Abstract
A competing risk framework occurs when individuals have the potential to experience only one of the several mutually exclusive outcomes. Standard survival methods often overestimate the cumulative incidence of events when competing events are censored. Mixture distributions have been previously applied to the competing risk framework to obtain inferences regarding the subdistribution of an event of interest. Often the competing event is treated as a nuisance, but it may be of interest to compare adverse events against the beneficial outcome when dealing with an intervention. In this paper, methods for using a mixture model to estimate an adverse-benefit ratio curve (ratio of the cumulative incidence curves for the two competing events) and the ratio of the subhazards for the two competing events are presented. A parametric approach is described with some remarks for extending the model to include uncertainty in the event type that occurred, left truncation in order to allow for time-dependent analyses, and uncertainty in the timing of the event resulting in interval censoring. The methods are illustrated with data from an HIV clinical cohort examining whether individuals initiating effective antiretroviral therapy have a greater risk of antiretroviral discontinuation or switching compared with HIV RNA suppression.