The Generalized F Distribution: an Umbrella for Parametric Survival Analysis

In a recent tutorial my colleagues and I advocated the generalized gamma (GG) distribution as a platform for parametric survival analysis. The GG family includes all four of the common types of hazard functions, making it particularly useful for estimating individual hazard functions as well as both relative hazards and relative times. In addition, the GG includes most of the commonly used parametric survival distributions. Survival analysis based on the GG distribution is practical since regression models are available in commonly used statistical packages. It is well known that the GG is contained in an even larger family, the generalized F (GF) distribution, which also includes the log logistic. The GF thus provides additional flexibility for parametric modeling. In this paper we discuss the GF family from this perspective. We provide a characterization of the hazard functions of the GF, showing that, except for the GG, the available hazard functions are limited to decreasing and arc-shaped hazards and, in particular, that the hazard function can be decreasing but not monotone. We also discuss fitting the GF with an alternative parameterization using standard statistical software and refine a description of the hazard functions for death after a diagnosis of clinical AIDS in four different eras of HIV therapy.