Accelerated hazards mixture cure model

We propose a new cure model for survival data with a surviving or cure fraction. The new model is a mixture cure model where the covariate effects on the proportion of cure and the distribution of the failure time of uncured patients are separately modeled. Unlike the existing mixture cure models, the new model allows covariate effects on the failure time distribution of uncured patients to be negligible at time zero and to increase as time goes by. Such a model is particularly useful in some cancer treatments when the treat effect increases gradually from zero, and the existing models usually cannot handle this situation properly. We develop a rank based semiparametric estimation method to obtain the maximum likelihood estimates of the parameters in the model. We compare it with existing models and methods via a simulation study, and apply the model to a breast cancer data set. The numerical studies show that the new model provides a useful addition to the cure model literature.     

Mixture cure rate models with accelerated failures and nonparametric form of covariate effects

Two-component mixture cure rate model is popular in cure rate data analysis with the proportional hazards and accelerated failure time (AFT) models being the major competitors for modelling the latency component. [Wang, L., Du, P., and Liang, H. (2012), ‘Two-Component Mixture Cure Rate Model with Spline Estimated Nonparametric Components’, Biometrics, 68, 726–735] first proposed a nonparametric mixture cure rate model where the latency component assumes proportional hazards with nonparametric covariate effects in the relative risk. Here we consider a mixture cure rate model where the latency component assumes AFTs with nonparametric covariate effects in the acceleration factor. Besides the more direct physical interpretation than the proportional hazards, our model has an additional scalar parameter which adds more complication to the computational algorithm as well as the asymptotic theory. We develop a penalised EM algorithm for estimation together with confidence intervals derived from the Louis formula. Asymptotic convergence rates of the parameter estimates are established. Simulations and the application to a melanoma study shows the advantages of our new method.     

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.     

Bayesian methods for generalized linear models with covariates missing at random

The authors propose methods for Bayesian inference for generalized linear models with missing covariate data. They specify a parametric distribution for the covariates that is written as a sequence of one‐dimensional conditional distributions. They propose an informative class of joint prior distributions for the regression coefficients and the parameters arising from the covariate distributions. They examine the properties of the proposed prior and resulting posterior distributions. They also present a Bayesian criterion for comparing various models, and a calibration is derived for it. A detailed simulation is conducted and two real data sets are examined to demonstrate the methodology.     

Bayesian Transformation Models for Multivariate Survival Data

In this paper, we propose a general class of Gamma frailty transformation models for multivariate survival data. The transformation class includes the commonly used proportional hazards and proportional odds models. The proposed class also includes a family of cure rate models. Under an improper prior for the parameters, we establish propriety of the posterior distribution. A novel Gibbs sampling algorithm is developed for sampling from the observed data posterior distribution. A simulation study is conducted to examine the properties of the proposed methodology. An application to a data set from a cord blood transplantation study is also reported.     

Marginalized transition random effect models for multivariate longitudinal binary data

Generalized linear models with random effects and/or serial dependence are commonly used to analyze longitudinal data. However, the computation and interpretation of marginal covariate effects can be difficult. This led Heagerty (1999, 2002) to propose models for longitudinal binary data in which a logistic regression is first used to explain the average marginal response. The model is then completed by introducing a conditional regression that allows for the longitudinal, within‐subject, dependence, either via random effects or regressing on previous responses. In this paper, the authors extend the work of Heagerty to handle multivariate longitudinal binary response data using a triple of regression models that directly model the marginal mean response while taking into account dependence across time and across responses. Markov Chain Monte Carlo methods are used for inference. Data from the Iowa Youth and Families Project are used to illustrate the methods.     

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.     

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.     

Modeling categorical covariates for lifetime data in the presence of cure fraction by Bayesian partition structures

In this paper, we propose a Bayesian partition modeling for lifetime data in the presence of a cure fraction by considering a local structure generated by a tessellation which depends on covariates. In this modeling we include information of nominal qualitative variables with more than two categories or ordinal qualitative variables. The proposed modeling is based on a promotion time cure model structure but assuming that the number of competing causes follows a geometric distribution. It is an alternative modeling strategy to the conventional survival regression modeling generally used for modeling lifetime data in the presence of a cure fraction, which models the cure fraction through a (generalized) linear model of the covariates. An advantage of our approach is its ability to capture the effects of covariates in a local structure. The flexibility of having a local structure is crucial to capture local effects and features of the data. The modeling is illustrated on two real melanoma data sets.     

