cancer from the Surveillance, Epidemiology, and End Results (SEER) Program using these Bayesian models.     

Opinion consensus of modified Hegselmann–Krause models

We consider the opinion consensus problem using a multi-agent setting based on the Hegselmann–Krause (H–K) Model. Firstly, we give a sufficient condition on the initial opinion distribution so that the system will converge to only one cluster. Then, modified models are proposed to guarantee convergence for more general initial conditions. The overall connectivity is maintained with these models, while the loss of certain edges can occur. Furthermore, a smooth control protocol is provided to avoid the difficulties that may arise due to the discontinuous right-hand side in the H–K model.     

The exponentiated exponential mixture and non-mixture cure rate model in the presence of covariates

This paper presents estimates for the parameters included in long-term mixture and non-mixture lifetime models, applied to analyze survival data when some individuals may never experience the event of interest. We consider the case where the lifetime data have a two-parameters exponentiated exponential distribution. The two-parameter exponentiated exponential or the generalized exponential distribution is a particular member of the exponentiated Weibull distribution introduced by [31]. Classical and Bayesian procedures are used to get point and confidence intervals of the unknown parameters. We consider a general survival model where the scale, shape and cured fraction parameters of the exponentiated exponential distribution depends on covariates.     

Random effects in promotion time cure rate models

In this paper, a survival model with long-term survivors and random effects, based on the promotion time cure rate model formulation for models with a surviving fraction is investigated. We present Bayesian and classical estimation approaches. The Bayesian approach is implemented using a Markov chain Monte Carlo (MCMC) based on the Metropolis-Hastings algorithms. For the second one, we use restricted maximum likelihood (REML) estimators. A simulation study is performed to evaluate the accuracy of the applied techniques for the estimates and their standard deviations. An example on an oropharynx cancer study is used to illustrate the model and the estimation approaches considered in the study.     

CANSURV: A Windows program for population-based cancer survival analysis

Patient survival is one of the most important measures of cancer patient care (the diagnosis and treatment of cancer). The optimal method for monitoring the progress of patient care across the full spectrum of provider settings is through the population-based study of cancer patient survival, which is only possible using data collected by population-based cancer registries. The probability of cure, “statistical cure”, is defined for a cohort of cancer patients as the percent of patients whose annual death rate equals the death rate of general cancer-free population. Mixture cure models have been widely used to model failure time data. The models provide simultaneous estimates of the proportion of the patients cured from cancer and the distribution of the failure times for the uncured patients (latency distribution). CANSURV (CAN-cer SURVival) is a Windows software fitting both the standard survival models and the cure models to population-based cancer survival data. CANSURV can analyze both cause-specific survival data and, especially, relative survival data, which is the standard measure of net survival in population-based cancer studies. It can also fit parametric (cure) survival models to the individual data. The program is available at http://srab.cancer.gov/cansurv. The colorectal cancer survival data from the Surveillance, Epidemiology and End Results (SEER) program [Surveillance, Epidemiology and End Results Program, The Portable Survival System/Mainframe Survival System, National Cancer Institute, Bethesda, 1999.] of the National Cancer Institute, NIH is used to demonstrate the use of CANSURV program.     

Parametric cure models of relative and cause-specific survival for grouped survival times

With parametric cure models, we can express survival parameters (e.g. cured fraction, location and scale parameters) as functions of covariates. These models can measure survival from a specific disease process, either by examining deaths due to the cause under study (cause-specific survival), or by comparing all deaths to those in a matched control population (relative survival). We present a binomial maximum likelihood algorithm to be used for actuarial data, where follow-up times are grouped into specific intervals. Our algorithm provides simultaneous maximum likelihood estimates for all the parameters of a cure model and can be used for cause-specific or relative survival analysis with a variety of survival distributions. Current software does not provide the flexibility of this unified approach.     

Modeling spot price dependence in Australian electricity markets with applications to risk management

We examine the dependence structure of electricity spot prices across regional markets in Australia. One of the major objectives in establishing a national electricity market was to provide a nationally integrated and efficient electricity market, limiting market power of generators in the separate regional markets. Our analysis is based on a GARCH approach to model the marginal price series in the considered regions in combination with copulae to capture the dependence structure between the marginals. We apply different copula models including Archimedean, elliptical and copula mixture models. We find a positive dependence structure between the prices for all considered markets, while the strongest dependence is exhibited between markets that are connected via interconnector transmission lines. Regarding the nature of dependence, the Student-t copula provides a good fit to the data, while the overall best results are obtained using copula mixture models due to their ability to also capture asymmetric dependence in the tails of the distribution. Interestingly, our results also suggest that for the four major markets, NSW, QLD, SA and VIC, the degree of dependence has decreased starting from the year 2008 towards the end of the sample period in 2010. Examining the Value-at-Risk of stylized portfolios constructed from electricity spot contracts in different markets, we find that the Student-t and mixture copula models outperform the Gaussian copula in a backtesting study. Our results are important for risk management and hedging decisions of market participants, in particular for those operating in several regional markets simultaneously.     

