Penalized hierarchical multi-parameter regression survival models

Standard survival models such as the proportional hazards (PH) model contain a single regression component, corresponding to the scale of the hazard. In contrast, we consider the so-called multi-parameter regression (MPR) approach whereby covariates enter the model through multiple distributional parameters simultaneously, e.g., scale and shape parameters. This approach has previously been shown to achieve enhanced flexibility with relatively low model complexity and computational cost. While there have been many recent developments regarding the use of such models in survival analysis, the literature remains limited. As such, the purpose of this thesis is to extend the methodology surrounding MPR modelling in survival analysis, more specifically, we consider extensions of MPR models to penalised variable selection and random effects modelling with a focus on models that have two regression components (shape and scale). 

Beyond a stepwise type selection method, variable selection methods are underdeveloped in the MPR survival modelling setting. Therefore, we propose penalized MPR estimation procedures using the following penalties: least absolute shrinkage and selection operator (LASSO), smoothly clipped absolute deviation (SCAD), and adaptive LASSO. We compare these procedures using extensive simulation studies and an application to data from an observational lung cancer study. 

It is very common to have clustered data arising from survival analysis studies. For such data, the assumption of independent event times on which the standard MPR model relies does not hold and to correctly model such data within the MPR framework requires extending the standard MPR theory. Hence, we extend MPR models to handle multivariate survival data by introducing random effects in both the scale and the shape regression components. We consider a variety of possible dependence structures for these random effects (independent, shared, and correlated), and estimation proceeds using a hierarchical likelihood approach which provides a computationally inexpensive and straightforward two step procedure to fit our frailty models while avoiding the often intractable integration of the random effects over the frailty distribution. The performance of our estimation procedure is investigated by a way of an extensive simulation study, and the merits of our modelling approach are illustrated through applications to two real data examples, lung cancer dataset and bladder cancer dataset.

Our final contribution combines the two aforementioned extensions to obtain a unified framework for the penalised variable selection in MPR models with random effects. The proposed penalised hierarchical likelihood procedures are developed for a generic penalty but we focus on the example of the adaptive LASSO penalty for the numerical studies. As per the other two extensions, performance of the proposed framework is evaluated by means of simulation studies and applications to real data examples.