Discrete and continuous bivariate lifetime models in presence of cure rate: a comparative study under Bayesian approach

The modeling and analysis of lifetime data in which the main endpoints are the times when an event of interest occurs is of great interest in medical studies. In these studies, it is common that two or more lifetimes associated with the same unit such as the times to deterioration levels or the times to reaction to a treatment in pairs of organs like lungs, kidneys, eyes or ears. In medical applications, it is also possible that a cure rate is present and needed to be modeled with lifetime data with long-term survivors. This paper presented a comparative study under a Bayesian approach among some existing continuous and discrete bivariate distributions such as the bivariate exponential distributions and the bivariate geometric distributions in presence of cure rate, censored data and covariates. In presence of lifetimes related to cured patients, it is assumed standard mixture cure rate models in the data analysis. The posterior summaries of interest are obtained using Markov Chain Monte Carlo methods. To illustrate the proposed methodology two real medical data sets are considered.     

The Poisson Generalized Linear Failure Rate Model

We formulate a new cure rate survival model by assuming that the number of competing causes of the event of interest has the Poisson distribution, and the time to this event has the generalized linear failure rate distribution. A new distribution to analyze lifetime data is defined from the proposed cure rate model, and its quantile function as well as a general expansion for the moments is derived. We estimate the parameters of the model with cure rate in the presence of covariates for censored observations using maximum likelihood and derive the observed information matrix. We obtain the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to perform global influence analysis. The usefulness of the proposed cure rate survival model is illustrated in an application to real data.     

Bayesian Semiparametric Cure Rate Model with an Unknown Threshold

Abstract.  We propose a Bayesian semiparametric model for survival data with a cure fraction. We explicitly consider a finite cure time in the model, which allows us to separate the cured and the uncured populations. We take a mixture prior of a Markov gamma process and a point mass at zero to model the baseline hazard rate function of the entire population. We focus on estimating the cure threshold after which subjects are considered cured. We can incorporate covariates through a structure similar to the proportional hazards model and allow the cure threshold also to depend on the covariates. For illustration, we undertake simulation studies and a full Bayesian analysis of a bone marrow transplant data set.     

Block and Basu bivariate lifetime distribution in the presence of cure fraction

This paper presents estimates for the parameters included in the Block and Basu bivariate lifetime distributions in the presence of covariates and cure fraction, applied to analyze survival data when some individuals may never experience the event of interest and two lifetimes are associated with each unit. A Bayesian procedure is used to get point and confidence intervals for the unknown parameters. Posterior summaries of interest are obtained using standard Markov Chain Monte Carlo methods in rjags package for R software. An illustration of the proposed methodology is given for a Diabetic Retinopathy Study data set.     

Generalized log-gamma regression models with cure fraction

In this paper, the generalized log-gamma regression model is modified to allow the possibility that long-term survivors may be present in the data. This modification leads to a generalized log-gamma regression model with a cure rate, encompassing, as special cases, the log-exponential, log-Weibull and log-normal regression models with a cure rate typically used to model such data. The models attempt to simultaneously estimate the effects of explanatory variables on the timing acceleration/deceleration of a given event and the surviving fraction, that is, the proportion of the population for which the event never occurs. The normal curvatures of local influence are derived under some usual perturbation schemes and two martingale-type residuals are proposed to assess departures from the generalized log-gamma error assumption as well as to detect outlying observations. Finally, a data set from the medical area is analyzed.     

Cure rate survival models with missing covariates: a simulation study

In this paper we study the cure rate survival model involving a competitive risk structure with missing categorical covariates. A parametric distribution that can be written as a sequence of one-dimensional conditional distributions is specified for the missing covariates. We consider the missing data at random situation so that the missing covariates may depend only on the observed ones. Parameter estimates are obtained by using the EM algorithm via the method of weights. Extensive simulation studies are conducted and reported to compare estimates efficiency with and without missing data. As expected, the estimation approach taking into consideration the missing covariates presents much better efficiency in terms of mean square errors than the complete case situation. Effects of increasing cured fraction and censored observations are also reported. We demonstrate the proposed methodology with two real data sets. One involved the length of time to obtain a BS degree in Statistics, and another about the time to breast cancer recurrence.     

