modelling the monetary value of a qaly: a new approach based on uk data

HEALTH ECONOMICSHealth Econ. 18: 933–950 (2009)Published online 14 October 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/hec.1416

MODELLING THE MONETARY VALUE OF A QALY: A NEWAPPROACH BASED ON UK DATAy

HELEN MASONa,�, MICHAEL JONES-LEEb and CAM DONALDSONa,b

aInstitute of Health and Society, Newcastle University, Newcastle upon Tyne, UKbBusiness School (Economics), Newcastle University, Newcastle upon Tyne, UK

SUMMARY

Debate about the monetary value of a quality-adjusted life year (QALY) has existed in the health economicsliterature for some time. More recently, concern about such a value has arisen in UK health policy. This paperreports on an attempt to ‘model’ a willingness-to-pay-based value of a QALY from the existing value of preventinga statistical fatality (VPF) currently used in UK public sector decision making. Two methods of deriving the valueof a QALY from the existing UK VPF are outlined: one conventional and one new. The advantages anddisadvantages of each of the approaches are discussed as well as the implications of the results for policy and healtheconomic evaluation methodology. Copyright r 2008 John Wiley & Sons, Ltd.

Received 21 July 2006; Revised 7 August 2008; Accepted 14 August 2008

KEY WORDS: value of a prevented fatality; value of a QALY; willingness to pay

1. INTRODUCTION

Debate about the monetary value of a quality-adjusted life year (QALY) has existed in the healtheconomics literature for some time (Phelps and Mushlin, 1991; Johannesson, 1995; Garber and Phelps,1997; Bleichrodt and Quiggin, 1999; Dolan and Edlin, 2002). More recently, concern about such a valuehas arisen in health technology assessment panels internationally. For example, in the UK, in additionto the appraisal of single interventions in terms of their incremental cost per QALY gained, it is alsoimportant for the National Institute for Health and Clinical Excellence (NICE) to know what monetaryvalue to attach to such QALY gains in order to be able to compute benefit–cost ratios that are directlycomparable to those afforded by other uses of National Health Service (NHS) funding.

In principle there would appear to be two ways in which willingness-to-pay (WTP)-based monetaryvalue of a QALY can be estimated. The first possibility would be to obtain a direct estimate using thecontingent valuation approach via survey research and, indeed, attempts to do this are already underway (Gyrd-Hansen, 2003). An alternative route has been to ‘model’ a WTP-based monetary value of aQALY from the existing ‘value of preventing a statistical fatality’ (VPF) currently used in public sectorsafety policy. Although such values are derived in a different context to that of health, there aresimilarities due to the values being based on surveys in which respondents are asked hypotheticalquestions about trade-offs between wealth and reduced risks of death. Also, as will be seen below, morerecent research has shown that such values do not differ greatly across different contexts. Indeed, at

*Correspondence to: Institute of Health and Society, Newcastle University, 21 Claremont Place, Newcastle upon Tyne NE2 4AA,UK. E-mail: [email protected] article was published online on 14 October 2008. An error was subsequently identified. This notice is included in the onlineand print versions to indicate that both have been corrected [04/03/2009].

Copyright r 2008 John Wiley & Sons, Ltd.

least in part, the basis of the VPF reflects the arguments made for taking an individual preference-basedapproach to valuing publicly funded health care whereby individual members of the community areinformed about probabilities and risks (e.g. of needing care and of treatment success) as part of thesurvey process (O’Brien and Gafni, 1996).

The appropriate way in which to derive an estimate of the money value of a QALY from a pre-determined VPF does, however, raise some fairly fundamental questions. Thus, for example, early(conventional) attempts to estimate the value of a QALY in this way simply involved dividing the VPFby the population mean remaining life expectancy or QALYs (Hirth et al., 2000). By contrast, morerecent work has focused on the way in which the VPF varies with the age of those who will enjoy areduction in the risk of premature death and, in particular, bases the estimate of the value of a QALYon the rate at which the VPF increases with increasing life expectancy (Baker et al., 2008).

