استنباط ناپارامتری از سهام بهره وری تجارت کردن در تجزیه و تحلیل هزینه های آب و برق

We performed an empirical elicitation of the equity-efficiency trade-off in cost-utility analysis using the rank-dependent quality-adjusted life-year (QALY) model, a model that includes as special cases many of the social welfare functions that have been proposed in the literature. Our elicitation method corrects for utility curvature and, therefore, our estimated equity weights are not affected by diminishing marginal utility. We observed a preference for equality in the allocation of health. The data suggest that the elicited equity weights were jointly determined by preferences for equality and by insensitivity to group size. A procedure is proposed to correct the equity weights for insensitivity to group size. Finally, we give an illustration how our method can be implemented in health policy.

The common procedure to aggregate health benefits in economic evaluations of health care is by unweighted aggregation, also referred to as quality-adjusted life-year (QALY)-utilitarianism. This procedure weights the health gains of each individual equally and leads to a maximization of health gains. Several authors have raised concerns about the equity implications of QALY-utilitarianism and have argued that it may be necessary to differentiate between individuals based on, for example, age, health status or previously enjoyed health (Harris, 1987, Nord, 1995, Williams, 1997 and Williams and Cookson, 2000). Empirical evidence supports these concerns and indicates that people, when choosing between different allocations of health gains, not only consider efficiency, the total amount of health gains, but also equity, the distribution of the health gains (e.g. Nord, 1993, Dolan, 1998 and Abellan and Pinto, 1999). These findings suggest that it may be preferable to replace QALY-utilitarianism by some sort of equity-weighted aggregation rule. Unfortunately, the available empirical research offers little guidance as to which rule should be used and how the equity weights could be elicited. Several authors have proposed theoretical models to incorporate equity considerations into cost-utility analysis (Wagstaff, 1991, Bleichrodt, 1997, Williams, 1997 and Dolan, 1998). Both Wagstaff, 1991 and Wagstaff, 1993 and Dolan (1998) proposed to use an iso-elastic social welfare function to allow for a trade-off between efficiency and equity. Within this class of social welfare functions, Dolan (1998) suggested, in particular, to use a Cobb–Douglas function. Wagstaff (1991) and Dolan (1998) did not derive the assumptions underlying their proposed social welfare functions, which complicates an assessment as to why the equity-efficiency trade-off should take the form they proposed. They did not explain either how the parameters in their social welfare functions could be assessed. Bleichrodt (1997) proposed a multiplicative social welfare function, derived the conditions on which it depends, and showed how its equity parameter could be elicited. The range of equity concerns that the multiplicative social welfare function can address is, however, limited. Williams (1997) suggested that individuals should be weighted according to their ‘fair innings’, the difference between the amount of health they already enjoyed and the amount of health they are entitled to over their lifetime. Williams’ proposal suggests that he had in mind some sort of weighted aggregation rule, but he did not specify what form this weighted rule should take nor did he explain how the equity weights could be elicited. Bleichrodt et al. (2004) recently proposed a new social welfare function to incorporate equity considerations into cost-utility analysis, the rank-dependent QALY model. Their model has several desirable characteristics. First, it is consistent with several social welfare functions that have been proposed in the literature, including QALY-utilitarianism, the Rawlsian social welfare function in which all weight goes to the worst-off individual, and the Gini social welfare function, which is widely used in inequality measurement. The rank-dependent QALY model can also accommodate Williams’ fair innings approach. Second, as Bleichrodt et al. (2004) showed, the rank-dependent QALY model depends on assumptions that have normative appeal. A third advantage of the model is that the elicitation of the equity weights is straightforward. Finally, the model is tractable: once the equity weights have been elicited, the model can easily be used in cost-utility analyses. The aim of this paper is to elicit the equity weights under the rank-dependent QALY model. For reasons explained in Section 2, we used a more general model than the model proposed in Bleichrodt et al. (2004). In Bleichrodt et al. (2004), the social utility function over QALYs is linear, whereas in this paper, we allow for a nonlinear social utility function over QALYs. We refer to this extended model as the nonlinear rank-dependent QALY model. A consequence of using a more general model is that its elicitation becomes more involved, because, in addition to the equity weights, the social utility function over QALYs must be determined. The structure of the rest of the paper is as follows. In Section 2, we describe the nonlinear rank-dependent QALY model. In Section 3, we explain the elicitation of the model. To elicit the model, we used an adjusted version of the trade-off method (Wakker and Deneffe, 1996), which was developed to measure utilities under risk. An advantage of the trade-off method is that it is nonparametric: it imposes no assumptions on the utility function or on the equity weighting function. We elicited the nonlinear rank-dependent QALY model both in a sample of students and in a sample of the general population. Section 4 describes the designs of the two experiments, Section 5 the results. Section 6 shows how our method can be implemented in health policy. Section 7 offers concluding remarks.

