# Multi-Parameter Regression Survival Modelling: An Alternative to Proportional Hazards

## TL;DR

This paper introduces multi-parameter regression (MPR) models for survival analysis, allowing covariates to influence multiple distributional parameters simultaneously, leading to more flexible models that relax proportional hazards assumptions.

## Contribution

It proposes a novel MPR modeling approach in survival analysis, including a new variable selection strategy and implementation in an R package.

## Key findings

- MPR models provide greater flexibility than traditional proportional hazards models.
- Time-dependent hazard ratios emerge naturally from two-parameter Weibull models.
- A new test for proportionality is motivated by MPR models.

## Abstract

It is standard practice for covariates to enter a parametric model through a single distributional parameter of interest, for example, the scale parameter in many standard survival models. Indeed, the well-known proportional hazards model is of this kind. In this paper we discuss a more general approach whereby covariates enter the model through more than one distributional parameter simultaneously (e.g., scale and shape parameters). We refer to this practice as "multi-parameter regression" (MPR) modelling and explore its use in a survival analysis context. We find that multi-parameter regression leads to more flexible models which can offer greater insight into the underlying data generating process. To illustrate the concept, we consider the two-parameter Weibull model which leads to time-dependent hazard ratios, thus relaxing the typical proportional hazards assumption and motivating a new test of proportionality. A novel variable selection strategy is introduced for such multi-parameter regression models. It accounts for the correlation arising between the estimated regression coefficients in two or more linear predictors -- a feature which has not been considered by other authors in similar settings. The methods discussed have been implemented in the mpr package in R.