A general frailty model to accommodate individual heterogeneity in the acquisition of multiple infections: An application to bivariate current status data

The analysis of multivariate time-to-event (TTE) data can become complicated due to the presence of clustering, leading to dependence between multiple event times. For a long time, (conditional) frailty models and (marginal) copula models have been used to analyze clustered TTE data. In this article, we propose a general frailty model employing a copula function between the frailty terms to construct flexible (bivariate) frailty distributions with the application to current status data. The model has the advantage to impose a less restrictive correlation structure among latent frailty variables as compared to traditional frailty models. Specifically, our model uses a copula function to join the marginal distributions of the frailty vector. In this article, we considered different copula functions, and we relied on marginal gamma distributions due to their mathematical convenience. Based on a simulation study, our novel model outperformed the commonly used additive correlated gamma frailty model, especially in the case of a negative association between the frailties. At the end of the article, the new methodology is illustrated on real-life data applications entailing bivariate serological survey data.     

A new parametric model for survival data with long-term survivors

We develop a new parametric model using the three-parameter Burr XII distribution for the analysis of survival data with long-term survivors, which includes the previous Weibull mixture model as a special case. The new model is applied to the analysis of a set of leukaemia data for which previous attempts in the literature using traditional parametric models were unsatisfactory due to lack of fit. It is shown that the new model improves the fit to the leukaemia data significantly and is thus capable of providing more credible answers to a variety of statistical inference problems that are of interest to medical researchers and practitioners.     

Estimation method of the semiparametric mixture cure gamma frailty model

Mixture cure frailty model has been proposed to analyze censored survival data with a cured fraction and unobservable information among the uncured patients. Different from a usual mixture cure model, the frailty model is employed to model the latency component in the mixture cure frailty model. In this paper, we extend the mixture cure frailty model by incorporating covariates into both the cure rate and the latency distribution parts of the model and propose a semiparametric estimation method for the model. The Expectation Maximization (EM) algorithm and the multiple imputation method are employed to estimate parameters of interest. In the simulation study, we show that both estimation methods work well. To illustrate, we apply the model and the proposed methods to a data set of failure times from bone marrow transplant patients.     

Impact of model misspecification in shared frailty survival models

Survival models incorporating random effects to account for unmeasured heterogeneity are being increasingly used in biostatistical and applied research. Specifically, unmeasured covariates whose lack of inclusion in the model would lead to biased, inefficient results are commonly modeled by including a subject-specific (or cluster-specific) frailty term that follows a given distribution (eg, gamma or lognormal). Despite that, in the context of parametric frailty models, little is known about the impact of misspecifying the baseline hazard or the frailty distribution or both. Therefore, our aim is to quantify the impact of such misspecification in a wide variety of clinically plausible scenarios via Monte Carlo simulation, using open-source software readily available to applied researchers. We generate clustered survival data assuming various baseline hazard functions, including mixture distributions with turning points, and assess the impact of sample size, variance of the frailty, baseline hazard function, and frailty distribution. Models compared include standard parametric distributions and more flexible spline-based approaches; we also included semiparametric Cox models. The resulting bias can be clinically relevant. In conclusion, we highlight the importance of fitting models that are flexible enough and the importance of assessing model fit. We illustrate our conclusions with two applications using data on diabetic retinopathy and bladder cancer. Our results show the importance of assessing model fit with respect to the baseline hazard function and the distribution of the frailty: misspecifying the former leads to biased relative and absolute risk estimates, whereas misspecifying the latter affects absolute risk estimates and measures of heterogeneity.     

A multiparameter regression model for interval-censored survival data

We develop flexible multiparameter regression (MPR) survival models for interval-censored survival data arising in longitudinal prospective studies and longitudinal randomised controlled clinical trials. A multiparameter Weibull regression survival model, which is wholly parametric, and has nonproportional hazards, is the main focus of the article. We describe the basic model, develop the interval-censored likelihood, and extend the model to include gamma frailty and a dispersion model. We evaluate the models by means of a simulation study and a detailed reanalysis of data from the Signal Tandmobiel study. The results demonstrate that the MPR model with frailty is computationally efficient and provides an excellent fit to the data.     

