Paired Comparison

Concept

Theoretically, Paired Comparison is a preference elicitation method in which respondents choose the preferred option from two alternatives presented simultaneously. It is founded on comparative judgement theory and random utility theory, assuming that observed choices reflect underlying preferences between competing alternatives. In health economics, paired comparison is used to derive relative preferences for health states, health outcomes or interventions and forms the basis of several modern valuation methods.

Mathematically, paired comparison has no universally recognised canonical mathematical formula. Responses are commonly analysed using probabilistic choice models that estimate the latent utility associated with each alternative from observed pairwise choices. The mathematical framework converts repeated binary comparisons into a preference scale.

In practice, paired comparison is implemented by presenting respondents with successive pairs of health states or interventions and recording the preferred option in each pair. The resulting choice data are analysed using recognised statistical models, such as the Bradley?Terry or Thurstone models, to estimate relative preference weights. These estimates may subsequently be anchored to generate utility values for economic evaluation.

---

Purpose

Used to estimate relative preferences between health states or interventions through repeated binary choices, supporting health state valuation and preference measurement for economic evaluation.

---

Mathematical Formulae

Primary Formula

There is no universally recognised canonical mathematical formula.

Supporting Formulae

A commonly used Bradley?Terry model is:

P(i ? j) = e????????(e??????? + e???????)

where:

• P(i ? j) = probability that alternative i is preferred to alternative j
• ?? = latent preference parameter for alternative i
• ?? = latent preference parameter for alternative j

Related Mathematical Methods

• Bradley?Terry model
• Thurstone Case V model
• Random utility theory
• Logistic regression
• Maximum likelihood estimation

---

Example

A respondent is presented with two health states. Health State A describes moderate pain with no mobility problems, while Health State B describes severe mobility problems with no pain. The respondent chooses Health State A. After many respondents complete similar paired comparisons, a Bradley?Terry model is fitted to estimate the relative preference values for all health states.

---

Excel Implementation

Function	Example Formula	Health Economics Application
COUNTIF	=COUNTIF(B2:B201,""A"")	Counts the number of times Health State A is preferred.
COUNTIFS	=COUNTIFS(B2:B201,""A"",C2:C201,""B"")	Summarises pairwise comparison outcomes.
SUM	=SUM(D2:D201)	Totals the number of observed preferences for an alternative.
LOG	=LN(B2/C2)	Calculates log-odds during exploratory analysis of paired comparison data.

---

VBA (Optional)

Automate the import and summarisation of paired comparison responses before exporting datasets for preference model estimation.

---

Sources

• Bradley RA, Terry ME. Rank Analysis of Incomplete Block Designs: I. The Method of Paired Comparisons. Biometrika. 1952.
• Thurstone LL. A Law of Comparative Judgment. Psychological Review. 1927.
• 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.
• ISPOR Good Practice Reports on health preference research.