The hazard function of time-related events, such as death or reoperation following heart valve replacement, often is time-varying in a structured fashion, as is the influence of risk factors associated with the events. A completely parametric system is presented for the decomposition of time-varying patterns of risk into additive, overlapping phases, descriptively labeled as early, constant-hazard, and late. Each phase is shaped by a different generic function of time constituting a family of nested equations and is scaled by a separate logit-linear or log-linear function of concomitant information. Model building uses maximum likelihood estimation. The resulting parametric equations permit hazard function, survivorship function, and probability estimates and their confidence limits to be portrayed and adjusted for concomitant information. These provide a comprehensive analysis of time-related events from which inferences may be drawn to improve, for example, the management of patients with valvar heart disease.