Generalized Gamma Frailty and Symmetric Normal Random Effects Model for Repeated Time-to-Event Data

Clustered time-to-event data are quite common in survival analysis and finding a suitable model to account for dispersion as well as censoring is an important issue. In this article, we present a flexible model for repeated, overdispersed time-to-event data with right-censoring. We present here a general model by incorporating generalized gamma and normal random effects in a Weibull distribution to accommodate overdispersion and data hierarchies, respectively. The normal random effect has the property of being symmetrical, which means its probability density function is symmetric around its mean. While the random effects are symmetrically distributed, the resulting frailty model is asymmetric in its survival function because the random effects enter the model multiplicatively via the hazard function, and the exponentiation of a symmetric normal variable leads to lognormal distribution, which is right-skewed. Due to the intractable integrals involved in the likelihood function and its derivatives, the Monte Carlo approach is used to approximate the involved integrals. The maximum likelihood estimates of the parameters in the model are then numerically determined. An extensive simulation study is then conducted to evaluate the performance of the proposed model and the method of inference developed here. Finally, the usefulness of the model is demonstrated by analyzing a data on recurrent asthma attacks in children and a recurrent bladder data set known in the survival analysis literature.