Survival Analysis: Optimize the Partial Likelihood of the Cox Model

Finding the coefficients that maximize the log-partial likelihood in Python

Negative log-partial likelihood of the Cox model with local optimum. Image by author.

Table of contents

1. Introduction
2. The Cox proportional hazard model
3. Optimization problem
4. Implementation
5. Conclusions
6. References

1. Introduction

Survival analysis encompasses a collection of statistical methods for describing time to event data.

In this post, we introduce a popular survival analysis algorithm, the Cox proportional hazards model¹. Then, we define its log-partial likelihood and the gradient, and optimize it to find the best set of model parameters through a practical Python example.

2. The Cox proportional hazard model

We define the survival rate as the percentage of patients who have not experienced the adverse event (e.g. death) after a certain period of time.

The Cox proportional hazards model can assess the association between variables and survival rate. Given a set of covariates x, it defines the hazard function as: