Example 7.30: Simulate censored survival data

To simulate survival data with censoring, we need to model the hazard functions for both time to event and time to censoring.

We simulate both event times from a Weibull distribution with a scale parameter of 1 (this is equivalent to an exponential random variable). The event time has a Weibull shape parameter of 0.002 times a linear predictor, while the censoring time has a Weibull shape parameter of 0.004. A scale of 1 implies a constant (exponential) baseline hazard, but this can be modified by specifying other scale parameters for the Weibull random variables.

First we'll simulate the data, then we'll fit a Cox proportional hazards regression model (section 4.3.1) to see the results.

Simulation is relatively straightforward, and is helpful in concretizing the notation often used in discussion survival data. After setting some parameters, we generate some covariate values, then simply draw an event time and a censoring time. The minimum of these is "observed" and we record whether it was the event time or the censoring time.

SAS

data simcox;
  beta1 = 2;
  beta2 = -1;
  lambdat = 0.002; *baseline hazard;
  lambdac = 0.004; *censoring hazard;
  do i = 1 to 10000;
    x1 = normal(0);
    x2 = normal(0);
    linpred = exp(-beta1*x1 - beta2*x2);
    t = rand("WEIBULL", 1, lambdaT * linpred);
    * time of event;
    c = rand("WEIBULL", 1, lambdaC);
           * time of censoring;
    time = min(t, c);    * which came first?;
    censored = (c lt t);
    output;
  end;
run;

The phreg procedure (section 4.3.1) will show us the effects of the censoring as well as the results of fitting the regression model. We use the ODS system to reduce the output.

ods select censoredsummary parameterestimates;
proc phreg data=simcox;
  model time*censored(1) = x1 x2;
run;

The PHREG Procedure

Summary of the Number of Event and Censored Values
 
                                     Percent
   Total       Event    Censored    Censored
   10000        5971        4029       40.29

          Analysis of Maximum Likelihood Estimates
 
                Parameter    Standard
Parameter  DF    Estimate       Error  Chi-Square  Pr > ChiSq

x1          1     1.98628     0.02213   8059.0716      <.0001
x2          1    -1.01310     0.01583   4098.0277      <.0001

Analysis of Maximum Likelihood Estimates
 
             Hazard
Parameter     Ratio

x1            7.288
x2            0.363

R

n = 10000
beta1 = 2; beta2 = -1
lambdaT = .002 # baseline hazard
lambdaC = .004  # hazard of censoring

x1 = rnorm(n,0)
x2 = rnorm(n,0)
# true event time
T = rweibull(n, shape=1, scale=lambdaT*exp(-beta1*x1-beta2*x2)) 
C = rweibull(n, shape=1, scale=lambdaC)   #censoring time
time = pmin(T,C)  #observed time is min of censored and true
event = time==T   # set to 1 if event is observed

Having generated the data, we assess the effects of censoring with the table() function (section 2.2.1) and load the survival() library to fit the Cox model.

> table(event)
event
FALSE  TRUE 
 4083  5917 

> library(survival)
> coxph(Surv(time, event)~ x1 + x2, method="breslow")
Call:
coxph(formula = Surv(time, event) ~ x1 + x2, method = "breslow")


    coef exp(coef) se(coef)     z p
x1  1.98     7.236   0.0222  89.2 0
x2 -1.02     0.359   0.0160 -64.2 0

Likelihood ratio test=11369  on 2 df, p=0  n= 10000 

These parameters result in data where approximately 40% of the observations are censored. The parameter estimates are similar to the true parameter values.