On bootstrap testing inference in cure rate models

Survival models deal with the time until the occurrence of an event of interest. However, in some situations the event may not occur in part of the studied population. The fraction of the population that will never experience the event of interest is generally called cure rate. Models that consider this fact (cure rate models) have been extensively studied in the literature. Hypothesis testing on the parameters of these models can be performed based on likelihood ratio, gradient, score or Wald statistics. Critical values of these tests are obtained through approximations that are valid in large samples and may result in size distortion in small or moderate sample sizes. In this sense, this paper proposes bootstrap corrections to the four mentioned tests and bootstrap Bartlett correction for the likelihood ratio statistic in the Weibull promotion time model. Besides, we present an algorithm for bootstrap resampling when the data presents cure fraction and right censoring time (random and non-informative). Simulation studies are conducted to compare the finite sample performances of the corrected tests. The numerical evidence favours the corrected tests we propose. We also present an application in an actual data set.     

Smooth Semi‐nonparametric Analysis for Mixture Cure Models and Its Application to Breast Cancer

Mixture cure models are widely used when a proportion of patients are cured. The proportional hazards mixture cure model and the accelerated failure time mixture cure model are the most popular models in practice. Usually the expectation–maximisation (EM) algorithm is applied to both models for parameter estimation. Bootstrap methods are used for variance estimation. In this paper we propose a smooth semi‐nonparametric (SNP) approach in which maximum likelihood is applied directly to mixture cure models for parameter estimation. The variance can be estimated by the inverse of the second derivative of the SNP likelihood. A comprehensive simulation study indicates good performance of the proposed method. We investigate stage effects in breast cancer by applying the proposed method to breast cancer data from the South Carolina Cancer Registry.     

Boosting flexible functional regression models with a high number of functional historical effects

We propose a general framework for regression models with functional response containing a potentially large number of flexible effects of functional and scalar covariates. Special emphasis is put on historical functional effects, where functional response and functional covariate are observed over the same interval and the response is only influenced by covariate values up to the current grid point. Historical functional effects are mostly used when functional response and covariate are observed on a common time interval, as they account for chronology. Our formulation allows for flexible integration limits including, e.g., lead or lag times. The functional responses can be observed on irregular curve-specific grids. Additionally, we introduce different parameterizations for historical effects and discuss identifiability issues.The models are estimated by a component-wise gradient boosting algorithm which is suitable for models with a potentially high number of covariate effects, even more than observations, and inherently does model selection. By minimizing corresponding loss functions, different features of the conditional response distribution can be modeled, including generalized and quantile regression models as special cases. The methods are implemented in the open-source R package FDboost. The methodological developments are motivated by biotechnological data on Escherichia coli fermentations, but cover a much broader model class.     

Likelihood methods for missing covariate data in highly stratified studies

Summary. The paper considers canonical link generalized linear models with stratum-specific nuisance intercepts and missing covariate data. This family includes the conditional logistic regression model. Existing methods for this problem, each of which uses a conditioning argu- ment to eliminate the nuisance intercept, model either the missing covariate data or the missingness process. The paper compares these methods under a common likelihood framework. The semiparametric efficient estimator is identified, and a new estimator, which reduces dependence on the model for the missing covariate, is proposed. A simulation study compares the methods with respect to efficiency and robustness to model misspecification.     

Survival analysis with long‐term survivors and partially observed covariates

The authors describe a method for fitting failure time mixture models that postulate the existence of both susceptibles and long‐term survivors when covariate data are only partially observed. Their method is based on a joint model that combines a Weibull regression model for the susceptibles, a logistic regression model for the probability of being a susceptible, and a general location model for the distribution of the covariates. A Bayesian approach is taken, and Gibbs sampling is used to fit the model to the incomplete data. An application to clinical data on tonsil cancer and a small Monte Carlo study indicate potential large gains in efficiency over standard complete‐case analysis as well as reasonable performance in a variety of situations.     

Global prior distributions for the analysis of discrete graphical models

Summary We propose a new class of prior distributions for the analysis of discrete graphical models. Such a class, obtained following a conditional approach, generalizes the hyper Dirichlet distributions of Dawid and Lauritzen (1993), since it can be extended to non decomposable graphical models. The two classes are compared in terms of model selection, with an application to a medical data-set illustrating the performance of the two resulting procedures. The proposed class turns out to select simpler, more par-simonious structures.     

