[mice0]    Mice: Weibull regression

Dellaportas and Smith (1993) analyse data from Grieve (1987) on photocarcinogenicity in four groups, each containing 20 mice, who have recorded a survival time and whether they died or were censored at that time. A portion of the data, giving survival times in weeks, are shown below. A
* indicates censoring.


[mice1]

The survival distribution is assumed to be Weibull. That is

   
f (t i , z i ) = re b z i t i r - 1 exp(-e b z i t i r )
   
where t
i is the failure time of an individual with covariate vector z i and b is a vector of unknown regression coefficients. This leads to a baseline hazard function of the form

   
l 0 (t i ) = rt i r - 1


Setting
m i = e b z i gives the parameterisation

   t
i ~ Weibull( t , m i )

For censored observations, the survival distribution is a truncated Weibull, with lower bound corresponding to the censoring time. The regression
b coefficients were assumed a priori to follow independent Normal distributions with zero mean and ``vague'' precision 0.0001. The shape parameter r for the survival distribution was given a Gamma(1, 0.0001) prior, which is slowly decreasing on the positive real line.

Median survival for individuals with covariate vector
z i is given by m i = (log2e - b z i ) 1/r

The appropriate graph and BUGS language are below, using an undirected dashed line to represent a logical range constraint.

    [mice2]    

   model
   {   
      for(i in 1 : M) {
         for(j in 1 : N) {
            t[i, j] ~ dweib(r, mu[i])I(t.cen[i, j],)
            culmative.t[i, j] <- culmative(t[i, j], t[i, j])
         }
         mu[i] <- exp(beta[i])
         beta[i] ~ dnorm(0.0, 0.001)
         median[i] <- pow(log(2) * exp(-beta[i]), 1/r)
      }
      r ~ dexp(0.001)
      veh.control <- beta[2] - beta[1]
      test.sub <- beta[3] - beta[1]
      pos.control <- beta[4] - beta[1]
   }



We note a number of tricks in setting up this model. First, individuals who are censored are given a missing value in the vector of failure times t, whilst individuals who fail are given a zero in the censoring time vector t.cen (see data file listing below). The truncated Weibull is modelled using I(t.cen[i],) to set a lower bound. Second, we set a parameter beta[j] for each treatment group
j . The contrasts beta[j] with group 1 (the irradiated control) are calculated at the end. Alternatively, we could have included a grand mean term in the relative risk model and constrained beta[1] to be zero.


Data     ( click to open )

Inits ( click to open )

Results

A burn in of 1000 updates followed by a further 10000 updates gave the parameter estimates

[mice3]