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The family of power series cure rate models provides a flexible modeling framework for survival data of populations with a cure fraction. In this work, we present a simplified estimation procedure for the maximum likelihood (ML) approach. ML estimates are obtained via the expectation-maximization (EM) algorithm where the expectation step involves computation of the expected number of concurrent causes for each individual. It has the big advantage that the maximization step can be decomposed into separate maximizations of two lower-dimensional functions of the regression and survival distribution parameters, respectively. Two simulation studies are performed: the first to investigate the accuracy of the estimation procedure for different numbers of covariates and the second to compare our proposal with the direct maximization of the observed log-likelihood function. Finally, we illustrate the technique for parameter estimation on a dataset of survival times for patients with malignant melanoma.

The analysis of recurrent event data is of particular importance in medical statistics where patients suffering from chronic diseases often present with multiple recurring relapses or cancer patients experience several tumor recurrences. Whereas individual subjects can be assumed to be independent, the times between events of one subject are neither independent nor identically distributed. Apart from the marginal approach by Wei et al. (1989), the shared frailty model, see for example Duchateau and Janssen (2008), has been used extensively to analyze recurrent event data, where the correlation between sequential times is implicitly taken into account via a random effect. Oakes (1989) and Romeo et al. (2006) showed and exemplified the equivalence of frailty models for bivariate survival data to Archimedean copulas. Despite the fact that copula-based models have been used to model parallel survival data, their application to recurrent failure time data has only recently been suggested by Lawless and Yilmaz (2011) for the bivariate case. Here, we extend this to more than two recurrent events and model the joint distribution of recurrent events explicitly using parametric copulas within a Bayesian framework. This framework allows for parametric as well as a nonparametric modeling of the marginal baseline hazards and models the influence of covariates on the marginals via a proportional hazards assumption. Furthermore, the parameters of the copula may also depend on the covariates. We illustrate the flexibility of this approach using data from an asthma prevention trial in young children.

This work presents a Bayesian hierarchical model with the dual objective to analyze stratified survival data and to automatically classify each stratum into a finite number of groups. This is achieved by specifying parametric as well as piecewise stratum-specific baseline hazards and a finite mixture distribution for the stratum-specific shape parameters. A proportional hazards or accelerated failure time regression component allows to identify the influence of covariates on the survival distribution. We illustrate the model using a dataset of Atlantic salmon, stratified by families, that have been challenged with infectious pancreatic necrosis virus (IPNV). The main objectives are to model the survival time in terms of certain covariates as well as to classify the salmon families into either an IPNV susceptible or resistant group with the ultimate goal of improving resistance to IPNV through a selective breeding program. We compare the fit of different models that include stratum-specific baselines and covariate effects. The classifications show a certain degree of robustness with respect to model choice.

This work presents a Bayesian approach for estimating the limiting availability of an one-unit repairable system when the data are subject to right censorship. It is assumed that the failure and repair times of the components are independent exponential random variables. A conjugate Bayesian analysis is performed considering an informative and non informative prior distributions. Simulations are presented to study the performance of the Bayesian solutions. Some observations are made in relation to the maximum likelihood method. The Weibull case is also discussed.

This work is concerned with asymptotic properties of the bivariate survival function estimator using the functional relationship between marginal survival functions and a class of copulas for the dependence structure. Specifically, we study consistency and weak convergence of the bivariate survival function estimator obtained considering a two-step procedure of estimation. The obtained results are found from a key decomposition of the bivariate survival function in quantities that can be studied separately. In particular, we use relating results to almost sure and weak convergence of estimators, almost sure convergence of uniformly equicontinuous functions, and the delta method for functionals.

Copula models have become increasingly popular for modelling the dependence structure in multivariate survival data. The two-parameter Archimedean family of Power Variance Function (PVF) copulas includes the Clayton, Positive Stable (Gumbel) and Inverse Gaussian copulas as special or limiting cases, thus offers a unified approach to fitting these important copulas. Two-stage frequentist procedures for estimating the marginal distributions and the PVF copula have been suggested by Andersen (Lifetime Data Anal 11:333–350, 2005), Massonnet et al. (J Stat Plann Inference 139(11):3865–3877, 2009) and Prenen et al. (J R Stat Soc Ser B 79(2):483–505, 2017) which first estimate the marginal distributions and conditional on these in a second step to estimate the PVF copula parameters. Here we explore an one-stage Bayesian approach that simultaneously estimates the marginal and the PVF copula parameters. For the marginal distributions, we consider both parametric as well as semiparametric models. We propose a new method to simulate uniform pairs with PVF dependence structure based on conditional sampling for copulas and on numerical approximation to solve a target equation. In a simulation study, small sample properties of the Bayesian estimators are explored. We illustrate the usefulness of the methodology using data on times to appendectomy for adult twins in the Australian NH&MRC Twin registry. Parameters of the marginal distributions and the PVF copula are simultaneously estimated in a parametric as well as a semiparametric approach where the marginal distributions are modelled using Weibull and piecewise exponential distributions, respectively.

Since many authors have emphasized the need of asymmetric link functions to fit binary regression models, we propose in this work two new skew-probit link functions for the binary response variables. These link functions will be named power probit and reciprocal power probit due to the relation between them including the probit link as a special case. Also, the probit regressions are special cases of the models considered in this work. A Bayesian inference approach using MCMC is developed for real data suggesting that the link functions proposed here are more appropriate than other link functions used in the literature. In addition, simulation study show that the use of probit model will lead to biased estimate of the regression coefficient.

A nonparametric Bayesian approach is applied to estimate a survival curve by means of a functional of the subsurvival functions associated with censored and non-censored events. In order to actually compute the Bayesian estimator, a numerical algorithm based on the Runge-Kutta fourth-order method is introduced. It provides good accuracy and is simple to program. Using a simulated data set, the performance of the Bayesian estimator is compared to the Product-Limit. A descriptive analysis of the results from the simulations is presented. The conclusions are given in terms of the proportion of the censored data and sample size. Also, the predictive value of the estimations is investigated using a cross-validation measure, namely the mean square predictive value. The numerical methodology is illustrated considering the original Kaplan-Meier data. Finally, the Bayesian analysis is applied to a real case of cervix uterine cancer, where the elicitation of the prior distribution considers the high proportion of censoring in the sample.

Copula models have become increasingly popular for modeling multivariate survival data. In this paper we review some of the recent work that has been appeared for copula model for bivariate survival data and propose a Bayesian modeling. Our approach is very flexible with respect to the choice of marginal distributions and, depending on the copula model employed, it is possible to have a class of variation for the dependence parameter. We compare some of the copula models using a descriptive diagnostic method and three popular Bayesian model selection criteria. Our methodology is illustrated with the Diabetic Retinopathy Study (1976).