Survival models for developmental genetic data: Age of onset of puberty and antisocial behavior in twins

The use of survival analysis for developmental genetic data is discussed. The main requirements for models based on the decomposition of frailty distributions into shared and unshared components are outlined for the simple case of twins. Extending the earlier work of Clayton, Oakes, and Hougaard, among others, three forms of hazard model are presented, all of which can be applied to pedigree data with flexible baseline hazards without the use of numerical integration. The first two models use an additive decomposition of frailty, with either gamma or positive stable law distributed (PSL) components. The third model previously described by Hougaard involves a multiplicative PSL decomposition. The models are applied to data on the onset of puberty in male twins and illustrate the importance of correct specification of the baseline hazard for correct inference about genetic effects. The difficulty of assessing model specification using information only on the margins is also noted. Overall, the new model with additive PSL components appeared to fit these data best. A second application illustrates the use of a time-varying covariate in examining the impact of puberty on the onset of conduct disorder symptomotology. © 1994 Wiley-Liss, Inc.     

Dispersion frailty models and HGLMs

In medical research recurrent event times can be analysed using a frailty model in which the frailties for different individuals are independent and identically distributed. However, such a homogeneous assumption about frailties could sometimes be suspect. For modelling heterogeneity in frailties we describe dispersion frailty models arising from a new class of models, namely hierarchical generalized linear models. Using the kidney infection data we illustrate how to detect and model heterogeneity among frailties. Stratification of frailty models is also investigated.     

The correlated and shared gamma frailty model for bivariate current status data: An illustration for cross‐sectional serological data

Frailty models are often used to study the individual heterogeneity in multivariate survival analysis. Whereas the shared frailty model is widely applied, the correlated frailty model has gained attention because it elevates the restriction of unobserved factors to act similar within clusters. Estimating frailty models is not straightforward due to various types of censoring. In this paper, we study the behavior of the bivariate‐correlated gamma frailty model for type I interval‐censored data, better known as current status data. We show that applying a shared rather than a correlated frailty model to cross‐sectionally collected serological data on hepatitis A and B leads to biased estimates for the baseline hazard and variance parameters. Copyright © 2009 John Wiley & Sons, Ltd.     

Generalized gamma frailty model

In this article, we present a frailty model using the generalized gamma distribution as the frailty distribution. It is a power generalization of the popular gamma frailty model. It also includes other frailty models such as the lognormal and Weibull frailty models as special cases. The flexibility of this frailty distribution makes it possible to detect a complex frailty distribution structure which may otherwise be missed. Due to the intractable integrals in the likelihood function and its derivatives, we propose to approximate the integrals either by Monte Carlo simulation or by a quadrature method and then determine the maximum likelihood estimates of the parameters in the model. We explore the properties of the proposed frailty model and the computation method through a simulation study. The study shows that the proposed model can potentially reduce errors in the estimation, and that it provides a viable alternative for correlated data. The merits of proposed model are demonstrated in analysing the effects of sublingual nitroglycerin and oral isosorbide dinitrate on angina pectoris of coronary heart disease patients based on the data set in Danahy et al. (sustained hemodynamic and antianginal effect of high dose oral isosorbide dinitrate. Circulation 1977; 55:381-387).     

Modeling time-to-cure from severe acute malnutrition: application of various parametric frailty models

Background

In developing countries about 3.5% of children aged 0–5 years are victims of severe acute malnutrition (SAM). Once the morbidity has developed the cure process takes variable period depending on various factors. Knowledge of time-to-cure from SAM will enable health care providers to plan resources and monitor the progress of cases with SAM. The current analysis presents modeling time-to-cure from SAM starting from the day of diagnosis in Wolisso St. Luke Catholic hospital, southwest Ethiopia.