As will be argued below, it transpires that the essential difference between these two approaches isthat the first yields the aggregate WTP across a large group of people for small reductions in eachindividual’s current-period risk of death which, when summed over the affected group, gives a totalgain in life expectancy of one year. By contrast, the second (new) approach produces the aggregateWTP for a total gain in life expectancy of one year, where this gain can most naturally be thought ofas resulting from reductions in the risk of death that are spread over the remainder of each affectedindividual’s lifetime, but with the magnitude if the risk reduction also being an increasing function ofthe size of the individuals’ hazard rate at each age (i.e. the probability of death at that age conditionalon having survived to that point). Clearly therefore, the question of which approach is the morerelevant depends essentially on the manner in which a gain in life expectancy is generated – i.e.whether the gain is principally the result of reductions in the risk of death in early or in later years ofadulthood.

As the research reported uses UK data, in the first section of the paper the origins of the baseline UKVPF to be used in most of the calculations are presented, focusing in particular on the underlyingrationale for employing preference-based methods in this context and the relevance of the VPF tomonetary valuation of a QALY. The conventional and new approaches used to convert from a VPF toa value of a QALY are then outlined. This is followed by a section focusing on sources of data forconverting from the VPF to the value of a QALY according to these approaches. The results from theapproaches are then outlined along with sensitivity analyses adjusting for quality of life (QoL) anddiscounting. In the final section, caveats are discussed and some recommendations made as to futureresearch and possible values to use pending the completion of such research.

2. CONCEPTUAL AND THEORETICAL ISSUES: VALUES ACROSS CONTEXTS

The fundamental ethical precept underpinning conventional social cost–benefit analysis is that publicsector allocative decisions should, so far as possible, ‘mirror’ the operation of market forces and should,in particular, reflect the preferences (and, more significantly, the strength of preference) of thosemembers of society affected by the public sector decision concerned. Since an individual’s maximumWTP for a good or service is a clear indication of his or her strength of preference for it relative to otherpotential objects of expenditure, it is not surprising that the WTP approach to valuation has become thenorm for the monetary valuation of publicly funded goods in the UK and several other countries. Thisis particularly so in the case of decisions affecting the safety of human life.

Thus, in the UK, for example, the Department for Transport (DfT), the Rail Industry, theDepartment for the Environment, Food and Rural Affairs (DEFRA) and other government agencies allemploy WTP-based values of safety in their cost–benefit analyses. The approach is also recommendedby the Health and Safety Executive (HSE). More specifically, these agencies now employ a WTP-basedvalue for the prevention of a ‘statistical fatality’ (VPF) of some £1.42m in 2005 prices (Department for

H. MASON ET AL.934

Copyright r 2008 John Wiley & Sons, Ltd. Health Econ. 18: 933–950 (2009)

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Transport, 2007). The sole exception is the value recommended by the HSE for valuing the preventionof cancer deaths; a figure equal to twice the standard value is prescribed in order to reflect, inter alia, theprotracted period of pain and suffering that typically precedes cancer death.

Essentially, the WTP-based VPF is defined as the aggregate WTP across a large group of individualsfor small risk reductions which, taken over the whole group, can be expected to prevent one prematuredeath during a forthcoming period (or, prevent a ‘statistical’ fatality). What the VPF is mostemphatically not is the ‘value (or price) of life’ in the sense of a sum that any given individual wouldaccept in compensation for the certainty of his or her own death. For most of us, no finite sum wouldsuffice for this purpose so that, in this sense, life is literally priceless. Rather, the VPF is the aggregateWTP for typically very small reductions in individual risk of death (which, realistically, is what mostsafety improvements actually offer at the individual level).

Since 1988 the VPF has been based on a consistent WTP approach. This has been as a result of anumber of studies, which involved surveying members of the public about their WTP for a reduction in therisk of death from road safety improvements (Jones-Lee et al., 1985, 1995). The current VPF was derivedusing a contingent valuation/standard gamble (CV/SG) ‘chained’ approach, which breaks down thevaluation process into a series of more manageable steps involving chaining together responses to WTPand SG questions (Carthy et al., 1999). Using the CV/SG chained approach, respondents are initiallypresented with a question asking them about their WTP for the certainty of a complete cure for a givennon-fatal road injury and their willingness to accept compensation for the certainty of remaining in theimpaired health state (the combination of which, based on some reasonable assumptions about underlyingpreferences obeying minimal conditions of consistency and regularity, it is argued, gives a reasonableestimate of the marginal rate of substitution (MRS) of wealth for the risk of the non-fatal injury). In asecond stage, respondents are presented with an SG question aimed at determining the ratio of the healthstate value for death over that for the non-fatal injury. The monetary value from the first stage can then becombined with the ratio from the second stage to obtain a WTP for reduced risk of death.

