Abstract

1. Additively separable QALY model 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 = (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 UQ(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 Results: Under the averaging model UQ(q) = k1u1+k2u2, the standard of taking u1 = r1 and u2 = r2 is no longer valid.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 UQ(q), the resulting theoretical error in UQ(q) is DUQ(q) = (1-r1)k2 + (1-r2)k1. The error DUQ(q) is largest for attribute levels q1,q2farthest 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 averagingrule by the multiplicative combination rule UQ(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. Multiplicative QALY model A RECIPE FOR INCOHERENCE: AVERAGING TIME-TRADEOFF OR STANDARD-GAMBLE UTILITIES ACROSS HEALTH ATTRIBUTESGordon B. Hazen, IEMS Department, Northwestern University, Evanston IL QALY model Standard gamble assessment one attribute at a time To obtain the utility u1 of health state q1, a subject indicates what chance 1-p of immediate death is worth a p chance at improving health quality from q1 to q1* when health attribute 2 is fixed at some level q2. The level of q2 may not be identified, in which case the subject may implicitly assume that q2 = q2*. If the response is p, then the utility u1 of q1 may be derived as: Standard gamble assessment To obtain the utility UQ(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. Standard gamble assessment one attribute at a time Note: The inferred utility is u1 = p, justas it is in the single-attribute case. Note: The inferred utility u1 in general is notu1 = p as it is in the single-attribute case, but rather should depend on the utility u2 of the second attribute. Taking u1 = p overestimates u1 by an amount equal to Time-tradeoff assessment To obtain the utility UQ(q) of health state q, a subject indicates what reduction in lifetime t0 would be worth taking to increase health quality to full health q*. The ratio t/t0 of the response t to the base lifetime t0 is the utility of health state q. Time-tradeoff assessment one attribute at at time If both attributes are assessed this way with the other attribute fixed at its best level, the overestimate in UQ is given by Note: The inferred utility is u1 = t/t0, justas it is in the single-attribute case. The total error can be as large as 0.5 on a scale from 0 to 1 when k1 = k2 = ½. • Conclusion: Using standard gamble or time tradeoff assessments one attribute at a time • Is inconsistent with assuming that health state utility is additively separable across health attributes; • Is consistent with assuming that health state utility is multiplicatively separable across attributes. • 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. Time-tradeoff assessment one attribute at at time To obtain the utility u1 of health state q1, a subject indicates what reduction in lifetime t0 would be worth taking to increase health quality to full health q1* when health attribute 2 is fixed at some level q2. Again, the level of q2may not be identified, in which case the subject may implicitly assume that q2 = q2*. If the response is t < t0, then the utility u1 for state q1 may be derived as: Note: The inferred utility u1 in general is again notu1 = t/t0as it is in the single-attribute case, but rather should depend on the utility u2 of the second attribute. Taking u1 = t/t0 overestimates u1by an amount equal to If both attributes are assessed this way with the other attribute fixed at its best level, the overestimate in UQ is given by The total error can be as large as 0.5 on a scale from 0 to 1 when k1 = k2 = ½.