A general hazard model for lifetime data in the presence of cure rate

Historically, the cure rate model has been used for modeling time-to-event data within which a significant proportion of patients are assumed to be cured of illnesses, including breast cancer, non-Hodgkin lymphoma, leukemia, prostate cancer, melanoma, and head and neck cancer. Perhaps the most popular type of cure rate model is the mixture model introduced by Berkson and Gage [1]. In this model, it is assumed that a certain proportion of the patients are cured, in the sense that they do not present the event of interest during a long period of time and can found to be immune to the cause of failure under study. In this paper, we propose a general hazard model which accommodates comprehensive families of cure rate models as particular cases, including the model proposed by Berkson and Gage. The maximum-likelihood-estimation procedure is discussed. A simulation study analyzes the coverage probabilities of the asymptotic confidence intervals for the parameters. A real data set on children exposed to HIV by vertical transmission illustrates the methodology.     

Modeling survival in childhood cancer studies using two-stage non-mixture cure models

Non-mixture cure models (NMCMs) are derived from a simplified representation of the biological process that takes place after treatment for cancer. These models are intended to represent the time from the end of treatment to the time of first recurrence of cancer in studies when a proportion of those treated are completely cured. However, for many studies overall survival is also of interest. A two-stage NMCM that estimates the overall survival from a combination of two cure models, one from end of treatment to first recurrence and one from first recurrence to death, is proposed. The model is applied to two studies of Ewing's tumor in young patients. Caution needs to be exercised when extrapolating from cure models fitted to short follow-up times, but these data and associated simulations show how, when follow-up is limited, a two-stage model can give more stable estimates of the cure fraction than a one-stage model applied directly to overall survival.     

Analysis of ordinal outcomes with longitudinal covariates subject to missingness

We propose a mixture model for data with an ordinal outcome and a longitudinal covariate that is subject to missingness. Data from a tailored telephone delivered, smoking cessation intervention for construction laborers are used to illustrate the method, which considers as an outcome a categorical measure of smoking cessation, and evaluates the effectiveness of the motivational telephone interviews on this outcome. We propose two model structures for the longitudinal covariate, for the case when the missing data are missing at random, and when the missing data mechanism is non-ignorable. A generalized EM algorithm is used to obtain maximum likelihood estimates.     

Influence diagnostics for the Weibull-Negative-Binomial regression model with cure rate under latent failure causes

In this paper, we propose a flexible cure rate survival model by assuming that the number of competing causes of the event of interest follows the Negative Binomial distribution and the time to event follows a Weibull distribution. Indeed, we introduce the Weibull-Negative-Binomial (WNB) distribution, which can be used in order to model survival data when the hazard rate function is increasing, decreasing and some non-monotonous shaped. Another advantage of the proposed model is that it has some distributions commonly used in lifetime analysis as particular cases. Moreover, the proposed model includes as special cases some of the well-know cure rate models discussed in the literature. We consider a frequentist analysis for parameter estimation of a WNB model with cure rate. Then, we derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to perform global influence analysis. Finally, the methodology is illustrated on a medical data.     

Stochastic EM algorithm for generalized exponential cure rate model and an empirical study

In this paper, we consider two well-known parametric long-term survival models, namely, the Bernoulli cure rate model and the promotion time (or Poisson) cure rate model. Assuming the long-term survival probability to depend on a set of risk factors, the main contribution is in the development of the stochastic expectation maximization (SEM) algorithm to determine the maximum likelihood estimates of the model parameters. We carry out a detailed simulation study to demonstrate the performance of the proposed SEM algorithm. For this purpose, we assume the lifetimes due to each competing cause to follow a two-parameter generalized exponential distribution. We also compare the results obtained from the SEM algorithm with those obtained from the well-known expectation maximization (EM) algorithm. Furthermore, we investigate a simplified estimation procedure for both SEM and EM algorithms that allow the objective function to be maximized to split into simpler functions with lower dimensions with respect to model parameters. Moreover, we present examples where the EM algorithm fails to converge but the SEM algorithm still works. For illustrative purposes, we analyze a breast cancer survival data. Finally, we use a graphical method to assess the goodness-of-fit of the model with generalized exponential lifetimes.