Given the well-established differentials between people’s disquiet or degree of ‘dread’ concerningthe prospect of death by different causes (Slovic et al., 1981), it is natural to expect that the WTP-based VPF might well differ substantially between such causes. However, recent research suggeststhat for most common causes of premature death, such dread effects are, to all intents and purposes,offset by ‘baseline risk’ effects so that marked differentials in the VPFs do not emerge (Chilton et al.,2002). Hence, the tendency to use a common WTP-based VPF for public sector safety decisionmaking in the UK. In addition, recent work for DEFRA on the WTP-based valuation of reductionin the risk of premature death due to air pollution has produced values for gains in life expectancywhich, appropriately aggregated, produce a VPF equivalent that is broadly similar to the roads/railfigure.

In view of all this, pending the estimation of WTP-based values that are specific to health care, it doesnot seem entirely unreasonable to base tentative estimates of the WTP-based value of a QALY on theexisting roads/rail VPF.

3. POSSIBLE APPROACHES TO CONVERTING THE VPF TO A VALUE OF A QALY

Before outlining the conventional and new approaches (referred to in what follows as approaches 1 and2, respectively) that have been developed to estimate a value of a QALY using the current VPF, twopoints are worth noting.

First, in Highways Economics Note 1:2005 (Department for Transport, 2007), the current VPF net ofmedical and ambulance costs as used by the DfT is £1 427 340 in 2005 prices. Second, for ease ofexposition, in this section, it is demonstrated how the approaches convert the VPF into the value of

MODELLING THE MONETARY VALUE OF A QALY 935


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a life year. Later in the paper the approaches are further developed for estimation of the value of aQALY as well as incorporation of discounting and adjustments for net output.

3.1. Approach 1

Possibly the simplest and most natural way to think about this issue is to appreciate that the VPF is infact an aggregate WTP across a large number of people for a risk reduction that can be expected toresult in the prevention of one premature death during the coming year (Jones-Lee, 1989). On average,the avoidance of one premature death would result in the avoidance of the loss of 32 years of lifeexpectancy (See Section 4.1 for an explanation of how this is derived). The WTP-based monetary valueof each of these 32 years is, therefore, given by

V1a ¼£1 427 340

32¼ £44 018 ð1Þ

Under this approach the WTP-based value of a life year would therefore be £44 018.While this was the rationale that appeared to underpin the earliest attempts to derive the value of a

QALY (Hirth et al., 2000), the most obvious criticism of the above approach is that it assumes that eachfuture life year is of equal value, when it is more likely that, as far as any given individual is concerned, ayear in the distant future is valued less highly than one in the near future.

However, it is also possible to think of the resulting value as being based on an aggregation across alarge group of people of their individual WTP for small risk reductions, which will result in small gainsin life expectancy that, when summed over the whole group, give the aggregate WTP for a total gain inlife expectancy of one year. For example, consider a large, representative group of n individuals eachoffered a 1/n reduction in the risk of death during the coming year. To the extent that, taken over thewhole group, these risk reductions will prevent precisely one ‘statistical’ fatality, the group’s aggregateWTP for the risk reductions will constitute the WTP-based VPF. However, as shown in Appendix A,given the reduction of 1/n in risk, each individual in the group will enjoy an increase in life expectancythat is closely approximated by 1/n times his/her remaining life expectancy, Ei. With n being large, theseindividual gains in life expectancy will, by definition, be small (i.e. essentially marginal). However,summed over the whole affected group of n individuals, the total gain in life expectancy will be equal to

n

Pð1=nÞEi ¼ ð1=nÞ n

PEi, that is, by the mean remaining life expectancy for individuals in the group. It

then follows immediately that WTP per year of the aggregate gain in life expectancy is equal to the VPFdivided by mean life expectancy. A more formal proof of this result is provided in Appendix A. It istherefore clear that based on this line of reasoning exactly the same result (in this case, £44 018 per lifeyear gained) can be arrived at. This is an important point to recognise as although one can reasonablycriticise earlier estimates of the value of a QALY, which have all been implicitly based on the ‘equallyvalued life years’ assumption, the more sound reasoning underlying the group aggregation argumentthat has just been outlined, combined with the knowledge that the approaches lead to the same result,actually lends substantially greater credence to such earlier results.

