Cox proportional hazards regression is a semiparametric
method for adjusting survival rate estimates to remove the effect
of confounding variables and to quantify the effect of predictor variables.
The method represents the effects of explanatory and confounding variables
as a multiplier of a common baseline hazard function, *h*_{0}(*t*).

For a baseline relative to 0, this model corresponds to

$$h\left({X}_{i},t\right)={h}_{0}(t)\mathrm{exp}\left[{\displaystyle \sum _{j=1}^{p}{x}_{ij}{b}_{j}}\right],$$

where $${X}_{i}=({x}_{i1},{x}_{i2},\cdots ,{x}_{ip})$$ is
the predictor variable for the *i*th subject, *h*(*X*_{i},*t*)
is the hazard rate at time *t* for *X*_{i},
and *h*_{0}(*t*)
is the baseline hazard rate function. The baseline hazard function
is the nonparametric part of the Cox proportional hazards regression
function, whereas the impact of the predictor variables is a loglinear
regression. The assumption is that the baseline hazard function depends
on time, *t*, but the predictor variables do not
depend on time. See Cox Proportional Hazards Model for details, including
the extensions for stratification and time-dependent variables, tied
events, and observation weights.

## References

[1] Cox, D.R., and D. Oakes. *Analysis
of Survival Data*. London: Chapman & Hall, 1984.

[2] Lawless, J. F. *Statistical
Models and Methods for Lifetime Data*. Hoboken, NJ: Wiley-Interscience,
2002.

[3] Kleinbaum, D. G., and M. Klein. *Survival Analysis*.
Statistics for Biology and Health. 2nd edition. Springer, 2005.