Birnbaum–Saunders frailty regression models for clustered survival data

We present a novel frailty model for modeling clustered survival data. In particular, we consider the Birnbaum–Saunders (BS) distribution for the frailty terms with a new directly parameterized on the variance of the frailty distribution. This allows, among other things, compare the estimated frailty terms among traditional models, such as the gamma frailty model. Some mathematical properties of the new model are studied including the conditional distribution of frailties among the survivors, the frailty of individuals dying at time t, and the Kendall’s \(\tau \) $\tau$ measure. Furthermore, an explicit form to the derivatives of the Laplace transform for the BS distribution using the di Bruno’s formula is found. Parametric, non-parametric and semiparametric versions of the BS frailty model are studied. We use a simple Expectation-Maximization (EM) algorithm to estimate the model parameters and evaluate its performance under different censoring proportion by a Monte Carlo simulation study. We also show that the BS frailty model is competitive over the gamma and weighted Lindley frailty models under misspecification. We illustrate our methodology by using a real data sets.