Shared frailty models with baseline generalized Pareto distribution

Abstract

In this article, we have considered three different shared frailty models under the assumption of generalized Pareto Distribution as baseline distribution. Frailty models have been used in the survival analysis to account for the unobserved heterogeneity in an individual risks to disease and death. These three frailty models are with gamma frailty, inverse Gaussian frailty and positive stable frailty. Then we introduce the Bayesian estimation procedure using Markov chain Monte Carlo (MCMC) technique to estimate the parameters. We applied these three models to a kidney infection data and find the best fitted model for kidney infection data. We present a simulation study to compare true value of the parameters with the estimated values. Model comparison is made using Bayesian model selection criterion and a well-fitted model is suggested for the kidney infection data.     

Estimation in the Positive Stable Shared Frailty Cox Proportional Hazards Model

Shared frailty models are of interest when one has clustered survival data and when focus is on comparing the lifetimes within clusters and further on estimating the correlation between lifetimes from the same cluster. It is well known that the positive stable model should be preferred to the gamma model in situations where the correlated survival data show a decreasing association with time. In this paper, we devise a likelihood based estimation procedure for the positive stable shared frailty Cox model, which is expected to obtain high efficiency. The proposed estimator is provided with large sample properties and also a consistent estimator of standard errors is given. Simulation studies show that the estimation procedure is appropriate for practical use, and that it is much more efficient than a recently suggested procedure. The suggested methodology is applied to a dataset concerning time to blindness for patients with diabetic retinopathy.     

Gamma frailty models for bivariate survival data

In this paper, we introduce the shared gamma frailty models with two different baseline distributions namely, the generalized log-logistic and the generalized Weibull. We introduce the Bayesian estimation procedure to estimate the parameters involved in these models. We present a simulation study to compare the true values of the parameters with the estimated values. We apply these models to a real-life bivariate survival data set of McGilchrist and Aisbett related to the kidney infection data and a better model is suggested for the data.     

The Identifiability of Dependent Competing Risks Models Induced by Bivariate Frailty Models

In this paper, we propose to use a special class of bivariate frailty models to study dependent censored data. The proposed models are closely linked to Archimedean copula models. We give sufficient conditions for the identifiability of this type of competing risks models. The proposed conditions are derived based on a property shared by Archimedean copula models and satisfied by several well‐known bivariate frailty models. Compared with the models studied by Heckman and Honoré and Abbring and van den Berg, our models are more restrictive but can be identified with a discrete (even finite) covariate. Under our identifiability conditions, expectation–maximization (EM) algorithm provides us with consistent estimates of the unknown parameters. Simulation studies have shown that our estimation procedure works quite well. We fit a dependent censored leukaemia data set using the Clayton copula model and end our paper with some discussions. © 2014 Board of the Foundation of the Scandinavian Journal of Statistics     

Frailty Models for Arbitrarily Censored and Truncated Data

In this paper, we propose a frailty model for statistical inference in the case where we are faced with arbitrarily censored and truncated data. Our results extend those of Alioum and Commenges (1996), who developed a method of fitting a proportional hazards model to data of this kind. We discuss the identifiability of the regression coefficients involved in the model which are the parameters of interest, as well as the identifiability of the baseline cumulative hazard function of the model which plays the role of the infinite dimensional nuisance parameter. We illustrate our method with the use of simulated data as well as with a set of real data on transfusion-related AIDS.     

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.     

A Comparison of Frailty and Other Models for Bivariate Survival Data

Multivariate survival data arise when eachstudy subject may experience multiple events or when study subjectsare clustered into groups. Statistical analyses of such dataneed to account for the intra-cluster dependence through appropriatemodeling. Frailty models are the most popular for such failuretime data. However, there are other approaches which model thedependence structure directly. In this article, we compare thefrailty models for bivariate data with the models based on bivariateexponential and Weibull distributions. Bayesian methods providea convenient paradigm for comparing the two sets of models weconsider. Our techniques are illustrated using two examples.One simulated example demonstrates model choice methods developedin this paper and the other example, based on a practical dataset of onset of blindness among patients with diabetic Retinopathy,considers Bayesian inference using different models.     