This having been said, it should be noted that there is a variant of the ‘group aggregation’ approachwhich, although very closely related to the one that has just been developed, will nonetheless almostcertainly produce a different result. Thus, suppose that rather than setting all of the individual riskreductions equal, these reductions are instead allowed to vary across individuals and are in fact set sothat individual gains in life expectancy are themselves all equal to 1/n. In this case, as shown inAppendix A, the WTP-based value of an aggregate gain of one year of life expectancy is given by thearithmetic mean of individual ratios Mi=Ei, where Mi is the ith individual’s MRS of wealth for risk, themean of which is (under the standard argument) the VPF. Notice that the mean of this ratio willgenerally differ from the ratio of the means of Mi and Ei. Indeed, the two will be equal only underexceptional circumstances (e.g. when Mi is strictly proportional to Ei).

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Finally, it should be stressed that the gain in life expectancy that features in the ‘group aggregation’rationale that underpins both versions of approach 1 is essentially the result of reductions in the current-period risk of death. This contrasts markedly with the nature of the risk reduction that underpins thesecond broad avenue of approach to estimating the monetary value of a QALY and which will beexplained in the following subsection.

3.2. Approach 2

Consider a large representative group of 40 year olds and a corresponding group of 50 year olds.Intuition, theory and empirical evidence all strongly suggest that – controlling for other factors such asincome, health state and so on – the WTP-based VPF for the 40 year olds will significantly exceed thatfor the 50-year-old group quite simply because the 40 year olds have a larger remaining life expectancythan the older group and hence would experience a larger loss of expected lifetime utility as a result ofpremature death. To the extent that the value of a gain in life expectancy will depend directly on theresultant gain in expected lifetime utility, then the rate at which the VPF changes as we move from oneage group to another should precisely parallel the rate at which people would be willing to trade offwealth against gains in life expectancy. Under approach 2 the aim is therefore to base the value of ayear’s gain in life expectancy on the population mean of the annual rate at which the WTP-based VPFvaries with life expectancy as one moves from one age group to another, controlling for other factorsthat are likely to influence the VPF. In seeking to establish the nature of the changes in survivalprobability that underpin the variations in life expectancy that are being valued under this approach, itis important to appreciate that, ceteris paribus, the process of ageing by one year is tantamount to a unitrightward shift in the ‘starting point’ (i.e. the origin) of an individual’s ‘Gompertz curve’ (orequivalently, a unit leftward shift in the curve itself), this curve being the relationship that expresses anindividual’s hazard rate as a function of his/her age where the hazard rate is the probability of death at agiven age conditional on surviving up to that age (Haybittle, 1998). On the assumption that theGompertz curve takes an essentially exponential form, then such a unit rightward shift in the origin willbe tantamount to a proportionate increase in the hazard rate at each additional year of age for theindividual concerned and the absolute impact will therefore increase as the hazard rate itself increaseswith age along the exponential Gompertz curve.1 See Figure 1 for a diagrammatic presentation of thiseffect. Conversely, the effect of a one-year gain in life expectancy can be regarded as a proportionatereduction in each of the age-specific hazard rates, with the size of the reduction increasing in later yearsof life. In short, under approach 2 one can think of the principal source of the gain in life expectancy asbeing the proportionate reduction in the larger, later-life hazard rates. This clearly contrasts markedlywith approach 1 under which, as already noted, the gain in life expectancy is exclusively the result of areduction in the current-period hazard rate. This then raises two further important points, namely:

� To the extent that, ceteris paribus, total lifetime utility is an increasing but strictly concavefunction of length of life, then the utility gain from the current-period hazard rate reduction thatunderpins approach 1 will exceed the corresponding utility gain resulting mainly from the later-life hazard rate reductions that are implicit in approach 2, given that both approaches involve again of one year in life expectancy as a result of the hazard rate reductions.

