Cox Proportional Hazards Model for Survival Analysis

This article discusses the Cox Proportional Hazards Model for Survival Analysis using data from the 'Survival' dataset. The model is used to estimate the effect of various factors on survival time. The article also covers the calculation of cumulative risk and plotting of survival and cumulative hazard curves.

> Survival= read.csv('Survival.csv')
> head(Survival)
ID X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 time1 status1 time2
1 066-0556 0 0 1 1 1 0 1 0 1 0 1 0 0 1129 0 366
2 041-2483 0 1 1 0 0 0 0 0 1 0 1 0 0 823 0 187
3 014-0836 0 0 1 1 1 0 0 0 1 0 0 0 0 797 0 366
4 033-0648 0 1 1 1 1 1 0 0 1 0 1 0 0 764 0 366
5 067-0126 0 1 1 1 1 0 0 0 1 0 0 0 0 741 0 366
6 066-0522 1 0 1 1 1 0 0 0 1 0 0 0 0 720 0 366
status2
1 0
2 0
3 0
4 0
5 0
6 0
> dim(Survival)
[1] 10687 18
> library(survival)
> dim(Survival)
[1] 10687 18
> y = Surv(Survival$time1,Survival$status1==1)
> x =Surv(Survival$time2,Survival$status2==1)
> model1= coxph(y~X1+X2+X3+X4+X5+X6+X7+X8+X9+X10+X11+X12, data = Survival)
> summary(model1)
Call:
coxph(formula = y ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 +
X10 + X11 + X12, data = Survival)
n= 10687, number of events= 638
coef exp(coef) se(coef) z Pr(>|z|)

X1 -0.37767 0.68546 0.11201 -3.372 0.000747 ***
X2 0.40135 1.49385 0.05183 7.743 9.69e-15 ***
X3 -1.78953 0.16704 0.09085 -19.698 < 2e-16 ***
X4 -1.17933 0.30748 0.08792 -13.414 < 2e-16 ***
X5 -0.73531 0.47936 0.10084 -7.292 3.05e-13 ***
X6 0.74169 2.09947 0.08930 8.306 < 2e-16 ***
X7 0.47168 1.60269 0.08303 5.681 1.34e-08 ***
X8 0.53454 1.70666 0.10934 4.889 1.01e-06 ***
X9 -0.91524 0.40042 0.24974 -3.665 0.000248 ***
X10 0.43303 1.54192 0.12677 3.416 0.000636 ***
X11 -0.31732 0.72810 0.08507 -3.730 0.000191 ***
X12 0.33538 1.39847 0.08924 3.758 0.000171 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
exp(coef) exp(-coef) lower .95 upper .95
X1 0.6855 1.4589 0.5504 0.8537
X2 1.4938 0.6694 1.3495 1.6536
X3 0.1670 5.9866 0.1398 0.1996
X4 0.3075 3.2522 0.2588 0.3653
X5 0.4794 2.0861 0.3934 0.5841
X6 2.0995 0.4763 1.7624 2.5010
X7 1.6027 0.6240 1.3620 1.8859
X8 1.7067 0.5859 1.3775 2.1146
X9 0.4004 2.4974 0.2454 0.6533
X10 1.5419 0.6485 1.2027 1.9768
X11 0.7281 1.3734 0.6163 0.8602
X12 1.3985 0.7151 1.1741 1.6658
Concordance= 0.84 (se = 0.012 )
Rsquare= 0.142 (max possible= 0.663 )
Likelihood ratio test= 1632 on 12 df, p=<2e-16

Wald test = 1911 on 12 df, p=<2e-16
Score (logrank) test = 3280 on 12 df, p=<2e-16
> model2 = coxph(y~X1+X2+X8+X12+X13,data = Survival)
> summary(model2)
Call:
coxph(formula = y ~ X1 + X2 + X8 + X12 + X13, data = Survival)
n= 10687, number of events= 638
coef exp(coef) se(coef) z Pr(>|z|)
X1 -0.54984 0.57704 0.11042 -4.979 6.38e-07 ***
X2 0.54871 1.73102 0.05209 10.534 < 2e-16 ***
X8 1.18657 3.27582 0.10872 10.914 < 2e-16 ***
X12 0.56976 1.76785 0.09221 6.179 6.46e-10 ***
X13 -0.25291 0.77654 0.24649 -1.026 0.305
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
exp(coef) exp(-coef) lower .95 upper .95
X1 0.5770 1.7330 0.4647 0.7165
X2 1.7310 0.5777 1.5630 1.9171
X8 3.2758 0.3053 2.6471 4.0538
X12 1.7678 0.5657 1.4755 2.1181
X13 0.7765 1.2878 0.4790 1.2589
Concordance= 0.734 (se = 0.011 )
Rsquare= 0.045 (max possible= 0.663 )
Likelihood ratio test= 487.6 on 5 df, p=<2e-16
Wald test = 543 on 5 df, p=<2e-16
Score (logrank) test = 673 on 5 df, p=<2e-16

> #calculating cumulative risk from model1
> #we use survfit fits curves
> #Kaplan-Meier is also used
> fit_estimate <- survfit(y~1,type="kaplan-meier", conf.type="log-log")
> fit_estimate
Call: survfit(formula = y ~ 1, type = "kaplan-meier", conf.type = "log-log")
n events median 0.95LCL 0.95UCL
10687 638 NA NA NA
> #plot survival
> plot(fit_estimate, main="survival estimates", xlab="time1", ylab="x")
>

#can plot cumulative hazard
> plot(fit_estimate, fun="cumhaz",xlab ='time', ylab="x")
>