Acceptability Curves and Confidence Ellipses

Confidence Ellipse

A confidence ellipse provides a visual representation of the uncertainty surrounding costs and effects (or indeed any two variables). The ellipse provides a region on the cost-effectiveness plane that should contain x% (e.g., 95%) of the uncertainty. By varying x, a series of contour lines can be plotted on the cost-effectiveness plane, each containing the relevant proportion of the cost and effect pairs. Figure 1 illustrates 95%, 50%, and 5% confidence ellipses.

Construction of the confidence ellipse requires the assumption that the costs and effects follow a bivariate normal distribution, that is, for each value of cost, the corresponding values of effect are normally distributed (and vice versa).

The drawback with the confidence ellipse is that while it presents the uncertainty around the costs and effects, it does not deal with the uncertainty surrounding the incremental cost-effectiveness ratio (ICER). One solution to this is to use the boundaries of the relevant confidence ellipse to approximate confidence intervals (e.g., 95%) for the ICER. This interval is given by the slopes of the rays from the origin, which are just tangential to the relevant ellipse (identified in Figure 2). Note that these will be overestimates of the confidence interval.

The particular shape and orientation of the confidence ellipse will be determined by the covariance [Page 2]of the costs and effects. This will in turn affect the confidence intervals estimated from the ellipse. Figure 3 illustrates the influence of the covariance on the confidence ellipse and the confidence limits.

Cost-Effectiveness Acceptability Curves

In contrast, the acceptability curve (or cost-effectiveness acceptability curve [CEAC]) focuses on the uncertainty surrounding the cost-effectiveness. An acceptability curve provides a graphical presentation of the probability that the intervention is cost-effective (has an ICER below the cost-effectiveness threshold) compared with the alternative intervention(s), given the data, for a range of values for the cost-effectiveness threshold. It should be noted that this is essentially a Bayesian view of probability (probability that the hypothesis is true given the data) rather than a frequentist/classical view of probability (probability of getting the data, or data more extreme, given that the hypothesis is true). It has been argued that this is more appropriate to the decision maker, who is concerned with the probability that the intervention is cost-effective (hypothesis is correct) given the cost-effectiveness results. However, a frequentist interpretation of the acceptability curve has been suggested, as the 1 − p value of a one-sided test of significance.

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