Proportional hazards model

Proportional hazards models are a sub-class of survival models in statistics, in which the effect of a treatment under study has a multiplicative effect on the subject's hazard rate. For example, a drug may halve one's immediate probability of stroke. This is in contrast to additive hazards models, wherein a treatment may increase one's hazard by a fixed amount which is independent of other covariates.

For the purposes of this article, consider survival models to consist of two parts: the underlying hazard function, often denoted $\Lambda _{0}(t)$, describing how hazard (risk) changes over time at baseline levels of covariates; and the effect parameters, describing how the hazard varies in response to explanatory covariates. A typical medical example would include as covariates, treatment assignment as well as patient characteristics to reduce variability and/or control for confounding.

The proportional hazards assumption is the assumption that covariates multiply hazard. In the simplest case of stationary coefficients, for example, a treatment with a drug may, say, halve a subject's hazard at any given time $t$, while the baseline hazard may vary. Note however, that the covariate is not restricted to binary predictors; in the case of a continuous covariate $x$, the hazard responds logarithmically; each unit increase in $x$ results in proportional scaling of the hazard. Typically under the fully-general Cox model, the baseline hazard is "integrated out", or heuristically removed from consideration, and the remaining partial likelihood is maximized. The effect of covariates estimated by any proportional hazards model can thus 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.

The Cox model may be specialized if a reason exists to assume that the baseline hazard follows a parametric form. In this case, the baseline hazard $\Lambda _{0}(t)$ is replaced by a parametric density; typically one can then just maximize the full likelihood which greatly simplifies model-fitting and provides interpretability, at the cost of flexibility. 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 which is rescaled by the covariates).

Incidentally, the Weibull distribution for the baseline hazard is the only assumption under which a model satisfies both the proportional hazards, and accelerated failure time models.

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 semiparametric 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.

Time-varying coefficients

In addition to allowing time-varying covariates (i.e., predictors), the Cox model may be generalized to time-varying coefficients as well. That is, the proportional effect of a treatment may vary with time; e.g. a drug may be very effective if administered within one month of morbidity, and become less effective as time goes on. The hypothesis of no change with time (stationarity) of the coefficient may then be tested. Details and software are available in Martinussen and Scheike (2006).

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."

The book of generalized linear models by Nelder and McCullagh has a chapter of converting PH models to GLM models.

See also

• Survival analysis
• Weibull distribution
• Accelerated failure time model

References

• Cox, D. R. (1972). "Regression Models and Life Tables". Journal of the Royal Statistical Society Series B. 34 (2): 187–220. JSTOR 2985181. MR0341758.

• DR Cox & D Oakes (1984) Analysis of survival data (Chapman & Hall)

• D Collett (2003) Modelling survival data in medical research (Chapman & Hall/CRC)

• TM Therneau & PM Grambsch (2000) Modeling survival data: extending the Cox Model (Springer)

• Martinussen & Scheike (2006) Dynamic Regression Models for Survival Data (Springer)