The proportional hazard model with Weibull frailty is used for analyzing right-censoring longitudinal survival data. Complex marginal integrals are avoided by using Hierarchal-likelihood to estimate covariate coefficients and prediction of random effects

the adjusted profile hierarchical-likelihood is taken to estimate the parameters of random factors'distribution. Simulation studies of comparing with other frailties including lognormal frailty and Gamma frailty are provided and the result indicates that the Weibull frailty performs very well. The examples for analyzing two real data sets are also presented

and the results indicate that the Weibull frailty model is at least as good as lognormal and Gamma frailty model

and the Weibull frailty model is more helpful to diagnose the heterogeneity.