Gamma frailty model for survival risk estimation: an application to cancer data

Abstract

Abstract Objectives: In time to event analysis, the risk for an event is usually estimated using Cox proportional hazards (CPH) model. But CPH model has the limitation of biased estimate due to unobserved hidden heterogeneity among the covariates, which can be tackled using frailty models. The best models were usually being identified using Akaike information criteria (AIC). Apart from AIC, the present study aimed to assess predictability of risk models using survival concordance measure. Methods: CPH model and frailty models were used to estimate the risk for breast cancer patient survival, and the frailty variable was assumed to follow gamma distribution. Schoenfeld global test was used to check the proportionality assumption. Survival concordance, AIC and simulation studies were used to identify the significance of frailty. Results: From the univariate analysis it was observed that for the covariate age, the frailty has a significant role (θ = 2.758, p-value: 0.0004) and the corresponding hazard rate was 1.93 compared to that of 1.38 for CPH model (age > 50 vs. ≤ 40). Also the covariates radiotherapy and chemotherapy were found to be significant (θ = 5.944, p-value: <0.001 and θ = 16, p-value: <0.001 respectively). Even though there were only minor differences in hazard rates, the concordance was higher for frailty than CPH model for all the covariates. Further the simulation study showed that the bias and root mean square error (RMSE) obtained for both the methods was almost the same and the concordance measures were higher for frailty model by 12–15%. Conclusions: We conclude that the frailty model is better compared to CPH model as it can account for unobserved random heterogeneity, and if the frailty coefficient doesn’t have an effect it gives exactly the same risk as that of CPH model and this has been established using survival concordance.