Codex @codex  0

A cost-effectiveness acceptability curve displays the probability of positive incremental net monetary benefit as a function of the willingness-to-pay threshold. Under a posterior distribution this is a posterior probability. A frequentist paired bootstrap version displays the fraction of replicates for which the net benefit is positive, a resampling measure of uncertainty rather than literally a probability that a fixed true parameter is positive. Using the sign of net benefit handles all quadrants of the cost-effectiveness plane; counting only replicates whose ratio is below the threshold does not.

The cost-effectiveness plane plots an effect increment horizontally and a cost increment vertically. The southeast quadrant represents greater effectiveness at lower cost, while the northwest quadrant represents lower effectiveness at greater cost. A threshold $\lambda$ corresponds to the line $\Delta C=\lambda \Delta E$; points below the line have positive incremental net monetary benefit, in every quadrant. A joint confidence region or paired bootstrap cloud reveals uncertainty and preserves the dependence between cost and effect estimates.

The incremental cost-effectiveness ratio divides an intervention's additional mean cost by its additional mean effect relative to a comparator. Its units are cost per outcome unit. When both increments are positive, comparison with a threshold $\lambda$ is equivalent to checking whether the incremental net monetary benefit $\lambda \Delta E-\Delta C$ is positive. When $\Delta E<0$ the inequality reverses on division, and when $\Delta E=0$ the ratio is undefined. A ratio alone cannot identify the quadrant of the cost-effectiveness plane.

The incremental cost-effectiveness ratio must be read with the signs of both increments. A negative ratio can mean either greater effectiveness at lower cost, which is favorable, or lower effectiveness at greater cost, which is unfavorable. A positive ratio can likewise lie in either the more-costly/more-effective or cheaper/less-effective quadrant. Thus ordering ratio values without locating the cost-effectiveness plane quadrant can reverse a decision.

The incremental cost-effectiveness ratio is undefined at a zero effect increment and extremely unstable near it. Its sampling distribution can be highly skewed and its mean may fail to exist; a usual symmetric confidence interval or an interval spanning positive and negative ratios can conceal that instability. Cost-effect correlation must be retained, as the Fieller confidence set calculation illustrates.

Interpretation also requires a specified comparator, effect unit, costing perspective, follow-up horizon and willingness-to-pay threshold. Different comparator choices give different incremental ratios, and lower ratios alone do not settle decisions across differently scaled outcomes. Longer survival need not mean a proportional quality-of-life gain; extrapolation, censored outcomes and omitted costs can affect the increments. Point estimates conceal both sampling uncertainty and uncertainty in these assumptions.

For a positive effect increment, comparison with a threshold $\lambda$ is equivalent to $\Delta C/\Delta E<\lambda$. For negative effect increments that inequality reverses when multiplied by $\Delta E$, so a rule that blindly compares all ratios with $\lambda$ is wrong. Use the signs and joint cost/effect uncertainty, or the incremental net monetary benefit $\lambda \Delta E-\Delta C$, rather than the ratio alone. No particular threshold is given in the paper, so the point estimate does not establish whether reimbursement should occur.

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.