Background: To accurately predict the response to treatment, we need a stable and effective risk score that can be calculated from patient characteristics. When we evaluate such risks from time-to-event data with right-censoring, Cox's proportional hazards model is the most popular for estimating the linear risk score. However, the intrinsic heterogeneity of patients may prevent us from obtaining a valid score. It is therefore insufficient to consider the regression problem with a single linear predictor.
Methods: we propose the model with a quasi-linear predictor that combines several linear predictors. This provides a natural extension of Cox model that leads to a mixture hazards model. We investigate the property of the maximum likelihood estimator for the proposed model. Moreover, we propose two strategies for getting the interpretable estimates. The first is to restrict the model structure in advance, based on unsupervised learning or prior information, and the second is to obtain as parsimonious an expression as possible in the parameter estimation strategy with cross- L1 penalty. The performance of the proposed method are evaluated by simulation and application studies.
Results: We showed that the maximum likelihood estimator has consistency and asymptotic normality, and the cross- L1-regularized estimator has root-n consistency. Simulation studies show these properties empirically, and application studies show that the proposed model improves predictive ability relative to Cox model.
Conclusions: It is essential to capture the intrinsic heterogeneity of patients for getting more stable and effective risk score. The proposed hazard model can capture such heterogeneity and achieve better performance than the ordinary linear Cox proportional hazards model.
Keywords: Cox’s proportional hazards model; Generalized average; Heterogeneity; Mixture model; Survival analysis.