1. A RECIPE FOR INCOHERENCE: AVERAGING TIME-TRADEOFF OR STANDARD-GAMBLE UTILITIES ACROSS HEALTH ATTRIBUTES Gordon B. Hazen, IEMS Department, Northwestern University, Evanston IL Abstract 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 = (q 1,q 2 ) is described by two health attributes, and modelers wish to use TTO or SG techniques to assess utilities u 1, u 2, and then form a weighted average to obtain overall QALY coefficient U Q (q) = k 1 u 1 +k 2 u 2, where k 1 and k 2 = 1  k 1 are the weights. We assume that when a subject specifies a TTO or SG response r 1 for a level q 1 of one attribute, s/he by default assumes the other attribute q 2 is at its best level, and vice-versa. Results: Under the averaging model U Q (q) = k 1 u 1 +k 2 u 2, the standard of taking u 1 = r 1 and u 2 = r 2 is no longer valid. If the modeler does so and then averages as just described to obtain the QALY coefficient U Q (q), the resulting theoretical error in U Q (q) is  U Q (q) = (1  r 1 )k 2 + (1  r 2 )k 1. The error  U Q (q) is largest for attribute levels q 1,q 2 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 (q) = u 1 u 2. 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. QALY model Standard gamble assessment To obtain the utility U Q (q) of health state q, a subject indicates what chance 1-p of immediate death is worth a p chance at improving health quality from q to q*. The response p is the utility of health state q. Time-tradeoff assessment To obtain the utility U Q (q) of health state q, a subject indicates what reduction in lifetime t 0 would be worth taking to increase health quality to full health q*. The ratio t/t 0 of the response t to the base lifetime t 0 is the utility of health state q. Additively separable QALY model Standard gamble assessment one attribute at a time To obtain the utility u 1 of health state q 1, a subject indicates what chance 1-p of immediate death is worth a p chance at improving health quality from q 1 to q 1 * when health attribute 2 is fixed at some level q 2. The level of q 2 may not be identified, in which case the subject may implicitly assume that q 2 = q 2 *. If the response is p, then the utility u 1 of q 1 may be derived as: Note: The inferred utility u 1 in general is not u 1 = p as it is in the single-attribute case, but rather should depend on the utility u 2 of the second attribute. Taking u 1 = p overestimates u 1 by an amount equal to Time-tradeoff assessment one attribute at at time To obtain the utility u 1 of health state q 1, a subject indicates what reduction in lifetime t 0 would be worth taking to increase health quality to full health q 1 * when health attribute 2 is fixed at some level q 2. Again, the level of q 2 may not be identified, in which case the subject may implicitly assume that q 2 = q 2 *. If the response is t < t 0, then the utility u 1 for state q 1 may be derived as: Note: The inferred utility u 1 in general is again not u 1 = t/t 0 as it is in the single-attribute case, but rather should depend on the utility u 2 of the second attribute. Taking u 1 = t/t 0 overestimates u 1 by an amount equal to Multiplicative QALY model Standard gamble assessment one attribute at a time Note: The inferred utility is u 1 = p, just as it is in the single- attribute case. Time-tradeoff assessment one attribute at at time Note: The inferred utility is u 1 = t/t 0, just as it is in the single-attribute case. Conclusion: Using standard gamble or time tradeoff assessments one attribute at a time 1.Is inconsistent with assuming that health state utility is additively separable across health attributes; 2.Is consistent with assuming that health state utility is multiplicatively separable across attributes. 3.In fact, a multiplicatively separable health state utility function is the only utility function that allows standard gamble or time tradeoffs to be done one attribute at a time (Hazen 2004). Reference GB Hazen (2004), “Multiattribute Structure for QALYs”, forthcoming in Decision Analysis. The total error can be as large as 0.5 on a scale from 0 to 1 when k 1 = k 2 = ½. If both attributes are assessed this way with the other attribute fixed at its best level, the overestimate in U Q is given by The total error can be as large as 0.5 on a scale from 0 to 1 when k 1 = k 2 = ½. If both attributes are assessed this way with the other attribute fixed at its best level, the overestimate in U Q is given by