Proportional Hazards Model (Cox Model)

 

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 Λ0(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)

Tied times

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.

Time-varying predictor

If there is time-varying 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 time-varying 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 time-varying predictor.

Below is the sample screen shot of input window

 

SNAGHTML17e88758

 

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