Deriving quality-adjusted life years (QALYs) from constant proportional time tradeoff and risk posture conditions

Abstract

With foundations in decision theory, Quality-Adjusted Life Years (QALYs) are the most important measure of health outcome in medical decision making. Constant proportional time tradeoff (CP-TTO) is a condition on riskless value with empirical support in the QALY literature. The primary use of CP-TTO in previous axiomatizations of QALYs has been to guarantee that the utility of health state and survival duration is a power function of duration. We show that CP-TTO has other useful applications in QALY research. Under mild conditions CP-TTO implies an additive value function. Furthermore, CP-TTO can be combined with assumptions of risky utility to yield several different parametric QALY representations, including one that is inconsistent with utility independence. We find that CP-TTO simplifies axiomatizations of QALY representations in the sense that if CP-TTO is among the assumptions of a QALY axiomatization, other standard preference assumptions, e.g., double cancellation or utility independence, become redundant and can be dropped from the axiom system. This can be desirable if it allows one to drop from the axiomatization an assumption that is practically difficult to test, e.g., double cancellation.

Introduction

Suppose a patient tells you that she is indifferent between living 40 years in poor health and living 20 years in full health. If she is also indifferent between living 30 years in poor health and 15 years in full health, then she has exhibited behavior consistent with the assumption of constant proportional time tradeoff. More generally, under constant proportional time tradeoff (CP-TTO), if for some health states a and b, we find ax∼by is an indifference expressed by the patient between living x years in a health state a and y years in a health state b, then for all c>0, such that cx,cy⩽M, a[cx]∼b[cy].

Under mild conditions, CP-TTO implies the necessary assumptions for additive conjoint measurement and a multiplicative representation of riskless value. Because risky utility represents the same preferences over outcomes as riskless value, risky utility is a strictly increasing function of riskless value (Keeney & Raiffa, 1993). This fact will allow us to identify several useful characteristics of utility functions for which we know that preferences satisfy CP-TTO. Following a similar argument to that of Maas and Wakker (1994), we show that CP-TTO together with a few assumptions of risky choice imply specific parametric forms of the utility of duration, including in one case a model that is inconsistent with utility independence. We also find that CP-TTO simplifies axiomatizations of quality-adjusted life years (QALY) representations in the sense that if CP-TTO is among the assumptions of a QALY axiomatization, other standard preference assumptions, e.g., double cancellation or utility independence, become redundant and can be dropped from the axiom system. This line of argument is similar in spirit, if not in the formal details, to arguments in Wakker (1989), Ebert (1988), and Miyamoto and Wakker (1996).

It is well known that under CP-TTO, when utility independence is invoked, the utility function is a member of the log/power family. Characterizing the utility of survival duration reduces in this case to the problem of estimating a single risk parameter (Pliskin, Shepard, & Weinstein, 1980). We show that invoking constant relative risk aversion (CRRA) along with CP-TTO also implies a log/power utility function. Thus, CRRA and utility independence imply each other when CP-TTO is invoked (This equivalence occurs when the health domain satisfies a restricted solvability condition. When restricted solvability is not assumed, CRRA appears to be a stronger condition than utility independence (see Theorem 4.1)). When constant absolute risk aversion (CARA) is then added, the utility function becomes linear and the riskless value function represents risky choices as well. In another line of development, we show that the utility function is an exponentiated linear model when CARA and CP-TTO are assumed and utility independence is dropped. Under this QALY model, estimation of a risk parameter for one health state in combination with a riskless health value elicitation in relation to any other health state, determine the curvature of the utility function. The exponentiated linear QALY model (or expo-linear QALY model), is the only QALY model that is known to violate utility independence, but is tractable for decision analysis. Keeney and Raiffa (1993, p. 259, Eq. (3.69)) first put forth this model for use in cases when utility independence is violated, but until now it has not received an axiomatic treatment.