Weighted Lindley frailty model: estimation and application to lung cancer data

Abstract

In this paper, we propose a novel frailty model for modeling unobserved heterogeneity present in survival data. Our model is derived by using a weighted Lindley distribution as the frailty distribution. The respective frailty distribution has a simple Laplace transform function which is useful to obtain marginal survival and hazard functions. We assume hazard functions of the Weibull and Gompertz distributions as the baseline hazard functions. A classical inference procedure based on the maximum likelihood method is presented. Extensive simulation studies are further performed to verify the behavior of maximum likelihood estimators under different proportions of right-censoring and to assess the performance of the likelihood ratio test to detect unobserved heterogeneity in different sample sizes. Finally, to demonstrate the applicability of the proposed model, we use it to analyze a medical dataset from a population-based study of incident cases of lung cancer diagnosed in the state of São Paulo, Brazil.

Appendix A1—Simulation results for the WL frailty model with Gompertz baseline hazard function

Here, we fixed $\kappa =0.5$, $\rho =0.6$, $\sigma ^2=0.8$, and $\beta _1=0.7$. We also analyzed the performance of the ML estimates, considering the same criteria adopted for the model with the Weibull baseline hazard function. In general, we observed similar results. However, when comparing the two studies, we noticed that the metrics RMSE, SD, and CP presented a better performance when we adopted the model with the Weibull baseline hazard function.

Table 6 Bias, RMSE and SD of the ML estimates, and empirical CP of $95\%$ asymptotic confidence intervals for the simulated data of the WL frailty model with Gompertz baseline hazard function

We also repeated the simulation study to analyze the performance of the LR test for $H_0:\sigma ^2=0$. In this study, we set $\kappa =0.5$, $\rho =0.6$, $\beta _1=0.7$ and $\sigma ^2\in \{0, 0.01, 0.10, 0.20, 0.50, 0.75, 1.00, 1.50\}$. The sample size was configured to study the model with the Gompertz baseline hazard function. The censored times were generated from the $\text {Uniform}(0,14)$ distribution, with the proportion of censoring times varying from 5% to 17%. Again, the results are similar to the model with the Weibull baseline hazard function. However, to obtain test power greater than or equal to 0.9, $\sigma ^2 \ge 0.75$ and $n\ge 500$ are required.

Table 7 Rejection rates of the null hypothesis (absence of unobservable heterogeneity) at $5\%$ nominal significance level for several unobserved heterogeneity and sample sizes considering the WL frailty model with Gompertz baseline hazard function