Bayesian Methods for Missing Covariates in Cure Rate Models

We propose methods for Bayesian inference for missing covariate data with a novel class of semi-parametric survival models with a cure fraction. We allow the missing covariates to be either categorical or continuous and specify a parametric distribution for the covariates that is written as a sequence of one dimensional conditional distributions. We assume that the missing covariates are missing at random (MAR) throughout. We propose an informative class of joint prior distributions for the regression coefficients and the parameters arising from the covariate distributions. The proposed class of priors are shown to be useful in recovering information on the missing covariates especially in situations where the missing data fraction is large. Properties of the proposed prior and resulting posterior distributions are examined. Also, model checking techniques are proposed for sensitivity analyses and for checking the goodness of fit of a particular model. Specifically, we extend the Conditional Predictive Ordinate (CPO) statistic to assess goodness of fit in the presence of missing covariate data. Computational techniques using the Gibbs sampler are implemented. A real data set involving a melanoma cancer clinical trial is examined to demonstrate the methodology.     

Sample size calculation for a proportional hazards mixture cure model with nonbinary covariates

Sample size calculation is a critical issue in clinical trials because a small sample size leads to a biased inference and a large sample size increases the cost. With the development of advanced medical technology, some patients can be cured of certain chronic diseases, and the proportional hazards mixture cure model has been developed to handle survival data with potential cure information. Given the needs of survival trials with potential cure proportions, a corresponding sample size formula based on the log-rank test statistic for binary covariates has been proposed by Wang et al. [25]. However, a sample size formula based on continuous variables has not been developed. Herein, we presented sample size and power calculations for the mixture cure model with continuous variables based on the log-rank method and further modified it by Ewell's method. The proposed approaches were evaluated using simulation studies for synthetic data from exponential and Weibull distributions. A program for calculating necessary sample size for continuous covariates in a mixture cure model was implemented in R.     

A proportional-hazards model for survival analysis and long-term survivors modeling: application to amyotrophic lateral sclerosis data

The majority of survival data are affected by explanatory variables. We develop a new regression model for survival data analysis. As an alternative to standard mixture models, another model is proposed to describe the eventual presence of a surviving fraction. The proposed models are based on the Marshall–Olkin extended generalized Gompertz distribution. A maximum-likelihood inference is presented in the presence of covariates and a censorship phenomenon. Explanatory variables are incorporated into the model through proportional-hazards to evaluate the effect of risk factors on overall survival under different assumptions. Parametric, semi-parametric, and non-parametric methods are applied to survival analysis of patients treated for amyotrophic lateral sclerosis. Interesting results about riluzole use and other treatment effects on patients'' survival have been obtained.     

Semiparametric Estimation in Transformation Models with Cure Fraction

Semiparametric transformation model has been extensively investigated in the literature. The model, however, has little dealt with survival data with cure fraction. In this article, we consider a class of semi-parametric transformation models, where an unknown transformation of the survival times with cure fraction is assumed to be linearly related to the covariates and the error distributions are parametrically specified by an extreme value distribution with unknown parameters. Estimators for the coefficients of covariates are obtained from pseudo Z-estimator procedures allowing censored observations. We show that the estimators are consistent and asymptotically normal. The bootstrap estimation of the variances of the estimators is also investigated.     

Bayesian variable selection for multioutcome models through shared shrinkage

Variable selection over a potentially large set of covariates in a linear model is quite popular. In the Bayesian context, common prior choices can lead to a posterior expectation of the regression coefficients that is a sparse (or nearly sparse) vector with a few nonzero components, those covariates that are most important. This article extends the “global‐local” shrinkage idea to a scenario where one wishes to model multiple response variables simultaneously. Here, we have developed a variable selection method for a K‐outcome model (multivariate regression) that identifies the most important covariates across all outcomes. The prior for all regression coefficients is a mean zero normal with coefficient‐specific variance term that consists of a predictor‐specific factor (shared local shrinkage parameter) and a model‐specific factor (global shrinkage term) that differs in each model. The performance of our modeling approach is evaluated through simulation studies and a data example.     

