A missing cornerstone in the Norwegian Priority Commission’s weighting scheme – Sub-treatment balancedness is a necessary property for priority setting criteria

The Norwegian government recently put in place a priority commission tasked with suggesting a set of explicit criteria for priority setting in the health care sector. The commission suggested three criteria, the first two of which equate to cost-effectiveness, where, essentially, the gain is measured in terms of Quality Adjusted Life Years (QALYs). The third criteria specifies that the number of QALYs be multiplied by a factor depending on the total health loss – also measured in QALYs – without the treatment in question. In this paper, we will show that the suggested weighting scheme creates situations in which the priority of treatment programmes will change based on arbitrary bundling (where two or more treatments are combined into one) or sub-divisions (where a treatment is split up into two or more components.) We show that these types of problems can be avoided or ameliorated if the QALY weighting scheme satisfies a property which we call subtreatment balanced – informally, that the total weighted QALY gain is preserved when treatments are bundled or sub-divided. To our best knowledge, this property has not previously been discussed in the priority setting literature. We demonstrate that sub-treatment balance can easily be achieved in general, and in particular we show how to adapt the weighting scheme suggested by the Norwegian priority commission in order to satisfy this sub-treatment balance. Finally, we argue that any weighting scheme used in health care priority setting should be sub-treatment balanced with respect to any other attribute of a treatment which policy makers would want to take into account when making their decisions. At the time of writing, the Norwegian government has yet to conclude on a final set of criteria for prioritization, and a task-group, led by professor Jon Magnussen, is re-evaluating the severity criterion suggested by the priority commission. However, sub-treatment balance is still relevant, as it should be required of any weighing scheme, and is crucial given that (i) the criterion results in weighting QALYs, and (ii) if the selected measure of severity is affected by the administered treatment. JEL Classification: I1, H51


Introduction
Imagine two patients, Alice and Beth.At the age of 20, Alice was diagnosed with a condition that would be debilitating without treatment.The condition could be kept at bay through the continuous use of a costly treatment regimen, in which case Alice would remain asymptomatic.However, due to its high cost, the treatment for the condition did not reach the threshold for cost-effectiveness in terms of cost per quality-adjusted life year (QALYsee section 1.1).
Imagine that the criteria for priority setting specified that prior to computing the cost-effectiveness of a treatment, the expected health gain (in terms of QALYs gained) was to be multiplied by a severity factor based on the ex ante health loss in the absence of treatment, i.e. the health-loss (also measured in QALYs) that would be incurred if no treatment was administered 1 .Without the treatment, the expected health loss for Alice, beginning at the age of 20 and stretching to a life expectancy of more than 80 years, would be substantial.After applying this severity weight, the costly treatment for the condition was then considered cost-effective for Alice, and she was offered the treatment for the remainder of her life.
Fast-forward forty years.Alice and Beth are both 60 years old.Alice has been receiving treatment for her condition continuously, so that she has not experienced any health loss.Until now, Beth has been healthy, and has also had no health loss.Now, at the age of 60, Beth is diagnosed with the same condition, and the unweighted cost per QALY is still prohibitive.Without treatment, the ex ante health loss for Beth is substantial, but much less so than it was for Alice forty years previous.Due to this smaller expected healthloss, the health loss-weighted cost per QALY is still too high, and therefore Beth is not offered the treatment.
Importantly, at the age of 60, Beth and Alice have had the same experienced healthloss so far (none), and have the same future prospects with and without treatment.However, while Alice will continue to get treatment, Beth will remain untreated.This apparent difference occurs because the remaining QALY-gain from the treatment for Alice is still multiplied by the severity factor first calculated on the basis of her expected health loss at the age of 20, while the identical expected QALY-gain for Beth is multiplied by a severity factor that is calculated on the basis of her expected health loss at the age of 60.
This example is hypothetical.However, a set of priority criteria exhibiting this kind of problem have recently been suggested by a commission on priority settings in health care in Norway.The suggested implementation of the weighting scheme aimed at prioritizing treatment based on 'severity' implies that the priority of any specific treatment will change, often dramatically, if it is split into sub-treatments or if it is bundled with other treatments.In this particular case, the proposed scheme means that combining any two treatments will always result in a higher priority than those same treatments viewed separately, so that e.g. in the example, Alice gets treatment also after the age of 60, when her case is indistinguishable from Beth's, because, in the example, we have bundled together her lifelong treatment.If Alice's case were to be sub-divided into two distinct treatments considered separately, one before the age of 60 and one after, she would, like Beth, not receive treatment after the age of 60.
In this paper, we explore the properties and conditions under which severity weighting schemes result in such inconsistencies, and describe how such problems can be mitigated or avoided.
1 Ex ante health loss is a key concept in this work.It means the expected health loss incurred in the absence of a treatment.In contrast ex post health loss refers to the remaining health loss after a treatment that does not fully restore health.