Methods

With the aim of coming up with appropriate survival (time-to-event) model that describes the SAM dataset, various parametric clustered time-to-event (frailty) models were compared. Frailty model, which is an extension of the proportional hazards Cox survival model, was used to analyze time-to-cure from SAM. Kebeles (villages) of the children were considered as the clustering variable in all the models. We used exponential, weibull and log-logistic as baseline hazard functions and the gamma as well as inverse Gaussian for the frailty distributions and then based on AIC criteria, all models were compared for their performance.

Results

The median time-to-cure from SAM cases was 14 days with the maximum of 63 days of which about 83% were cured. The log-logistic model with inverse Gaussian frailty has the minimum AIC value among the models compared. The clustering effect was significant in modeling time-to-cure from SAM. The results showed that age of a child and co-infection were the determinant prognostic factors for SAM, but sex of the child and the type of malnutrition were not significant.

Conclusions

The log-logistic with inverse Gaussian frailty model described the SAM dataset better than other distributions used in this study. There is heterogeneity between the kebeles in the time-to-cure from SAM, indicating that one needs to account for this clustering variable using appropriate clustered time-to-event frailty models.     

Estimation and variable selection via frailty models with penalized likelihood

The penalized likelihood methodology has been consistently demonstrated to be an attractive shrinkage and selection method. It does not only automatically and consistently select the important variables but also produces estimators that are as efficient as the oracle estimator. In this paper, we apply this approach to a general likelihood function for data organized in clusters, which corresponds to a class of frailty models, which includes the Cox model and the Gamma frailty model as special cases. Our aim was to provide practitioners in the medical or reliability field with options other than the Gamma frailty model, which has been extensively studied because of its mathematical convenience. We illustrate the penalized likelihood methodology for frailty models through simulations and real data. Copyright © 2012 John Wiley & Sons, Ltd.     

A relaxation of the gamma frailty (Burr) model

Frailty models are used in univariate data to account for individual heterogeneity. In the popular gamma frailty model the marginal hazard has the form of a Burr model. Although the Burr model is very useful and can offer insight on the data, it is far from perfect. The estimation of the covariate effects is linked to the baseline hazard and this makes the model coefficients hard to interpret. At the same time, the frailties are assumed constant over time, while biological reasoning in some cases may indicate that frailties may be time dependent. In this paper we present a relaxation of the Burr model which is based on loosening the link between the estimation of the covariate effects and the baseline hazard. This can be achieved by replacing the cumulative baseline hazard in the Burr model by a set of time functions, and the frailty variance by a vector of coefficients directly estimated from the data using a partial likelihood. We illustrate the similarities of the model with the Burr model and a further extension of the latter, a model with an autoregressive stochastic process for the frailty. We compare the models on simulated data sets with constant and time-dependent frailties and show how the relaxed Burr models performs on two different real data sets. We show that the relaxed Burr model serves as a good approximation to the Burr model when the frailty is constant, and furthermore it gives better results when the frailty is time dependent.     

Nonproportional hazards and unobserved heterogeneity in clustered survival data: When can we tell the difference?

Multivariate survival data are frequently encountered in biomedical applications in the form of clustered failures (or recurrent events data). A popular way of analyzing such data is by using shared frailty models, which assume that the proportional hazards assumption holds conditional on an unobserved cluster-specific random effect. Such models are often incorporated in more complicated joint models in survival analysis. If the random effect distribution has finite expectation, then the conditional proportional hazards assumption does not carry over to the marginal models. It has been shown that, for univariate data, this makes it impossible to distinguish between the presence of unobserved heterogeneity (eg, due to missing covariates) and marginal nonproportional hazards. We show that time-dependent covariate effects may falsely appear as evidence in favor of a frailty model also in the case of clustered failures or recurrent events data, when the cluster size or number of recurrent events is small. When true unobserved heterogeneity is present, the presence of nonproportional hazards leads to overestimating the frailty effect. We show that this phenomenon is somewhat mitigated as the cluster size grows. We carry out a simulation study to assess the behavior of test statistics and estimators for frailty models in such contexts. The gamma, inverse Gaussian, and positive stable shared frailty models are contrasted using a novel software implementation for estimating semiparametric shared frailty models. Two main questions are addressed in the contexts of clustered failures and recurrent events: whether covariates with a time-dependent effect may appear as indication of unobserved heterogeneity and whether the additional presence of unobserved heterogeneity can be detected in this case. Finally, the practical implications are illustrated in a real-world data analysis example.     