� The question of which approach yields the more policy-relevant result clearly depends on thenature of the hazard rate reduction generated by the particular health-care or safety improvementprogramme under consideration. Thus, prima facie it would appear that if the value of a QALY

1More specifically, with the Gompertz function taking the form hðtÞ ¼/ ebt, where h(t) is the hazard rate at time t andp and b arepositive constants, a unit rightward shift in the origin would yield a new Gompertz function hðtÞ ¼/ ebðtþ1Þ, which is equivalent tomultiplying all of the original hazard rates by a factor eb [Correction made here after initial online publication].



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was required as an input to the assessment of the desirability of, say, a medical intervention whoseeffect would be more or less immediate, then approach 1 would seem to be the appropriate sourceof such a value. By contrast, if the focus was on a reduction in air pollution or health effect (suchas the beneficial impact of a protracted medical treatment), then to the extent that epidemiologicalevidence suggests that such effects will manifest themselves principally via proportionate changesin individual hazard rates (and will therefore have their main impact in later life years), thenapproach 2 would appear to be more relevant.

All of this having been said there is, however, one fairly serious complicating factor as far asapproach 2 is concerned and this relates to the way in which the VPF does in fact vary with age, andhence with remaining life expectancy. More specifically, growth of annual income over early years ofadult life together with market imperfections in the form of restrictions on borrowing against higherlevels of future income might reasonably be expected to yield an inverted-U over the life cycle for thetypical individual’s WTP for a current-period safety improvement. Nonetheless, it also seemsreasonable to suppose that if life-cycle variations in income and wealth were controlled for, then thisWTP for safety improvement would be an unambiguously increasing function of remaining lifeexpectancy (and hence a decreasing function of age) and to a large extent these intuitions are confirmedby theory (Shepard and Zeckhauser, 1982). However, several empirical studies (Jones-Lee et al., 1985)have found that even when factors such as income are controlled for, WTP for safety improvement stillfollows an inverted-U life cycle, so that at least over early years of adult life, individual valuation ofsafety is a strictly decreasing function of remaining life expectancy. To the extent that approach 2essentially treats the value of a life year as the rate at which the VPF increases with remaining lifeexpectancy then it is clear that, applied to younger adult age groups, this approach will yield a negativevalue of a life year which is, to say the least, simply beyond the bounds of credibility. Is approach 2therefore fundamentally flawed? We think not. Rather, we would argue that the tendency for thevaluation of safety to rise with age (and hence decline with remaining life expectancy) over early years ofadult life, even when income effects are controlled for, is in fact a reflection of a fundamental change inattitude to risk and awareness of vulnerability to physical harm that would appear to be a commonfeature of the process of maturation for many people over the period from their late teens to their mid-20s. It would therefore seem reasonable to argue that since the VPF–age relationship over early years of

0 1 2

Hazard rate

Gompertz curve for starting age x+1

Gompertz curve for starting age x

Effect of increasing starting age by 1 year

Figure 1. Gompertz curve showing the effect of increasing the starting age by one year

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adulthood is largely a result of a fundamental change in preferences and attitudes rather than a changein an individual’s future hazard rates, then estimation of the value of a gain in life expectancy should bebased only on the time interval over which the VPF is a decreasing function of age. In response to thisargument it might, of course, be objected that in empirical studies such as that reported in Jones-Leeet al. (1985), the ‘inverted-U’ WTP-vs-age relationship does not peak before about the age of 40, whichwould appear to sit rather uncomfortably with the suggestion that for the typical individual the‘recklessness of youth’ will have passed by his or her mid-20s. But in considering this issue it should beborne in mind that the point at which the inverted-U WTP-vs-age relationship peaked in the Jones-Leeet al. study was to all intents and purposes driven by the specification of the regression relationship inwhich the non-linear effect of age (which was evident in the raw data) was captured by an ‘age minusmean age all squared’ variable so that, setting aside the effect of the linear age variable (which wasgenerally insignificant), it was inevitably the case that the ‘inverted-U’ relationship reached a maximumat mean age. While it has to be conceded that this is something of a limitation of the estimated VPF–agerelationships, they are nonetheless arguably the best estimates currently available.

