An Implicitly Defined Parametric Model for Censored Survival Data and Covariates

Parametric models have much to recommend them for use in the analysis of censored survival data. They are efficient and simple and can sometimes account for features of the data that violate assumptions of partially parametric models such as proportional hazards. Many parametric models are also robust to departures from their own assumptions. Often, parametric models have precise mechanistic origins or interpretations, and many can be derived as members of a common family (Hazelrig, Turner, and Blackstone, 1982). This work was motivated by the need to model the effects of covariates on event times that appear to be distributed uniformly on some interval. To accomplish this, we introduce a new parametric survival model which describes cumulative distributions that are hybrids of the exponential and uniform. Hence, the cumulative distribution function can be nearly linear, depending on parameter values. The model is defined only implicitly, i.e., it satisfies the deriving equation, but cannot be written as an explicit function of time. In spite of this, the model can be used for maximum likelihood estimation of covariate effects in a familiar fashion. First, we derive the model equations and illustrate their behavior. Then the equations required for maximum likelihood estimation of parameters are presented. Finally, software is described and an example illustrating the use of the model is given.