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.     

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.     

Proportional hazards models with frailties and random effects

We discuss some of the fundamental concepts underlying the development of frailty and random effects models in survival. One of these fundamental concepts was the idea of a frailty model where each subject has his or her own disposition to failure, their so-called frailty, additional to any effects we wish to quantify via regression. Although the concept of individual frailty can be of value when thinking about how data arise or when interpreting parameter estimates in the context of a fitted model, we argue that the concept is of limited practical value. Individual random effects (frailties), whenever detected, can be made to disappear by elementary model transformation. In consequence, unless we are to take some model form as unassailable, beyond challenge and carved in stone, and if we are to understand the term 'frailty' as referring to individual random effects, then frailty models have no value. Random effects models on the other hand, in which groups of individuals share some common effect, can be used to advantage. Even in this case however, if we are prepared to sacrifice some efficiency, we can avoid complex modelling by using the considerable power already provided by the stratified proportional hazards model. Stratified models and random effects models can both be seen to be particular cases of partially proportional hazards models, a view that gives further insight. The added structure of a random effects model, viewed as a stratified proportional hazards model with some added distributional constraints, will, for group sizes of five or more, provide no more than modest efficiency gains, even when the additional assumptions are exactly true. On the other hand, for moderate to large numbers of very small groups, of sizes two or three, the study of twins being a well known example, the efficiency gains of the random effects model can be far from negligible. For such applications, the case for using random effects models rather than the stratified model is strong. This is especially so in view of the good robustness properties of random effects models. Nonetheless, the simpler analysis, based upon the stratified model, remains valid, albeit making a less efficient use of resources.     

Hierarchical likelihood inference on clustered competing risks data

The frailty model, an extension of the proportional hazards model, is often used to model clustered survival data. However, some extension of the ordinary frailty model is required when there exist competing risks within a cluster. Under competing risks, the underlying processes affecting the events of interest and competing events could be different but correlated. In this paper, the hierarchical likelihood method is proposed to infer the cause‐specific hazard frailty model for clustered competing risks data. The hierarchical likelihood incorporates fixed effects as well as random effects into an extended likelihood function, so that the method does not require intensive numerical methods to find the marginal distribution. Simulation studies are performed to assess the behavior of the estimators for the regression coefficients and the correlation structure among the bivariate frailty distribution for competing events. The proposed method is illustrated with a breast cancer dataset. Copyright © 2015 John Wiley & Sons, Ltd.     

Analyzing sickness absence with statistical models for survival data

OBJECTIVES: Sickness absence is the outcome in many epidemiologic studies and is often based on summary measures such as the number of sickness absences per year. In this study the use of modern statistical methods was examined by making better use of the available information. Since sickness absence data deal with events occurring over time, the use of statistical models for survival data has been reviewed, and the use of frailty models has been proposed for the analysis of such data. METHODS: Three methods for analyzing data on sickness absences were compared using a simulation study involving the following: (i) Poisson regression using a single outcome variable (number of sickness absences), (ii) analysis of time to first event using the Cox proportional hazards model, and (iii) frailty models, which are random effects proportional hazards models. Data from a study of the relation between the psychosocial work environment and sickness absence were used to illustrate the results. RESULTS: Standard methods were found to underestimate true effect sizes by approximately one-tenth [method i] and one-third [method ii] and to have lower statistical power than frailty models. CONCLUSIONS: An uncritical use of standard methods may underestimate the effect of work environment exposures or leave predictors of sickness absence undiscovered.     

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.     

Recurrent injury event-time analysis

Public health decision making based on data sources that are characterized by a lack of independence and other complicating factors requires the development of innovative statistical techniques. Studies of injuries in occupational cohorts require methods to account for recurrent injuries to workers over time and the temporary removal of workers from the 'risk set' while recuperating. In this study, the times until injury events are modelled in an occupational cohort of employees in a large power utility company where employees are susceptible to recurrent events. The injury history over a ten-year period is used to compare the hazards of specific jobs, adjusted for age when first hired, and race/ethnicity differences. Subject-specific random effects and multiple event-times are accommodated through the application of frailty models which characterize the dependence of recurrent events over time. The counting process formulation of the proportional hazards regression model is used to estimate the effects of covariates for subjects with discontinuous intervals of risk. In this application, subjects are not at risk of injury during recovery periods or other illness, changes in jobs, or other reasons. Previous applications of proportional hazards regression in frailty models have not needed to account for the changing composition of the risk set which is required to adequately model occupational injury data. Published in 1999 by John Wiley & Sons, Ltd. This article is a US Government work and is in the public domain in the United States.     

Bayesian transformation cure frailty models with multivariate failure time data

We propose a class of transformation cure frailty models to accommodate a survival fraction in multivariate failure time data. Established through a general power transformation, this family of cure frailty models includes the proportional hazards and the proportional odds modeling structures as two special cases. Within the Bayesian paradigm, we obtain the joint posterior distribution and the corresponding full conditional distributions of the model parameters for the implementation of Gibbs sampling. Model selection is based on the conditional predictive ordinate statistic and deviance information criterion. As an illustration, we apply the proposed method to a real data set from dentistry.     

