Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 41 / 5

Let $\Delta C$ and $\Delta E$ be the new-minus-standard mean cost and survival increments. The incremental cost-effectiveness ratio isUsing $365$ days per year, this is £14,600 per additional life-year. It is not cost per quality-adjusted life-year because no quality-of-life adjustment is supplied. Both point-estimated increments are positive, so the new intervention exchanges higher cost for longer survival; this is not dominance.

Assume the two trial arms are independent samples, their cost and survival means have a valid joint normal approximation, and the supplied standard errors and correlation coefficient estimate the joint uncertainty in their mean differences. This could follow from a central limit theorem for patient-level cost/survival pairs, with finite second moments and suitably handled survival follow-up. The supplied covariance matrix for $(\widehat{\Delta C},\widehat{\Delta E})$ isThe off-diagonal entry has units pounds times days, and must not be discarded. Independence of trial arms does not mean cost and survival increments are independent: patient-level cost and survival are paired within each arm.

For a proposed ratio $r$, the null assertion is $\Delta C-r\Delta E=0$. Its estimated variance is $90000-4500r+225r^2$. Fieller's theorem inverts the associated two-sided normal tests. With $q={\Phi }^{-1}(0.975)\approx 1.96$, retain $r$ whenExpanding givesThe leading coefficient is positive, since $45>q(15)$; the effect increment is separated from zero at this level. With $q^2=3.841459$, the polynomial is approximatelyIt is nonpositive between its two real roots. The resulting approximate $95\%$ Fieller confidence set is therefore a bounded confidence interval:On the $365$-day life-year scale the endpoints are approximately £9,133 and £36,376. Exact coverage would need an exact pivot with its corresponding critical value; using estimated covariance and a central limit theorem makes this an asymptotic interval. In other data, a poorly determined denominator can give unbounded or disconnected Fieller confidence sets, which should not be replaced by an artificially bounded interval.

An alternative is a paired bootstrap: independently resample patients within each randomized arm, keeping each patient's cost and survival together; recompute both mean differences and their ratio in each replicate. The $2.5\%$ and $97.5\%$ empirical ratio quantiles give a percentile bootstrap confidence interval. The raw paired data are needed to calculate it, so the numerical endpoints cannot be obtained from this summary table alone. Near-zero bootstrap effect increments or sign changes warn that ordinary ratio-percentile intervals may be unreliable; one should retain the joint cost/effect uncertainty or invert a suitable bootstrap test instead.

A simpler first-order alternative is the delta method. For $r=\Delta C/\Delta E$, the gradient is $(1/\Delta E,-\Delta C/\Delta E^2)$, givingThe resulting approximate Wald confidence interval is $40\pm 1.96\sqrt{133.333}=(17.37,62.63)$ pounds per day. Its symmetric first-order form misses the denominator's nonlinear effect, explaining its marked difference from the Fieller confidence set; it is not the preferred ratio interval here.

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.