**Base-case. ** After constructing the variance–covariance matrix for the analysis by Goeree et al. [1], the template was used to replicate their analysis. The cost and effect pairs for each of the 1000 iterations, for each of the seven options, were first plotted on the cost-effectiveness plane (Fig. 1a). After determining which options were dominated (none) and subject to extended dominance (options C, D, and G), it was possible to estimate the efficiency frontier. In Figure 1a, it can be seen that this is composed of options B, A (ICER = $7755, compared with B), F (ICER = $12,183, compared with A), and E (ICER = $110,845, compared with F)—the efficiency frontier is presented for the mean costs and mean effects, rather than for each of the 1000 simulations.

The corresponding CEACs are also plotted in Figure 1b, where it can be seen that one of the optimal options (A) never had the highest probability of being cost-effective. Instead, B initially had the highest probability of being cost-effective (λ < $7970), followed by D ($7970 = λ < $12,540), C ($12,540 = λ < $14,850), F ($14,850 = λ < $112,120), and E (λ = $112,120). Thus, options that were subject to extended dominance (C and D) were also associated with the highest probability of being cost-effective at certain levels of ë. This provides an illustration of why CEACs should not and cannot be used to identify the optimal option, as has been previously shown by Fenwick et al. [17].

In order to simultaneously present the optimal option and the level of uncertainty associated with that option, the CEAF was plotted (Fig. 1c). The ICERs for options A, F, and E are plotted as vertical lines in order to demonstrate how the optimal option was identified. Prior to the first vertical line (λ < $7755), option B was optimal, between the first two vertical lines ($7755 = λ < $12,183), option A was optimal, followed by option F ($12,183 = λ < $110,845), and finally, option E (λ = $110,845). It should also be noted that the CEAF may be disjointed. This can be seen at the first vertical line (λ = $7755), where the probability of option B being cost-effective was 32.7%, whereas the corresponding probability for option A was 25.8%. Moreover, at this point, as well as the optimal option (A) not having the highest probability of being cost-effective, there was a 74.2% probability that option A was not the most cost-effective option. This arose because the optimal option was determined by the highest expected net benefit, whereas the CEAC simply represents the proportion of iterations over which each option had the highest net benefit.

By replicating the Goeree et al. [1] analysis, it was also possible to estimate the EVPI for an individual patient (Fig. 1c). The EVPI was relatively low when there was a high probability that the optimal option was cost-effective, with local maxima when the optimal option changed (and the probability of a wrong decision was relatively high), e.g., at a λ of approximately $12,000 and $111,000. However, this was not always the case, as when the optimal option changed from B to A there was a point of inflexion, but the EVPI continued to increase. This arose, despite there being a local maxima in the error probability, because the falling probability of making an incorrect decision was outweighed by the increasing consequences of a wrong decision. Finally, it should be noted that at high levels of the cost-effectiveness threshold, the EVPI fell to virtually zero. This was because of virtually nonexistent uncertainty at these levels of λ, where option E had a high probability of being cost-effective because of strong evidence that it was the most effective option.

**Sensitivity analysis. ** When the analyses by Goeree et al. [1] were replicated, with the assumption that the costs and effects between options were independent, the cost-effectiveness plane (Fig. 2a) was visually the same as in the base-case analyses (see Fig. 1a), because near identical cost and effect pairs (within options) were drawn in both of these probabilistic analyses. Thus, the ICERs were very similar to that in the base-case analyses. Conversely, the CEACs changed, with the curves tending to converge—those options that previously had the highest probability of being cost-effective, for a particular λ, tended to have a lower probability of being cost-effective, and other options tended to have a higher probability of being cost-effective. Importantly, this meant that option A (which was the optimal option between a λ of $7755 and $12,183) now had the highest probability of being cost-effective for a small range of λ-values, and option C (which was subject to extended dominance) no longer had the highest probability of cost-effectiveness over any threshold value. Here, for reasons of brevity, rather than plotting the CEACs in the traditional manner (as in Fig. 1b), they have been plotted on the same diagram as the CEAF and EVPI. In Figure 2b, the CEACs are plotted, where the (bold) continuous section of the CEAC denotes the range of λ-values over which each option was optimal, and the (lighter) dashed section of the CEAC denotes the range of λ-values over which each option did not provide the highest expected net benefit.

