Chapter 3 Exercises
Heart transplant cost-effectiveness

1. In Example 3.5.1 (heart transplants), extend the model to estimate the cost-effectiveness of transplantation, as in Example 2.6.1.

2. Suppose the policy-maker is willing to pay q = £25,000 for each year of life gained. What is the posterior probability that transplantation is cost-effective according to this policy?
[Hint: calculate this by summarising the "incremental net benefit" of transplantation, defined as q*Is - Ic, where Is is the incremental survival and Ic is the incremental cost]

3. Compute the posterior probability that the incremental cost divided by the incremental survival is less than the willingness-to-pay amount of £25,000. Compare this with the answer to part 2, and explain any difference. Why is the method in part 2 a more appropriate way to calculate the probability of cost-effectiveness?

Solutions

model {
yT ~ dbin(pT, nT)
pT ~ dunif(0, 1)
for (i in 1:8) {
sP[i] ~ dexp(theta)
}
theta ~ dgamma(0.001, 0.001)
surv.t <- pT/theta # expected survival with transplant
Is <- surv.t - 2
   
Ic   <- 20000 + 3000*surv.t # incremental cost (part 1)
   CE <- step(25000*Is - Ic) # indicator for cost-effectiveness (part 2)

   CE2 <- step(25000 - Ic / Is) # indicator for ICER < amount willing to pay (part 3)
}

Data:
list(yT=8, nT=10, sP=c(2,3,4,4,6,7,10,12))

Inits:
list(theta=1)

   node   mean   sd   MC error   2.5%   median   97.5%   start   sample
   node   mean   sd   MC error   2.5%   median   97.5%   start   sample
   CE   0.841   0.3657   5.252E-4   0.0   1.0   1.0   1   501000
   CE2   0.8547   0.3524   4.992E-4   0.0   1.0   1.0   1   501000
   Ic   35410.0   6811.0   9.406   26600.0   34010.0   52500.0   1   501000

   Is   3.136   2.27   0.003135   0.2015   2.669   8.835   1   501000
   pT   0.7496   0.1202   1.655E-4   0.4821   0.7639   0.9396   1   501000
   surv.t   5.136   2.27   0.003135   2.201   4.669   10.83   1   501000


1. The expected cost per life year gained from transplantation (incremental cost-effectiveness ratio) is obtained from dividing the posterior mean incremental cost by the posterior mean incremental survival, in this case, £35410 / 5.136 = £6894.

2. The posterior probability of cost-effectiveness is the posterior probability that the incremental net benefit is positive, in this case 0.841.

3. The probability that Ic/Is is less than the cost-effectiveness threshold of £25,000 is 0.8547. This is slightly different to step 2 (we can check by running more iterations that both have converged to 4 significant figures) and an inappropriate way to calculate the probability of cost-effectiveness.

This is because the interpretation of Ic/Is changes according to the sign of Ic and Is. If Ic > 0 and Is < 0 then the new treatment (transplantation in this case) is more expensive and is associated with worse expected survival, therefore it is neither effective nor cost-effective, even though Ic / Is < £20,000. If Ic < 0 and Is < 0 then the treatment is cheaper though is associated with worse survival, then if (-Ic)/(-Is) = Ic/Is < £25000 then it will be more cost effective to not perform the transplant!

However, if the incremental net benefit q*Is - Ic > 0, then the new treatment is cost-effective whatever the sign of Is and Ic.