All things considered, therefore, it was felt appropriate to apply approach 2 not only to the fullavailable data set, but also to a subset of the data with those in the younger ‘problematic’ age grouptrimmed out.

More specifically, approach 2a looks at the relationship between Mi and Ei from age 18 onwards,therefore incorporating the values for those aged between 18 and 40 as well as those in the older agegroup.

By contrast, approach 2b focuses exclusively on those aged over 40 on the basis that the lower valuesof life among the under 40s do, indeed, reflect a problematic ‘recklessness of youth’.

3.2.1. Approach 2a. Since neither theory nor empirical evidence will allow us to specify the preciserelationship between Mi vs Ei, we assume the following simple functional form:

Mi ¼ aEbi ð2Þ

With the Mi-vs-age relationship estimated in three UK studies being essentially an inverted-U shape,peaking in middle age (Jones-Lee et al., 1985, 1995; Carthy et al., 1999), then, based on these studies,and as a first approximation, it would seem appropriate to take the average of Mi=M, where M is thepopulation mean of Mi, as being about 0.45 for an 80 year old (for whom average remaining lifeexpectancy is 8 years), about 1.36 for a 40 year old (with an average remaining life expectancy of 39years) and about 0.7 for an 18 year old (with an average remaining life expectancy of 60 years). Giventhis, and taking M ¼ 1 427 340 along with a zero discount rate with no adjustment for declining QoL inlater years, would give, from Equation (2):

ln Mi ¼ ln aþ b ln Ei ð3Þ

Incorporating the data for an 18, 40 and 80 year old, it is necessary to calculate the b using the formulaPxy=

Px2 where x ¼ ln Ei � ln Ei and y ¼ ln Mi � ln Mi. Table I presents the calculations used to

estimate x and y for each of the three ages. Using the values that are presented in Table I, b can becalculated using the following equation:

b ¼P

xyPx2

ð4Þ

b ¼ð0:836 � �0:074Þ þ ð0:406þ 0:590Þ þ ð�1:243 � �0:515Þ

0:3862 þ 0:4062 þ 1:2432¼

0:819

2:41ð5Þ

b ¼ 0:339 ð6Þ

Substituting (6) into (4)



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ln Mi ¼ ln aþ 0:339 ln Ei ð7Þ

ln a ¼ ln Mi � 0:339 ln Ei ð8Þ

ln a ¼ 13:88� ð0:339 � 3:273Þ ð9Þ

a ¼ 353 655 ð10Þ

Hence, from Equations (4), (6) and (10)

Mi ¼ 353 655E0:339i ð11Þ

from which it follows that

dMi

dEi¼ 120 216E�0:66i ð12Þ

Given that Ei varies across the UK population, there exists a density function giving the proportion ofthe population having each particular value of Ei. The general form of this density function can bederived from statistical life tables. For the sake of simplicity it will be assumed that the density functionfor Ei takes the form

rðEiÞ ¼ gE1=2i ð13Þ

which entails that Z 60

8

gE1=2i dEi ¼ 1 ð14Þ

so that

g½23E

3=2i �

608 ¼ 1 ð15Þ

Hence,

294:75g ¼ 1 ð16Þ


g ¼ 0:00339 ð17Þ

From Equations (12), (14) and (17) we therefore have

V2a¼Z 60

8

ð120 216E�0:66i Þ0:00336E1=2i dEi ð18Þ

¼Z 60

8

396E�0:160i dEið19Þ

Table I. Calculation of x and y

Age lnEi lnMi ln Ei � ln Ei ln Mi � ln Mi

18 4.109 13.814 0.836 �0.07440 3.680 14.478 0.406 0.59080 2.029 13.372 �1.243 �0.515Mean 3.273 13.888

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¼h472E0:839

i

i608 ð20Þ

¼ 23 199 ð21Þ

Hence, V2a ¼ £23 199

Although across the whole age range the WTP per QALY value is positive, it is low because of therelatively small VPFs’ that arise from those under the age of 40.

