Nuisance Parameter Estimation in Survival Models

Abstract

The Cox proportional hazards model is the most commonly used method for right-censored data. Unfortunately, the interpretation of the hazard ratio is non-intuitive and it is commonly misinterpreted as a relative risk. This motivates alternative methods with more intuitive interpretations that avoid a reliance on the proportional hazards assumption. We explore censored quantile regression and restricted mean treatment effects as potential alternatives to the Cox proportional hazards model. However, both alternatives require a conditional survival function as a nuisance parameter. This thesis focuses on the impact of the conditional survival function on the estimation of censored quantile regression and restricted means. In particular, we illustrate that a non-parametric estimator of the conditional survival function improves the estimation of censored quantile regression when the semi-parametric assumptions of current methods are badly violated. Unfortunately, the non-parametric estimator is inefficient when the semi-parametric assumptions are satisfied. Rather than rely on either parametric assumptions or a non-parametric model, we instead pursue an estimator that performs well in both situations. We propose estimating the conditional survival function with stacked survival models. By minimizing prediction error, stacked survival models estimate an optimally weighted combination of several survival models. This allows stacking to span parametric, semiparametric, and non-parametric models to estimate the conditional survival function. As such, stacking can give weight to approximately correct parametric models, but shifts weight to non-parametric models when assumptions are badly violated. We demonstrate that the stacked survival model improves estimation of conditional survival functions and found it to always outperform the model selected by cross-validation. In addition, we illustrate that stacked survival models improve the estimation of restricted mean treatment effects in a wide variety of situations, while maintaining the efficiency of current methods.