---
tag: N029V3JX60
kind: concept
name: Proportional hazards model
msc: [62N01, 62N02, 62J12]
sources:
  - url: https://en.wikipedia.org/wiki/Proportional_hazards_model
same_as:
  wikidata: Q223218
  wikipedia: Proportional hazards model
---

## Summary

A proportional hazards model describes how covariates act on the [[hazard function|Y959NKQ3XW]] of an event time.
Each covariate multiplies the hazard by a factor that does not change with time.
In the Cox model, introduced by David Cox, the hazard is a baseline hazard times $e^{\beta^T x}$, and the baseline is left completely unspecified.
The coefficients $\beta$ can be estimated by maximising the partial likelihood, which does not involve the baseline hazard.
It is the most used regression model of [[survival analysis|443ZWDX88P]].

## Explanations

### L1 human

Suppose smokers have, at every age, twice the risk of a heart attack of non-smokers.
The risk itself may rise with age in a complicated way, but the ratio stays $2$.
A proportional hazards model says exactly this: each factor scales the whole hazard curve up or down by a fixed amount.
The doubling corresponds to a coefficient $\beta = \log 2$ for the smoking indicator.
The method can estimate this ratio without ever estimating the shape of the age curve.