1.1
Background -Priority in health care The legitimacy of the publicly financed health care systems of Northern Europe is under pressure: there is an ever-increasing gap between the medical frontier and budgetary constraints, and new beneficial treatments are not necessarily cost-effectiveness.To ameliorate this situation, clear policy practises for the priority setting of treatments are needed.Furthermore, these must both be acceptable and be perceived as fair by the general public.
In June 2013, a body of experts, politicians and representatives of other stake holders' organizations were appointed by the Norwegian government to deliver a so-called Norwegian Official Report2 (NOU).This bodycalled the Priority Commission3 had as its mandate to clarify the existing (and, where needed, to chisel out new) guidelines for priority setting in the Norwegian public health care system.The work behind the NOU was led by Professor Ole Frithjof Norheim, and the resulting NOU Transparent and Fair -Priority Setting in the health services (Norheim et al., 2014), hereafter referred to as The NOU, was submitted to the Norwegian Ministry of Health on November 12 th 2014.
The final legal status of a Norwegian Official Report is not determined until the policy process of which it is part has concluded; it may result in a new legislation (enactment), or it may be abandoned (Ringard et al., 2012, p. 24).When an NOU is enacted, it is an important source source for interpreting the resulting actin particular during an initial period, when the intention behind the law is to be interpreted by bureaucrats, lawyers, and judges.
The NOU discusses a range of subjects and issues related to priority setting in the health care sector.The mandate of the commission was to describe and set out principles of, criteria for, and the means of implementing priority settings within the Norwegian health care system.
The commission was also tasked with revising the criteria for the priority settings currently in place.The criteria originate from the last commission on priority setting, the Lønningutvalget II's report (Lønning et al., 1997).These criteria were, to put it bluntly, costeffectiveness with a glance towards 'severity'.
The NOU recommended substituting the existing severity criterion with an explicit weighting of the severity of a condition in terms of the health loss the condition incurs.
Both gains and losses are measured in quality adjusted life years (QALYs)4 .For the work presented here, only a superficial understanding of the QALY-paradigm is needed: the QALY integrates longevity and Health Related Quality of Life into a single numerical measure of health.The QALY thus reflects both improvements in survival time and improvements in health status.The interested reader may see e.g.Pliskin et al. (1980) or Weinstein et al. (2009) for an introduction to the topic.
Unfortunately, the way the NOU has suggested to implement the severity weighting will lead to inconsistencies of the type presented in the initial example of Alice and Beth.In this paper we explain how and why these problems occur, and present a simple solution.

The Priority Commission's criteria for priority setting
Chapter seven of the NOU details the suggested criteria for priority setting.The NOU proposes three criteria for setting priorities for different treatments5   offered by the Norwegian health care system:  Resources: the cost   of treatment   ;  QALY gain: the expected health gain   associated with treatment   ; and  QALY loss: the expected ex ante QALY loss ℓ  suffered by the recipient of treatment   .
Taken together, the first two items are standard cost-effectiveness: in the choice between otherwise equally resource-intensive treatments, the one which procures the greatest QALY gain should be preferred.
The third criterion states that some weight should be attributed to the magnitude of the QALY loss suffered by the recipient of the QALY gain.This means that if two individuals stand to obtain equal QALY gains , then the one with the largest QALY loss should be given priority.