3.2.2. Approach 2b. As previously noted above, impacts of wealth and age such as possible risk-seekingbehaviour by younger individuals result in the inverted-U-shaped relationship between Mi and Ei. If apolicy maker were to wish to discount such aspects, then in trying to overcome them empirically, it ispossible to focus only on those members of society at or above middle age. As in approach 2a, it isassumed that Mi=M as being about 0.45 for an 80 year old and about 1.36 for a 40 year old (for whomaverage remaining life expectancy is 39 years).Therefore, under approach 2b the aim will be to arrive at an estimate of the mean of dMi=dEi taken overthe subset of the population within this age group.

As in approach 2a we take this relationship as having the following form:

Mi ¼ aEbi ð22Þ

In order to save on space in the main text, for approach 2b, the sequence of calculations equivalent tothose shown for approach 2a have been relegated to Appendix B. From the calculations in Appendix Bit can been seen that the resultant valuation of 2b is £40 029.

4. DATA SOURCES

4.1. Estimating remaining life expectancy

All the approaches described above require an estimate of remaining life expectancy. This is used toreflect the life years lost from a fatality and facilitate the conversion of the VPF into a value per life year.

Approach 1 compares the VPF with a representative life expectancy for the sample population usedto elicit the VPF. Details of the ages and gender of the sample’s members are not available. However, itis known that the sample was randomly drawn from the adult population. The mix of the samplepopulation is assumed to reflect the adult population for Great Britain as given by data from the Officefor National Statistics (Office for National Statistics). This suggests an average age for males of 46, anaverage age for females of 48 and a gender mix of 52:48%, the majority being female. Weighted bypopulation shares, the means of male life expectancy at 46 and the female life expectancy at 48 give thelife expectancies used in approach 1.

Approach 2 assesses the value per life year in relation to three specific ages – 18, 40 and 80. The lifeexpectancy at 18 is the average of the male and female life expectancies at 18 weighted by the gendershares of the population at 18. Life expectancy was similarly calculated for ages 40 and 80.

Life expectancy by age and gender is calculated using the Interim Life Tables produced by theGovernment Actuary’s Department (GAD) (Government Actuary Department, 2004). These tablesgive, by gender and age, the mortality rate (represented by GAD using the notation qx) between age xand age x11 (the probability that a person aged x will die before reaching age x11).



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It would be simpler to use GAD’s directly published life expectancies rather than calculating themindirectly from GAD mortality rates. However, this method would cause problems when theapproaches are modified to take into account QoL and discounting. For instance, the section belowexplains how the approaches can produce a value per QALY. In this context, using the direct lifeexpectancy figure would effectively assume a 100% chance of people experiencing the relatively high-quality younger life years up to that age and then a 0% chance of experiencing the relatively low-qualityolder life years beyond that age. In fact, not everyone will live to the age implied by the direct lifeexpectancy and some will live beyond it. This method would therefore overestimate expected QALYsand, thus, underestimate the QALY value.

4.2. Estimating a value per QALY

The method for calculating expected QALYs is very similar to the life expectancy calculations explainedabove. The only modification to the methodology is to apply a QoL weight to the probability of eachpotential year of life. The sum of the quality-adjusted probabilities of living through each potential yearof life gives the expected QALYs.

The QoL weights, along a scale from zero for death to ‘1’ for full health, is taken from UK populationnorms for EQ-5D (Kind et al., 1999). The EQ-5D tariff does allow for states worse than death, whichwould receive a value below zero. However, the UK population norms were estimated from arepresentative sample who considered themselves to be generally healthy. This gives utility values,elicited using the EQ-5D, by age and gender.

4.3. Discounting and adjusting for net output

Both the value per life year approaches and the value per QALY approaches have also been modified totake into account discounting at a rate of 1.5% (as per the recommended rate of pure time preferencegiven in Her Majesty’s Treasury Green Book) (Great Britain HM Treasury, 2004). This 1.5% discountrate is one part of the overall discount rate of 3.5% recommended in the Green Book. The full discountrate is made up of two parts: a rate of pure time preference (estimated as 1.5%) and a part that reflectsdiminishing marginal utility of consumption (estimated as 2%). However, theory shows that a WTP-based VPF can be expected to grow at much the same rate as the marginal utility of consumptiondeclines and so is excluded from the calculations for reasons of double counting. However, the rate ofpure time preference is included as it incorporates uncertainty about the future.

