Cost-Effectiveness Plane

Cost-Effectiveness Plane in Economic Evaluation

The cost-effectiveness plane, also called the incremental cost-effectiveness plane, shows the result of a comparison as a point instead of a ratio. A single ICER hides which of the two differences behind it is negative, so a negative ICER can mean a dominant result in the south-east quadrant or a dominated one in the north-west. A threshold line and the cloud of points from a probabilistic analysis can be drawn on the same plane, and net benefit and the acceptability curve are read from it.

Axes, origin and the direction of subtraction

Black described the "CE plane" in 1990, plotting each strategy by its effectiveness and cost relative to a reference standard. The comparator sits at the origin, and every point is a difference. The two coordinates are defined as follows.

$$\Delta E = E_1 - E_0, \qquad \Delta C = C_1 - C_0$$

where $E_1$ and $C_1$ are the expected effect and expected cost of the intervention, $E_0$ and $C_0$ are those of the comparator, $\Delta E$ is the incremental effectiveness and $\Delta C$ is the incremental cost. Both must come from the same population, perspective and time horizon, and the effect is usually measured in QALYs. In the usual convention $\Delta E$ runs along the horizontal axis and $\Delta C$ up the vertical axis. A new treatment costing GBP 12,500 per person with 3.4 QALYs, compared with current care at GBP 10,000 and 3.0 QALYs, plots at $\Delta E = 0.4$ and $\Delta C = 2{,}500$.

The ICER is the ratio of the two coordinates, so it is the slope of the line from the origin to the plotted point, as below.

$$\text{ICER} = \frac{\Delta C}{\Delta E}$$

where the symbols are as defined above. For the example, $\text{ICER} = 2{,}500 / 0.4 = 6{,}250$ GBP per QALY gained.

Some software, such as TreeAge Pro, can draw cost on the horizontal axis, which swaps north-west and south-east, so the axis labels and subtraction direction should be read first.

The four quadrants and what each one means

The two axes split the plane into four quadrants, named by compass direction or numbered 1 to 4 anticlockwise from the north-east.

Fenwick and colleagues describe south-east results as always cost-effective and north-west results as never cost-effective. The Canadian National Advisory Committee on Immunization (NACI) guide notes that "cost-saving" and "dominant" are often wrongly used as synonyms; cost-saving covers the whole southern half of the plane, whereas dominant means the south-east only.

The threshold line, its rotation and net benefit

The cost-effectiveness threshold, written $\lambda$, is the cost per unit of effect at or below which the intervention is judged worthwhile. Black drew it as a line through the origin with slope equal to the maximum acceptable cost-effectiveness ratio, and classed points to the right of the line as cost-effective. The line and the measure that goes with it are written as follows.

$$\Delta C = \lambda \, \Delta E, \qquad \text{INMB} = \lambda \, \Delta E - \Delta C$$

where INMB is the incremental net monetary benefit in currency. A point below and to the right of the line has positive INMB and is cost-effective at $\lambda$; a point on the line is a tie. The horizontal distance to the right of the line, $\Delta E - \Delta C / \lambda$, is the incremental net health benefit. Unlike the ICER, net benefit keeps the same meaning in all four quadrants.

Changing the threshold rotates the line about the origin. At $\lambda = 0$ it lies along the horizontal axis, so only cost-saving points count; as $\lambda$ rises it turns anticlockwise, so more north-east points fall below it and fewer south-west points do. A straight line values a QALY lost at the same rate as a QALY gained. O'Brien and colleagues found willingness to accept exceeded willingness to pay in every study they reviewed and suggested a downward kink through the origin for the south-west quadrant; the NACI guide likewise says decision makers may consider a steeper threshold there.

Why a negative ICER cannot be read without the quadrant

A ratio is negative in both the south-east and the north-west. Abbott and colleagues call this ambiguity a potential pitfall, since "from the ratio alone" the reader cannot tell which difference is negative. Positive ratios carry the same trap. Paulden notes that a positive ICER below the threshold is cost-effective in the north-east but not in the south-west, where the ratio is money saved per QALY lost and a higher figure is the better result.

In the illustrative figures below, four hypothetical interventions are each compared with the same comparator at a threshold of GBP 20,000 per QALY.

1. Same positive ratio, opposite decisions. A and B both have an ICER of 15,000. For A, $\text{INMB} = 20{,}000 \times 0.20 - 3{,}000 = 1{,}000$, so A is cost-effective. For B, $\text{INMB} = 20{,}000 \times (-0.20) + 3{,}000 = -1{,}000$, because the GBP 3,000 saved is worth less than the 0.20 QALYs given up.

