Applying a marginalized frailty model to competing risks

The marginalized frailty model is often used for the analysis of correlated times in survival data. When only two correlated times are analyzed, this model is often referred to as the Clayton–Oakes model [7,22]. With time-to-event data, there may exist multiple end points (competing risks) suggesting that an analysis focusing on all available outcomes is of interest. The purpose of this work is to extend the single risk marginalized frailty model to the multiple risk setting via cause-specific hazards (CSH). The methods herein make use of the marginalized frailty model described by Pipper and Martinussen [24]. As such, this work uses the martingale theory to develop a likelihood based on estimating equations and observed histories. The proposed multivariate CSH model yields marginal regression parameter estimates while accommodating the clustering of outcomes. The multivariate CSH model can be fitted using a data augmentation algorithm described by Lunn and McNeil [21] or by fitting a series of single risk models for each of the competing risks. An example of the application of the multivariate CSH model is provided through the analysis of a family-based follow-up study of breast cancer with death in absence of breast cancer as a competing risk.