Diagnostics of Cox Proportional Hazards Models
In Cox model, there is a baseline hazard function h0(t) which is modified multiplicatively by covariates
(including group indicators), so hazard function for any individual case is
h(t)=h0(t)exp(βTx).
Log-Log Survival Plot
Plotting log-log transformation of the K-M survival curve, namely, log-log(S(t)), versus time t provides a
way of assessing the PH model assumption. Since
log[H( t | X*)] - log [H( t | X)] = βT ( X* - X) by PH model assumption, we would see the difference between
the two log-log K-M curves is constant over t . This method works when we have categorical covariates
involved. Remember that you can always categorize a continuous variable using cut-points. If PH holds, we
should get parallel curves for different groups.
We make this plot first for a data set generated from a PH model and then for a data set not satisfying PH
assumption.
par(mfrow=c(2,1), oma=rep(4,4))
# “data1” is generated from rtime2. Note that the default setting for rtimr2() will generate data
from exp(1) model
# z2 is ordinal with 2 categories.
data1 <- rtime2(N=300, p.censor=0.3)
data1.surv<-survfit(Surv(time, status)~factor(z2), data=data1)
plot(data1.surv, lty=c(3,2), fun="cloglog", xlab="log(time)", ylab="log-log
S(t)",main="Exponential(PH) Model")
legend(10, -3, c("z2=0", "z2=1"), lty=c(3,2))
# Here “data” is simulated using rtime3 – the bimodal lognormal model
rtime3 <- function(N=100, beta=c(2, log(2)), mu=1, p.censor=0.3)
{
w <- 1:N
for (i in 1:N) {
w[i] <- sample(c(rnorm(1, 1, 0.15), rnorm(1, 3, .15)), 1, replace=F)
}
z1 <- sample(c(0,1),N, replace=T)
z2 <- runif(N, 0, 1)
if (p.censor==0) status <- 1
else if (0 < p.censor && p.censor < 1) {
status <- sample(c(0,1), size=N, replace=T,
prob=c(p.censor,1-p.censor))}
else stop("Wrong Argument in p.censor.")
data.frame(time=exp(mu + beta[1]*z1 + beta[2]*z2 + w),
status=status,
z1 = z1, z2=z2)
}
# z1 is bianry
data2 <- rtime3(N=100, beta=c(2, log(2)), mu=1, p.censor=0.3)
data2.surv <- survfit(Surv(time, status)~factor(z1), data=data2)
plot(data2.surv, lty=c(3,2), fun="cloglog", xlab="log(time)", ylab="log-log S(t)", main="Bimodal
log-Normal (NON-PH) Model")
legend(400, -3, c("z1=0", "z1=1"), lty=c(3,2))
title(main="Bimodal Log-Normal (NON-PH) Model")
Residuals
Most diagnostic tools in survival analysis are based on various residuals.
1. The martingale residual, which is the default, is used for discovering the correct functional form
for a predictor. For example, consider the lung cancer data.
> nlung <- na.omit(lung[, c("time", "status", "sex",
+ "ph.ecog", "ph.karno", "pat.karno", "wt.loss")])
> par(mfrow = c(2,2))
> attach(nlung)
> fit.1 <- coxph(Surv(time,status) ~ strata(sex) +
+ ph.karno + pat.karno + wt.loss, data = nlung)
> scatter.smooth(ph.ecog, resid(fit.1))
> fit.2 <- coxph(Surv(time,status) ~ strata(sex) +
+ ph.ecog + pat.karno + wt.loss, data = nlung)
> scatter.smooth(ph.karno, resid(fit.2))
> fit.3 <- coxph(Surv(time,status) ~ strata(sex) +
+ ph.ecog + ph.karno + wt.loss, data = nlung)
> scatter.smooth(pat.karno, resid(fit.3))
> fit.4 <- coxph(Surv(time,status) ~ strata(sex) +
+ ph.ecog + ph.karno + pat.karno, data = nlung)
>scatter.smooth(wt.loss,resid(fit.4))
> mtext("Detecting Functional Form with Martingale Residuals",
outer=T, side=1)
2. The deviance residual, which is a normalized transform of martingale residual, can be used for
identifying poorly predicted subjects. However, it has shown that deviance residuals do not work
well and cannot be recommended.
fit <- coxph(Surv(time, status)~z1+z2, data=data1)
plot(resid(fit, type="deviance"), ylab="Deviance Residuals")
title(sub="(Data from Weibull Model)")
abline(0)
3.
The Cox-Snell residual.
Same as in parametric models, plotting the cumulative hazards function of Cox-Snell residuals provides a
way of checking goodness of fit.
cox.snell <- data1$status - resid(fit)
sv <- survfit(Surv(cox.snell, data1$status)~1)
plot(sv$time, -log(sv$surv), ylab="H.hat(ri)", xlab="ri")
abline(0,1)
title(main="Cumulative Hazards of Cox-Snell Residuals",
sub="A way of checking goodness of fit for Cox models")
> cox.zph(fit)
rho
z1
0.0729
z2
-0.0374
GLOBAL NA
chisq
1.143
0.318
1.479
p
0.285
0.573
0.477