7.1. Main findings In this paper, we have elicited, both in a sample of students and in a sample from the general population, the trade-off between equity and efficiency in the allocation of health. We assumed the nonlinear rank-dependent QALY model, a model that encompasses many of the social welfare functions that have been proposed in the literature. A correction for utility curvature was applied but we found that, on the aggregate level, social preferences were approximately linear in QALYs. People were generally inequality averse, except when the better-off group was small. The reason why we found no global inequality aversion may be insensitivity to group size. Global inequality aversion was observed when we corrected for insensitivity to group size. Few differences were observed between the sample of students and the sample from the general population. 7.2. Possible biases As noted in Section 5, the exclusions due to violations of rank-dependency may have affected the results. We tested for the effect of these exclusions by making the extreme assumption that the excluded subjects violated rank-dependency in every question. This assumption means that these subjects had the highest equity weights of all subjects in the questions for p1, p2 and p3, and the lowest equity weights in the questions for p4 and p5. Such a preference pattern is unlikely and the assumption is almost certainly too extreme, which means that the actual bias will be smaller, but the analysis gives an indication of the maximum effect of the exclusions on the median equity weights. Under the assumption, the median equity weights for 1/6, 1/3, 1/2, 2/3 and 5/6 changed by 0, +0.003, +0.006, −0.007 and 0, respectively, in the student sample and by 0, +0.015, +0.026, −0.022, −0.025 in the general population sample. Hence, even under an extreme assumption about the effect of the exclusions due to violations of rank-ordering, the effect of these exclusions was small. A frequently encountered problem in preference assessment tasks is that people have a tendency to respond in round numbers, often multiples of five, which can lead to bias. Because a choice-based procedure was used, round answers were less likely in our study. In fact, the proportion of round answers (multiples of five) was 21.7% in the student sample and 21.3% in the general population sample, which are not significantly different from 20%, the proportion of round answers expected when people do not have a tendency to use round answers. It may have been possible that some subjects did not understand the concept of a QALY properly, leading to additional response error. It would have been easier to perform the experiment with years of life instead of QALYs. We opted to use QALYs, because policy makers and researchers are most interested in the trade-off between equity and efficiency as measured by QALYs. Upon questioning by the experimenter, most subjects seemed to understand the concept of a QALY well. To complete the exercise, they generally assumed that people in the cohort lived in relatively good health for the largest part of their life, and that the largest QALY loss was related to life-years lost. Our findings depend on the validity of the nonlinear rank-dependent QALY model, Expression (2). Even though Expression (2) is quite general, it may in some cases be too restrictive. The model assumes, in particular, that the equity weights depend only on individuals’ relative positions, their rank, and not on absolute differences between the amounts of QALYs received. If this assumption does not hold then our results may no longer be valid. Another violation would occur if there is no separability between the equity weights and the utility for QALYs. In that case, the elicitation of the utility for QALYs might depend on the proportion used. We could have used a more general model than Expression (2) to take these possible violations into account. This would, however, have led to a model that is more difficult to apply in practice. The question is whether violations of the nonlinear rank-dependent QALY model, if any, are sufficiently widespread and serious to justify giving up the tractability of the model. Finally, it is possible that, even though we tried to control for it, asymmetric errors may still have affected the results. If this were true, then these errors will have had most effect on w(1/6)w(1/6) and w(5/6)w(5/6), biasing w(1/6)w(1/6) upwards and w(5/6)w(5/6) downwards. The effect on the other three weights that we elicited is probably negligible, because in these estimations the stimuli were not close to the bounds and there was enough room for error “on both sides”. Our main finding of a generally convex equity weighting function, i.e. aversion to inequality, is confirmed when we only look at w(1/3)w(1/3), w(1/2)w(1/2) and w(2/3)w(2/3), giving grounds for confidence in the results. 7.3. Final remarks Our study suggests that people are averse to inequalities in health. If people's societal preferences ought to have a place in health policy, then our findings connote that QALYs should be weighted for equity concerns. We have shown that the rank-dependent QALY model can be used for this: we have presented a method to elicit the equity weights under the model and we have shown how these equity weights can be implemented in health policy. We repeat that the purpose of the latter exercise was illustrative; before more robustness checks are performed, restraint should be exercised in using the data we presented in actual policy making. Finally, a few words about the equity concept we used are in order. Because we studied people's preferences over allocations of lifetime QALYs, our study focused on differences in lifetime health expectancy between groups of newborns. This setup implicitly assumed that the desirability of a distribution depends on people's (expected) lifetime health. In that sense, our approach is close to Williams’ fair innings approach. Several authors have discussed other concepts of equity and have argued that equity may also be concerned with other issues, such as patients’ actual health state and when and how health losses occur (Culyer and Wagstaff, 1993, Cuadras et al., 2001 and Dolan and Olsen, 2001). Our empirical results have little bearing in case such equity concerns are adopted. How these other equity concerns can be operationalized, remains, therefore, an open question.