Extended Dominance

Extended dominance identifies an alternative that is not directly dominated by one other option but is still inefficient relative to a combination of more and less effective alternatives. The sections below explain how extended dominance is detected through sequential ICERs, how the cost-effectiveness frontier changes when an option is removed, and why every removal must be followed by recalculation.

Extended dominance applies only when several mutually exclusive alternatives are compared. It differs from strict dominance because the inefficient option may cost more and produce more health than the option immediately below it, yet still provide health less efficiently than the surrounding alternatives.

What extended dominance means

An alternative is subject to extended dominance when a combination of two other alternatives can produce the same expected health outcome at a lower expected cost. The extendedly dominated option therefore lies above the efficient cost-effectiveness frontier.

The comparison is not based on one alternative being both cheaper and more effective. Instead, inefficiency is revealed by the pattern of sequential incremental cost-effectiveness ratios across several ordered alternatives.

• Extended dominance requires at least three relevant alternatives.
• An extendedly dominated option is not necessarily strictly dominated.
• Extended dominance indicates that an option is less efficient than a combination of alternatives.
• An extendedly dominated option should be removed before final sequential ICERs are interpreted.
• Removing an option changes the relevant comparisons and requires recalculation.

How extended dominance differs from strict dominance

Strict dominance is identified through a direct comparison between individual alternatives. An option is strictly dominated when another alternative costs no more and produces no less health, with at least one strict improvement.

Extended dominance depends on the relationship among several alternatives. An option may cost more and produce more health than its immediate comparator but still be inefficient because its sequential ICER is higher than the ICER of a more effective option.

• Strict dominance compares the expected cost and effectiveness of individual alternatives directly.
• Extended dominance examines the efficiency pattern across multiple alternatives.
• Strictly dominated options are removed before testing for extended dominance.
• Both types of removal are required before constructing the final cost-effectiveness frontier.

How sequential ICERs reveal extended dominance

After strictly dominated alternatives have been removed, the remaining options are ordered from lowest to highest expected effectiveness. A sequential ICER is calculated between each alternative and the next less effective remaining option.

Sequential ICERs on the efficient frontier should increase as effectiveness increases. If an option’s sequential ICER is greater than the ICER of the next more effective option, the middle option is subject to extended dominance.

Sequential ICER = Change in cost ÷ Change in effectiveness

For three consecutively ordered alternatives A, B and C, alternative B is subject to extended dominance when:

ICER from A to B > ICER from B to C

This pattern means resources could move from A toward C more efficiently without selecting B as a separate option.

A corrected four-option example

Consider four mutually exclusive alternatives after any strictly dominated options have been removed. The costs and QALYs below are synthetic and use the same population, perspective and time horizon.

Alternative	Expected cost	Expected QALYs
A	£10,000	4.0
B	£13,000	4.3
C	£18,000	4.5
D	£22,000	4.8

The alternatives are already ordered from lowest to highest expected effectiveness. Sequential comparisons are therefore calculated between A and B, B and C, and C and D.

A to B: (£13,000 − £10,000) ÷ (4.3 − 4.0) = £10,000 per QALY

B to C: (£18,000 − £13,000) ÷ (4.5 − 4.3) = £25,000 per QALY

C to D: (£22,000 − £18,000) ÷ (4.8 − 4.5) = £13,333 per QALY

The ICER rises from £10,000 to £25,000 and then falls to £13,333 per QALY. Because the ICER from B to C exceeds the ICER from C to D, alternative C is subject to extended dominance.

Why alternative C is inefficient

Alternative C produces 4.5 QALYs at an expected cost of £18,000. A combination of alternatives B and D can produce the same expected health outcome at a lower expected cost.

Moving 40% of the way from B to D increases expected effectiveness from 4.3 to 4.5 QALYs. The corresponding expected cost is £16,600, which is £1,400 less than the cost of C.

Required proportion = (4.5 − 4.3) ÷ (4.8 − 4.3) = 0.40

Combined expected cost = £13,000 + [0.40 × (£22,000 − £13,000)] = £16,600

Cost of alternative C = £18,000

Alternative C lies above the line connecting B and D on the cost-effectiveness plane. This geometric position is why C does not belong on the efficient frontier.

Recalculate after removing C

Removing an extendedly dominated alternative changes which options are adjacent. The sequential ICERs must therefore be recalculated using the remaining alternatives rather than copied from the earlier analysis.

After C is removed, alternatives A, B and D remain. The recalculated comparisons are:

A to B: (£13,000 − £10,000) ÷ (4.3 − 4.0) = £10,000 per QALY

B to D: (£22,000 − £13,000) ÷ (4.8 − 4.3) = £18,000 per QALY

The recalculated ICERs increase from £10,000 to £18,000 per QALY. No further extended dominance is present, so A, B and D form the final efficient frontier.

The complete elimination process

Fully incremental analysis is iterative because every removal changes the relevant comparisons. Analysts should continue the process until no strict or extended dominance remains and the sequential ICERs increase with effectiveness.

The final frontier contains only the alternatives needed for threshold-based interpretation. The complete order of operations should remain visible in the audit trail.

1. Order all alternatives from lowest to highest expected effectiveness.
2. Remove any exact duplicates or alternatives with the same effectiveness but greater cost.
3. Remove all strictly dominated alternatives.
4. Reorder the remaining alternatives if necessary.
5. Calculate sequential incremental costs, effects and ICERs.
6. Remove an alternative when its ICER exceeds the ICER of the next more effective alternative.
7. Recalculate every affected sequential comparison.
8. Repeat the dominance checks until sequential ICERs increase with effectiveness.
9. Apply the cost-effectiveness threshold to the final frontier.