Health lossthe devil is in the details
In the following section, we first set down some notation to use in discussing this issue, and summarize what the NOU writes about it.
The NOU's chapter on the health loss criterion (Norheim et al., 2014, Chap. 7.5 pp. 93-96), discusses several ways of approaching the health loss issue.The NOU decided on health losses being defined as the life-long expected health loss, ex ante the treatment being considered; ex post treatment already received.
The NOU (Norheim et al., 2014, section 7.6.2) thus recommends that in evaluating a treatment   in a priority setting context, the standard CUA-fraction6     should be replaced with an expression of the form (ℓ  ,  )   which incorporates the health loss-parameter ℓ  in addition to the health gain-parameter   to yield a health loss-weighted QALY-gain (ℓ  ,   ).The health loss ℓ  is composed of two parts: ℓ  = ℓ  0 + ℓ  1 where ℓ  0 represent the health loss already experienced in the past, while ℓ  1 represents the expected future health loss ex ante the treatment   .For example, the total health loss ℓ associated with a kidneytransplant operation, is the sum of the estimated health loss ℓ 0 already experienced at the time of the operation, plus the expected future health loss ℓ 1 without (i.e.ex ante) the operation.
In all fairness, the NOU does hint at problems with ex ante health loss, and suggests that 'the most precise way [of weighting health gains] consists in weighting each unit of health gain (e.g., a good life year) by the size of the health loss at the time the beneficiary receives the health gain'8 , but asserts that to weight the health gains directly on the margin would be demanding9 (Norheim et al., 2014, p. 97).In this work, we show that this is not the case.

The introduction formalized
Definition 1 (QALY weighting function) Let  be some available treatment, which is expected to provide a QALY gain  to a patient with expected ex ante QALY loss ℓ.We always assume ℓ ≥ , and both  and ℓ refer to, respectively, QALYs gained with the treatment, and total, lifelong QALYs lost, ex-ante the treatment.
This weighted QALY gain is to be used in connection with cost-effectiveness analysis.More specifically, it is common to operate with a monetary threshold value, representing the maximal (societal) willingness-to-pay (WTP) for one additional QALY gained.For example, in Norway, an unofficial threshold value of NOK 500 000 is often used; thus, a gain of 2 QALYs would be considered cost-effective below a total cost of NOK 1 000 000.
The interpretation of the terms in Definition 1 is that if  is an available treatment, so that  yields an unweighted QALY gain of  and is such that the expected life-long ex ante QALY loss of beneficiaries of  is ℓ, then the loss-weighted QALY gain is (ℓ, ).Furthermore, given a WTP-value for one additional loss-weighted QALY, and if we adopt the convention of expressing the cost  of treatment  in units of the societal WTP threshold value per QALY, then  is considered to be cost-effective when (ℓ, ) ≥ , since this inequality obtains exactly when 1 ≥ /(ℓ, ), i.e. when the cost per weighted QALY is less than the threshold WTP.For example, using the unofficial Norwegian threshold value of NOK 500 000 as the unit for the cost , a treatment  costing NOK 2.5 million (equals 5 times the WTP-threshold, i.e.  = 5) and providing a weighted gain of 3 QALYs, we have 3 < 5, and  would not be considered cost-effective.If  is instead provided at the cost of NOK 1.25 million ( = 2.5), we see by 3 ≥ 2.5 that  achieves costeffective relative to the NOK 500 000 WTP threshold.

