Strict Dominance

Strict Dominance and the Negative ICER Trap

Strict dominance is the plainest of the three dominance rules used to narrow a cost-effectiveness comparison before any ratio is interpreted. It compares one real option with another, involves no tie and no blend of options, and its exclusion holds even when health is given no monetary value. It is usually the first check made in an incremental analysis. A negative incremental cost-effectiveness ratio, on its own, cannot tell a reader whether a new option dominates or is dominated.

The condition and its place on the cost-effectiveness plane

Strict dominance compares two options on their expected costs and expected health effects, and requires the dominating option to be better on both at once. Neither difference may be zero. Written formally, option A strictly dominates option B when the following holds.

$$C_A < C_B \quad \text{and} \quad E_A > E_B$$

where $C_A$ and $C_B$ are the expected costs of options A and B, and $E_A$ and $E_B$ are their expected health effects, for example in quality-adjusted life years (QALYs). The same condition can be expressed as the increments of B over A.

$$\Delta C = C_B - C_A > 0, \qquad \Delta E = E_B - E_A < 0$$

where $\Delta C$ is the incremental cost and $\Delta E$ the incremental effect of choosing B instead of A. On the cost-effectiveness plane, with the difference in effect on the horizontal axis and the difference in cost on the vertical axis, B plotted against A lies strictly inside the north-west quadrant, off both axes. Seen from the other side, A plotted against B lies strictly inside the south-east quadrant, and A is then described as dominant.

Points on the axes, where the options are tied on cost or on effect, are the boundary cases handled by weak dominance. Under the convention used here, weak dominance is the inclusive pairwise rule, so every strictly dominated option is also weakly dominated, while extended dominance is the separate rule that tests an option against a blend of two others.

Names used for the same rule

The condition is stable across sources, but its name is not. The NICE technology appraisal manual (PMG36) simply calls such a technology "dominated", defining it as more costly and less effective than another technology in the analysis. The guide to the dampack package for R calls it "strongly dominated", the Health Economics Resource Center of the US Department of Veterans Affairs refers to "strong dominance", and the TreeAge Pro documentation calls an option above and to the left of another on the plane "absolutely dominated".

Why strict dominance is the least contestable exclusion

An exclusion rule is only safe if the excluded option could never have been chosen by the decision-maker the analysis is meant to serve. Net monetary benefit shows this most clearly, because it converts health into money at a threshold $\lambda$. The difference in net monetary benefit between A and B is given below.

$$NMB_A - NMB_B = \lambda (E_A - E_B) + (C_B - C_A)$$

where $NMB_A$ and $NMB_B$ are the net monetary benefits of the two options, $\lambda$ is the value placed on one unit of health effect, and the other terms are as defined above. Under strict dominance both bracketed terms are positive, so the difference is positive at every threshold of zero or above. A therefore beats B whatever value the decision-maker places on health, including a value of zero, when only cost matters.

This sets strict dominance apart from the other two rules. An option tied on cost with a more effective one is only worse when health has a positive value, and an extendedly dominated option is excluded by reference to two other options rather than one. Like weak dominance, strict dominance compares one option with one other.

As with every dominance exclusion, adding further options does not bring a strictly dominated option back, since new options cannot change the comparison with A. It can only return if A, and any other option that dominates it, is withdrawn from the comparison set.

The rule still rests on assumptions. The options must be mutually exclusive, the costs and effects must be measured from the same perspective and over the same time horizon, and the comparison is made on expected values. Anything the cost and effect measures leave out, such as the distribution of health across groups, sits outside the comparison; one published reader's guide points out that even a cheaper and more effective option may be set aside for reasons such as worsening existing inequity.

