Frailty models and copulas: similarities and differences

Copulas and frailty models are important tools to model bivariate survival data. Equivalence between Archimedean copula models and shared frailty models, e.g. between the Clayton-Oakes copula model and the shared gamma frailty model, has often been claimed in the literature. In this note we show that, in both the models, there is indeed a well-known equivalence between the copula functions; the modeling of the marginal survival functions, however, is quite different. The latter fact leads to different joint survival functions.     

Fisher Information for Two Gamma Frailty Bivariate Weibull Models

The asymptotic properties of frailty models for multivariate survival data are not well understood. To study this aspect, the Fisher information is derived in the standard bivariate gamma frailty model, where the survival distribution is of Weibull form conditional on the frailty. For comparison, the Fisher information is also derived in the bivariate gamma frailty model, where the marginal distribution is of Weibull form.     

A Composite Likelihood Approach to Multivariate Survival Data

This paper is about the statistical analysis of multivariate survival data. We discuss the additive and multiplicative frailty models which have been the most popular models for multivariate survival data. As an alternative to the additive and multiplicative frailty models, we propose basing inference on a composite likelihood function that only requires modelling of the marginal distribution of pairs of failure times. Each marginal distribution of a pair of failure times is here assumed to follow a shared frailty model. The method is illustrated with a real-life example.     

Estimation in Semiparametric Marginal Shared Gamma Frailty Models

The semiparametric marginal shared frailty models in survival analysis have the non–parametric hazard functions multiplied by a random frailty in each cluster, and the survival times conditional on frailties are assumed to be independent. In addition, the marginal hazard functions have the same form as in the usual Cox proportional hazard models. In this paper, an approach based on maximum likelihood and expectation–maximization is applied to semiparametric marginal shared gamma frailty models, where the frailties are assumed to be gamma distributed with mean 1 and variance θ. The estimates of the fixed–effect parameters and their standard errors obtained using this approach are compared in terms of both bias and efficiency with those obtained using the extended marginal approach. Similarly, the standard errors of our frailty variance estimates are found to compare favourably with those obtained using other methods. The asymptotic distribution of the frailty variance estimates is shown to be a 50–50 mixture of a point mass at zero and a truncated normal random variable on the positive axis for θ₀ = 0. Simulations demonstrate that, for θ₀ < 0, it is approximately an x −(100 − x )%, 0 ≤ x ≤ 50, mixture between a point mass at zero and a truncated normal random variable on the positive axis for small samples and small values of θ₀; otherwise, it is approximately normal.     

Applying a marginalized frailty model to competing risks

The marginalized frailty model is often used for the analysis of correlated times in survival data. When only two correlated times are analyzed, this model is often referred to as the Clayton–Oakes model [7,22]. With time-to-event data, there may exist multiple end points (competing risks) suggesting that an analysis focusing on all available outcomes is of interest. The purpose of this work is to extend the single risk marginalized frailty model to the multiple risk setting via cause-specific hazards (CSH). The methods herein make use of the marginalized frailty model described by Pipper and Martinussen [24]. As such, this work uses the martingale theory to develop a likelihood based on estimating equations and observed histories. The proposed multivariate CSH model yields marginal regression parameter estimates while accommodating the clustering of outcomes. The multivariate CSH model can be fitted using a data augmentation algorithm described by Lunn and McNeil [21] or by fitting a series of single risk models for each of the competing risks. An example of the application of the multivariate CSH model is provided through the analysis of a family-based follow-up study of breast cancer with death in absence of breast cancer as a competing risk.     

Exploring the Clustering Effect of the Frailty Survival Model by Means of the Brier Score

In this article, the Brier score is used to investigate the importance of clustering for the frailty survival model. For this purpose, two versions of the Brier score are constructed, i.e., a “conditional Brier score” and a “marginal Brier score.” Both versions of the Brier score show how the clustering effects and the covariate effects affect the predictive ability of the frailty model separately. Using a Bayesian and a likelihood approach, point estimates and 95% credible/confidence intervals are computed. The estimation properties of both procedures are evaluated in an extensive simulation study for both versions of the Brier score. Further, a validation strategy is developed to calculate an internally validated point estimate and credible/confidence interval. The ensemble of the developments is applied to a dental dataset.     

Modelling Converging Hazards in Survival Analysis

The Cox proportional hazards model has become the standard model for survival analysis. It is often seen as the null model in that "... explicit excuses are now needed to use different models" (Keiding, Proceedings of the XIXth International Biometric Conference, Cape Town, 1998). However, converging hazards also occur frequently in survival analysis. The Burr model, which may be derived as the marginal from a gamma frailty model, is one commonly used tool to model converging hazards. We outline this approach and introduce a mixed model which extends the Burr model and allows for both proportional and converging hazards. Although a semi-parametric model in its own right, we demonstrate how the mixed model can be derived via a gamma frailty interpretation, suggesting an E-M fitting procedure. We illustrate the modelling techniques using data on survival of hospice patients.     

