A Bivariate Power Generalized Weibull Distribution: a Flexible Parametric Model for Survival Analysis

Abstract: We are concerned with the flexible parametric analysis of bivariate survival data. Elsewhere, we have extolled the virtues of the "power generalized Weibull" (PGW) distribution as an attractive vehicle for univariate parametric survival analysis: it is a tractable, parsimonious, model which interpretably allows for a wide variety of hazard shapes and, when adapted (to give an adapted PGW, or APGW, distribution), covers a wide variety of important special/limiting cases. Here, we additionally observe a frailty relationship between a PGW distribution with one value of the parameter which controls distributional choice within the family and a PGW distribution with a smaller value of the same parameter. We exploit this frailty relationship to propose a bivariate shared frailty model with PGW marginal distributions: these marginals turn out to be linked by the so-called BB9 or "power variance function" copula. This particular choice of copula is, therefore, a natural one in the current context. We then adapt the bivariate PGW distribution, in turn, to accommodate APGW marginals. We provide a number of theoretical properties of the bivariate PGW and APGW models and show the potential of the latter for practical work via an illustrative example involving a well-known retinopathy dataset, for which the analysis proves to be straightforward to implement and informative in its outcomes. The novelty in this article is in the appropriate combination of specific ingredients into a coherent and successful whole.