Intermittent state-wise survival risk estimation in breast cancer patients using a multi-state frailty model

In cancer survival studies, most of the patients are followed up for a longer period of time, which results in multiple time-to-events (recurrence, metastasis, death). These multiple events affect survival differently, and hence the hazard needs to be estimated simultaneously. Also, there may exist random heterogeneity (frailty) that affects the simultaneous risk prediction. Hence, a multi-state frailty model (MFM) is more useful. The present study highlights the importance of incorporating shared frailty into the multi-state model (MSM). We assumed frailty parameters to follow a gamma and log-normal distribution and derived the estimation procedure for MFM using the semi-parametric penalized partial likelihood method. The derived model is illustrated using breast cancer (BC) data, and found that each covariate, along with recurrence as a catalyst, was found to be a significant predictor for survival. Further, we developed a procedure to generate a four-state MSM having six transitions. From both real data modeling and simulation studies, we observed that frailty had a significant impact on predicting the risk simultaneously. We found that both the MFM models had the highest Concordance (C-index). Thus, the study recommends using a multi-state frailty model for simultaneous risk prediction in longitudinal study designs.