Analysing multicentre competing risks data with a mixed proportional hazards model for the subdistribution

In the competing-risks setting, to test the effect of a covariate on the probability of one particular cause of failure, the Fine and Gray model for the subdistribution hazard can be used. However, sometimes, competing risks data cannot be considered as independent because of a clustered design, for instance in registry cohorts or multicentre clinical trials. Frailty models have been shown useful to analyse such clustered data in a classical survival setting, where only one risk acts on the population. Inclusion of random effects in the subdistribution hazard has not been assessed yet. In this work, we propose a frailty model for the subdistribution hazard. This allows first to assess the heterogeneity across clusters, then to incorporate such an effect when testing the effect of a covariate of interest. Based on simulation study, the effect of the presence of heterogeneity on testing for covariate effects was studied. Finally, the model was illustrated on a data set from a registry cohort of patients with acute myeloid leukaemia who underwent bone marrow transplantation.     

Methodology for analyzing censored correlated data: application of marginal and frailty approaches in human genetics. The European Community Alport Syndrome Concerted Action Group (ECASCA)

BACKGROUND: Statistical analysis for correlated censored data allows to study censored events in clustered structure designs. Considering a possible correlation among failure times of the same group, standard methodology is no longer applicable. We investigated proposed models in this context to study familial data about a genetic disease, Alport syndrome. Alport syndrome is a severe hereditary disease due to abnormal collagenous chains. Renal failure is the main symptom of the disease. It progresses toward end-stage renal failure (IRT) according to a high time variability. As shown by genetic studies, mutations of COL4A5 gene are involved in the X-linked Alport Syndrome. Due to the large range of the mutation types, the aim of this study was to search for a possible genetic origin of the heterogeneity of the disease severity. METHODS: Marginal survival models and mixed effects survival models (so-called frailty models) were proposed to take into account the possible non independence of the observations. In this study, time until end-stage renal failure is a rightly censored end point. Possible intra-familial correlations due to shared environmental and/or genetic factors could induce dependence among familial failure times. In this paper, we fit marginal and frailty proportional hazards models to evaluate the effect of mutation type on the risk of IRT and an interfamilial heterogeneity of failure times. RESULTS: In this study, the use of these models allows to show the presence of an interfamilial heterogeneity of the failure times to IRT. Moreover, the results suggest that some mutation types are linked to a higher risk of fast evolution to IRT, which explains partially the interfamilial heterogeneity of the failure times. CONCLUSIONS: This paper shows the interest of marginal and frailty models to evaluate the heterogeneity of censored responses and to study relationships between a censored criterion and covariables. This study puts forward the importance of characterizing the mutation at a molecular level to understand the relationship between genotype and phenotype.     

Nested frailty models using maximum penalized likelihood estimation

The frailty model is a random effect survival model, which allows for unobserved heterogeneity or for statistical dependence between observed survival data. The nested frailty model accounts for the hierarchical clustering of the data by including two nested random effects. Nested frailty models are particularly appropriate when data are clustered at several hierarchical levels naturally or by design. In such cases it is important to estimate the parameters of interest as accurately as possible by taking into account the hierarchical structure of the data. We present a maximum penalized likelihood estimation (MPnLE) to estimate non-parametrically a continuous hazard function in a nested gamma-frailty model with right-censored and left-truncated data. The estimators for the regression coefficients and the variance components of the random effects are obtained simultaneously. The simulation study demonstrates that this semi-parametric approach yields satisfactory results in this complex setting. In order to illustrate the MPnLE method and the nested frailty model, we present two applications. One is for modelling the effect of particulate air pollution on mortality in different areas with two levels of geographical regrouping. The other application is based on recurrent infection times of patients from different hospitals. We illustrate that using a shared frailty model instead of a nested frailty model with two levels of regrouping leads to inaccurate estimates, with an overestimation of the variance of the random effects. We show that even when the frailty effects are fairly small in magnitude, they are important since they alter the results in a systematic pattern.     