Main aim and outline of this work
It seems as if a false dichotomy has been set up by the NOU, in which one must either compute QALY loss ex ante or ex post: by pointing to problems associated with computing ex post, they conclude that ex ante is the better solution.It is not; the better solution is computing losses on the margin.This means that weight assigned to the health gain changes continuously because the remaining ex ante health loss is reduced as health-improvement is achieved.Another way to view this is to say that the treatment is subdivided into infinitesimal improvements, each of which is weighted by the ex ante health loss of that infinitesimal 'slice' of the treatment.Contrary to the NOU's comments, we will show that this calculation is neither difficult nor impractical using simple integrals.
In the rest of this paper, we first give some examples demonstrating serious problems with the QALY loss weighting scheme as proposed in the NOU.We will show that unless the weighting scheme satisfies a simple propertyto be dubbed sub-treatment balancedthe consequences will be undesirable: health care providers will have perverse incentives, and the absence of this property will imply what we regard as unfair priority setting policies.Next we will show that if the weighting scheme is a function of health loss alone, (ℓ, ) = (ℓ) ⋅ , it cannot be balanced.We will then provide a very simple, albeit general, solution to some of these problems.
The relationship between the loss-weighting in the NOU and the solutions developed here is described, and we also prove a simple result about cost-effective solutions under any weighting scheme.
We have not aimed to review the severity literature in any detail.The NOU has been our primary source of references, and we will not go beyond that in this paper.However, although we have not read all of the works cited in the NOU (the reference list covers 16 pages), we have not been able to find our solution presented in any other work from this literature.

Potentially problematic properties of QALY weighting functions
In this section, we start by providing two short examples illustrating problems with the weighting scheme suggested in the NOU, before we describe the problem more formally.
Example 1 Consider treatments  1 and  2 targeting a patient group providing unweighted QALY gains of  1 = 15 and  2 = 15 .For this example, let us require that  2 cannot be administered unless  1 has been provided first.Figure 1 illustrates this example.The initial ex ante QALY loss for the patient group is ℓ = 30 .Using the weighting scheme  suggested in the NOU, treatment  1 will provide 15 ⋅ (30) = 15 ⋅ 2 = 30 weighted QALYs.After  1 have been performed, the remaining QALY loss (ex ante for  2 ) is only 15, and  2 will provide 15 ⋅ (15) = 15 ⋅ 1 = 15 weighted QALYs.Let the WTP per weighted QALY be 1, and let the cost for each of the treatments be 25.Since  1 provides 30 weighted QALYs, the cost per weighted QALY falls below 1, and it is considered cost-effective. 2 provides only 15 weighted QALYs, each at a cost of more than 1, and it is therefore not considered cost-effective.
Consider what happens if we bundle  1 and  2 into one larger treatment unit, , providing 30 unweighted QALYs at a combined cost of 50.With an ex ante QALY loss of 30, the weighted QALY benefit from  is 30 ⋅ (30) = 30 ⋅ 2 = 60.Since the cost for each of the weighted QALYs is below 1,  is considered to be cost-effective.Thus, by the mere act of bundling  1 and  2 , the sub-treatment  2 has suddenly become costeffective.As a second observation, consider the situation if  1 had a total cost of 35 and  2 had a total cost of 25.Considered separately, neither treatment would reach the threshold value for cost-effectiveness, but bundled into , thereby providing 60 weighted QALYs, both would be accepted.

Figure 1: Graphical illustration of the sub-treatment problem
Note: Graphical illustration of the sub-treatment problem: Prior to any treatment, the health loss (ℓ) is the whole right area of the figure, including  1 and  2 .Following sub-treatment  1 , the remaining health loss is reduced to ℓ −  1 .If  is considered as a whole, the benefit of the two sub-treatments will be weighted by ℓ, but if considered separately,  1 will be weighted by ℓ and  2 will be weighted by ℓ −  1 .
The first observation is problematic, while the second borders on the absurd.Clearly these phenomena provide perverse incentives for private health care providers.
The problem with assigning weights to QALY gains based on the ex ante magnitude of a non-health gain quality, has also been noted by Hope et al. (2010) in a paper focusing on various strategies for the weighting of a primary health outcome by orthogonal considerations (Hope et al. (2010) studies resource allocations in a context of needs), and they specifically point to the problem of value-boosting by bundling treatments.It is worth noting that the present work can also be adapted to a solution to this problem in the context of Hope et al. (2010).
This kind of weighting scheme would result in over-investment in treatments when bundled together and under-investments in treatments targeting patients with lower healthlosses.These effects may be amplified by not loss-weighting on the margin.However, the situations described above can be ameliorated if the QWF satisfy what we call sub-treatment balancedness.

