Abstract
The generalized gamma (GG) distribution is a widely used, flexible tool for parametric survival analysis. Many alternatives and extensions to this family have been proposed. This paper characterizes the flexibility of the GG by the quartile ratio relationship, log(Q2/Q1)/log(Q3/Q2), and compares the GG on this basis with two other three-parameter distributions and four parent distributions of four or five parameters. For most parameter combinations of other distributions, a very similar GG, as assessed by the Kullback-Liebler distance, can be found by matching the three quartiles; extreme cases where this fails are examined. Limited additional flexibility is observed, supporting the basic GG family as an ideal platform for parametric survival analysis.
Highlights
Parametric survival analysis has been the source for the development of distributions with richness and flexibility for modeling time-to-event data
We have investigated competing distributions including the three-parameter Exponentiated Weibull (EW; Cox and Matheson 2014) and a family that includes the GG as a special case, the five-parameter Beta-Generalized Gamma (Matheson and Cox 2017)
The quartile ratio relationship (QRR) curves for several combinations of (θ, τ) are shown in Fig. 2, panel a to illustrate this; panel b compares the hazard functions of the Beta-Generalized Gamma distribution (BGG)(0, 1, −2, 0.5, 0.5) to the closest approximating GG, GG(−1, 0.85, −4)
Summary
Parametric survival analysis has been the source for the development of distributions with richness and flexibility for modeling time-to-event data. Matching a GG to a Competitor Distribution Given any parametric family, one can choose parameter values, calculate the three quartiles of the resulting distribution, evaluate the QRR, and determine whether there is a GG with the same QRR as described above This process is entirely independent of data, simulation, or considerations of censoring; it is a purely theoretical exercise for matching two distributions. Competitors to the Generalized Gamma Alternate Three-Parameter Distributions Cox and Matheson (2014) previously investigated the exponentiated Weibull, another family having all four of the basic hazard shapes, as a competitor to the GG They found that given any member of the EW family, a matching GG can be found whose survival and hazard functions are indistinguishable. Another three-parameter family having the four basic hazard shapes is the Generalized Weibull (GW), which is most defined by its CDF:
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