Multi-Attribute Utility

Multi-Attribute Utility in Health State Valuation

Multi-attribute utility is a way of scoring health states described on several dimensions at once, such as vision, mobility and pain, by valuing each dimension on its own and joining the parts with a formula that stays consistent with utility theory. In health economics it is the theory behind the scoring of the Health Utilities Index (HUI), whose questionnaire answers become one number on the scale where dead is 0 and full health is 1, ready for a QALY. No valuation survey can cover every combination of levels, so the combining formula decides how impairments add up when a person has more than one. A multiplicative function and a simple sum can give the same patient different utilities, and so different incremental QALYs in a cost-utility analysis.

Why preference-based instruments need a combining function

Horsman and colleagues describe two main ways of measuring utilities, direct measurement and the use of multi-attribute systems. In direct elicitation a respondent values a described state with a task such as the standard gamble. In the multi-attribute route the respondent only completes a questionnaire, and a scoring function derived from community preference measures turns the answers into a utility, the route known as indirect utility assessment.

The number of states makes the second route unavoidable. The HUI2 classification describes 24,000 unique health states and the HUI3 classification 972,000, and the HUI3 function was built to score every one of them.

Furlong and colleagues, in the HUI3 technical report, describe multi-attribute utility theory (MAUT) as an extension of expected utility theory to outcomes described by several attributes. The theory says how preferences should be measured, consistent with the axioms of expected utility, which is why the standard gamble is the HUI's method of choice and visual analogue scale scores are converted to standard gamble equivalents. It also says which functional forms are admissible. Torrance and colleagues add that the measurement instrument determines whether the score obtained is a value or a utility.

Additive, multiplicative and multilinear forms

MAUT allows three basic forms, additive, multiplicative and multilinear, and Torrance and colleagues explain that the type of preference independence among the attributes determines which one is required. Kuspinar and colleagues, following the HUI3 manual, write the multiplicative form on a disutility scale, where 0 is full health and 1 is the lower anchor, dead in that study and the all-worst state in the HUI3.

$$1 + c\,u' = \prod_{j=1}^{n} \left( 1 + c\, c_j\, u'_j \right)$$

where $u'$ is the overall disutility of the health state, $n$ is the number of attributes, $c_j$ is the mean disutility of the corner state for attribute $j$ (that attribute at its worst level and all others at their best), $u'_j$ is the single-attribute disutility of the state's level on attribute $j$, equal to 0 at the best level and 1 at the worst, and $c$ is the scaling constant that captures the interaction between attributes.

The scaling constant is fixed by the corner disutilities, as the non-zero root of the equation that gives the state with every attribute at its worst level a disutility of 1.

$$1 + c = \prod_{j=1}^{n} \left( 1 + c\, c_j \right)$$

where $c$ and $c_j$ are as defined above. If the corner disutilities sum to 1, $c$ is 0 and the function is additive. If they sum to more than 1, $c$ lies between minus 1 and 0, and if they sum to less than 1, $c$ is positive. The HUI3 report used the same rule to choose its form. When $c$ is 0 the function reduces to a weighted sum.

$$u' = \sum_{j=1}^{n} c_j\, u'_j$$

where the corner disutilities $c_j$ act as attribute weights that sum to 1 and $u'_j$ is the single-attribute disutility.

The multilinear form lets some pairs of attributes interact one way and other pairs the other way, but it needs many more parameters and preference measurements, and Furlong and colleagues note that few studies have estimated it, especially in health.

How health attributes interact

The choice of form is a claim about how impairments combine. The HUI3 report states that the additive model is appropriate only if preference interactions among attributes are not important, and that in health status applications the data have, in general, strongly rejected it. Its test asks whether the disutility of all attributes at their worst equals the sum of the separate disutilities, and the usual finding was that it is smaller.

When the combined loss is smaller than the sum of the separate losses, the attributes are preference complements, so each functioning attribute is worth more when the others also function. When it is larger, they are preference substitutes. Furlong and colleagues report that HUI1 and HUI2 data also rejected the additive form and showed strong complementarity, and their HUI3 pretest found most attributes acting as complements, with vision and hearing often seen as substitutes. A multiplicative function imposes one type of interaction on every attribute, the price of a single interaction parameter.

Building a function from corner states

The HUI3 study used a person-mean approach. Corner states were valued on a scale anchored at the all-worst state (called Pits) as 0 and perfect health as 1, and each single-attribute function was scaled so that the attribute's lowest level scored 0 and its normal level 1. Preference measures for 43 states were needed to fit the function. Two further steps then placed dead on the scale, so final scores put dead at 0 and perfect health at 1, with negative values for states worse than dead.

Kuspinar and colleagues followed the same manual for a multiple sclerosis index with 5 items of 3 levels each, so 243 states, of which each patient valued 12. With standard gamble values the corner disutilities summed to 0.85, so the function was multiplicative with a positive scaling constant, solved in Excel Solver. For rating-scale responses where dead was not at the bottom of the scale, they converted each score as below.

$$u^{*} = \frac{x - d}{1 - d}$$

where $u^{*}$ is the utility on the dead equals 0, full health equals 1 scale, $x$ is the score on the original scale and $d$ is the score that dead receives on that original scale.

Worked example with three illustrative attributes

All numbers here are illustrative and come from no published instrument. A condition-specific instrument has three attributes, mobility, pain and mood, each with a best, an intermediate and a worst level. A valuation survey gives corner disutilities of 0.50 for mobility, 0.45 for pain and 0.35 for mood, with the all-worst state at disutility 1, and every intermediate level has a single-attribute disutility of 0.5.

