Confidence Ellipse

A graphical region, typically on a cost-effectiveness plane, showing where the true joint value of incremental cost and effect is expected to fall.

Source: Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006. doi:10.1093/oso/9780198526629.001.0001.

Incremental cost and incremental effect are usually correlated, since the same patients and events drive both, so their joint uncertainty is not captured by treating each separately. A confidence ellipse reflects this by tilting and stretching along the direction of their correlation, marking the region where the true joint value is likely to lie. A rectangle formed from each measure's separate interval would wrongly imply they vary independently and would misstate the joint uncertainty. The ellipse's shape encodes the correlation. Briggs and colleagues (2006) describe it.

Source: Briggs et al. 2006

A confidence ellipse is constructed from the estimated means, variances, and covariance of incremental cost and incremental effect, drawing the region that contains a stated proportion, such as ninety-five per cent, of the joint distribution. The ellipse is centred on the point estimate, its axes determined by the variances and its tilt by the correlation between cost and effect. Assuming an approximately bivariate normal joint distribution, the ellipse marks the boundary within which the true joint value is expected to lie with the chosen probability.

Source: O'Brien 1996

A confidence ellipse is used on the cost-effectiveness plane because incremental cost and effect are jointly uncertain and correlated, so their uncertainty is best shown as a two-dimensional region rather than separate one-dimensional intervals, which would ignore the correlation and the joint nature of the uncertainty. The ellipse displays where the true incremental cost and effect jointly lie, conveying how the uncertainty spreads across the quadrants of the plane. This gives a fuller picture of the uncertainty in a cost-effectiveness comparison than marginal intervals alone.

Source: O'Brien 1996

A confidence ellipse reflects the correlation between incremental cost and effect through its orientation: a positive correlation tilts the ellipse so higher costs accompany greater effects, and a negative correlation tilts it the other way, while zero correlation gives axes aligned with the plane. The ellipse's width along each direction reflects the variances. So its shape and tilt encode the joint distribution, and ignoring the correlation, by using separate intervals, would misrepresent the region where the true values jointly lie.

Source: Briggs, Claxton & Sculpher 2006

A confidence ellipse typically assumes an approximately bivariate normal joint distribution of incremental cost and effect, which may not hold, especially with skewed costs or small samples, so the elliptical region may misrepresent the true joint uncertainty. It summarises uncertainty at one point in the analysis and does not directly give the probability of cost-effectiveness at a threshold, for which acceptability curves are used. These limitations mean the ellipse is one way to display joint uncertainty, complemented by other representations of decision uncertainty.

Source: Briggs, Claxton & Sculpher 2006