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It has recently been shown that the relative risks of the order of 2 to 4 that are frequently found for cancer among relatives of affected cases are unlikely to be explainable by shared environmental risk factors. Classical methods of epidemiological analysis are not well suited to such analysis because they assume that the outcomes of each individual are independent. Classical methods of genetic analysis, on the other hand, are limited in their handling of environmental factors and variable ages of onset. The recent development of random effects models for survival analysis, however, appears to bridge this gap. Specifically, a proportional hazards model is postulated for the effects of measured covariates and of one or more components of frailty that are unmeasured but assumed to have some common distribution and known covariance structure within each family. From these assumptions, the posterior expectation of the hazard for each individual can be derived, given the covariate value and the observed and expected disease history of the family. These are then treated as known in a standard partial likelihood analysis; this is essentially a form of expectation-maximization algorithm. However, this does not provide a valid estimate of the covariance matrix because it fails to take account of the variability in the estimates of the frailties; an alternative approach using the imputation-posterior algorithm is suggested. This paper describes extensions of this approach to multivariate frailty distributions, modifications for application to pedigree and case-control studies, some simulation results, and applications to studies of breast cancer in twins and of lung cancer in relation to family smoking habits.

Classic methods in genetics for the analysis of binary attributes, based on an assumption of a "threshold" on a normally distributed latent variable called "liability," estimate the strength of genetic and environmental effects from differences in correlations between relatives of differing genetic relatedness. Two problems that are not easily addressed by these methods are the need to take the age of onset into account (particularly in chronic diseases in which incidence rates vary considerably with age and the lengths of time at risk can vary between individuals) and the desirability of incorporating measured covariates (genetic or environmental). The standard methods of cohort analysis used in epidemiology allow for both of these features, but until recently have been restricted to independent individuals. Recent developments in survival analysis have extended the widely used "proportional hazards" model of Cox by the addition of latent variable, epsilon, reflecting the shared susceptibility of related subjects because of their shared genes or shared environment. We show how this approach can be combined with more traditional models of gene-environment interaction to allow the main effects of measured genetic markers and environmental variables to be estimated, as well as the residual variance of genetic and environment and their interactions. The approaches are applied to a cohort of female twin births in Sweden from 1886 to 1958, linked with the Swedish cancer registry from 1961 to 1982.