Equity weights in the allocation of health care: the rank-dependent QALY model

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Abstract

This paper introduces the rank-dependent quality-adjusted life-years (QALY) model, a new method to aggregate QALYs in economic evaluations of health care. The rank-dependent QALY model permits the formalization of influential concepts of equity in the allocation of health care, such as the fair innings approach, and it includes as special cases many of the social welfare functions that have been proposed in the literature. An important advantage of the rank-dependent QALY model is that it offers a straightforward procedure to estimate equity weights for QALYs. We characterize the rank-dependent QALY model and argue that its central condition has normative appeal.

Introduction

The use of quality-adjusted life-years (QALY)1 has become standard practice in the analysis of the cost-effectiveness of medical interventions. The use of the QALY measure is commonly associated with the assumption that health care resources should be allocated so as to achieve the maximal health gain as measured by additional QALYs. Many authors Broome, 1988, Harris, 1988, Lockwood, 1988, Culyer, 1989, Wagstaff, 1991, Dolan, 1998 have raised concerns about the equity implications of this allocation rule.

The basic problem is presented by Williams (1997) in terms of the ‘fair innings’ argument. Is it equitable that an increase of, say, one QALY should be valued equally whether it accrues to someone who is already ‘rich’ in years or to someone whose life in the absence of treatment will be short and miserable? Williams suggests that an appropriate response may be to use ‘equity weights’, but expresses the concern (p. 28) that:

there is a danger such weights become arbitrary and capricious and come to be used to fudge outcomes in ways that would not be acceptable if their basis were exposed. One safeguard against this is to have some underlying (or ‘over-arching’) general principle enunciated, which can be confronted with evidence so that its various implications can be explored in a quantitative way.

Williams argues that the fair-innings principle may provide the basis for such an approach, but observes that much work needs to be done in developing the conceptual basis of a ‘fair innings’. In particular, it is important to determine the implications of any general principle, considered in isolation or in combination with other principles which may seem desirable. A set of principles may appear unexceptionable when considered separately, but may produce unpalatable implications when considered jointly, or may be mutually inconsistent. The best way to avoid such undesirable outcomes is through the derivation of equity weights from clearly stated conditions. The acceptability or otherwise of these conditions may then be assessed both in isolation and with respect to their joint implications.

The object of this paper is to present a general allocation rule which incorporates equity weights and to derive the conditions on which this rule depends. Under this rule, the equity weight assigned to an individual depends on his rank when individuals are ranked in terms of their expected lifetime QALYs. That is, the equity weights depend on the relative positions of the individuals. We refer to this allocation rule as the rank-dependent QALY model. Models of rank dependence are widely used in decision under uncertainty Quiggin, 1981, Yaari, 1987, Schmeidler, 1989, Tversky and Kahneman, 1992 and the measurement of inequality Weymark, 1981, Ebert, 1988, Yaari, 1988. Rank dependence was used to give a preference foundation for the QALY model in Bleichrodt and Quiggin, 1997, Bleichrodt and Quiggin, 1999 and in Miyamoto (1999). Here we introduce the concept of rank dependence for the social evaluation of QALY profiles.

We give a preference foundation for the rank-dependent QALY model and we will argue that the model depends on reasonable conditions. An important advantage of the rank-dependent QALY model for empirical research and practical applications is that it provides a simple way to estimate equity weights from individual choices, as we show in Section 2.5. Williams and Cookson (2000, p. 1905) argue that the “great challenge [for health economists] is to bridge the gap between the economic requirement to estimate precisely targeted equity–efficiency trade-offs, and the psychological capabilities of respondents to think about equity and efficiency in such a tightly defined manner.” The rank-dependent QALY model can bridge the gap to which Williams and Cookson refer: the model can incorporate both efficiency concerns and a wide array of equity concerns and its empirical elicitation is straightforward and not cognitively demanding.

The paper is organized as follows. Section 2 is the central section of the paper. It describes the rank-dependent QALY model and shows that the model can incorporate equity concerns into QALY-based decision making and that it has several important social welfare functions as special cases, including unweighted aggregation, generally referred to as QALY utilitarianism, and Rawls (1971) maximin rule. We also show how the equity weights can be elicited in the rank-dependent QALY model. An interesting feature of the rank-dependent QALY model is that it can be decomposed into an efficiency term and an equity term. This decomposition may be used to provide a welfare economic foundation for the measurement of equity in health and health care (Van Doorslaer et al., 1993).

Sections 3 and 4 are more technical and contain a derivation of the rank-dependent QALY model. Our analysis relies to a large extent on the observation, based on the work of Harsanyi, 1953, Harsanyi, 1955, Atkinson, 1970, and Rawls (1971), that there is a close connection between models of choice under uncertainty and models of social choice. Section 3 presents a translation of de Finetti’s (1931) famous book-making principle to the context of social choices over QALY profiles. We use the book-making principle to derive conditions for the optimality of QALY utilitarianism. Section 4 argues that the full form of the book-making principle is too restrictive in social choice. We introduce a weaker version, the comonotonic book-making principle. The comonotonic book-making principle is the central condition in our derivation of the rank-dependent QALY model. We argue that the comonotonic book-making principle has normative appeal, and thereby give a defense for the use of the rank-dependent QALY model in the economic evaluation of health care. Section 5 concludes. Proofs of the results presented throughout the paper are given in Appendix A.

Section snippets

Background

We consider a policy maker who has to choose between health care programs leading to different allocations of expected lifetime QALYs among the individuals in society. Let n be the number of individuals in society. Let Qi denote the expected number of QALYs received by individual i throughout his lifetime. A QALY profile (Q1,…,Qn) specifies the expected number of lifetime QALYs received by each individual. We denote the set of all QALY profiles by Q. Occasionally we use the notation (p1,Q1;…;pm,

The book-making principle

Let us now turn to the characterization of the rank-dependent QALY model. Crucial in our characterization is a consistency condition, which is a reformulation of de Finetti’s book-making principle (de Finetti, 1931) for social evaluations of QALY profiles. The book-making principle is based on the idea that a number of good decisions, when taken together, should still be good. In terms of the economic evaluation of health care, it means the following. Suppose that the policy maker has performed

The comonotonic book-making principle

Theorem 1 is perhaps a surprising result. The assumptions of weak order, anonymity, and the existence of a constant equivalent are widely accepted, while the book-making principle seems a necessary consistency requirement for economic evaluations of health care. However, Theorem 1 shows that these four, seemingly innocuous, conditions imply that the social welfare function must take the QALY-utilitarian form. Hence, Theorem 1 provides a rationale for the use of QALY utilitarianism in the

Conclusion

This paper has introduced the rank-dependent QALY model, a new model to evaluate QALY profiles. It provides a way to incorporate equity concerns into cost-utility analysis and is consistent with many types of social welfare function that have been proposed in the literature, including QALY utilitarianism, Rawls’ maximin, and the Gini social welfare function.

The central preference condition in the characterization of the rank-dependent QALY model is the comonotonic book-making principle. We have

Acknowledgements

Helpful remarks were received from two anonymous referees and from Philippe Delquié, Carmen Herrero, Magnus Johannesson, Fernando Vega, Antonio Vilar, and seminar participants at the Stockholm School of Economics, the University of Alicante, the University of Oslo, and INSEAD, Fontainebleau. Han Bleichrodt’s research was made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences and by a grant from the Netherlands Organisation for Scientific Research (NWO). John

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