Population-based family case-control study designs are important in epidemiology as they enable efficient investigation of rare diseases and the evaluation of the contributing risk factors in terms of clustered subjects. Such studies are subject to methodological complications since the case-control sets include: family members with and without the disease, varying ages of disease onset and intra-family correlation. This thesis addresses the paucity of published work on the analysis of competing risks data in this area by developing methods for the analysis of age of onset with multiple disease types arising from follow-up (or prospective) studies and by extending prospective methods to incorporate data from case-control (or retrospective) studies. The prospective and retrospective models are presented in terms of both the cause specific hazards and the subdistribution hazards. A three stage estimation procedure is developed to obtain estimates of the frailty term, the cumulative baseline hazard function, as well as the regression and dependence parameters. This estimation procedure results in regression coefficients with a marginal interpretation while allowing for the correlation structure to be specified by a frailty term. Monte Carlo simulation is used to evaluate the performance of the proposed models. Results suggest that the proposed models generally offer improved performance in terms of relative bias and efficiency.