2. Same negative ratio, opposite decisions. C and D both have an ICER of −10,000. C dominates, with $\text{INMB} = 2{,}000 + 1{,}000 = 3{,}000$, while D is dominated, with $\text{INMB} = -2{,}000 - 1{,}000 = -3{,}000$.

The ICER is undefined when $\Delta E = 0$. As a point approaches the vertical axis the ratio's absolute value grows without limit (when $\Delta C$ is not zero), and a small change in $\Delta E$ across zero switches it from a very large positive number to a very large negative one. Paulden calls the ICER a discontinuous measure, and O'Brien and Briggs note that confidence intervals for the ratio are problematic when the numerator or the denominator can be zero. The 2026 HEMA report on uncertainty by Steuten and colleagues notes that presenting results as net benefit may avoid some of the difficulties of interpreting ratios, and recommends reporting expected net health benefit alongside the ICER.

From one point to a cloud of simulations

Probabilistic sensitivity analysis draws parameter values, runs the model and records one pair of $\Delta C$ and $\Delta E$; repeating this many times, or bootstrapping trial data, gives a cloud that approximates the joint distribution of the two increments. The mean of the cloud is the expected-value point used for the decision. The HEMA report lists the plane, the acceptability curve and the tornado diagram as common outputs for communicating uncertainty, and states that none gives a complete account of uncertainty when presented alone.

The cost-effectiveness acceptability curve, introduced by van Hout and colleagues in 1994, is read from the cloud. Fenwick and colleagues define the probability of cost-effectiveness at $\lambda$ as the proportion of points south and east of a ray of slope $\lambda$ through the origin. South-east points are always counted and north-west points never, so the curve starts at the share of cost-saving points and tends to the share with positive $\Delta E$. The HEMA report recommends that expected loss accompany the curve, and that the acceptability frontier also be presented when several options are compared, because the option most likely to be optimal is not necessarily the one with the highest expected net benefit.

Worked example with ten simulated draws and their acceptability curve

The figures are illustrative; a real analysis would use thousands of draws. Ten draws compare a new treatment with current care at GBP 20,000 per QALY.

1. Expected point. The mean incremental effect is $\overline{\Delta E} = 1.50 / 10 = 0.15$ QALYs and the mean incremental cost is $\overline{\Delta C} = 20{,}000 / 10 = 2{,}000$ GBP, an ICER of about 13,333 GBP per QALY in the north-east.

2. Expected net benefit. The ten INMBs sum to 10,000, a mean of 1,000, the same as $\text{INMB} = 20{,}000 \times 0.15 - 2{,}000 = 1{,}000$ from the expected point, so the new treatment is preferred.

3. Points on the acceptability curve. At $\lambda = 0$ only draw 7 saves money, so the probability is 0.1. At GBP 20,000 six draws have positive INMB, a probability of 0.6. At GBP 30,000 draws 3 and 8 cross the line and the probability is 0.8. As $\lambda$ grows without limit it tends to 0.9, because draw 6 is never cost-effective.

4. Interpretation. Expected net benefit is positive, yet in 4 of the 10 draws the comparator would have been the better choice. In an illustrative HEMA example with a right-skewed distribution the two measures point opposite ways, with a new medicine best in only 30% of simulations yet a positive expected incremental net health benefit of 0.10 QALYs per patient.

Uncertainty regions and several options

A cloud can be summarised by a region containing a stated share of the joint distribution. Van Hout and colleagues calculated a 95% probability ellipse on the plane, but Geenen and colleagues note that an ellipse assumes an elliptical distribution and did not suit either of their two real-world case studies. A region that crosses the threshold line shows uncertainty about the decision, and one that crosses the vertical axis leaves the ICER unbounded, a problem the Fieller method page discusses.

The HEMA report notes that because the plane represents pairwise comparisons, it becomes less useful when several comparators are relevant. Slopes from a common baseline are not the sequential ICERs of a fully incremental analysis, which the cost-effectiveness frontier shows.

Drawing and reporting the plane

The plane is an XY scatter chart with both axes crossing at zero, axis titles giving the effect unit, currency and price year, and the expected-value point shown as a separate series from the draws. A threshold line runs through two points satisfying $\Delta C = \lambda \Delta E$, with $\lambda$ stated in the caption. Among the minimum reporting expectations the HEMA report proposes are the expected incremental costs, health effects and net health benefit from the probabilistic analysis, the deterministic result with an explanation of any material difference, and the number of simulations with evidence that the probabilistic results are stable. Budget impact is reported separately, since the plane shows value per person, not total spending.

Limitations and common misreadings

The cloud reflects only the uncertainties included in the probabilistic analysis. Most misreadings come from reading the ratio before the picture, such as treating every negative ICER as good news or judging a south-west ratio by the north-east rule.

Sources

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