**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.