Log-Burr XII regression models with censored data

In survival analysis applications, the failure rate function may frequently present a unimodal shape. In such case, the log-normal or 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 Burr XII distribution is proposed for modeling data with a unimodal failure rate function as an alternative to the log-logistic regression model. Assuming censored data, we consider a classic analysis, a Bayesian analysis and a jackknife estimator for the parameters of the proposed model. For different parameter settings, sample sizes and censoring percentages, various simulation studies are performed and compared to the performance of the log-logistic and log-Burr XII regression models. Besides, we use sensitivity analysis to detect influential or outlying observations, and residual analysis is used to check the assumptions in the model. Finally, we analyze a real data set under log-Burr XII regression models.     

Semiparametric analysis of clustered interval-censored survival data with a cure fraction

A generalization of the semiparametric Cox’s proportional hazards model by means of a random effect or frailty approach to accommodate clustered survival data with a cure fraction is considered. The frailty serves as a quantification of the health condition of the subjects under study and may depend on some observed covariates like age. One single individual-specific frailty that acts on the hazard function is adopted to determine the cure status of an individual and the heterogeneity on the time to event if the individual is not cured. Under this formulation, an individual who has a high propensity to be cured would tend to have a longer time to event if he is not cured. Within a cluster, both the cure statuses and the times to event of the individuals would be correlated. In contrast to some models proposed in the literature, the model accommodates the correlations among the observations in a more natural way. A multiple imputation estimation method is proposed for both right-censored and interval-censored data. Simulation studies show that the performance of the proposed estimation method is highly satisfactory. The proposed model and method are applied to the National Aeronautics and Space Administration’s hypobaric decompression sickness data to investigate the factors associated with the occurrence and the time to onset of grade IV venous gas emboli under hypobaric environments.     

Lifetime analysis based on the generalized Birnbaum-Saunders distribution

In this paper, we consider a family of generalized Birnbaum-Saunders distributions and present a lifetime analysis based mainly on the hazard function of this model. In addition, we carry out maximum likelihood estimation by using an iterative algorithm, which produces robust estimates. Asymptotic inference is also presented. Next, the quality of the estimation method is examined by means of Monte Carlo simulations. We then provide a practical example to illustrate the obtained results. From this example and based on goodness-of-fit methods, we show that the GBS distribution results in a more appropriate model for modeling fatigue data than other models commonly used to model this type of data. Finally, we estimate the hazard function and the critical point of this function.     

Semiparametric bivariate Archimedean copulas

While parametric copulas often lack expressive capacity to capture the complex dependencies that are usually found in empirical data, non-parametric copulas can have poor generalization performance because of overfitting. A semiparametric copula method based on the family of bivariate Archimedean copulas is introduced as an intermediate approach that aims to provide both accurate and robust fits. The Archimedean copula is expressed in terms of a latent function that can be readily represented using a basis of natural cubic splines. The model parameters are determined by maximizing the sum of the log-likelihood and a term that penalizes non-smooth solutions. The performance of the semiparametric estimator is analyzed in experiments with simulated and real-world data, and compared to other methods for copula estimation: three parametric copula models, two semiparametric estimators of Archimedean copulas previously introduced in the literature, two flexible copula methods based on Gaussian kernels and mixtures of Gaussians and finally, standard parametric Archimedean copulas. The good overall performance of the proposed semiparametric Archimedean approach confirms the capacity of this method to capture complex dependencies in the data while avoiding overfitting.     

Prediction of survival probabilities with Bayesian Decision Trees

Practitioners use Trauma and Injury Severity Score (TRISS) models for predicting the survival probability of an injured patient. The accuracy of TRISS predictions is acceptable for patients with up to three typical injuries, but unacceptable for patients with a larger number of injuries or with atypical injuries. Based on a regression model, the TRISS methodology does not provide the predictive density required for accurate assessment of risk. Moreover, the regression model is difficult to interpret. We therefore consider Bayesian inference for estimating the predictive distribution of survival. The inference is based on decision tree models which recursively split data along explanatory variables, and so practitioners can understand these models. We propose the Bayesian method for estimating the predictive density and show that it outperforms the TRISS method in terms of both goodness-of-fit and classification accuracy. The developed method has been made available for evaluation purposes as a stand-alone application.     

Comparison of semiparametric maximum likelihood estimation and two-stage semiparametric estimation in copula models

We consider bivariate distributions that are specified in terms of a parametric copula function and nonparametric or semiparametric marginal distributions. The performance of two semiparametric estimation procedures based on censored data is discussed: maximum likelihood (ML) and two-stage pseudolikelihood (PML) estimation. The two-stage procedure involves less computation and it is of interest to see whether it is significantly less efficient than the full maximum likelihood approach. We also consider cases where the copula model is misspecified, in which case PML may be better. Extensive simulation studies demonstrate that in the absence of covariates, two-stage estimation is highly efficient and has significant robustness advantages for estimating marginal distributions. In some settings, involving covariates and a high degree of association between responses, ML is more efficient. For the estimation of association, PML does not offer an advantage.