Weak Dominance

Weak Dominance, Ties and Exclusion Rules

Most dominance checks in cost-effectiveness analysis look for an option that is both cheaper and more effective than another. Weak dominance extends that check to the boundary cases, where two options are tied on cost or tied on health effect, so one of them can be ruled out even though it is not worse on both. Some software and teaching material use the same term for a different rule, so a reader needs to check which meaning a report applies.

The condition on the cost-effectiveness plane

Weak dominance compares two options on their expected costs and expected health effects. One option weakly dominates another when it is no more costly, no less effective, and strictly better on at least one of the two. Written formally, the condition is as follows.

$$C_A \le C_B \quad \text{and} \quad E_A \ge E_B, \quad \text{with} \quad C_A < C_B \ \text{or} \ E_A > E_B$$

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). When both inequalities are strict, A is cheaper and more effective, which is strict dominance. Weak dominance therefore includes strict dominance and adds the two tied cases, in which B costs more for the same effect or B achieves less for the same cost. Under this inclusive convention every strictly dominated option is also weakly dominated, but not the reverse.

On the cost-effectiveness plane, with the difference in effect on the horizontal axis and the difference in cost on the vertical axis, a weakly dominated option B plotted against A lies in the north-west quadrant or on one of its two edges, excluding the origin. A tie on effect places B on the vertical axis above the origin, and a tie on cost places it on the horizontal axis to the left of the origin.

A parallel in game theory

The same term is used in game theory. Osborne and Rubinstein call an action weakly dominated when another action is at least as good whatever the other players do and better for at least some of their choices, and they allow the dominating option to be a mixture of actions. The cost-effectiveness version is closer to dominance across two attributes. The dominating option has to be at least as good on cost and on effect, and better on one. The idea of being beaten by a mixture has its counterpart in extended dominance, where a blend of two other options does better than the one being tested.

Why a weakly dominated option can be removed safely

Removing an option before the main analysis is only safe if that option could never have been the one chosen. The clearest way to show this is to compare the options in terms of net monetary benefit, which converts health into money at a stated threshold. For any option the net monetary benefit is defined as follows.

$$NMB_i = \lambda E_i - C_i$$

where $NMB_i$ is the net monetary benefit of option $i$, $\lambda$ is the value placed on one unit of health effect (the threshold), $E_i$ is the expected health effect and $C_i$ is the expected cost of that option. The difference between A and B is as follows.

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

where the terms are as defined above. When A weakly dominates B, the first term is zero or positive and the second subtracts a cost difference that is zero or negative, with at least one of the two strictly favouring A, so the difference is positive at every positive threshold. In the two tied cases the difference reduces to $C_B - C_A$, positive whatever the threshold, or to $\lambda (E_A - E_B)$, positive at every positive threshold. A weakly dominated option can only match its dominator at a threshold of zero, when health has no value, so it can be taken out before incremental ratios are calculated.

Why tied options break incremental ratios

The incremental cost-effectiveness ratio divides the difference in cost by the difference in effect. When two options are tied on one of those differences, the ratio stops carrying information about value for money. Ties make the division either misleading or impossible, as the ratio below shows.

$$ICER_{B,A} = \frac{C_B - C_A}{E_B - E_A}$$

where $ICER_{B,A}$ is the extra cost per extra unit of effect of moving from option A to option B, and the other terms are as defined above. If B costs more than A for the same effect, the denominator is zero and the ratio is undefined; a spreadsheet returns a division error. If B costs the same as a more effective option C, the ratio of C against B is zero, which looks like excellent value but only reflects that B should not be in the comparison at all.

For this reason, dominance checks, including the tied cases, come before the sequential ratios of a fully incremental analysis. Options are ranked by cost, with ties in cost broken by effect, and weakly dominated options are removed. Sequential ratios are then calculated and checked for extended dominance, and after each removal the ratios are recalculated and the checks repeated until the remaining ratios rise with cost.

Worked example with four options and two ties

The figures below are illustrative. Four mutually exclusive options are compared over a lifetime horizon, with costs and QALYs already discounted.

1. Check for weak dominance. B has the same QALYs as A but costs £2,000 more, so A weakly dominates B. B also has the same cost as C but 0.40 fewer QALYs, so C weakly dominates B as well. B is removed.

2. Calculate the sequential ratios for the remaining options. Moving from A to C gives the following ratio.

$$ICER_{C,A} = \frac{12000 - 10000}{5.40 - 5.00} = \frac{2000}{0.40} = 5000$$

where $ICER_{C,A}$ is the cost in pounds per QALY gained by moving from A to C. Moving from C to D gives the next ratio.

$$ICER_{D,C} = \frac{16000 - 12000}{5.60 - 5.40} = \frac{4000}{0.20} = 20000$$

where $ICER_{D,C}$ is the cost in pounds per QALY gained by moving from C to D.

