A class of Bayesian shared gamma frailty models with multivariate failure time data

For multivariate failure time data, we propose a new class of shared gamma frailty models by imposing the Box-Cox transformation on the hazard function, and the product of the baseline hazard and the frailty. This novel class of models allows for a very broad range of shapes and relationships between the hazard and baseline hazard functions. It includes the well-known Cox gamma frailty model and a new additive gamma frailty model as two special cases. Due to the nonnegative hazard constraint, this shared gamma frailty model is computationally challenging in the Bayesian paradigm. The joint priors are constructed through a conditional-marginal specification, in which the conditional distribution is univariate, and it absorbs the nonlinear parameter constraints. The marginal part of the prior specification is free of constraints. The prior distributions allow us to easily compute the full conditionals needed for Gibbs sampling, while incorporating the constraints. This class of shared gamma frailty models is illustrated with a real dataset.     

Maximum likelihood methods for cure rate models with missing covariates

We propose maximum likelihood methods for parameter estimation for a novel class of semiparametric survival models with a cure fraction, in which the covariates are allowed to be missing. We allow the 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 propose a novel EM algorithm for maximum likelihood estimation and derive standard errors by using Louis's formula (Louis, 1982, Journal of the Royal Statistical Society, Series B 44, 226-233). Computational techniques using the Monte Carlo EM algorithm are discussed and implemented. A real data set involving a melanoma cancer clinical trial is examined in detail to demonstrate the methodology.     

A general class of Bayesian survival models with zero and nonzero cure fractions

We propose a new class of survival models which naturally links a family of proper and improper population survival functions. The models resulting in improper survival functions are often referred to as cure rate models. This class of regression models is formulated through the Box-Cox transformation on the population hazard function and a proper density function. By adding an extra transformation parameter into the cure rate model, we are able to generate models with a zero cure rate, thus leading to a proper population survival function. A graphical illustration of the behavior and the influence of the transformation parameter on the regression model is provided. We consider a Bayesian approach which is motivated by the complexity of the model. Prior specification needs to accommodate parameter constraints due to the non-negativity of the survival function. Moreover, the likelihood function involves a complicated integral on the survival function, which may not have an analytical closed form, and thus makes the implementation of Gibbs sampling more difficult. We propose an efficient Markov chain Monte Carlo computational scheme based on Gaussian quadrature. The proposed method is illustrated with an example involving a melanoma clinical trial.     

Estimation of multivariate frailty models using penalized partial likelihood

There exists a growing literature on the estimation of gamma distributed multiplicative shared frailty models. There is, however, often a need to model more complicated frailty structures, but attempts to extend gamma frailties run into complications. Motivated by hip replacement data with a more complicated dependence structure, we propose a model based on multiplicative frailties with a multivariate log-normal joint distribution. We give a justification and an estimation procedure for this generally structured frailty model, which is a generalization of the one presented by McGilchrist (1993, Biometrics 49, 221-225). The estimation is based on Laplace approximation of the likelihood function. This leads to estimating equations based on a penalized fixed effects partial likelihood, where the marginal distribution of the frailty terms determines the penalty term. The tuning parameters of the penalty function, i.e., the frailty variances, are estimated by maximizing an approximate profile likelihood. The performance of the approximation is evaluated by simulation, and the frailty model is fitted to the hip replacement data.     

Estimating regression parameters and degree of dependence for multivariate failure time data

Multivariate failure time data are frequently encountered in longitudinal studies when subjects may experience several events or when there is a grouping of individuals into a cluster. To take into account the dependence of the failure times within the unit (the individual or the cluster) as well as censoring, two multivariate generalizations of the Cox proportional hazards model are commonly used. The marginal hazard model is used when the purpose is to estimate mean regression parameters, while the frailty model is retained when the purpose is to assess the degree of dependence within the unit. We propose a new approach based on the combination of the two aforementioned models to estimate both these quantities. This two-step estimation procedure is quicker and more simple to implement than the EM algorithm used in frailty models estimation. Simulation results are provided to illustrate robustness, consistency, and large-sample properties of estimators. Finally, this method is exemplified on a diabetic retinopathy study in order to assess the effect of photocoagulation in delaying the onset of blindness as well as the dependence between the two eyes blindness times of a patient.     

