Correlated Gamma-frailty Model Research Articles

The Heritability of Mortality Due to Heart Diseases: A Correlated Frailty Model Applied to Danish Twins

AbstractData of the Danish Twin Registry on monozygotic and dizygotic twins are used to analyse genetic and environmental influences on susceptibility to heart diseases for males and females, respectively. The sample includes 7955 like-sexed twin pairs born between 1870 and 1930. Follow-up was from 1 January 1943 to 31 December 1993 which results in truncation (twin pairs were included in the study if both individuals were still alive at the beginning of the follow-up) and censoring (nearly 40% of the study population was still alive at the end of the follow-up). We use the correlated gamma-frailty model for the genetic analysis of frailty to account for this censoring and truncation. During the follow-up 9370 deaths occurred, 3393 deaths were due to heart diseases in general, including 2476 deaths due to coronary heart disease (CHD). Proportions of variance of frailty attributable to genetic and environmental factors were analyzed using the structural equation model approach. Different standard biometric models are fitted to the data to evaluate the magnitude and nature of genetic and environmental factors on mortality. Using the best fitting model heritability of frailty (liability to death) was found to be 0.55 (0.07) and 0.53 (0.11) with respect to heart diseases and CHD, respectively, for males and 0.52 (0.10) and 0.58 (0.14) for females in a parametric analysis. A semi-parametric analysis shows very similar results. These analyses may indicate the existence of a strong genetic influence on individual frailty associated with mortality caused by heart diseases and CHD in both, males and females. The nature of genetic influences on frailty with respect to heart diseases and CHD is probably additive. No evidence for dominance and shared environment was found.

Dependent Hazards in Multivariate Survival Problems

A new class of bivariate survival distributions is constructed from a given family of survival distributions. The properties of these distributions are analyzed. It is shown that the same bivariate survival function can be derived using two radically different concepts: one involves transformation of the well-known bivariate survival function; the other involves correlated stochastic hazards. The new conditions that guarantee negative associations of life spans are derived. An exponential representation of the survival function for two related individuals is derived in terms of the conditional distribution of the stochastic hazards among survivors. Versions of the multivariate correlated gamma-frailty model are investigated.

Likelihood Ratio Test in the Correlated Gamma-Frailty Model

The correlated gamma-frailty model is a generalization of Cox' proportional hazard model, which allows for correlation between individuals within the same group. The nonparametric maximum likelihood estimator in this model has previously been studied by Murphy (1994, 1995) and Parner (1998). Here we show that the likelihood ratio test can be performed with the cf2 distribution as asymptotic law, with the degrees of freedom f equal to the number of Euclidean parameters fixed under the hypothesis. As a side effect we also have a new proof for asymptotic normality and efficiency of the Euclidean component of the maximum likelihood parameter. Finally, we show how standard errors can be computed.

Half of the variation in susceptibility to mortality is genetic: findings from Swedish twin survival data.

Molecular epidemiological studies confirm tremendous variability in genetic and environmental susceptibility to disease and death for humans. This variability as well as the roles of genetic and environmental factors in susceptibility to death can be estimated in the analysis of survival data on related individuals (e.g., twins). In this paper, correlated gamma-frailty models are applied to survival data on Swedish twins to estimate genetic parameters in six models of susceptibility. It is shown that the frailty model with additive genetic and nonshared environmental components fits the data best. The estimate of narrow-sense heritability in gamma frailty is about 50%. The results of genetic analysis confirm our earlier findings from the studies of Danish twins that about 50% of individual susceptibility approximated by gamma-distributed frailty is heritable.

How heritable is individual susceptibility to death? The results of an analysis of survival data on Danish, Swedish and Finnish twins.

Molecular epidemiological studies confirm a substantial contribution of individual genes to variability in susceptibility to disease and death for humans. To evaluate the contribution of all genes to susceptibility and to estimate individual survival characteristics, survival data on related individuals (eg twins or other relatives) are needed. Correlated gamma-frailty models of bivariate survival are used in a joint analysis of survival data on more than 31,000 pairs of Danish, Swedish and Finnish male and female twins using the maximum likelihood method. Additive decomposition of frailty into genetic and environmental components is used to estimate heritability in frailty. The estimate of the standard deviation of frailty from the pooled data is about 1.5. The hypothesis that variance in frailty and correlations of frailty for twins are similar in the data from all three countries is accepted. The estimate of narrow-sense heritability in frailty is about 0.5. The age trajectories of individual hazards are evaluated for all three populations of twins and both sexes. The results of our analysis confirm the presence of genetic influences on individual frailty and longevity. They also suggest that the mechanism of these genetic influences may be similar for the three Scandinavian countries. Furthermore, results indicate that the increase in individual hazard with age is more rapid than predicted by traditional demographic life tables.

How heritable is individual susceptibility to death? The results of an analysis of survival data on Danish, Swedish and Finnish twins

AbstractMolecular epidemiological studies confirm a substantial contribution of individual genes to variability in susceptibility to disease and death for humans. To evaluate the contribution of all genes to susceptibility and to estimate individual survival characteristics, survival data on related individuals (eg twins or other relatives) are needed. Correlated gamma-frailty models of bivariate survival are used in a joint analysis of survival data on more than 31 000 pairs of Danish, Swedish and Finnish male and female twins using the maximum likelihood method. Additive decomposition of frailty into genetic and environmental components is used to estimate heritability in frailty. The estimate of the standard deviation of frailty from the pooled data is about 1.5. The hypothesis that variance in frailty and correlations of frailty for twins are similar in the data from all three countries is accepted. The estimate of narrow-sense heritability in frailty is about 0.5. The age trajectories of individual hazards are evaluated for all three populations of twins and both sexes. The results of our analysis confirm the presence of genetic infiuences on individual frailty and longevity. They also suggest that the mechanism of these genetic infiuences may be similar for the three Scandinavian countries. Furthermore, results indicate that the increase in individual hazard with age is more rapid than predicted by traditional demographic life tables.

Asymptotic theory for the correlated gamma-frailty model

The frailty model is a generalization of Cox's proportional hazard model, where a shared unobserved quantity in the intensity induces a positive correlation among the survival times. Murphy showed consistency and asymptotic normality of the nonparametric maximum likelihood estimator (NPMLE) for the shared gamma-frailty model without covariates. In this paper we extend this result to the correlated gamma-frailty model, and we allow for covariates. We discuss the definition of the nonparametric likelihood function in terms of a classical proof of consistency for the maximum likelihood estimator, which goes back to Wald. Our proof of the consistency for the NPMLE is essentially the same as the classical proof for the maximum likelihood estimator. A new central limit theorem for processes of bounded variation is given. Furthermore, we prove that a consistent estimator for the asymptotic variance of the NPMLE is given by the inverse of a discrete observed information matrix.