2.1
Formal examples and description of the problem Recall that the weighting of a QALY gain obtained through a treatment should be weighted with the QALY loss ex ante this treatment.This means that if the patient stands to loose ℓ QALYs without treatment, and to gain  ≤ ℓ QALYs with the treatment, then the weighted QALY gain become (ℓ, ).By extension, if a patient stands to gain, say 3 and 9 QALYs from two subsequent treatments  1 and  2 , and stands to lose 20 QALYs in total without any treatment, the weighted QALY gain should be (20,3) ⏟ T 1 with ex ante QALY loss 20 + (17,9) ⏟ T 2 with ex ante QALY loss 20−3=17 (1) We also see that if the two treatments were bundled into one treatment with a total unweighted QALY gain of 12 = 3 + 9 , then the weighted QALY gain would be computed as  (20,12).In this example the total unweighted QALY gain is  = 12, and the two partial gains are 0.25 and 0.75 , respectively.
More generally, if two treatments  1 and  2 provide QALY gains of  and (1 − ) respectively (for 0 <  < 1), yielding a total QALY gain of  =  + (1 − ), then it is interesting to compare the various total weighted QALY gains respective to some QWF .
Combined treatment: (ℓ, ) (2) First T 1 , then T 2 : (ℓ, ) + (ℓ − , (1 − )) Since both values (2) and (3) in a sense represent the same total QALY gain relative to the same ex ante QALY loss, one could argue that the weighted QALY totals should be equal.If this is not the case, then health care providers could manipulate the willingness to pay by arbitrarily bundling (combining treatments), splitting up treatments, or changing the order in which the treatments are administered.
The most simple QWF is (ℓ, ) = .This QWF depends only on  (it is trivial), and it is clear that with this QWF the three different weighted totals above are all equal.In fact when the unweighted QALY gain is used, (ℓ, ) =  =  ⋅ (ℓ, ).We next briefly discuss some consequences for situations when equality between (2) and (3) does not hold.
Assume that a treatment  provides a QALY gain of  QALYs, to someone with an ex ante QALY loss of ℓ ≥  .This implies that  could be priced at   = (ℓ, ) and therefore be adopted.Assume next that (ℓ − (1 − ), ) <  ⋅ (ℓ, ) for some  ∈ [0,1].Say  = 0.5; then if  is split up into treatments  1 and  2 providing subsequent gains of 0.5 QALYs each, then  2 would not be considered cost-effective if priced at 0.5  .This might be acceptable; indeed, all the weighting schemes we consider here will have this property.However, we may hope that if  is cost-effective at cost   , then for any  ∈ [0,1], if  is spilt up into two subsequent treatments  1 and  2 providing QALY gains of  and (1 − ), then the cost   can be allocated between  1 and  2 so that both subtreatments remain cost-effective.
Assume next that (ℓ − (1 − ), ) ≥  ⋅ (ℓ, ) for all ℓ,  and  ∈ [0,1].By non-triviality, we may then choose ℓ,  and  such that (ℓ, ) > (ℓ − (1 − ), ).Then, we see that In other words, one may increase the total weighted QALY gain of a treatment by splitting it into sub-treatments.This seems like an even more undesirable property than the one discussed immediately above.It should be noted that none of the QWFs proposed either below or in the NOU have this property.

Solution to the problem
In this section we first describe a criterion for QWFs which ensure that some of the problems discussed in Section 2 are either solved or ameliorated.We next describe how QWFs satisfying this criterion are constructed.

