Cox regression with frailties

freireich:etal:63's data actually arise via a paired design, although this information has been ignored in most published analyses. Patients were matched according to their remission status (partial or complete). One patient from each pair received the drug 6-MP whilst the other received the placebo. We may introduce a fourth column (called pair) in the BUGS data file ( leukfr.dat) to indicate each of the 21 pairs of patients:

  1 1  0.5  1
  1 1  0.5  2
  2 1  0.5  3
  . .   .   .
  . .   .   .
 17 1  0.5 19
 22 1  0.5 20
 23 1  0.5 21
  6 1 -0.5 19
  6 1 -0.5 18
  6 1 -0.5  8
  6 0 -0.5  1
  . .   .   .
  . .   .   .
 35 0 -0.5 21

We model the potential `clustering' of failure times within pairs of patients by introducing a group-specific random effect or frailty term into the proportional hazards model. Using the counting process notation introduced in the Leuk example, this gives

A non-informative Gamma prior is assumed for , the precision of the frailty parameters. Note that the above `additive' formulation of the frailty model is equivalent to assuming multiplicative frailties with a log-Normal population distibution. Alternatively clayton:91 discusses the Cox proportional hazards model with multiplicative frailties, but assumes a Gamma population distribution. However, this formulation does not lead conclusively to log-concave full conditional distributions, and so cannot currently be implemented in BUGS.

The modifications to the BUGS code needed to include a frailty term in the leuk example are shown below, and may be found in the file leukfr.bug.

for(j in 1:T) {
  for(i in 1:N) { 
    dN[i,j]   ~ dpois(Idt[i,j]);                         # Likelihood
    Idt[i,j] <- Y[i,j]*exp(beta*Z[i]+b[pair[i]])*dL0[j]; # Intensity  
  }
  ......... 
}                           
for(k in 1:Npairs) {
    b[k] ~ dnorm(0.0, tau); 
}
tau ~ dgamma(0.001, 0.001); sigma <- sqrt(1/tau);

Analysis

2000 iterations took 3 minutes after a 1000 iteration burn-in. The posterior mean (standard error) of the regression coefficient was 1.58 (0.43). This is slightly larger than the estimate of when the clustering was ignored. However, the posterior mean of was only 0.18 (95% credible interval 0.02-0.59). This suggests that population variation in frailty was small, and hence there is relatively little clustering or association of failure times within matched pairs.