A Weibull Regression Model with Gamma Frailties for Multivariate Survival Data

Frequently in the analysis of survival data, survival times within the same group are correlated due to unobserved co-variates. One way these co-variates can be included in the model is as frailties. These frailty random block effects generate dependency between the survival times of the individuals which are conditionally independent given the frailty. Using a conditional proportional hazards model, in conjunction with the frailty, a whole new family of models is introduced. By considering a gamma frailty model, often the issue is to find an appropriate model for the baseline hazard function. In this paper a flexible baseline hazard model based on a correlated prior process is proposed and is compared with a standard Weibull model. Several model diagnostics methods are developed and model comparison is made using recently developed Bayesian model selection criteria. The above methodologies are applied to the McGilchrist and Aisbett (1991) kidney infection data and the analysis is performed using Markov Chain Monte Carlo methods. This revised version was published online in July 2006 with corrections to the Cover Date.     

Multilevel Mixed Linear Models for Survival Data

For the analysis of correlated survival data mixed linear models are useful alternatives to frailty models. By their use the survival times can be directly modelled, so that the interpretation of the fixed and random effects is straightforward. However, because of intractable integration involved with the use of marginal likelihood the class of models in use has been severely restricted. Such a difficulty can be avoided by using hierarchical-likelihood, which provides a statistically efficient and fast fitting algorithm for multilevel models. The proposed method is illustrated using the chronic granulomatous disease data. A simulation study is carried out to evaluate the performance.     

On the Weibull-Gamma frailty model, its infinite moments, and its connection to generalized log-logistic, logistic, Cauchy, and extreme-value distributions

It is shown that the commonly used Weibull-Gamma frailty model has a finite number of finite moments only and that its marginal distribution generalizes the log-logistic distribution. In some cases there is not even a finite variance, and there are cases without a single finite moment. Upon transformation to the entire real line, generalized logistic and generalized Cauchy distributions are introduced and their connection with the previous ones established, as well as with the extreme-value distribution. Apart from intrinsic and classroom value, the family can be of use when formulating non-informative priors in Bayesian data analysis. Also, gauging the amount of finite moments is important when checking regularity conditions in the Weibull-Gamma model. Our findings are illustrated using data from survival in cancer patients.     

Conditional and marginal estimates in case-control family data – extensions and sensitivity analyses

This work considers two specific estimation techniques for the family-specific proportional hazards model and for the population-averaged proportional hazards model. So far, these two estimation procedures were presented and studied under the gamma frailty distribution mainly because of its simple interpretation and mathematical tractability. Modifications of both procedures for other frailty distributions, such as the inverse Gaussian, positive stable and a specific case of discrete distribution, are presented. By extensive simulations, it is shown that under the family-specific proportional hazards model, the gamma frailty model appears to be robust to frailty distribution mis-specification in both bias and efficiency loss in the marginal parameters. The population-averaged proportional hazards model, is found to be robust under the gamma frailty model mis-specification only under moderate or weak dependency within cluster members.     

A Nonparametric Frailty Model for Clustered Survival Data

Clayton-type counting process formulations for survival data and parametric gamma models for cluster-specific frailty quantities are now routinely applied in analyses of clustered survival data. On the other hand, although nonparametric frailty models have been studied, they are not used much in practice. In this article, the distribution of the frailty terms is assumed to be an unknown random variable. The unknown frailty distribution is then modelled completely with a Dirichlet process prior. This prior assigns cluster units into sub-classes whose members have the same random frailty effect. The Gibbs sampler algorithm is used for computing posterior parameter estimates of the fixed effect hazards regression and the frailty distribution. The methodology is used to analyze community-clustered child survival in sub-Saharan Africa. The results show that the communities could be separated into fewer distinct classes of risk of childhood mortality; the fewer classes could be studied easily in order to provide useful guidance on the more effective use of resources for child health intervention programmes.     

Weighted estimation for multivariate shared frailty models for complex surveys

Multivariate frailty models have been used for clustered survival data to characterize the relationship between the hazard of correlated failures/events and exposure variables and covariates. However, these models can introduce serious biases of the estimation for failures from complex surveys that may depend on the sampling design (informative or noninformative). In order to consistently estimate parameters, this paper considers weighting the multivariate frailty model by the inverse of the probability of selection at each stage of sampling. This follows the principle of the pseudolikelihood approach. The estimation is carried out by maximizing the penalized partial and marginal pseudolikelihood functions. The performance of the proposed estimator is assessed through a Monte Carlo simulation study and the 4 waves of data from the 1998–1999 Early Childhood Longitudinal Study. Results show that the weighted estimator is consistent and approximately unbiased.

     

Small Sample Bias in the Gamma Frailty Model for Univariate Survival

The gamma frailty model is a natural extension of the Cox proportional hazards model in survival analysis. Because the frailties are unobserved, an E-M approach is often used for estimation. Such an approach is shown to lead to finite sample underestimation of the frailty variance, with the corresponding regression parameters also being underestimated as a result. For the univariate case, we investigate the source of the bias with simulation studies and a complete enumeration. The rank-based E-M approach, we note, only identifies frailty through the order in which failures occur; additional frailty which is evident in the survival times is ignored, and as a result the frailty variance is underestimated. An adaption of the standard E-M approach is suggested, whereby the non-parametric Breslow estimate is replaced by a local likelihood formulation for the baseline hazard which allows the survival times themselves to enter the model. Simulations demonstrate that this approach substantially reduces the bias, even at small sample sizes. The method developed is applied to survival data from the North West Regional Leukaemia Register.     

