 Solution

One presentation is the cost-effectiveness plane, with $\Delta E$ horizontally and $\Delta C$ vertically. Plot the point estimate $(45,1800)$ with a joint confidence region or paired bootstrap cloud. Its position shows the tradeoff quadrant, while the cloud shows uncertainty, dependence and possible dominance. A willingness-to-pay threshold $\lambda$ is the line $\Delta C=\lambda \Delta E$ through the origin. Points below that line have positive incremental net monetary benefit, including points in quadrants where a ratio rule would fail. The point-estimated switch occurs at $\lambda =40$ pounds per day.

A second presentation is a cost-effectiveness acceptability curve, giving an uncertainty-based measure of cost-effectiveness for a range of thresholds. For paired bootstrap increments $(\Delta E^{\ast (b)},\Delta C^{\ast (b)})$, plotThis counts positive net benefits, not merely ratios below the threshold, so it handles all quadrants. Using Bayesian inference the curve is the posterior probability that incremental net monetary benefit is positive; a frequentist bootstrap curve is a resampling summary, not literally a probability that a fixed parameter is positive.

Here the estimated incremental net monetary benefit and its standard error areThus a normal approximation uncertainty curve isIt passes through one half at £40 per day. Alternatively display the incremental net monetary benefit curve itself with approximate pointwise $95\%$ bands $45\lambda -1800\pm 1.96s_{\mathrm{INMB}}(\lambda )$. The plane preserves the joint uncertainty; the acceptability curve shows how the threshold changes the cost-effectiveness assessment. Neither presentation chooses society's willingness-to-pay threshold.