Prediction of individual long-term outcomes in smoking cessation trials using frailty models

In smoking cessation clinical trials, subjects commonly receive treatment and report daily cigarette consumption over a period of several weeks. Although the outcome at the end of this period is an important indicator of treatment success, substantial uncertainty remains on how an individual's smoking behavior will evolve over time. Therefore it is of interest to predict long-term smoking cessation success based on short-term clinical observations. We develop a Bayesian method for prediction, based on a cure-mixture frailty model we proposed earlier, that describes the process of transition between abstinence and smoking. Specifically we propose a two-stage prediction algorithm that first uses importance sampling to generate subject-specific frailties from their posterior distributions conditional on the observed data, then samples predicted future smoking behavior trajectories from the estimated model parameters and sampled frailties. We apply the method to data from two randomized smoking cessation trials comparing bupropion to placebo. Comparisons of actual smoking status at one year with predictions from our model and from a variety of empirical methods suggest that our method gives excellent predictions.     

Maximum likelihood estimation in random effects cure rate models with nonignorable missing covariates

We introduce a method of parameter estimation for a random effects cure rate model. We also propose a methodology that allows us to account for nonignorable missing covariates in this class of models. The proposed method corrects for possible bias introduced by complete case analysis when missing data are not missing completely at random and is motivated by data from a pair of melanoma studies conducted by the Eastern Cooperative Oncology Group in which clustering by cohort or time of study entry was suspected. In addition, these models allow estimation of cure rates, which is desirable when we do not wish to assume that all subjects remain at risk of death or relapse from disease after sufficient follow-up. We develop an EM algorithm for the model and provide an efficient Gibbs sampling scheme for carrying out the E-step of the algorithm.     

A proportional hazards cure model for the analysis of time to event with frequently unidentifiable causes

We propose a semiparametric method for the analysis of masked-cause failure data that are also subject to a cure. We present estimators for the failure time distribution, the cure rate, and the covariate effect on each of these, assuming a proportional hazards cure model for the time to event of interest and we use the expectation-maximization algorithm to conduct the likelihood maximization. The method is applied to data from a breast cancer clinical trial.     

Analysis of clustered recurrent event data with application to hospitalization rates among renal failure patients

End-stage renal disease (commonly referred to as renal failure) is of increasing concern in the United States and many countries worldwide. Incidence rates have increased, while the supply of donor organs has not kept pace with the demand. Although renal transplantation has generally been shown to be superior to dialysis with respect to mortality, very little research has been directed towards comparing transplant and wait-list patients with respect to morbidity. Using national data from the Scientific Registry of Transplant Recipients, we compare transplant and wait-list hospitalization rates. Hospitalizations are subject to two levels of dependence. In addition to the dependence among within-patient events, patients are also clustered by listing center. We propose two marginal methods to analyze such clustered recurrent event data; the first model postulates a common baseline event rate, while the second features cluster-specific baseline rates. Our results indicate that kidney transplantation offers a significant decrease in hospitalization, but that the effect is negated by a waiting time (until transplant) of more than 2 years. Moreover, graft failure (GF) results in a significant increase in the hospitalization rate which is greatest in the first month post-GF, but remains significantly elevated up to 4 years later. We also compare results from the proposed models to those based on a frailty model, with the various methods compared and contrasted.     

A frailty mixture cure model with application to hospital admission data [corrected

Mixture cure models have been utilized to analyze survival data with possible cure. This paper considers the inclusion of frailty into the mixture cure model to model recurrent event data with a cure fraction. An attractive feature of the proposed model is the allowance for heterogeneity in risk among those individuals experiencing the event of interest in addition to the incorporation of a cured component. Maximum likelihood estimates can be obtained using the Expectation Maximization algorithm and standard errors are calculated from the Bootstrap method. The model is applied to hospital readmission data among colorectal cancer patients.     

Two‐stage estimation for multivariate recurrent event data with a dependent terminal event

Recurrent event data arise in longitudinal follow‐up studies, where each subject may experience the same type of events repeatedly. The work in this article is motivated by the data from a study of repeated peritonitis for patients on peritoneal dialysis. Due to the aspects of medicine and cost, the peritonitis cases were classified into two types: Gram‐positive and non‐Gram‐positive peritonitis. Further, since the death and hemodialysis therapy preclude the occurrence of recurrent events, we face multivariate recurrent event data with a dependent terminal event. We propose a flexible marginal model, which has three characteristics: first, we assume marginal proportional hazard and proportional rates models for terminal event time and recurrent event processes, respectively; second, the inter‐recurrences dependence and the correlation between the multivariate recurrent event processes and terminal event time are modeled through three multiplicative frailties corresponding to the specified marginal models; third, the rate model with frailties for recurrent events is specified only on the time before the terminal event. We propose a two‐stage estimation procedure for estimating unknown parameters. We also establish the consistency of the two‐stage estimator. Simulation studies show that the proposed approach is appropriate for practical use. The methodology is applied to the peritonitis cohort data that motivated this study.     