Finding strictly dominated options in a fully incremental analysis

In a fully incremental analysis strict dominance is checked before any sequential ratio is interpreted. The NICE manual (PMG36) asks for results to be presented in this form with dominated and extendedly dominated technologies removed, and the Health Economics Resource Center describes the same order, with strong dominance first, then extended dominance on the trimmed list. Once the options are ranked by expected cost, the strict dominance check reduces to one comparison per option. With options indexed in ascending order of expected cost, and no two options tied on cost, option $k$ is strictly dominated when the following holds.

$$E_k < \max_{j < k} E_j$$

where $E_k$ is the expected effect of option $k$ and the maximum is taken over the expected effects of all options $j$ that cost less than option $k$. In words, an option is strictly dominated when any cheaper option, including one several places down the cost ranking, produces more health. Checking an option only against its immediate neighbour in the cost ranking can miss strict dominance, as the example below shows. Ties in cost or effect are then resolved under weak dominance, sequential ratios are calculated for the options that remain, and any option whose ratio is higher than that of the next more costly option is removed under extended dominance, with ratios recalculated after each removal.

Worked example with five screening strategies

The figures below are illustrative. Five mutually exclusive screening strategies, S0 (no screening) to S4, are compared per person over a lifetime horizon, with costs and QALYs already discounted and the strategies listed in order of expected cost.

1. Check for strict dominance. The highest QALY total among strategies cheaper than S2 is 10.08, from S1, and S2 achieves 10.05, so S1 strictly dominates S2 because it costs £300 less and gives 0.03 more QALYs. The highest QALY total among strategies cheaper than S3 is still 10.08 from S1, and S3 achieves 10.06, so S1 also strictly dominates S3, by £500 in cost and 0.02 QALYs. S4 achieves 10.20, above every cheaper strategy, so it is retained. S2 and S3 are removed.

2. Note the neighbour trap. S3 is more effective than S2, its immediate neighbour in the cost ranking. Compared with S2 alone, S3 shows an extra £200 for 0.01 QALYs, a ratio of £20,000 per QALY that looks like an ordinary result, even though S3 is worse than S1 on both counts.

3. Calculate the sequential ratios. The ratio for moving from S0 to S1 is as follows.

$$ICER_{S1,S0} = \frac{1500 - 1000}{10.08 - 10.00} = \frac{500}{0.08} = 6250$$

where $ICER_{S1,S0}$ is the cost in pounds per QALY gained by moving from S0 to S1. The ratio for moving from S1 to S4 is as follows.

$$ICER_{S4,S1} = \frac{3000 - 1500}{10.20 - 10.08} = \frac{1500}{0.12} = 12500$$

where $ICER_{S4,S1}$ is the cost in pounds per QALY gained by moving from S1 to S4. The ratios rise from £6,250 to £12,500 per QALY, so no remaining strategy is extendedly dominated and S0, S1 and S4 form the cost-effectiveness frontier.

4. Confirm with net monetary benefit. At an illustrative threshold of £20,000 per QALY the net monetary benefits are £199,000 for S0 (20,000 × 10.00 minus 1,000), £200,100 for S1 (20,000 × 10.08 minus 1,500), £199,200 for S2 (20,000 × 10.05 minus 1,800), £199,200 for S3 (20,000 × 10.06 minus 2,000) and £201,000 for S4 (20,000 × 10.20 minus 3,000). S4 has the highest value, which matches the ratio method, and both S2 and S3 fall below S1. At a threshold of zero, where net monetary benefit is simply minus the cost, S1 still beats S2 and S3, as the strict dominance condition requires.

Strict dominance under uncertainty

Strict dominance is established on expected costs and effects, and the exclusion belongs to that comparison. In a probabilistic sensitivity analysis, each simulation draws a new set of parameter values, and in some draws the dominated option can be cheaper, more effective or both. An option that is strictly dominated on expected values can therefore have a positive probability of having the highest net monetary benefit at a given threshold, and it can appear with a non-zero line on a cost-effectiveness acceptability curve.