Estimating the causal effects in randomized trials for survival data with a cure fraction and non compliance

We consider causal inference in randomized studies for survival data with a cure fraction and all-or-none treatment non compliance. To describe the causal effects, we consider the complier average causal effect (CACE) and the complier effect on survival probability beyond time t (CESP), where CACE and CESP are defined as the difference of cure rate and non cured subjects’ survival probability between treatment and control groups within the complier class. These estimands depend on the distributions of survival times in treatment and control groups. Given covariates and latent compliance type, we model these distributions with transformation promotion time cure model whose parameters are estimated by maximum likelihood. Both the infinite dimensional parameter in the model and the mixture structure of the problem create some computational difficulties which are overcome by an expectation-maximization (EM) algorithm. We show the estimators are consistent and asymptotically normal. Some simulation studies are conducted to assess the finite-sample performance of the proposed approach. We also illustrate our method by analyzing a real data from the Healthy Insurance Plan of Greater New York.     

Defective models induced by gamma frailty term for survival data with cured fraction

In this paper, we propose a defective model induced by a frailty term for modeling the proportion of cured. Unlike most of the cure rate models, defective models have advantage of modeling the cure rate without adding any extra parameter in model. The introduction of an unobserved heterogeneity among individuals has bring advantages for the estimated model. The influence of unobserved covariates is incorporated using a proportional hazard model. The frailty term assumed to follow a gamma distribution is introduced on the hazard rate to control the unobservable heterogeneity of the patients. We assume that the baseline distribution follows a Gompertz and inverse Gaussian defective distributions. Thus we propose and discuss two defective distributions: the defective gamma-Gompertz and gamma-inverse Gaussian regression models. Simulation studies are performed to verify the asymptotic properties of the maximum likelihood estimator. Lastly, in order to illustrate the proposed model, we present three applications in real data sets, in which one of them we are using for the first time, related to a study about breast cancer in the A.C.Camargo Cancer Center, São Paulo, Brazil.     

A simplified estimation procedure based on the EM algorithm for the power series cure rate model

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.     

Joint analysis of longitudinal measurements and survival times with a cure fraction based on partly linear mixed and semiparametric cure models

In a joint analysis of longitudinal quality of life (QoL) scores and relapse-free survival (RFS) times from a clinical trial on early breast cancer conducted by the Canadian Cancer Trials Group, we observed a complicated trajectory of QoL scores and existence of long-term survivors. Motivated by this observation, we proposed in this paper a flexible joint model for the longitudinal measurements and survival times. A partly linear mixed effect model is used to capture the complicated but smooth trajectory of longitudinal measurements and approximated by B-splines and a semiparametric mixture cure model with the B-spline baseline hazard to model survival times with a cure fraction. These two models are linked by shared random effects to explore the dependence between longitudinal measurements and survival times. A semiparametric inference procedure with an EM algorithm is proposed to estimate the parameters in the joint model. The performance of proposed procedures are evaluated by simulation studies and through the application to the analysis of data from the clinical trial which motivated this research.     

Bivariate Birnbaum-Saunders accelerated lifetime model: estimation and diagnostic analysis

In this paper, we discuss the bivariate Birnbaum-Saunders accelerated lifetime model, in which we have modeled the dependence structure of bivariate survival data through the use of frailty models. Specifically, we propose the bivariate model Birnbaum-Saunders with the following frailty distributions: gamma, positive stable and logarithmic series. We present a study of inference and diagnostic analysis for the proposed model, more concisely, are proposed a diagnostic analysis based in local influence and residual analysis to assess the fit model, as well as, to detect influential observations. In this regard, we derived the normal curvatures of local influence under different perturbation schemes and we performed some simulation studies for assessing the potential of residuals to detect misspecification in the systematic component, the presence in the stochastic component of the model and to detect outliers. Finally, we apply the methodology studied to real data set from recurrence in times of infections of 38 kidney patients using a portable dialysis machine, we analyzed these data considering independence within the pairs and using the bivariate Birnbaum-Saunders accelerated lifetime model, so that we could make a comparison and verify the importance of modeling dependence within the times of infection associated with the same patient.     

