Weibull frailty for modelling heterogeneity
This thesis will investigate the effect of differing assumed frailty models on the corresponding estimated regression coefficients. The main purpose of the Cox model in survival analysis is to measure the effects of covariates by estimating their coefficients. Sometimes the covariates fail to fully account for the true differences in risk because there is another variable which is omitted in the model. This failure may often be explained by unaccounted random heterogeneity or frailty. The non-negative random variable called frailty should be included in the model in order to measure correctly the effects on the response variable and in order to avoid underestimation or overestimation of the parameters. This thesis presents both the Cox proportional hazard frailty model and the parametric failure time frailty model. In a Cox proportional hazard frailty model, the log-normal and Weibull distributions are used as frailty models in the linear predictor while the baseline hazard is unspecified. By contrast, in a parametric failure time frailty model, the Weibull distribution is used as the baseline hazard function with log-normal and Weibull frailty. This thesis examines maximum hierarchical likelihood (MHL) and restricted maximum hierarchical likelihood (RMHL) estimation procedures. This thesis also confirms that the Best Linear Unbiased Prediction (BLUP) estimator of the variance component is biased and inconsistent. Two simulation studies were carried out: the first simulation evaluates the performance of the parameter estimates in the parametric failure time frailty model and the second simulation evaluates the Cox proportional hazard frailty model. In the first simulation, the MHL and RMHL results were close to the true values, but the MHL results were slightly more accurate. In the second simulation, Weibull frailty and Weibull survival times were generated so both survival times and frailty come from Weibull distribution. Then, the Cox model with log-normal and the Weibull frailty was fitted to the simulated dataset. A simulation to compare the maximum adjusted profile hierarchical likelihood (MAPHL) and the MHL by applying the Weibull frailty was carried out. Then, the mean squared errors of MAPHL and MHL estimates were calculated. The results show that both procedures have the same accuracy.