When are person tradeoffs valid?
Introduction
The person tradeoff (PTO) method was first devised by Patrick et al. (1973) (they call it “the equivalence [of numbers] technique”) and later promoted by several other researchers as a means of mitigating fairness concerns in health allocation (Nord, 1995, Nord et al., 1999, Murray and Lopez, 1996, Pinto-Prades, 1997, Baron and Ubel, 2002, Ubel et al., 2000). The PTO method has been used in important health policy applications. For example, the World Bank uses the PTO in the estimation of the quality weights in its burden of disease studies.
Despite its long history and popularity, the PTO has, to our knowledge, no theoretical basis to support its use. As of yet, it is unknown under which conditions the PTO adequately reflects societal preferences over allocations of health. The purpose of this paper is to identify and test these conditions. The identification of these preference conditions will facilitate an understanding of the rules that govern societal health allocation decisions when the PTO is employed. It will thus help clarify assumptions associated with applications of PTO values in modeling and analysis (e.g., cost-effectiveness and cost-value analysis). Further, identifying preference conditions will allow for empirical testing of the validity of the PTO. Towards the end of this paper we report the results of a first empirical test of the conditions underlying the PTO. The experiment sheds light on the question of whether or not using the PTO is justified. The enterprise of testing the validity of the PTO is of significant importance for the obvious reason that if we ignore the preference implications of the PTO, we may end up recommending policies that conflict with society's best interests.
Patrick et al. (1973) is consistent with the following:
- (1)
Assume four health state improvements , “y to z”, and “y to y” (for brevity we denote these by , (y, z), and (y, y) respectively) and natural numbers m and n.
- (2)
Policy 1 is such that m persons receive and n persons are left untreated (i.e., they receive (y, y)).
- (3)
Policy 2 is such that m persons receive (i.e., they are left untreated) and n persons receive (y, z)).
- (4)
Policy 1 is equivalent to Policy 2 if and only if , where V is a function that assigns a numeric value to health states.
In the Patrick et al. (1973) analysis, z is “optimal functioning” and assigned a value of 1, y and are both set equal to death, which is (implicitly) given a value 0. Finally, x is some suboptimal health state which is to be assigned a value between 0 and 1. So for example, by the Patrick et al. (1973) measurement scheme a decision maker is indifferent between the following policies:
Policy A = “Save 500 lives (m), but these people are left with blindness (x)”, and
Policy B = “Save 100 lives (n), returning all one hundred to full health (z)”
Since the work by Patrick et al. (1973), the general method given in (1) to (4) above has been used numerous times as a means of evaluating person tradeoffs (see for example, Nord, 1999, Murray and Lopez, 1996, Pinto-Prades, 1997, Baron and Ubel, 2002). Ubel et al. (2000) have objected to (4) above, because it treats health state improvements as value differences. While Ubel et al. (2000) did not state an alternative formula for expressing health state improvements as a combination of health state utilities, their position is consistent with a more general approach given here as (4′):
- (4′)
Policy 1 is equivalent to Policy 2 if and only if .
In this paper, we identify the assumptions underlying (4) and (4′). The comparison of assumptions will permit better insight into how much more restrictive (4) is than (4′). Several empirical studies have examined implications of the PTO formula (Damschroder et al., 2004, Dolan and Green, 1998, Dolan and Tsuchiya, 2003) and generally obtained negative results. Unfortunately, it is not clear from these studies why PTO measurements give inconsistent results, because several assumptions are tested simultaneously. The advantage of our approach is that it allows for exact tests of (4) and (4′) and their difference, in the sense that no confounding assumptions have to be made. Hence, our analysis defines a new empirical framework for testing the validity of PTO measurements. We pick up this framework in Sections 6 and 7 where we present the results of an experiment aimed at testing the conditions underlying (4) and (4′). This analysis provides insight into the issue of bias in PTO measurement. Theorem 1, Theorem 2 offer a full understanding of the preference conditions implied by PTO formulas. Thus, when a preference condition is violated we are assured that PTO measurement yields biased results.