3.1
The sub-treatment balanced priority setting criterion We always assume that 0 ≤  ≤ ℓ; the QALY loss ex ante ℓ is at least as great as the potential QALY gain  of the treatment under consideration.
Definition 2 (Sub-treatment balanced) A QWF (ℓ, ) is sub-treatment balanced when, for all  ∈ [0,1]: For example,  1 and  2 may be two doses of a medicine, taken at two future points in time (e.g., pain killers for rheumatic disorder), or they could be two surgical interventions performed at the same time independently of each other (e.g., mastectomy  1 possibly followed by reconstructive surgery  2 (5) If the QWF is balanced, it is impossible to make a treatment more valuable by bundling.It is also always possible to take a cost-effective treatment  and relocate the costs between any two sub-treatments  1 and  2 in such a way that both become cost-effective.Indeed, when  is sub-treatment balanced, then In less mathematical terms, balance is the requirement that any treatment that can be construed as consisting of more than one sub-treatment will have the same overall priority as the combined priority of its constituent sub-treatments.Without this property, arbitrary grouping or sub-division of treatment options will have a separate impact on priority setting.
We have not been able to come up with any set of ethical guidelines that are congruent with the notion that bundling is in itself an ethically relevant factor to consider for priority setting.
The next result shows that any weighting-scheme like the one proposed by the NOU fails to be sub-treatment balanced.This both illustrates a serious shortcoming of the proposed health-loss criterion, and justifies the development of a family of balanced QWFs.
Proposition 1 Let  be a positive monotonically increasing function, so that is a QWF.Then  is either not sub-treatment balanced, or  is constant.
We end this section with a theorem which shows that a sub-treatment balanced QWF deals with the problems from Example 1 insofar as possible: (i) if a treatment is costeffective, then it remains so when it is split up into sub-treatmentspossibly after distributing the costs according to weightand, (ii) there is no cost-effectivene bundling of cost-ineffective treatments, Theorem 1 Let  be a sub-treatment balanced QWF.Then When  is balanced, then the order of treatments is also irrelevant for the total weighted QALY gain.

3.2
A family of sub-treatment balanced QALY weighting functions Thus far we have shown that balancedness removes or ameliorates problems, and that any weighting scheme as proposed in the NOU cannot be balanced.In this sub-section we therefore define a family of QWFs   (ℓ, ), which, for any suitable (to be defined later) function , satisfies the following criteria: 1.   incorporates health-loss ex ante at the margin of health gains.
2.   is as such independent of the QALY paradigm; any ratio-scale utility assigned to 'good life years' can be used to measure the health loss and the health gains 10 .3.   is sub-treatment balanced.
Definition 3 (Sub-treatment balanced QALY ′-weighted function) Let ′ be any positive and integrable function, so that  is well-defined 11 by: We call ′ a loss-weight function (LWF).We next define, uniformly in ′, a function   (ℓ, ) by It is immediate from the that and thus that for  ∈ [0,1] we have   (ℓ, ) =   (ℓ, ) +   (ℓ − , (1 − )) .Also,   (ℓ, 0) = 0 is obvious.Since the LWF ′ is required to be positive,  is strictly increasing in  when  ≤ ℓ.If ′ is non-negative, then   is also monotonic in ℓ, whence   is a QWF as per Defintion 1.We have also shown Proposition 2 Let   be defined by Equation 8; then, if the LWF ′ is an increasing function, then   is a sub-treatment balanced QWF.QED The exact properties of the QWF will obviously depend strongly on which ′ is chosen as the LWF; consequently, finding the 'right' LWF ′ will be a matter of some importance.
Given point estimates for   (ℓ, ) for different values of ℓ and  , it will be a rather straightforward task to fit an acceptable ′ to such observations.The requirement that ′ > 0 only means that a greater QALY gain always means a larger loss-weighted gain.Similarly, if the weighting function ′ is also non-decreasingi.e.never gives less importance to greater ex ante QALY lossesthen it is also increasing in ℓ.
But are such QWFsas asserted in the NOUoverly complicated to understand and difficult to apply?Quite the contrary, as we show in the next section.10with  = 0.64 and  = 0.06.Hence,   (ℓ, ) is interpretable as the area  under the curve (shaded), which is bounded by the QALY loss and the QALY loss minus the health gain.It is evident that the the same gain  leads to a larger total utility when the loss ℓ increases, as long as ′ is monotonically increasing.