On the preservation of copula structure under truncation

The author characterizes the copula associated with the bivariate survival model of Clayton (1978) as the only absolutely continuous copula that is preserved under bivariate truncation.     

Parametric Analysis for Matched Pair Survival Data

Hougaard's (1986) bivariate Weibull distribution with positive stable frailties is applied to matched pairs survival data when either or both components of the pair may be censored and covariate vectors may be of arbitrary fixed length. When there is no censoring, we quantify the corresponding gain in Fisher information over a fixed-effects analysis. With the appropriate parameterization, the results take a simple algebraic form. An alternative marginal (independence working model) approach to estimation is also considered. This method ignores the correlation between the two survival times in the derivation of the estimator, but provides a valid estimate of standard error. It is shown that when both the correlation between the two survival times is high, and the ratio of the within-pair variability to the between-pair variability of the covariates is high, the fixed-effects analysis captures most of the information about the regression coefficient but the independence working model does badly. When the correlation is low, and/or most of the variability of the covariates occurs between pairs, the reverse is true. The random effects model is applied to data on skin grafts, and on loss of visual acuity among diabetics. In conclusion some extensions of the methods are indicated and they are placed in a wider context of Generalized Estimating Equation methodology.     

Bivariate degradation analysis of products based on Wiener processes and copulas

Modern highly reliable products usually have complex structure and many functions. This means that they may have two or more performance characteristics. All the performance characteristics can reflect the product's performance degradation over time, and they may be independent or dependent. If the performance characteristics are independent, they can be modelled separately. But if they are not independent, it is very important to find the joint distribution function of the performance characteristics for estimating the reliability of the product as accurately as possible. Here, we suppose that a product has two performance characteristics and the degradation paths of these two performance characteristics can be governed by a Wiener process with a time-scale transformation, and that the dependency of the performance characteristics can be described by a copula function. The parameters of the two performance characteristics and the copula function can be estimated jointly. The model in such a situation is very complicated and analytically intractable and becomes cumbersome from a computational viewpoint. For this reason, the Bayesian Markov chain Monte Carlo method is developed for this problem that allows the maximum-likelihood estimates of the parameters to be determined in an efficient manner. For an illustration of the proposed model, a numerical example about fatigue cracks is presented.     

Ascertainment-Adjusted Maximum Likelihood Estimation for the Additive Genetic Gamma Frailty Model

The additive genetic gamma frailty model has been proposed for genetic linkage analysis for complex diseases to account for variable age of onset and possible covariates effects. To avoid ascertainment biases in parameter estimates, retrospective likelihood ratio tests are often used, which may result in loss of efficiency due to conditioning. This paper considers when the sibships are ascertained by having at least two affected sibs with the disease before a given age and provides two approaches for estimating the parameters in the additive gamma frailty model. One approach is based on the likelihood function conditioning on the ascertainment event, the other is based on maximizing a full ascertainment-adjusted likelihood. Explicit forms for these likelihood functions are derived. Simulation studies indicate that when the baseline hazard function can be correctly pre-specified, both approaches give accurate estimates of the model parameters. However, when the baseline hazard function has to be estimated simultaneously, only the ascertainment-adjusted likelihood method gives an unbiased estimate of the parameters. These results imply that the ascertainment-adjusted likelihood ratio test in the context of the additive genetic gamma frailty may be used for genetic linkage analysis.     

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.     