In the Goeree et al. [1] analysis, these two potentially counterintuitive results arose because all the options were positively correlated. Consequently, when, for example, option B had a low net benefit in a particular iteration, it was likely that in the same iteration, other options would also have a low net benefit, and at low levels of λ, this meant that option B had the highest net benefit in a high proportion of iterations (and thereby a high CEAC). Conversely, when the options were assumed to be independent, when option B had a low net benefit, then there was a higher chance of another option having a higher net benefit, in the same iteration, and option B thereby had the highest net benefit in a lower proportion of iterations.

The assumption that the costs and effects between options were independent also changed the CEAF (Fig. 2b), where the optimal options tended to have a lower probability of being cost-effective, and the EVPI (Fig. 2b), which also tended to be higher. Here, as in Figure 1c, the CEAF was again constructed by plotting the CEAC, as a continuous line, over the range of λ-values over which each option had the highest expected net benefit. It should also be noted that the EVPI even increased when the probability of the optimal option being cost-effective increased, e.g., at a λ of $12,000; in the base-case the EVPI was 41.4 when option A had a 17.9% probability of being cost-effective, yet it rose to 107.4 when the options were assumed to be independent and the probability of option A being cost-effective increased to 19.8%. This change in the EVPI can be explained by the fact that when option B, for example, had a high net benefit in a particular iteration in the base-case (independent) analysis, it was more (less) likely that other options would also have a high net benefit. Thus, in the base-case the consequences of the wrong decision (represented by the incremental net benefit) were estimated to be less than in the analysis, where the options were assumed to be independent.

The final two analyses had the same correlation structure as the base-case, but had differing levels of variance. In both analyses, the cost and effect pairs for each option were centered around the same means, and thereby ICERs, but the pairs were more condensed when the level of variation was reduced by a factor of five (Fig. 3a) and more spread out when the variation was inflated by a factor of five (Fig. 4a). Looking at the CEACs, reducing the level of variation tended to increase the probability of optimal options being cost-effective, at the expense of the probability associated with other options (Fig. 3b). Indeed, option A now had the highest probability of being cost-effective over a reasonable range of λ-values, and options C/D only briefly did so ($11,690 ≤ λ < $12,950).

In the Goeree et al. [1] analysis, the two potentially counterintuitive results arose because options C and D, which were subject to extended dominance, had relatively high levels of variation in net benefit, compared with option A. For example, at a λ of $12,000, the mean net benefits for options A, C, and D were $9844, $9841, and $9840, and the variances were $7426, $10,967, and $11,027. Thus, as the probability distributions surrounding the mean net benefit of options C/D were more spread out than for option A within a particular iteration, despite the marginally lower mean values of net benefit for C/D, there was a higher probability of a higher net benefit being drawn for options C/D than for option A. When the level of variation was reduced, the distributions became more tightly centered around the mean, and within a particular iteration, option A had a higher probability of having a higher net benefit than options C/D (see CEACs in Fig. 3b). Increasing the level of variation had the opposite effect on the CEACs (Fig. 4b), as those options that had a relatively high level of variation (e.g., options C and D) had the highest net benefit in a greater proportion of iterations, and thereby a greater probability of being cost-effective than in the base-case, at the expense of the those options that had a relatively low level of variation (e.g., options A and B).

Reducing the level of variation increased the proportion of iterations where optimal options had the highest net benefit. Thus, as optimal options tended to have the highest probability of being cost-effective, the CEAF was more often equivalent to the uppermost CEAC, and the probability of making a wrong decision tended to be lower (Fig. 3b). This, combined with the lower level of variation in net benefit, which reduced the consequences of making a wrong decision, meant that the EVPI was also lower when the variation was reduced (Fig. 3b). However, the converse occurred when the level of variation increased—the CEAF was less frequently equivalent to the uppermost CEAC, the probability of making the wrong decision tended to increase, as did the consequences of making a wrong decision, and in turn the EVPI also increased (Fig. 4b).