Single Attribute Utility

Concept

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Theoretically, Single Attribute Utility is a utility representation in which preferences are defined over variations in one attribute while all other relevant attributes are held constant. It is grounded in expected utility theory and multi-attribute utility theory, where individual attributes such as pain, mobility or survival are assigned utility functions reflecting the decision-maker?s preferences. In health economics, single attribute utility functions are used as components of broader multi-attribute health utility models and decision-analytic evaluations.

Mathematically, a single attribute utility function maps each feasible level of an attribute x to a utility value u(x), commonly bounded between 0 and 1. The functional form may be linear, concave or convex depending on the individual?s attitude towards changes in the attribute. Parameters are estimated using preference elicitation methods such as standard gamble, time trade-off, rating scales or certainty-equivalent tasks.

In practice, respondents evaluate different levels of a single health or treatment attribute while other conditions remain fixed. The resulting preference data are used to estimate a utility function, which may then be incorporated into an additive or multiplicative multi-attribute utility model. Applications include valuing treatment convenience, symptom severity, adverse-event risk and dimensions of health-related quality of life.

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Purpose

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Used to quantify preferences over different levels of one health-related attribute and to provide a component utility function for multi-attribute decision and health valuation models.

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Mathematical Formulae

Primary Formula

u = u(x)

with normalisation:

u(xworst) = 0

u(xbest) = 1

Supporting Formulae

Linear utility function:

u(x) = (x ? xworst) / (xbest ? xworst)

Exponential utility function:

u(x) = [1 ? exp(?�(x ? xworst))] / [1 ? exp(?�(xbest ? xworst))]

Additive multi-attribute utility model:

U(x?, ..., x?) = ?k?u?(x?)

where:

• k? = scaling constant for attribute i
• u?(x?) = single attribute utility function

Related Mathematical Methods

• Expected Utility Theory
• Multi-Attribute Utility Theory
• Standard Gamble
• Time Trade-Off
• Utility Function Estimation
• Preference Elicitation

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Example

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Pain severity is measured on a scale from 0, representing no pain, to 10, representing the worst possible pain. Utility is defined so that no pain has utility 1 and the worst pain has utility 0.

For a linear single attribute utility function:

u(x) = 1 ? x / 10

For pain severity x = 4:

u(4) = 1 ? 4 / 10 = 0.60

The estimated utility of 0.60 can be incorporated into a broader multi-attribute model describing treatment outcomes.

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Excel Implementation

Function	Example Formula	Health Economics Application
LET	=LET(x,B2,Worst,$B$10,Best,$B$11,(x-Worst)/(Best-Worst))	Calculates a normalised linear single attribute utility.
EXP	=(1-EXP(-$F$1*(B2-$B$10)))/(1-EXP(-$F$1*($B$11-$B$10)))	Calculates an exponential utility function.
SUMPRODUCT	=SUMPRODUCT(WeightRange,UtilityRange)	Combines single attribute utilities in an additive utility model.
MAX	=MAX(0,MIN(1,C2))	Restricts estimated utility values to the interval from 0 to 1.
XLOOKUP	=XLOOKUP(B2,AttributeLevelRange,UtilityValueRange)	Retrieves an empirically elicited utility for a specified attribute level.

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VBA (Optional)

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VBA can automate estimation and application of single attribute utility functions across respondents, attributes and health states.

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Sources

• von Neumann J, Morgenstern O. Theory of Games and Economic Behavior. Princeton University Press.
• Keeney RL, Raiffa H. Decisions with Multiple Objectives: Preferences and Value Trade-Offs. Cambridge University Press.
• Torrance GW, Feeny D, Furlong W. Visual analog scales: do they have a role in the measurement of preferences for health states? Medical Decision Making.
• Brazier J, Ratcliffe J, Salomon JA, Tsuchiya A. Measuring and Valuing Health Benefits for Economic Evaluation. Oxford University Press.
• Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.