That probability can even exceed one half when the distribution of differences is skewed. Suppose that A has a lower mean cost and a higher mean QALY total than B across the simulations, so that B is strictly dominated on expected values, and that at a chosen threshold B has £1,000 more net monetary benefit than A in 70% of simulations and £5,000 less in the other 30%. The mean difference is calculated as follows.

$$\overline{NMB_A - NMB_B} = 0.30 \times 5000 + 0.70 \times (-1000) = 800$$

where $\overline{NMB_A - NMB_B}$ is the mean, across simulations, of A's net monetary benefit minus B's, in pounds. A has the higher expected value and is the better choice, although B is ahead in most draws. Barton, Briggs and Fenwick show related results for multiple options, including an extendedly dominated option with the highest probability of being cost-effective, and conclude that acceptability curves describe decision uncertainty but should not be used to pick the optimal option. The NICE manual (PMG36) asks for acceptability curves to be accompanied by the cost-effectiveness acceptability frontier, which follows the option with the highest expected net benefit. A strictly dominated option is removed from the incremental table of expected values, but it still belongs in simulation outputs such as acceptability curves and expected value of perfect information, since removing it can change them.

The negative ICER trap

The incremental cost-effectiveness ratio divides the incremental cost by the incremental effect, so its sign depends on both. When the two differences have opposite signs the ratio is negative, and that happens in two quadrants with opposite meanings. One is the south-east, where the new option is cheaper and more effective and dominates, and the other is the north-west, where it costs more, achieves less and is dominated. The ratio for the new option N against comparator R is shown below.

$$ICER_{N,R} = \frac{C_N - C_R}{E_N - E_R}$$

where $C_N$ and $C_R$ are the expected costs and $E_N$ and $E_R$ the expected effects of the new option and the comparator. Paulden notes that a strategy with a negative ratio is cost-effective if it lies in the south-east quadrant but not if it lies in the north-west, and a published reader's guide warns that the ratio alone cannot show which of the two applies.

The worked example shows the problem directly. Treating S2 as the new option against S1 gives the following ratio.

$$ICER_{S2,S1} = \frac{1800 - 1500}{10.05 - 10.08} = \frac{300}{-0.03} = -10000$$

where $ICER_{S2,S1}$ is the ratio for moving from S1 to S2, in pounds per QALY. Reversing the roles so that S1 is the new option against S2 gives the following ratio.

$$ICER_{S1,S2} = \frac{1500 - 1800}{10.08 - 10.05} = \frac{-300}{0.03} = -10000$$

where $ICER_{S1,S2}$ is the ratio for moving from S2 to S1. Both comparisons give minus £10,000 per QALY, yet in the first the new option is strictly dominated and in the second it strictly dominates.

The size of a negative ratio carries no ranking information either. A saving of £100 with a gain of 0.01 QALYs gives minus £10,000 per QALY, a saving of £500 with the same gain gives minus £50,000, and a saving of £100 with a gain of only 0.002 QALYs also gives minus £50,000. A larger saving, or a smaller health gain, can produce the same figure, so a more negative ratio does not mean a better option. Clear reporting therefore gives the incremental cost and incremental effect alongside any ratio, describes a south-east result as dominant and a north-west result as dominated rather than quoting a negative number, or presents incremental net monetary benefit, which keeps its meaning in every quadrant.

Where strict dominance is misread

Several misreadings follow from the points above. The first is to compare each option only with its immediate neighbour in the cost ranking, which produces ratios such as the £20,000 per QALY for S3 against S2 that should never have been calculated. The second is to treat the dominating option as the recommended one. S1 dominates two strategies in the example, yet S4 is preferred at £20,000 per QALY.

A third error is to treat a dominance finding as certain. A small cost saving combined with a small QALY gain can place a point estimate in the south-east quadrant while the simulations spread across all four, so a claim of dominance needs the accompanying uncertainty analysis. A fourth is to read a negative ratio as good news without checking the quadrant. Finally, strict dominance is defined only on the costs and effects that the model measures, so a change in perspective, time horizon or outcome measure can reverse it, and considerations outside those measures, such as equity, are not settled by it.

Sources

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