1. Choose the form. The corner disutilities sum to 0.50 + 0.45 + 0.35 = 1.30. That exceeds the all-worst disutility of 1, so the attributes are complements, the additive form is ruled out and $c$ lies between minus 1 and 0.

2. Solve for the scaling constant. Expanding the scaling equation with these three values, subtracting 1 + $c$ from both sides and dividing by $c$ leaves a quadratic.

$$0.07875\,c^{2} + 0.5575\,c + 0.30 = 0$$

where $c$ is the scaling constant. Its root in the permitted range is $c$ = minus 0.5867. As a check, the three factors 1 + $c c_j$ are 0.7067, 0.7360 and 0.7947, and 0.7067 × 0.7360 × 0.7947 = 0.4133, which equals 1 + $c$.

3. Score a patient with intermediate mobility, worst pain and best mood. For mobility 0.5867 × 0.50 × 0.5 = 0.1467, so its factor is 0.8533. For pain 0.5867 × 0.45 × 1 = 0.2640, so its factor is 0.7360, and the mood factor is 1. The product 0.8533 × 0.7360 = 0.6280, so the disutility is 0.3720 / 0.5867 = 0.6340 and the utility on the all-worst scale is 0.3660.

4. Place dead on the scale. Suppose dead scores 0.20 on the all-worst scale. The utility becomes 0.1660 / 0.80 = 0.2075, and the all-worst state scores minus 0.20 / 0.80, or minus 0.25, worse than dead.

5. Value a treatment. A treatment improves pain to the intermediate level while mobility stays intermediate. For pain 0.5867 × 0.45 × 0.5 = 0.1320, so its factor is 0.8680. The product 0.8533 × 0.8680 = 0.7407 gives a disutility of 0.2593 / 0.5867 = 0.4420, an all-worst utility of 0.5580 and a dead-anchored utility of 0.3580 / 0.80 = 0.4475. The gain over 0.2075 is 0.2400 a year, and 0.2400 × 3 = 0.72 QALYs over three years without discounting.

6. Compare a simple sum. Adding the disutilities gives 0.25 + 0.45 = 0.70 before and 0.25 + 0.225 = 0.475 after, so dead-anchored utilities of 0.125 and 0.40625, a gain of 0.28125 a year and about 0.84 QALYs over three years. The sum overstates the gain by about 0.12 QALYs and gives the all-worst state a disutility of 1.30, not the 1 respondents gave it.

The HUI3 function as published

The HUI3 report presents the final function in a form that needs only a look-up table. Each attribute level has a coefficient $b_j$, equal to 1.00 at the normal level, and the utility of a state is calculated as below.

$$u^{*} = 1.371 \left( b_1 \times b_2 \times \cdots \times b_8 \right) - 0.371$$

where $u^{*}$ is the utility on the dead equals 0, perfect health equals 1 scale and $b_1$ to $b_8$ are the coefficients for the levels of vision, hearing, speech, ambulation, dexterity, emotion, cognition and pain.

The report's example has vision, ambulation and emotion at level 2 and pain at level 3, with coefficients 0.98, 0.93, 0.95 and 0.90. Their product is 0.7792, and 1.371 × 0.7792 = 1.068, which less 0.371 gives 0.697, or 0.70 rounded. Every attribute at its worst level gives a product of about 0.0087 and a score of about minus 0.36, the lowest HUI3 score Horsman and colleagues report. The AQoL developers describe the HUI and AQoL as the only utility instruments using a flexible multiplicative model, and the 15D valuation system also applies multi-attribute utility theory.

Multi-attribute functions and statistical value sets

Not every preference-based instrument is scored this way. Dolan built the UK EQ-5D tariff from time trade-off valuations of 42 states, with a model using the severity level of each dimension, an intercept for any move away from full health and a term for any dimension at its most severe level. The SF-6D model was estimated from standard gamble values for 249 states given by 611 members of the UK public and predicts all 18,000 states. Such a value set is estimated statistically from directly valued states, not assembled from corner states, and the HUI developers describe the EQ-5D form as implicit and ad hoc.

Use in cost-utility analysis and the NICE reference case

Horsman and colleagues note that HUI utility scores are used in cost-utility and cost-effectiveness analyses. A multi-attribute utility enters a model as a health state utility value for each state, multiplied by time to give QALYs.

In England, the NICE manual (PMG36) states in section 4.3.4 that valuation should use public preferences from a representative sample of the UK population with a choice-based method, and NICE PMG51 (published 27 August 2026) rewrites section 4.3.6 so that the EQ-5D-5L is the preferred measure of health-related quality of life in adults. Under PMG36 section 4.3.10, an analysis that uses another instrument, such as the HUI3, must show with evidence that the EQ-5D lacks content validity and performs poorly in that population.

Limits and common misreadings

Most errors come from treating the scoring formula as a neutral lookup.

1. Assuming additivity by default. A weighted sum is valid only if the corner disutilities sum to 1, and the HUI data rejected it.
2. Reading level codes as scores. The HUI3 report stresses that attribute level codes lack interval scale properties, while utility scores have them.
3. Confusing single-attribute and overall scores. Single-attribute HUI scores give 0 to the worst level and 1 to full function, and are not anchored at dead.
4. Treating one interaction as universal. A multiplicative function makes all attributes complements or all substitutes, even where pairs differ.
5. Mixing values and utilities. In the multiple sclerosis study the standard gamble gave much higher values than the rating scale.
6. Comparing instruments directly. HUI3, AQoL and EQ-5D scores for one patient can differ because their descriptive systems, valuation methods and combining rules differ.

Sources

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• Dolan P. Modeling valuations for EuroQol health states. Medical Care. 1997;35(11):1095-1108. https://doi.org/10.1097/00005650-199711000-00002
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