3. Check for extended dominance and choose. The ratios rise from £5,000 to £20,000 per QALY, so no remaining option is extendedly dominated and A, C and D form the cost-effectiveness frontier. At an illustrative threshold of £25,000 per QALY, D is preferred, because the £20,000 per QALY needed to move from C to D is below the threshold.

4. Confirm with net monetary benefit. At £25,000 per QALY the net monetary benefits are £115,000 for A (25,000 × 5.00 − 10,000), £113,000 for B (25,000 × 5.00 − 12,000), £123,000 for C (25,000 × 5.40 − 12,000) and £124,000 for D (25,000 × 5.60 − 16,000). D has the highest value and B is below both options that dominate it, which matches the result of the ratio method.

Had B been left in, the ratio of B against A would have been undefined because the QALY difference is zero, and the ratio of C against B would have been zero.

Assumptions behind the rule

The rule rests on a small number of assumptions that are usually left unstated. The options are mutually exclusive, so only one is chosen, and the choice is made on expected costs and effects. Health is given a positive value, since at a threshold of zero an option tied on cost with a more effective one is no longer worse. In practice, exact ties rarely come from separate estimates; they usually come from an assumption, such as equal effectiveness between two treatments or identical costs for two delivery routes, so a weak dominance finding is only as sound as the evidence behind that assumption.

Weak dominance in appraisal decisions

Appraisal methods create tied comparisons when they treat outcomes or costs as equivalent. The tied cases arise most naturally in cost-minimisation analysis, where options judged to give equivalent health outcomes are compared on cost alone, so the cheaper option weakly dominates the others in the sense used on this page.

NICE's cost-comparison approach in its manual (PMG36) resembles this rule, with judgements of equivalence in place of exact ties. The committee usually recommends a technology when there is enough certainty that it has at least equivalent benefits compared with current management and overall uses less resources, or that it has significantly greater benefits and overall uses similar resources. Conclusions on the similarity of benefits rest on a pragmatic view of all the available evidence, and the manual states that, ideally, a non-inferiority or equivalence study with appropriate margins should be presented. Because "at least equivalent" and "similar" are judgements rather than exact equalities, and the wording "similar resources" does not exclude a slightly higher cost, the NICE conditions are best read as an analogy to weak dominance.

Weak, strict and extended dominance compared

The three rules differ in what they compare and in the options they remove. Keeping them apart avoids two common reporting errors, which are removing too little, so that undefined or zero ratios appear, and treating a pairwise rule and a blend rule as the same thing.

The terminology is not consistent across sources. The guide to the dampack package for R, for example, uses "strongly dominated" for what this page calls strict dominance and uses "weak dominance" as another name for extended dominance. Because the two meanings remove different options, a report should state which rule it applied. On this page, weak dominance means the inclusive pairwise rule above and extended dominance means the blend rule.

Where weak dominance is misapplied

A frequent error is to calculate ratios before removing tied options and then to report an undefined ratio as a very high cost per QALY, or a zero ratio as a sign of good value. Another is to treat weak and extended dominance as the same rule because some sources use the same name for both. The difference lies in what each exclusion rests on. Weak dominance needs one real option that is at least as good on both cost and effect and better on one, whereas extended dominance rests on two other options, with the blend serving as a geometric device that shows that, at every positive threshold, at least one of the two has higher net monetary benefit than the excluded option. Both exclusions depend on the comparison set, so either can change if the dominating option, or one of the options in the blend, is withdrawn.

Weak dominance is also applied to expected values, and the exclusion belongs to that comparison only. In a probabilistic sensitivity analysis the options may differ in individual simulations when the tied inputs are sampled independently, although a tie created by a shared parameter remains a tie in every draw. Outputs that are computed across simulations, such as acceptability curves and value of information estimates, can change if an option is removed, so an option excluded on expected values still belongs in those outputs.

Sources

• Alarid-Escudero F, Enns E, Knowlton G, DARTH Team. Basic cost effectiveness analysis. Vignette for the dampack package for R. 2024. https://cran.r-project.org/web/packages/dampack/vignettes/basic_cea.html
• Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed. Oxford: Oxford University Press; 2015.
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; 2022, last updated 13 October 2026. Chapter 6.
• Osborne MJ, Rubinstein A. A Course in Game Theory. Cambridge, MA: MIT Press; 1994. Section 4.3.