Shrinkage estimators for semi-parametric proportional hazards mixture cure models

Survival analysis is essential for modelling time-to-event data, particularly in medical research. Mixture cure models are widely used methods to study patients' latency and incidence components. This research focuses on mixture model properties in the semi-parametric estimation of the Cox proportional hazard models in the presence of the multicollinearity problem, where the explanatory variables are linearly dependent so that the input design matrix is ill-conditioned. In the mixture of cure models, the multicollinearity issue can happen in both latency and incidence components, where the commonly used least squares (LS) method may lead to unreliable estimates for the coefficients of the underlying model. To address this issue, we propose shrinkage methods to estimate the coefficient of the underlying model. To do so, we developed new expectation-maximization (EM) algorithms to incorporate the shrinkage methods for both components. Through various simulations, we show that the proposed shrinkage methods cope with the multicollinearity problem in latency and incidence components and lead to more reliable estimates in semi-parametric settings. Our findings indicate that Ridge and Liu-type (LT) shrinkage methods provide more reliable parameter estimates and outperform the LS estimation method in scenarios with high multicollinearity. The developed methods are finally applied to a dataset on breast cancer, analyzing the disease prognosis and survival rates of patients with 10 or more positive lymph nodes. The results consistently show that the Ridge and LT methods offer better estimation and survival results compared to the LS method. Our numerical studies show the practical advantages of our proposed shrinkage methods in medical research.