Wienke, A. Correlated frailty models in ageing THE ROLE OF CORRELATED FRAILTY MODELS IN STUDIES OF HUMAN HEALTH, AGEING AND LONGEVITY

This talk gives an overview about the analysis of twin data on diseases, aging and longevity. One goal of our research in this area is the evaluation of the genetic contribution to life span and mortality from different causes. Such evaluation is performed using the idea of correlated frailty. We apply different correlated frailty models to the analysis of bivariate survival data from Danish, Finnish, and Swedish twin registers to test for the presence of genetic influences on longevity. These models allow us to test whether susceptibility to major human diseases, such as cancer and cardiovascular disease, are heritable. The approach allows us to take observed covariates into account. Frailty models have become very popular during the last two decades and their applications are numerous. Shared frailty explains correlations between lifetimes within clusters (here, a cluster can consist of individuals from the same group, say a family, litter, clinic, community; or of multiple or recurrent events from the same individual). However, it does have some limitations. First, it forces the unobserved factors to be the same within the cluster, which is often not appropriate. Second, the dependence between survival times within the cluster is based on marginal distributions of survival times. To see this, when covariates are present in a Wienke, A. Correlated frailty models in ageing proportional hazards model with a gamma distributed frailty, the dependence parameter and the population heterogeneity are confounded [1], meaning that the joint distribution can be identified from the marginal distributions [2]. Third, in most cases univariate frailty will only induce positive association within the cluster. However, in some situations the survival times for subjects within the same cluster are negatively associated. To avoid these limitations, correlated frailty models have been developed for the analysis of multivariate failure time data. Yashin and Iachine [3] established the correlated gamma-frailty model. With special focus on the bivariate case advantages and limitations of different variants of the correlated gamma-frailty model are discussed and illustrated by applications to twin data. The distribution of frailty may, in principle, have an arbitrary form. There is usually no a priori belief about the distribution of frailty. Thus, it is often a matter of mathematical convenience to specify such a distribution. New asymptotic results support the choice of gamma distributed frailty [4]. The gamma distribution has become very popular in frailty models since it provides explicit formulae for the population survival function. The estimation procedure is relatively simple and feasible in this case. However, the true distribution of frailty in a population may not necessarily be the desired "mathematically convenient" one. It may prove to be, say, lognormal. A traditional maximum likelihood method is not valid in such a case. Some other methods like MCMC or numerical likelihood integration needs to be used to estimate parameters in the lognormal frailty models [5]. We compare different approaches to the estimation of parameters in correlated frailty models. Whereas these models are becoming more and more popular, less is known how dealing with various estimation strategies. Extensive simulation study is necessary to understand possible obstacles in estimation methods and to make conclusions about the validity of the results obtained using the methods. Both parametric and semi-parametric models will be investigated. In our analysis we focus on all-cause mortality, cause-specific mortality from cancer, coronary heart disease, stroke and respiratory diseases. [6,7] Models for survival analysis typically assume that everybody in the study population is susceptible to the event under study and will eventually experience this event if the follow-up is sufficiently long. This is often a questionable assumption of the widely used proportional hazard models and their extensions frailty models. However, there are situations when a Wienke, A. Correlated frailty models in ageing fraction of individuals are not expected to experience the event of interest; that is, those individuals are cured or insusceptible. For example, researchers may be interested in analyzing the recurrence of a disease. Many individuals may never experience a recurrence; therefore, a cured fraction of the population exists. Cure models extend the understanding of time-to-event data by allowing for the formulation of more accurate and informative conclusions. These conclusions are otherwise unobtainable from an analysis which fails to account for a cured or insusceptible fraction of the population. If a cured component is not present, the analysis reduces to standard approaches of survival analysis. Use of cure models has been popular for joint modelling of the overall risk of a disease and the age-at-onset distribution of the diseased individuals. In cure models the population is divided into two subpopulations so that an individual either is cured or has a proper survival function. Traditional cure models assume that those individuals experiencing the event of interest are homogeneous in risk. During the last fifteen years, extensions of cure models were developed in order to allow for heterogeneity among the fraction under risk by using frailty models where the frailty distribution is a mixture of a discrete and a continuous part. The frailty mixture distribution has point mass at zero while heterogeneity among those experiencing the event of interest is modelled via a continuous distribution. We suggest the use of the copula of the correlated gamma-frailty model and show by using simulations that all the parameters are estimable in a one-step ML estimation procedure [8]. Identifying correlations of durations is a requirement for successfully analysing genetic factors. In survival analysis there is a recurring problem of censored data, which complicates observations far more than does complete data. Using a survival model to estimate correlations among lifetimes can solve this problem. In this paper, instead of treating life spans directly, we wish to analyse both genetic and environmental factors acting on frailty for cause-specific mortality. The correlated gamma frailty model can be used to fit the lifetime data and provide a specific parameter for the correlation among frailties. If censoring can be assumed to be non-informative with regard to all different causes, then the censoring times can be taken as the minimum of the hypothetical censoring times arising from the different causes of censoring. For estimating the marginal survival function the Kaplan-Meier estimator is appropriate. However, the situation becomes much more difficult if the censoring arising from at least one of the different causes can be assumed to be informative. Above we discussed cause-specific mortality data analysis using the correlated gamma-frailty Wienke, A. Correlated frailty models in ageing model, assuming independence among causes of death in a `competing risk' scenario. In this paragraph, we investigate the effect of removing this limitation. The model allows us to test the hypothesis on dependence between the competing risks. The class of multivariate distributions presented is characterised by the association parameters, using arbitrary marginal distributions. The multivariate distribution is specified in full by the association and variance parameters and the marginal distribution functions. In the present analysis, we focus on the mortality rates of coronary heart disease (CHD). To simplify description, in this paper we consider models limited to two competing risks (death as a result of CHD and death arising from other causes). The model can be extended to the case of multiple competing risks or multivariate lifetimes. Results of a simulation study are included. Both limitations and future uses for this model are discussed. [9]