Impact of misspecified residual correlation structure on the parameter estimates in a shared spatial frailty model

In practice, survival data are often collected over geographical regions. Shared spatial frailty models have been used to model spatial variation in survival times, which are often implemented using the Bayesian Markov chain Monte Carlo method. However, this method comes at the price of slow mixing rates and heavy computational cost, which may render it impractical for data-intensive application. Alternatively, a frailty model assuming an independent and identically distributed (iid) random effect can be easily and efficiently implemented. Therefore, we used simulations to assess the bias and efficiency loss in the estimated parameters, if residual spatial correlation is present but using an iid random effect. Our simulations indicate that a shared frailty model with an iid random effect can estimate the regression coefficients reasonably well, even with residual spatial correlation present, when the percentage of censoring is not too high and the number of clusters and cluster size are not too low. Therefore, if the primary goal is to assess the covariate effects, one may choose the frailty model with an iid random effect; whereas if the goal is to predict the hazard, additional care needs to be given due to the efficiency loss in the parameter(s) for the baseline hazard.     

Consistency of the NPML Estimator in the Right-Censored Transformation Model

Abstract.  This paper studies the representation and large-sample consistency for non-parametric maximum likelihood estimators (NPMLEs) of an unknown baseline continuous cumulative-hazard-type function and parameter of group survival difference, based on right-censored two-sample survival data with marginal survival function assumed to follow a transformation model, a slight generalization of the class of frailty survival regression models. The paper's main theoretical results are existence and unique a.s. limit, characterized variationally, for large data samples of the NPMLE of baseline nuisance function in an appropriately defined neighbourhood of the true function when the group difference parameter is fixed, leading to consistency of the NPMLE when the difference parameter is fixed at a consistent estimator of its true value. The joint NPMLE is also shown to be consistent. An algorithm for computing it numerically, based directly on likelihood equations in place of the expectation-maximization (EM) algorithm, is illustrated with real data.     

Posterior consistency in frailty models and simulation studies to test the presence of random effects

The estimation of random effects in frailty models is an important problem in survival analysis. Testing for the presence of random effects can be essential to improving model efficiency. Posterior consistency in dispersion parameters and coefficients of the frailty model was demonstrated in theory and simulations using the posterior induced by Cox’s partial likelihood and simple priors. We also conducted simulation studies to test for the presence of random effects; the proposed method performed well in several simulations. Data analysis was also conducted. The proposed method is easily tractable and can be used to develop various methods for Bayesian inference in frailty models.     

A class of semiparametric transormation frailty models with application to estimating treatment effects and heterogenity in aids clinical traials

Muitivariate failure time data are common in medical research; com¬monly used statistical models for such correlated failure-time data include frailty and marginal models. Both types of models most often assume pro¬portional hazards (Cox, 1972); but the Cox model may not fit the data well This article presents a class of linear transformation frailty models that in¬cludes, as a special case, the proportional hazards model with frailty. We then propose approximate procedures to derive the best linear unbiased es¬timates and predictors of the regression parameters and frailties. We apply the proposed methods to analyze results of a clinical trial of different dose levels of didansine (ddl) among HIV-infected patients who were intolerant of zidovudine (ZDV). These methods yield estimates of treatment effects and of frailties corresponding to patient groups defined by clinical history prior to entry into the trial.     

A Multivariate Model for Repeated Failure Time Measurements

A parametric multivariate failure time distribution is derived from a frailty-type model with a particular frailty distribution. It covers as special cases certain distributions which have been used for multivariate survival data in recent years. Some properties of the distribution are derived: its marginal and conditional distributions lie within the parametric family, and association between the component variates can be positive or, to a limited extent, negative. The simple closed form of the survivor function is useful for right-censored data, as occur commonly in survival analysis, and for calculating uniform residuals. Also featured is the distribution of ratios of paired failure times. The model is applied to data from the literature     

A comparative study of tests for paired lifetime data

In this paper, we investigate different procedures for testing the equality of two mean survival times in paired lifetime studies. We consider Owen’s M-test and Q-test, a likelihood ratio test, the paired t-test, the Wilcoxon signed rank test and a permutation test based on log-transformed survival times in the comparative study. We also consider the paired t-test, the Wilcoxon signed rank test and a permutation test based on original survival times for the sake of comparison. The size and power characteristics of these tests are studied by means of Monte Carlo simulations under a frailty Weibull model. For less skewed marginal distributions, the Wilcoxon signed rank test based on original survival times is found to be desirable. Otherwise, the M-test and the likelihood ratio test are the best choices in terms of power. In general, one can choose a test procedure based on information about the correlation between the two survival times and the skewness of the marginal survival distributions.