Nested frailty models using maximum penalized likelihood estimation.

The frailty model is a random effect survival model, which allows for unobserved heterogeneity or for statistical dependence between observed survival data. The nested frailty model accounts for the hierarchical clustering of the data by including two nested random effects. Nested frailty models are particularly appropriate when data are clustered at several hierarchical levels naturally or by design. In such cases it is important to estimate the parameters of interest as accurately as possible by taking into account the hierarchical structure of the data. We present a maximum penalized likelihood estimation (MPnLE) to estimate non-parametrically a continuous hazard function in a nested gamma-frailty model with right-censored and left-truncated data. The estimators for the regression coefficients and the variance components of the random effects are obtained simultaneously. The simulation study demonstrates that this semi-parametric approach yields satisfactory results in this complex setting. In order to illustrate the MPnLE method and the nested frailty model, we present two applications. One is for modelling the effect of particulate air pollution on mortality in different areas with two levels of geographical regrouping. The other application is based on recurrent infection times of patients from different hospitals. We illustrate that using a shared frailty model instead of a nested frailty model with two levels of regrouping leads to inaccurate estimates, with an overestimation of the variance of the random effects. We show that even when the frailty effects are fairly small in magnitude, they are important since they alter the results in a systematic pattern.