Most PTO exercises are carried out in a riskless context. This means that the respondent makes decisions between two alternatives that deliver health state improvements to a cohort with certainty. However, it is clear that health policy decisions inherently involve risk or uncertainty (Doctor and Miyamoto, 2005). We know, for example, that policy makers face uncertainty as to the number of persons who will be afflicted with disease or injury in any given year. Therefore, PTO exercises draw from a subset of a larger (and perhaps more realistic) set of policy decision making scenarios that involve risk. It behooves us then to model person tradeoffs in the most general way possible so as to accommodate potential future modifications that might involve risky choices. Therefore, in what follows we describe the basic preference objects with which the policy maker is confronted as risky policies. The outcomes of these risky policies are distributions of health state improvements and the goal of the policy maker is utility maximization of health state improvements. The model we develop governs preferences over riskless policies as well, as we will explain in the next section. Thus, current practice is a special case of our approach.
Section snippets
Background
Let Ci be the set of all possible health state improvements that are experienced by the ith person in society over some pre-specified period of time (e.g., 1 year)1 and n the number of individuals in society affected by a particular health program. The set of health state improvements, C, is a set of n-tuples of the form c = 〈c1, c2, …, cn〉,
Definitions
We now provide conditions that will provide a formal justification for the use of person tradeoff measurement. Definition 1 Marginality (Fishburn, 1965). (1/2:c1;c2) ∼ (1/2:c3;c4) whenever c1, c2, c3, c4 ∈ C, and any ci ∈ Ci that appears once (twice) in (1/2:c1;c2) also appears once (twice) in (1/2:c3;c4) and vice versa.
Marginality says that preferences depend only on marginal probability distributions over health state improvements. As an illustration, consider the following two policies (Table 1) involving two
Main result
The following theorem characterizes statement 4′, the PTO model suggested by Ubel et al. (2000). Theorem 1 Structural Assumption 1 holds. Then for all health improvements and (y, z) in H × H, for all positive integers m and n, and for all policies and c2 = 〈01, …, 0m, (ym+1, zm+1), …, (ym+n, zm+n)〉 in C for which c1 ≽ c2, the statements (i) and (ii) are equivalent: The preference relation, ≽, satisfies marginality (Definition 1) and anonymity (Definition 2). There exists
Extension to non-expected utility
Thus far we have assumed that the social decision maker behaves according to expected utility. Our main motivation to do so was that social policy is normative in nature and expected utility is still the dominant normative theory of decision making. However, if we want to test whether people, when put in the role of a social planner, actually make choices that are consistent with the PTO, a problem arises. Testing preference conditions is a descriptive activity and there exists abundant
Design
We used the above theory in an experimental test of the validity of the two versions of the PTO, (4) and (4′). The experiment tested the following four questions:
- (1)
Assuming expected utility, does marginality hold?
- (2)
Does additivity hold?
- (3)
Is the assumption of expected utility appropriate or do people deviate from expected utility in choosing between health policies?
- (4)
If people deviate from expected utility, does the generalization of marginality, Definition 1′ hold?
Of the 113 subjects in the experiment
Marginality
Fig. 1 shows the results for the two tests of marginality. Remember that marginality predicts indifference. Approximately 40% of the subjects are indifferent in each test separately. Seventy percent of the subjects who satisfy marginality in the first test also satisfy marginality in the second test. Consequently, about 25% of the subjects satisfy marginality in both tests. Recall that Theorem 1, Theorem 2 require that marginality holds in both tests.
Among the subjects who strictly prefer one
Discussion
It is worth evaluating critically the preference conditions that give rise to PTO measurement so as to determine the appropriate (and valid) scope of its use. If the assumptions outlined in Theorem 2 hold, then the results obtained from PTO measurements would not conflict with the objectives of CEA. Indeed such measurements of health state values were the original intent of Patrick et al. (1973). Beyond use of the PTO formula given by Patrick et al. (1973) for valuation of outcomes in CEA, the
Concluding remarks
We began this paper with the question, “When are person tradeoffs valid?” The answer, we have shown, is that they are valid when respondents are indifferent to social choices for which the expected benefit to each person in society is the same (i.e., marginality, see Definition 1), when decisions are not affected by a persons identity (i.e., anonymity, see Definition 2), and when health improvements are additive in preference (i.e., additivity, see Definition 3). As we explained, marginality
Acknowledgements
Jason Doctor's research was made possible by a grant from the United States Department of Health and Human Services, National Institutes of Health, National Library of Medicine (NIH-R01-LM009157-01). Han Bleichrodt's research was made possible by a grant from the Netherlands Organization for Scientific Research (NWO).
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