11.14 Cox Proportional Hazards Model

  • Because \(h_0(t)\) is unknown we cannot just plug \(h(t|x_i)\) into the likelihood function and apply maximum likelihood

  • Cox proportionalhazards model (Cox, 1972) estimates \(\beta\) without having to specify the form of \(h_0(t)\) by using a partial (relative) likelihood where \(h_0(t)\) cancels out. (Details in text)

Let’s fit the Cox proportional hazards models using the coxph() function. To begin, we consider a model that uses sex as the only predictor.

fit.cox <- coxph(Surv(time, status) ~ sex, data = BrainCancer)
summary(fit.cox)
## Call:
## coxph(formula = Surv(time, status) ~ sex, data = BrainCancer)
## 
##   n= 88, number of events= 35 
## 
##           coef exp(coef) se(coef)     z Pr(>|z|)
## sexMale 0.4077    1.5033   0.3420 1.192    0.233
## 
##         exp(coef) exp(-coef) lower .95 upper .95
## sexMale     1.503     0.6652     0.769     2.939
## 
## Concordance= 0.565  (se = 0.045 )
## Likelihood ratio test= 1.44  on 1 df,   p=0.2
## Wald test            = 1.42  on 1 df,   p=0.2
## Score (logrank) test = 1.44  on 1 df,   p=0.2

Regardless of which test we use, we see that there is no clear evidence for a difference in survival between males and females.

Now we fit a model that makes use of additional predictors.

fit.all <- coxph(
Surv(time, status) ~ sex + diagnosis + loc + ki + gtv +
   stereo, data = BrainCancer)
fit.all
## Call:
## coxph(formula = Surv(time, status) ~ sex + diagnosis + loc + 
##     ki + gtv + stereo, data = BrainCancer)
## 
##                        coef exp(coef) se(coef)      z        p
## sexMale             0.18375   1.20171  0.36036  0.510  0.61012
## diagnosisLG glioma  0.91502   2.49683  0.63816  1.434  0.15161
## diagnosisHG glioma  2.15457   8.62414  0.45052  4.782 1.73e-06
## diagnosisOther      0.88570   2.42467  0.65787  1.346  0.17821
## locSupratentorial   0.44119   1.55456  0.70367  0.627  0.53066
## ki                 -0.05496   0.94653  0.01831 -3.001  0.00269
## gtv                 0.03429   1.03489  0.02233  1.536  0.12466
## stereoSRT           0.17778   1.19456  0.60158  0.296  0.76760
## 
## Likelihood ratio test=41.37  on 8 df, p=1.776e-06
## n= 87, number of events= 35 
##    (1 observation deleted due to missingness)

  • The diagnosis variable has been coded so that the baseline corresponds to meningioma.

  • Results indicate that the risk associated with HG glioma is more than eight times (i.e. \(e^{2.15}=8.62\)) the risk associated with meningioma. In other words, after adjusting for the other predictors, patients with HG glioma have much worse survival compared to those with meningioma.

  • In addition, larger values of the Karnofsky index, ki, are associated with lower risk, i.e. longer survival.