Cox Distributions and Multiphase Distributions

Abstract

The aim of this work is to present the relationships between the Cox proportional hazards model and phase-type distributions, particularly Cox distributions. Phase-type distributions form a flexible class of lifetime distributions that can be interpreted as distributions of time to absorption in a Markov chain with a finite number of phase states. This allows for accurate approximations of a wide range of empirical survival time distributions, even in situations where classical parametric distributions (e.g., exponential or Weibull) do not provide a satisfactory fit. The work discusses the basic concepts of survival analysis: survival function, distribution function, density function, and hazard function. Special attention is given to the Cox model, which is a standard tool for analyzing the impact of multiple prognostic factors on survival time. Its assumptions, interpretation of regression coefficients, and the concept of hazard ratios are presented. In the following sections, the class of phase-type distributions is characterized, and it is shown how they can be used to model hazard functions in medical applications (among others.oncology, hepatology) and cybernetics – such as the analysis of the reliability of complex technical systems and computer networks. The theoretical considerations are illustrated by examples drawn from the classic works of Cox and Christensen regarding the survival of patients with cancer and liver diseases. Additionally, a cybernetic example is presented, in which the Cox model and multi-phase distributions are applied to describe the time to failure in an information processing system.