Frailty models: Applications to biomedical and genetic studies

In survival analysis, frailty models are potential choices for modeling unexplained heterogeneity in a population. This tutorial presents an overview and general framework of frailty modeling and estimation for multiplicative hazards models in the context of biomedical and genetic studies. Other topics in frailty models, such as diagnostic methods for model adequacy and inference in frailty models, are also discussed. Examples of analyses using multivariate frailty models in a non-parametric hazards setting on biomedical datasets are provided, and the implications of choosing to use frailty and relevance to genetic applications are discussed.     

Cure rate model with interval censored data

In cancer trials, a significant fraction of patients can be cured, that is, the disease is completely eliminated, so that it never recurs. In general, treatments are developed to both increase the patients' chances of being cured and prolong the survival time among non-cured patients. A cure rate model represents a combination of cure fraction and survival model, and can be applied to many clinical studies over several types of cancer. In this article, the cure rate model is considered in the interval censored data composed of two time points, which include the event time of interest. Interval censored data commonly occur in the studies of diseases that often progress without symptoms, requiring clinical evaluation for detection (Encyclopedia of Biostatistics. Wiley: New York, 1998; 2090-2095). In our study, an approximate likelihood approach suggested by Goetghebeur and Ryan (Biometrics 2000; 56:1139-1144) is used to derive the likelihood in interval censored data. In addition, a frailty model is introduced to characterize the association between the cure fraction and survival model. In particular, the positive association between the cure fraction and the survival time is incorporated by imposing a common normal frailty effect. The EM algorithm is used to estimate parameters and a multiple imputation based on the profile likelihood is adopted for variance estimation. The approach is applied to the smoking cessation study in which the event of interest is a smoking relapse and several covariates including an intensive care treatment are evaluated to be effective for both the occurrence of relapse and the non-smoking duration.     

Cancer incidence for Swedish twins studied by means of bivariate frailty models

Cancer incidence rates for Swedish twins born between 1928 and 1965 and who both were alive at age 30 are studied by means of bivariate frailty models. Altogether, 7,280 fraternal (DZ) and 4,699 identical (MZ) twin pairs were followed up through December 31, 1995, for cancer status. The association between cancer incidence rates was statistically greater among the MZ than among the DZ pairs and stronger between women than between men; however, the magnitude of this association is relatively small and decreases over time. The relative decrease in dependency (association) is most easily detected using shared frailty models but may also be demonstrated, at least for women, using correlated frailty models. We also demonstrate that estimates of the correlation coefficient are similar when using any correlated frailty models derived from the power variance family but that these estimates disagree regarding the age at which the dependence is most important. The relative importance of dependence across age may sometimes be more interesting than the correlation coefficient itself. The latter may usually be estimated using alternative methods. Furthermore, when estimating correlation coefficients close to the boundary of the parameter space, simulation studies indicate that the correlated inverse Gaussian frailty model is more robust than the gamma frailty model.     

The use of Gaussian quadrature for estimation in frailty proportional hazards models

In this paper, we propose a novel Gaussian quadrature estimation method in various frailty proportional hazards models. We approximate the unspecified baseline hazard by a piecewise constant one, resulting in a parametric model that can be fitted conveniently by Gaussian quadrature tools in standard software such as SAS Proc NLMIXED. We first apply our method to simple frailty models for correlated survival data (e.g. recurrent or clustered failure times), then to joint frailty models for correlated failure times with informative dropout or a dependent terminal event such as death. Simulation studies show that our method compares favorably with the well-received penalized partial likelihood method and the Monte Carlo EM (MCEM) method, for both normal and Gamma frailty models. We apply our method to three real data examples: (1) the time to blindness of both eyes in a diabetic retinopathy study, (2) the joint analysis of recurrent opportunistic diseases in the presence of death for HIV-infected patients, and (3) the joint modeling of local, distant tumor recurrences and patients survival in a soft tissue sarcoma study. The proposed method greatly simplifies the implementation of the (joint) frailty models and makes them much more accessible to general statistical practitioners.