Cure Models with Modified Log-Logistic Distribution: An Application to Oncology Studies

The log-logistic distribution has been widely used in survival analysis, particularly in modeling survival times and event data in healthcare and biological studies. This study investigates the parameter estimation of the Log-Logistic Tangent (LLT) distribution using Maximum Likelihood Estimation (MLE), focusing on the consistency, bias, and precision of the estimated parameters. The simulation study results reveal that the estimated values of the parameters α and β deviate from their true values, indicating some bias in the estimation process. The mean values of the estimated parameters are found to be 0.623 for α and 1.433 for β, with respective standard deviations of 0.072 and 0.147, highlighting the variability across iterations. Further analysis of the asymptotic properties of the LLT model shows that the parameter estimates converge to stable values as sample size increases, demonstrating consistency in the estimation process. Additionally, asymptotic normality is confirmed through the calculation of the observed Fisher information matrix and derived standard errors. The LLT model was successfully applied to real-life data, yielding survival probability estimates, which were further validated through statistical testing. The study concludes that while the LLT model is effective in capturing survival patterns, improvements can be made to reduce bias, refine optimization techniques, and explore alternative estimation methods. Recommendations for future research include expanding the model to handle covariates and time-varying effects, thereby enhancing its applicability in diverse fields such as healthcare, finance, and engineering.