Purpose: Health outcomes are often specified using multiple health attributes.  Procedures for assigning QALY coefficients to multiattribute health states include, for example, the Health Utilities Index and the EuroQol.  For some cost-effectiveness analyses, the HUI or EuroQol attributes are not specific enough to address important issues.  In such cases, modelers may be tempted to assess time-tradeoff or standard gamble utilities one attribute at a time, and then combine the assessed utilities by averaging over attributes.  We point out why this procedure is mathematically incoherent, and show what errors in the inferred QALY coefficients may occur as a result.

Methods: We consider the case in which health status q = (q₁,q₂) is described by two health attributes, and modelers wish to use TTO or SG techniques to assess utilities u₁, u₂, and then form a weighted average to obtain overall QALY coefficient U(q) = k₁u₁+k₂u₂, where k₁ and k₂ = 1-k₁ are the weights.  We assume that when a subject specifies a TTO or SG response r₁ for a level q₁ of one attribute, s/he by default assumes the other attribute q₂ is at its best level, and vice-versa. 

Results: Under the averaging model U(q) = k₁u₁+k₂u₂, the standard of taking u₁ = r₁ and u₂ = r₂ is no longer valid.  If the modeler does so and then averages as just described to obtain the QALY coefficient U(q), the resulting theoretical error in U(q) is DU(q) = (1-r₁)k₂ + (1-r₂)k₁.  The error DU(q) is largest for attribute levels q₁,q₂ farthest below their best possible levels, and can be as large as 0.5 on a scale from 0 to 1 when attributes are equally weighted.  The only way to avoid such errors is to replace the averaging rule by the multiplicative combination rule U(q) = u₁u₂.

Conclusions: Assessing time-tradeoff or standard gamble utilities one attribute at a time, and then averaging the assessed utilities to obtain an overall QALY coefficient is mathematically incoherent and can lead to large errors in the resulting QALY coefficients.