Cox model relates the time that passes before
some event occurs to one or more covariates that may be
associated with that
quantity.
Cox model can be viewed as consisting of two
parts: the underlying hazard function, often denoted Λ_{0}(t),
describing how the hazard (risk) changes over time at baseline levels of
covariates; and the effect parameters, describing how the hazard varies in
response to explanatory covariates.
The proportional hazards condition
states that covariates are multiplicatively related to the hazard. For example,
a treatment with a drug may decrease a subject's hazard at any given time t 20%, while the baseline hazard may vary.
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.
Outcome variable: which is
coded as 1 (had the endpoint) and 0 (end point is unknown).
Time variable: the time
to the event (if had the endpoint), or time to drop out (if endpoint is
unknown)
Several methods have been proposed to handle
situations in which there are ties in the time data. Breslow's
method uses unmodified approach, even when ties are present. An alternative
approach that is considered to give better results is Efron's
method.
If there is timevarying predictor, separates
the observations by the status of the predictor and includes the start
time and stop time for the status of predictor. Below is an example showing observations for
timevarying predictor passive smoke, the time variable is menstrual cycle,
event is pregnant.
Subject ID 
Start
cycle 
Stop cycle 
Passive
smoke 
Pregnant 
… 




100 
1 
2 
0 
0 
100 
3 
5 
1 
0 
100 
6 
7 
0 
1 
In this example, ID=100 takes 7 menstrual
cycle to become pregnant, during the first 2 cycles and last 2 cycles, she had
none passive smoke exposure, during cycle 3 to 5, she had passive smoke
exposure.
Start time and stop
time variables need to be specified if there are timevarying
predictor.
Below is the sample
screen shot of input window
A sample
output and explanation:
Proportional Harzard
Model
Call:
coxph(formula = Surv(CYCLE,
PREG) ~ AGE + BMI + factor(EDU) +
HSMK, data = WD, method = "efron")
n= 199, number of events= 185
coef exp(coef) se(coef) z Pr(>z)
AGE 0.008683 0.991355 0.024749 0.351 0.726
BMI 0.012254 0.987821 0.053866 0.227 0.820
factor(EDU)1 0.086291
1.090124 0.183462 0.470
0.638
factor(EDU)2 0.268964
1.308609 0.256657 1.048
0.295
HSMK 0.135586 1.145208 0.160831
0.843 0.399
exp(coef) exp(coef) lower .95 upper
.95
AGE 0.9914 1.0087
0.9444 1.041
BMI 0.9878 1.0123
0.8888 1.098
factor(EDU)1 1.0901
0.9173 0.7609 1.562
factor(EDU)2 1.3086
0.7642 0.7913 2.164
HSMK 1.1452 0.8732
0.8356 1.570
Rsquare= 0.011 (max possible= 1 )
Likelihood ratio test= 2.18 on 5 df, p=0.8241
Wald test = 2.21 on 5 df, p=0.8196
Score (logrank)
test = 2.22 on
5 df,
p=0.8186
The exp(coef)
is the hazard ratio (HR). Exp(coef)
is 1 over hazard ratio.
Empower also output a survival curve and
95% CI