170 CHAPTER 10 Survival Analysis10.5 COX PROPORTIONAL HAZARD MODELS
The Cox proportional hazards regression model is called semi -
parametric because it includes regression parameters for covariates
(which may or may not be time dependent), but in terms of the baseline
hazard function, it is completely general (hence not parametric). So part
of the modeling is parametric, and another part is nonparametric, hence
the term semi - parametric. In SAS ® , the model can be implemented
using the procedure PHREG, or STCOX in STATA. An excellent and
detailed treatment with SAS applications can be found in Walker and
Shostak ( 2010 , pp. 413 – 428). A similar treatment using the STATA
software package can be found in Cleves et al. (2008).
The purpose of the model is to test for the effects of a specifi c set
of k covariates on the event times. These covariates can be numerical
or categorical. In the case of categorical variables, such as treatment
groups, the model can estimate relative risks for the occurrence of an
event in a fi xed interval when the patient gets treatment A versus when
the patient gets treatment B.
For example, in the RE - LY trial to compare three treatments, two
doses of dabigatran and warfarin as a control, the Cox model was used
to estimate the relative risk of the patient getting a stroke during the
trial while on one treatment versus another. This ratio was used to test
for superiority or noninferiority of the dabigatran doses versus warfarin
with respect to stroke or systemic embolism as the event. The model
was also used for other types of event, with major bleeding being a
primary safety endpoint.
The model is defi ned by its hazard function h ( t ) = λ ( t )exp( β 1 X 1 +
β 2 X 2 +... + β (^) m X m ), where m is the number of covariates the X i are the
covariates and is the baseline hazard function ( t represents time). We
only consider t ≥ 0. It is called a proportional hazards model because
h ( t ) is proportional to λ ( t ), since h ( t )/ λ ( t ) is a constant (does not depend
on t ) that is determined by the covariates. The parameters β (^) i are esti-
mated by maximizing the partial likelihood. The estimation procedure
will not be described here, but its computation requires the use of
numerical methods and high - speed computers.
There are many books on survival analysis that cover the Cox
model, and even some solely dedicate to the method. A recent text
providing an up - to - date theoretical treatment is O ’ Quigley (2008) ,
which includes over 700 references. Other texts worthy of mention are