Examples
For ′() =  +  we obtain () =  +  2 2 .This form gives the first unit of gain a weight proportional to the remaining QALY loss in addition to a fixed weight proportional to the gain.
For  ≃ 0.64 and  ≃ 0.06 this LWF isat the marginthe weighting scheme suggested in the NOU.The value   (ℓ, ) corresponds to the area under the graph of  between ℓ −  and ; see Figure 2, panel B.
According to the NOU's suggested algorithm, any treatment targeting a patient group with a very high expected QALY losssay, of 35 QALYswould imply a societal WTP threshold three times higher than for treatments targeting patients with an expected QALY loss of below 15 QALYs.A treatment expected to restore these patients to full health would justify costs up to 6.2 million Euro (assuming base rate WTP of 0.060 million Euro per QALY).Under the LWF above, the WTP with marginal weighting yields a WTP for the full restoration of health of 3.5 million Euro.If a treatment for the same group has an expected QALY gain of only 5 QALY, the WTP would be 0.88 million Euro with the NOU's weights, and 0.76 million Euro with the marginal weighting above instead.As expected, the difference is much smaller for the latter example of restoring only 5 of 35, since most of that QALY gain is weighted with a substantial health loss.11with  = 2) is interpretable as the areas under the curve.The left plus the middle areas correspond to   (ℓ 2 , ), while the middle plus the right areas correspond to   (ℓ 1 , ); even if the gain is equal, the health losses are different, resulting in greater weight to the first quantity.
Looking at panel B of Figure 2, it is clear that the weighting is linear in loss.This may not reflect public preferences.Another possible ′ is  +  • (), which when  (11) then averaged, and if average health gain is weighted by average health lossa problem akin to the ecological fallacy.

Directions for further research
If the framework of health-loss weighting QALY-gains proposed in the NOU is adopted, we encourage further efforts aimed at identifying the LWFs which yield the QWFs that most closely reflect the social welfare function of the adopting jurisdiction.This should be effected through elicitations from experts and/or the public.In general, for any weighting scheme in which the property or properties on which the weights are calculated will be altered by the act of treatment, adjustments similar to the ones discussed here will be required to ensure sub-treatment balance.

Conclusion
If it is decided that ex ante health loss should be used to weight gains in health economic cost-utility calculations for priority setting, as suggested by the NOU, our solution (Equation 10) can be used directly for cost-utility analyss.If the health-loss notion of severity is abandoned, other severity weighting of the QALY should also adhere to the sub-treatment balanced property; suitably rephrased, it is not an approximation to the loss-weight-on-themargin problem only, but an exact and easy solution to other types of QALY-weighting schemes.
Since this paper was first drafted, it has become clear that the ex ante health-loss as it was proposed by the priority commission will not be adopted in its suggested form.The Norwegian government has appointed a special task-group, lead by professor Jon Magnussen, to suggest a new operationalization of severity for use in a priority setting 14 .Severity weighting of QALYs can take a number of different forms.However, as the title of this paper indicates, we suggest that any operationalization of a priority-weighting scheme by a measure of severity (or other similar concepts) should satisfy sub-treatment balancedness.Any weighting scheme using a measure of severity that is influenced by the treatment in question will have the potential for lack of sub-treatment balance, as well as problems such as the ones discussed by Hope and colleagues (2010).These problems can be ameliorated by emplying the methods presented herethat is, by use of a suitable integral.
so that the standard cost-utility expression reappears.This ′ coincides with the only type possible where the QWF does have a multiplicative decomposition: (ℓ, ) = (ℓ) ⋅  and  is identically 1.