# Regression Analysis Using the Proportional Hazards Model

• Dirk F. Moore
Chapter
Part of the Use R! book series (USE R)

## Abstract

In the previous chapter we saw how to compare two survival distributions without assuming a particular parametric form for the survival distributions, and we also introduced a parameter ψ that indexes the difference between the two survival distributions via the Lehmann alternative, $$S_{1}(t) = \left [S_{0}(t)\right ]^{\psi }$$. Using Eq. 2.2.1 we can see that we can re-express this relationship in terms of the hazard functions, yielding the proportional hazards assumption ,
$$\displaystyle{ h_{1}(t) =\psi h_{0}(t). }$$
(5.1.1)
This equation is the key to quantifying the difference between two hazard functions, and the proportional hazards model is widely used. (Later we will see how to assess the validity of this assumption, and ways to relax it when necessary.) Furthermore, we can extend the model to include covariate information in a vector z as follows:
$$\displaystyle{ \psi = e^{z\beta }. }$$
(5.1.2)
While other functional relationships between the proportional hazards constant ψ and covariates z are possible, this is by far the most common in practice. This proportional hazards model will allow us to fit regression models to censored survival data, much as one can do in linear and logistic regression. However, not assuming a particular parametric form for h0(t), along with the presence of censoring, makes survival modeling particularly complicated. In this chapter we shall see how to do this using what we shall call a partial likelihood . This modification of the standard likelihood was developed initially by D.R. Cox [12], and hence is often referred to as the Cox proportional hazards model.

## Keywords

Score Function Failure Time Baseline Hazard Survival Distribution Partial Likelihood
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Aalen, O.: Nonparametric inference for a family of counting processes. Ann. Stat. 701–726 (1978)Google Scholar
2. 3.
Andersen, P.K., Borgan, O., Gill, R.D., Keiding, N.: Statistical Models Based on Counting Processes, corrected edition. Springer, New York (1996)Google Scholar
3. 12.
Cox, D.R.: Regression models and life-tables. J. R. Stat. Soc. Ser. B Methodol. 187–220 (1972)Google Scholar
4. 19.
Fleming, T.R., Harrington, D.P.: Counting Processes and Survival Analysis. Wiley, Hoboken (2011)Google Scholar
5. 28.
Harrell, F.E.: Regression Modeling Strategies: With Applications to Linear Models, Logistic Regression, and Survival Analysis, 2nd edn. Springer Science & Business Media, New York (2015)Google Scholar