Proportional hazards models

General
Proportional hazards models are a sub-class of survival models in statistics.

For the purposes of this article, consider survival models to consist of two parts: the underlying hazard function, describing how hazard (risk) changes over time, and the effect parameters, describing how hazard relates to other factors - such as the choice of treatment, in a typical medical example. The proportional hazards assumption is the assumption that effect parameters multiply hazard: for example, if taking drug X halves your hazard at time 0, it also halves your hazard at time 1, or time 0.5, or time $$t$$ for any value of $$t$$. The effect parameter(s) estimated by any proportional hazards model can be reported as hazard ratios.

Sir David Cox observed that if the proportional hazards assumption holds (or, is assumed to hold) then it is possible to estimate the effect parameter(s) without any consideration of the hazard function. This approach to survival data is called application of the Cox proportional hazards model, sometimes abbreviated to Cox model or to proportional hazards model.

Other proportional hazards models exist. Another approach to survival data is to assume that the proportional hazards assumption holds, but in addition to assume that the hazard function follows a known form. For example, assuming the hazard function to be the Weibull hazard function gives the Weibull proportional hazards model (in which the survival times follow a Weibull distribution).

Note on terminology
The generic term parametric proportional hazards models can be used to describe proportional hazards models in which the hazard function is specified. The Cox proportional hazards model is sometimes called a semi-parametric model by contrast.

Some authors (e.g. Bender, Augustin and Blettner, Statistics in Medicine 2005) use the term Cox proportional hazards model even when specifying the underlying hazard function, to acknowledge the debt of the entire field to David Cox.

The term Cox regression model (omitting proportional hazards) is sometimes used to describe the extension of the Cox model to include time-dependent factors. However, this usage is potentially ambiguous since the Cox proportional hazards model can itself be described as a regression model.

Relationship to Poisson models
There is a relationship between proportional hazards models and Poisson regression models which is sometimes used to fit approximate proportional hazards models in software for Poisson regression. The usual reason for doing this is that calculation is much quicker. This was more important in the days of slower computers but can still be useful for particularly large data sets or complex problems. Authors giving the mathematical details include Laird and Olivier (Journal of the American Statistical Association, 1981), who remark "Note that we do not assume [the Poisson model] is true, but simply use it as a device for deriving the likelihood."