Incorporating Imprecision into the Proportional Hazards Model

Abstract

This thesis explores the potential and implications of incorporating imprecision into the Cox proportional hazards (PH) model, a widely endorsed method for examining the effects of covariates in survival analysis. Despite the fact that the PH model does not impose any parametric assumptions regarding the distribution of the baseline hazard function, it relies on the assumption of proportional hazards over time, which is often not sustainable in real-world scenarios. The research highlights the inherent limitations of the PH model, notably its vulnerability to deviations arising from factors like time-dependent covariates, which can compromise the integrity of statistical analyses in vital areas such as clinical research and public health. By introducing imprecision into the PH model, this thesis establishes advanced methodologies that effectively balance the trade-off between imprecision and validity when the conventional assumptions of the model are compromised. In response to these limitations, the thesis introduces novel methodologies to address the non-proportionality of hazards, proposing two innovative imprecise proportional hazards models: the individual-based model and the group-based model. These models offer a robust alternative to the conventional PH model by accommodating variability within the hazard functions and enabling the estimation of more reliable survival functions.
The thesis introduces another robust PH model designed for survival data with continuous covariates. Diverging from traditional measurement error approaches, the robust PH model integrates errors directly into covariate values as a strategy to mitigate the proportional hazards assumption. This shifts the focus from merely diminishing estimation bias to enhancing model adaptability. The proposed model incorporates additive errors into continuous covariates which are distribution-free, but fluctuate strictly within a predefined small interval. Consequently, imprecise estimates can be derived for individuals’ survival functions which enhances the flexibility and reliability of the robust PH model, particularly when the validity of the proportional hazards assumption is questioned.
This thesis concludes by introducing a novel imprecise estimation technique referred to as the Most Likely Data method (MLD) as an alternative to the well-known maximum likelihood estimation (MLE). Unlike the MLE, which offers precise point estimates through optimizing the likelihood function, the MLD focuses on interval estimates derived from the most likely observed data configurations. In this method, the parameter space is partitioned into intervals based on data that is most likely to be observed compared to others, resulting in a distinct interval for each possible observations. For discrete distributions, the MLD method can be applied seamlessly to both binomial and Poisson distributions, allowing partitioning the parameter space for different observations and providing a close-form technique for identifying imprecise estimates. In the context of the PH model, the MLD methodology revealed promising results as a means of relaxing the PH assumption. The objective of introducing the MLD method in this thesis is to pave the way for further investigation and development in the field of statistical inference.
Besides challenging the conventional application of the PH model in specific research contexts, the findings of this thesis offer significant methodological advancements that can enhance the robustness of conclusions drawn from survival data, thus influencing future research and practices in similar fields.