A Non-parametric Frailty Model for Temporally Clustered Multivariate Failure Times

Abstract A model is introduced here for multivariate failure time data arising from heterogenous populations. In particular, we consider a situation in which the failure times of individual subjects are often temporally clustered, so that many failures occur during a relatively short age interval. The clustering is modelled by assuming that the subjects can be divided into ‘internally homogenous’ latent classes, each such class being then described by a time‐dependent frailty profile function. As an example, we reanalysed the dental caries data presented earlier in Härkänen et al. [Scand. J. Statist. 27 (2000) 577], as it turned out that our earlier model could not adequately describe the observed clustering.     

Confidence intervals for the difference between two median survival times for clustered survival data

Clustered survival data arise often in clinical trial design, where the correlated subunits from the same cluster are randomized to different treatment groups. Under such design, we consider the problem of constructing confidence interval for the difference of two median survival time given the covariates. We use Cox gamma frailty model to account for the within-cluster correlation. Based on the conditional confidence intervals, we can identify the possible range of covariates over which the two groups would provide different median survival times. The associated coverage probability and the expected length of the proposed interval are investigated via a simulation study. The implementation of the confidence intervals is illustrated using a real data set.     

Frailty regression models in mixture distributions

We propose frailty regression models in mixture distributions and assume the distribution of frailty as gamma or positive stable or power variance function distribution. We consider Weibull mixture as an example. There are some interesting situations like survival times in genetic epidemiology, dental implants of patients and twin births (both monozygotic and dizygotic) where genetic behavior (which is unknown and random) of patients follows a known frailty distribution. These are the situations which motivate to study this particular model.     

Non-mixture cure correlated frailty models in Bayesian approach

In this article, we develop a Bayesian approach for the estimation of two cure correlated frailty models that have been extended to the cure frailty models introduced by Yin [34]. We used the two different type of frailty with bivariate log-normal distribution instead of gamma distribution. A likelihood function was constructed based on a piecewise exponential distribution function. The model parameters were estimated by the Markov chain Monte Carlo method. The comparison of models is based on the Cox correlated frailty model with log-normal distribution. A real data set of bilateral corneal graft rejection was used to compare these models. The results of this data, based on deviance information criteria, showed the advantage of the proposed models.     

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.     

Multi-state Models: A Review

Multi-state models are models for a process, for example describing a life history of an individual, which at any time occupies one of a few possible states. This can describe several possible events for a single individual, or the dependence between several individuals. The events are the transitions between the states. This class of models allows for an extremely flexible approach that can model almost any kind of longitudinal failure time data. This is particularly relevant for modeling different events, which have an event-related dependence, like occurrence of disease changing the risk of death. It can also model paired data. It is useful for recurrent events, but has limitations. The Markov models stand out as much simpler than other models from a probability point of view, and this simplifies the likelihood evaluation. However, in many cases, the Markov models do not fit satisfactorily, and happily, it is reasonably simple to study non-Markov models, in particular the Markov extension models. This also makes it possible to consider, whether the dependence is of short-term or long-term nature. Applications include the effect of heart transplantation on the mortality and the mortality among Danish twins.     

Sample size determination for clustered right-censored data using copula models

Determination of an adequate sample size is critical to the design of research ventures. For clustered right-censored data, Manatunga and Chen [Sample size estimation for survival outcomes in cluster-randomized studies with small cluster sizes. Biometrics. 2000;56(2):616–621] proposed a sample size calculation based on considering the bivariate marginal distribution as Clayton copula model. In addition to the Clayton copula, other important family of copulas, such as Gumbel and Frank copulas are also well established in multivariate survival analysis. However, sample size calculation based on these assumptions has not been fully investigated yet. To broaden the scope of Manatunga and Chen [Sample size estimation for survival outcomes in cluster-randomized studies with small cluster sizes. Biometrics. 2000;56(2):616–621]'s research and achieve a more flexible sample size calculation for clustered right-censored data, we extended the work by assuming the marginal distribution as bivariate Gumbel and Frank copulas. We evaluate the performance of the proposed method and investigate the impacts of the accrual times, follow-up times and the within-clustered correlation effect of the study. The proposed method is applied to two real-world studies, and the R code is made available to users.