How the cost-effectiveness frontier represents efficiency

The cost-effectiveness frontier connects the alternatives that provide the greatest expected effectiveness achievable at each relevant cost level. Options lying above the frontier are inefficient because another option or combination can provide the same health outcome at lower cost.

Strictly dominated and extendedly dominated alternatives do not appear on the final frontier. The slopes between the remaining frontier points are the final sequential ICERs.

• A frontier point represents an efficient alternative among the options evaluated.
• A point above the frontier is inefficient relative to another option or combination.
• Increasing frontier slopes correspond to increasing sequential ICERs.
• A falling sequential ICER signals that an intermediate option may be subject to extended dominance.
• The frontier depends on the alternatives included in the analysis.

Why pairwise comparisons can give the wrong answer

Comparing every alternative only with one common baseline does not establish the efficient sequence among mutually exclusive options. A baseline comparison can make an inefficient alternative appear acceptable because it ignores the next relevant alternative on the frontier.

Fully incremental analysis instead compares each remaining option with the next less effective efficient alternative. This preserves the opportunity-cost logic needed to identify the correct frontier.

• Pairwise ICERs against a common baseline should not replace sequential analysis.
• An alternative can have an apparently acceptable baseline ICER and still be extendedly dominated.
• Adding or removing an alternative can change the frontier and the relevant incremental comparisons.
• The complete set of mutually exclusive alternatives should be included whenever feasible.

How the threshold is applied after dominance checks

The cost-effectiveness threshold should be applied only after strictly and extendedly dominated alternatives have been removed. Applying the threshold earlier can select an inefficient option or interpret an ICER that will not appear on the final frontier.

Once the final frontier has been established, the preferred option is the most effective alternative whose sequential ICER remains below the applicable threshold. Expected net benefit provides an equivalent direct ranking when the same evidence and threshold are used.

• Dominance screening determines which alternatives remain efficient candidates.
• The threshold determines how far along the final frontier the decision should move.
• Expected net benefit can verify the preferred option at the stated threshold.
• A cost-effectiveness result remains separate from affordability and the final institutional recommendation.

How uncertainty affects the frontier

The expected-cost and expected-effectiveness values used to construct the base-case frontier are uncertain. Changes in model inputs or structural assumptions can alter which alternatives are strictly dominated, extendedly dominated or efficient.

Probabilistic analysis should not attempt to create a definitive frontier from every simulation and then average the resulting ICERs. Expected net benefit is generally more stable for decision-making under uncertainty, while frontier analysis remains useful for explaining the expected-value comparison.

• The base-case frontier is constructed from expected costs and expected effects.
• Sensitivity analysis can show whether dominance classifications change under alternative assumptions.
• Expected net benefit should determine the preferred option under probabilistic uncertainty.
• The probability that an option appears on a simulated frontier is not itself the decision rule.
• Structural changes that alter the available alternatives require the frontier to be rebuilt.

How to check extended dominance in Excel

Excel can support fully incremental analysis when the worksheet preserves the ordered alternatives and recalculates every comparison after a removal. A single formula should not be treated as a complete automated solution because extended dominance is an iterative process.

Suppose expected cost is in column B and expected effectiveness is in column C. After sorting the alternatives, use separate calculation columns for incremental cost, incremental effectiveness and the sequential ICER.

• Use =B3-B2 to calculate the incremental cost between adjacent alternatives.
• Use =C3-C2 to calculate the incremental effectiveness between adjacent alternatives.
• Use =IF(E3=0,NA(),D3/E3) to calculate a sequential ICER while flagging a zero effect difference.
• Compare each sequential ICER with the ICER immediately below it.
• Flag an intermediate option when its ICER exceeds the next sequential ICER.
• Remove the flagged option from the working frontier and recalculate the affected rows.
• Preserve the original alternative table and a removal log for audit purposes.

Common mistakes and safeguards

Extended-dominance errors usually arise from incorrect ordering, failure to remove strict dominance first or failure to recalculate after an option is removed. These mistakes can produce a frontier that contains inefficient alternatives.

The safeguard is to make each step visible and repeat the checks until the remaining ICERs increase with effectiveness. Calculations should be verified independently when the decision is sensitive to a small difference.

• Ordering alternatives by cost without checking effectiveness can obscure dominated options.
• Testing for extended dominance before removing strict dominance produces unreliable sequential comparisons.
• Comparing every alternative with a common baseline can miss extended dominance.
• Removing an option without recalculating adjacent ICERs leaves the analysis incomplete.
• Treating a falling sequential ICER as acceptable retains an inefficient intermediate option.
• Calling extended dominance direct dominance misstates why the option was removed.
• Averaging ICERs across probabilistic simulations produces a misleading summary.
• Treating an efficient-frontier position as proof of affordability confuses efficiency with budget impact.

What should be reported

A transparent extended-dominance analysis should allow readers to reproduce the ordering, removals and recalculated frontier. Reporting only the final set of alternatives hides the reasoning that determined which options were excluded.

The following information makes the analysis auditable and helps prevent incorrect reuse of intermediate ICERs:

• Report every mutually exclusive alternative initially considered.
• Report expected costs and expected health outcomes using consistent units.
• Report the original ordering of alternatives.
• Report every strictly dominated alternative and the option that dominates it.
• Report the initial sequential incremental costs, effects and ICERs.
• Report every alternative removed through extended dominance and the comparison establishing its inefficiency.
• Report all recalculated ICERs after each removal.
• Report the final cost-effectiveness frontier.
• Report the threshold and expected-net-benefit result used to identify the preferred option.
• Report uncertainty and structural assumptions that could change the frontier.