Graphical Techniques for Selecting Explanatory Variables for Time Series Data

Bayesian model building techniques are developed for data with a strong time series structure and possibly exogenous explanatory variables that have strong explanatory and predictive power. The emphasis is on finding whether there are any explanatory variables that might be used for modelling if the data have a strong time series structure that should also be included. We use a time series model that is linear in past observations and that can capture both stochastic and deterministic trend, seasonality and serial correlation. We propose the plotting of absolute predictive error against predictive standard deviation. A series of such plots is utilized to determine which of several nested and non-nested models is optimal in terms of minimizing the dispersion of the predictive distribution and restricting predictive outliers. We apply the techniques to modelling monthly counts of fatal road crashes in Australia where economic, consumption and weather variables are available and we find that three such variables should be included in addition to the time series filter. The approach leads to graphical techniques to determine strengths of relationships between the dependent variable and covariates and to detect model inadequacy as well as determining useful numerical summaries.     

Semi-parametric survival analysis via Dirichlet process mixtures of the First Hitting Time model

Time-to-event data often violate the proportional hazards assumption inherent in the popular Cox regression model. Such violations are especially common in the sphere of biological and medical data where latent heterogeneity due to unmeasured covariates or time varying effects are common. A variety of parametric survival models have been proposed in the literature which make more appropriate assumptions on the hazard function, at least for certain applications. One such model is derived from the First Hitting Time (FHT) paradigm which assumes that a subject’s event time is determined by a latent stochastic process reaching a threshold value. Several random effects specifications of the FHT model have also been proposed which allow for better modeling of data with unmeasured covariates. While often appropriate, these methods often display limited flexibility due to their inability to model a wide range of heterogeneities. To address this issue, we propose a Bayesian model which loosens assumptions on the mixing distribution inherent in the random effects FHT models currently in use. We demonstrate via simulation study that the proposed model greatly improves both survival and parameter estimation in the presence of latent heterogeneity. We also apply the proposed methodology to data from a toxicology/carcinogenicity study which exhibits nonproportional hazards and contrast the results with both the Cox model and two popular FHT models.

     

The generalized log-gamma mixture model with covariates: local influence and residual analysis

In a sample of censored survival times, the presence of an immune proportion of individuals who are not subject to death, failure or relapse, may be indicated by a relatively high number of individuals with large censored survival times. In this paper the generalized log-gamma model is modified for the possibility that long-term survivors may be present in the data. The model attempts to separately estimate the effects of covariates on the surviving fraction, that is, the proportion of the population for which the event never occurs. The logistic function is used for the regression model of the surviving fraction. Inference for the model parameters is considered via maximum likelihood. Some influence methods, such as the local influence and total local influence of an individual are derived, analyzed and discussed. Finally, a data set from the medical area is analyzed under the log-gamma generalized mixture model. A residual analysis is performed in order to select an appropriate model. The authors would like to thank the editor and referees for their helpful comments. This work was supported by CNPq, Brazil.     

Structural identification and variable selection in high-dimensional varying-coefficient models

Varying-coefficient models have been widely used to investigate the possible time-dependent effects of covariates when the response variable comes from normal distribution. Much progress has been made for inference and variable selection in the framework of such models. However, the identification of model structure, that is how to identify which covariates have time-varying effects and which have fixed effects, remains a challenging and unsolved problem especially when the dimension of covariates is much larger than the sample size. In this article, we consider the structural identification and variable selection problems in varying-coefficient models for high-dimensional data. Using a modified basis expansion approach and group variable selection methods, we propose a unified procedure to simultaneously identify the model structure, select important variables and estimate the coefficient curves. The unique feature of the proposed approach is that we do not have to specify the model structure in advance, therefore, it is more realistic and appropriate for real data analysis. Asymptotic properties of the proposed estimators have been derived under regular conditions. Furthermore, we evaluate the finite sample performance of the proposed methods with Monte Carlo simulation studies and a real data analysis.