Three-state frailty model for age at onset of dementia and death in Swedish twins

We present a frailty model to estimate the relative importance of genetic and environmental factors on age at onset of dementia in a twin design. We use modern survival methodology to define a model that accounts simultaneously for longitudinal aspects, e.g., left truncation and right censoring in data, and the multivariate nature of twin data. Additionally, we present a novel three-state frailty model, with nondemented, demented, and dead states, describing variation in the onset of disease and mortality simultaneously in one model, while accounting for possible dependence for the two competing events. The frailty structure, i.e., the latent random effects structure, mimics the traditional twin model for continuous variables used in quantitative genetics, and as such describes within-pair dependence. This in turn leads to estimates for intrapair correlations, as well as for additive genetic, and shared and nonshared environmental components of variance. A hierarchical Bayesian model formulation and Gibbs sampling are used to estimate posterior distributions of the parameters. The models are applied to Swedish Twin Registry data on the onset of dementia in elderly twins. Based on the three-state frailty model, we estimate the intrapair correlations for dementia to be 0.87 [90% credible interval: 0.61,0.98] and 0.68[0.18,0.91] for MZ and DZ twins, respectively. Based on our model, we estimate that genetic effects account for about one third, and shared environmental effects for almost a half, of the variation in dementia hazards between individuals. More data, however, are needed to gain precision in these estimates.     

Estimating and testing for center effects in competing risks

The problems of fitting Gaussian frailties proportional hazards models for the subdistribution of a competing risk and of testing for center effects are considered. In the analysis of competing risks data, Fine and Gray proposed a proportional hazards model for the subdistribution to directly assess the effects of covariates on the marginal failure probabilities of a given failure cause. Katsahianbiet al. extended their model to clustered time to event data, by including random center effects or frailties in the subdistribution hazard. We first introduce an alternate estimation procedure to the one proposed by Katsahian et al. This alternate estimation method is based on the penalized partial likelihood approach often used in fitting Gaussian frailty proportional hazards models in the standard survival analysis context, and has the advantage of using standard survival analysis software. Second, four hypothesis tests for the presence of center effects are given and compared via Monte-Carlo simulations. Statistical and numerical considerations lead us to formulate pragmatic guidelines as to which of the four tests is preferable. We also illustrate the proposed methodology with registry data from bone marrow transplantation for acute myeloid leukemia (AML).     

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.     

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.     

Reduced-rank hazard regression for modelling non-proportional hazards

The Cox proportional hazards model is the most common method to analyse survival data. However, the proportional hazards assumption might not hold. The natural extension of the Cox model is to introduce time-varying effects of the covariates. For some covariates such as (surgical)treatment non-proportionality could be expected beforehand. For some other covariates the non-proportionality only becomes apparent if the follow-up is long enough. It is often observed that all covariates show similar decaying effects over time. Such behaviour could be explained by the popular (gamma-) frailty model. However, the (marginal) effects of covariates in frailty models are not easy to interpret. In this paper we propose the reduced-rank model for time-varying effects of covariates. Starting point is a Cox model with p covariates and time-varying effects modelled by q time functions (constant included), leading to a pxq structure matrix that contains the regression coefficients for all covariate by time function interactions. By reducing the rank of this structure matrix a whole range of models is introduced, from the very flexible full-rank model (identical to a Cox model with time-varying effects) to the very rigid rank one model that mimics the structure of a gamma-frailty model, but is easier to interpret. We illustrate these models with an application to ovarian cancer patients.     

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.     

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).     

Incorporation of nested frailties into semiparametric multi‐state models

Proportional hazards models are among the most popular regression models in survival analysis. Multi‐state models generalize them by jointly considering different types of events and their interrelations, whereas frailty models incorporate random effects to account for unobserved risk factors, possibly shared by clusters of subjects. The integration of multi‐state and frailty methodology is an interesting way to control for unobserved heterogeneity in the presence of complex event history structures and is particularly appealing for multicenter clinical trials. We propose the incorporation of correlated frailties in the transition‐specific hazard function, thanks to a nested hierarchy. We studied a semiparametric estimation approach based on maximum integrated partial likelihood. We show in a simulation study that the nested frailty multi‐state model improves the estimation of the effect of covariates, as well as the coverage probability of their confidence intervals. We present a case study concerning a prostate cancer multicenter clinical trial. The multi‐state nature of the model allows us to evidence the effect of treatment on death taking into account intermediate events. Copyright © 2015 JohnWiley & Sons, Ltd.     

Bayesian inference for recurrent events data using time-dependent frailty

In medical studies, we commonly encounter multiple events data such as recurrent infection or attack times in patients suffering from a given disease. A number of statistical procedures for the analysis of such data use the Cox proportional hazards model, modified to include a random effect term called frailty which summarizes the dependence of recurrent times within a subject. These unobserved random frailty effects capture subject effects that are not explained by the known covariates. They are typically modelled constant over time and are assumed to be independently and identically distributed across subjects. However, in some situations, the subject-specific random frailty may change over time in the same manner as time-dependent covariate effects. This paper presents a time-dependent frailty model for recurrent failure time data in the Bayesian context and estimates it using a Markov chain Monte Carlo method. Our approach is illustrated by a data set relating to patients with chronic granulomatous disease and it is compared to the constant frailty model using the deviance information criterion.     

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.     

Model selection for multi-component frailty models

Various frailty models have been developed and are now widely used for analysing multivariate survival data. It is therefore important to develop an information criterion for model selection. However, in frailty models there are several alternative ways of forming a criterion and the particular criterion chosen may not be uniformly best. In this paper, we study an Akaike information criterion (AIC) on selecting a frailty structure from a set of (possibly) non-nested frailty models. We propose two new AIC criteria, based on a conditional likelihood and an extended restricted likelihood (ERL) given by Lee and Nelder (J. R. Statist. Soc. B 1996; 58:619-678). We compare their performance using well-known practical examples and demonstrate that the two criteria may yield rather different results. A simulation study shows that the AIC based on the ERL is recommended, when attention is focussed on selecting the frailty structure rather than the fixed effects.