Promotion time models with time-changing exposure and heterogeneity: application to infectious diseases

Promotion time models have been recently adapted to the context of infectious diseases to take into account discrete and multiple exposures. However, Poisson distribution of the number of pathogens transmitted at each exposure was a very strong assumption and did not allow for inter-individual heterogeneity. Bernoulli, the negative binomial, and the compound Poisson distributions were proposed as alternatives to Poisson distribution for the promotion time model with time-changing exposure. All were derived within the frailty model framework. All these distributions have a point mass at zero to take into account non-infected people. Bernoulli distribution, the two-component cure rate model, was extended to multiple exposures. Contrary to the negative binomial and the compound Poisson distributions, Bernoulli distribution did not enable to connect the number of pathogens transmitted to the delay between transmission and infection detection. Moreover, the two former distributions enable to account for inter-individual heterogeneity. The delay to surgical site infection was an example of single exposure. The probability of infection was very low; thus, estimation of the effect of selected risk factors on that probability obtained with Bernoulli and Poisson distributions were very close. The delay to nosocomial urinary tract infection was a multiple exposure example. The probabilities of pathogen transmission during catheter placement and catheter presence were estimated. Inter-individual heterogeneity was very high, and the fit was better with the compound Poisson and the negative binomial distributions. The proposed models proved to be also mechanistic. The negative binomial and the compound Poisson distributions were useful alternatives to account for inter-individual heterogeneity.     

Bayesian Semiparametric Models for Survival Data with a Cure Fraction

We propose methods for Bayesian inference for a new class of semiparametric survival models with a cure fraction. Specifically, we propose a semiparametric cure rate model with a smoothing parameter that controls the degree of parametricity in the right tail of the survival distribution. We show that such a parameter is crucial for these kinds of models and can have an impact on the posterior estimates. Several novel properties of the proposed model are derived. In addition, we propose a class of improper noninformative priors based on this model and examine the properties of the implied posterior. Also, a class of informative priors based on historical data is proposed and its theoretical properties are investigated. A case study involving a melanoma clinical trial is discussed in detail to demonstrate the proposed methodology.     

Analysis of failure time data with multilevel clustering, with application to the child vitamin a intervention trial in Nepal

We consider the problem of estimating covariate effects in the marginal Cox proportional hazard model and multilevel associations for child mortality data collected from a vitamin A supplementation trial in Nepal, where the data are clustered within households and villages. For this purpose, a class of multivariate survival models that can be represented by a functional of marginal survival functions and accounts for hierarchical structure of clustering is exploited. Based on this class of models, an estimation strategy involving a within-cluster resampling procedure is proposed, and a model assessment approach is presented. The asymptotic theory for the proposed estimators and lack-of-fit test is established. The simulation study shows that the estimates are approximately unbiased, and the proposed test statistic is conservative under extremely heavy censoring but approaches the size otherwise. The analysis of the Nepal study data shows that the association of mortality is much greater within households than within villages.     

Robust Linear Mixed Models with Normal/Independent Distributions and Bayesian MCMC Implementation

Linear mixed effects models have been widely used in analysis of data where responses are clustered around some random effects, so it is not reasonable to assume independence between observations in the same cluster. In most biological applications, it is assumed that the distributions of the random effects and of the residuals are Gaussian. This makes inferences vulnerable to the presence of outliers. Here, linear mixed effects models with normal/independent residual distributions for robust inferences are described. Specific distributions examined include univariate and multivariate versions of the Student‐ t, the slash and the contaminated normal. A Bayesian framework is adopted and Markov chain Monte Carlo is used to carry out the posterior analysis. The procedures are illustrated using birth weight data on rats in a toxicological experiment. Results from the Gaussian and robust models are contrasted, and it is shown how the implementation can be used for outlier detection. The thick‐tailed distributions provide an appealing robust alternative to the Gaussian process in linear mixed models, and they are easily implemented using data augmentation and MCMC techniques.     

A version of the EM algorithm for proportional hazard model with random effects

Proportional hazard models with multivariate random effects (frailties) acting multiplicatively on the baseline hazard have recently become a topic of an intensive research. One of the main practical problems related to the models is the estimation of parameters. To this aim, several approaches based on the EM algorithm have been proposed. The major difference between these approaches is the method of the computation of conditional expectations required at the E-step. In this paper an alternative implementation of the EM algorithm is proposed, in which the expected values are computed with the use of the Laplace approximation. The method is computationally less demanding than the approaches developed previously. Its performance is assessed based on a simulation study and compared to a non-EM based estimation approach proposed by Ripatti and Palmgren (2000).