**
**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.

The survival distribution is assumed to be Weibull. That is

f (t

where t

l

Setting m

t

For censored observations, the survival distribution is a truncated Weibull, with lower bound corresponding to the censoring time. The regression

Median survival for individuals with covariate vector

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

{

for(i in 1 : M) {

for(j in 1 : N) {

t[i, j] ~ dweib(r, mu[i])C(t.cen[i, j],)

cumulative.t[i, j] <- cumulative(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 C(t.cen[i],) to set a lower bound. Second, we set a parameter beta[j] for each treatment group

Results

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