The 1.5% discount rate has been applied to the series of (quality-adjusted) probabilities of livingthrough each potential year of life. The discounted life expectancies and discounted QALY expectanciesthen replace life expectancies and QALY expectancies in the calculations.

It is the convention of the DfT to add net output to the WTP-based VPF as it is assumed that theWTP value includes the value of consumption forgone due to death. Net output is usually assumed tobe 10% of gross output (Hopkin and Simpson, 1995). Therefore, as part of the sensitivity analysis, thevalue of a QALY is estimated using both the VPF as referenced in the current Highways EconomicsNote 1 and as this value minus 10% of gross output, reducing the VPF to £1 382 229.

5. RESULTS

The results from each of the approaches are given in Table II. Values were estimated for each approachusing a number of different assumptions as to whether QoL and discounting were incorporated, alongwith the basic methods presented in Sections 3.1 and 3.2.

Looking at the final column of Table II, the results from approach 2 are lower than those estimatedusing approach 1. This is what would be expected as approach two takes into account the inverted-U-

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shaped relationship between the VPF and age, thus incorporating negative values for younger people inapproach 2a and lower values for older people in approach 2b.

Discounting and QoL adjustments lead to higher values than those that are based only onundiscounted life years. This is because the initial value is based on the ratio of VPF to the number oflife years. As we introduce quality adjustments or discounting, the denominator falls (i.e. the number ofdiscounted QALYs is less than the number of life years) and the ratio increases.

The results calculated using the VPF excluding net output are lower than those using the full VPFbut follow the same pattern when discounting and QoL adjustments are made.

6. DISCUSSION

6.1. Further interpretation of results

Two main approaches have been used to estimate the value of a QALY. Approach 1 is based on theassumption that each future life year is of equal value. Approach 2 takes into account the relationshipbetween the VPF and age, which in a number of empirical studies has been found to be an inverted-U-shaped relationship peaking in middle age (Jones-Lee et al., 1985; Carthy et al., 1999). Thus,the values arising from approach 2 are lower than those from approach 1. This is especially so forapproach 2a which is particularly affected by the lower VPF of younger adults. The values produced byapproach 2 (again, especially 2a) are closer to the £30 000 threshold set by NICE than those fromapproach 1 and, indeed, are closer still given that the £30 000 threshold was set at the establishment ofNICE in 1999.

Another way of looking at these results is in terms of QALY types. The values elicited underapproach 1 can be viewed as ‘rule-of-rescue’ type or life-saving QALYs as the rationale that underpinsthis approach is the result of reductions in ‘current-period’ risk of death. This is life saving in theimmediate sense, such as through emergency treatment as, of course, life cannot be prolongedindefinitely. In contrast, the QALY gain that arises from approach 2 comes from the proportionatereduction in risk of death later in life. Thus, this may be more reflective of longer-term medicaltreatments that extend life but not immediately.

In practical terms, these arguments would imply that QALYs arising from emergency serviceswould receive a higher value than QALYs arising from, say, elective surgery such as a coronaryartery bypass graft, which extends life but is not immediately life saving. This raises the possibilityof a third type, which is QALY arising from QoL enhancement only. The third approach, which ispresented in Appendix C, provides a first attempt at trying to estimate the value of a QALY that

Table II. Estimates of the value of a life year and a QALY

Value pery

Life year QALY

Undiscounted Discounted Undiscounted Discounted

Full VPFApproach 1 44 018 56 331 55 856 70 896Approach 2a 23 199 29 691 27 286 34 925Approach 2b 40 029 57 569 48 022 67 470

VPF excluding net outputApproach 1 42 626 54 550 54 091 68 656Approach 2a 20 422 26 070 24 219 30 745Approach 2b 38 764 55 749 46 504 65 338



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would arise from QoL only gains. This approach uses the value of serious injuries (VSI) to infer suchvalues and based on this we get, as one might expect, an even lower value of a QALY, if due to the highdegree of uncertainty surrounding how the VSI was calculated, we focus on the median values presentedin Table III.

6.2. Transferring values between contexts

Underlying the conduct of this research is the implicit assumption that individuals’ preferences forhealth gains from health-care interventions are the same as their preferences for reductions in the risk ofdeath through road safety improvements. Making this assumption allows for a direct conversion of theVPF into the value of a QALY. However, there is no empirical evidence available, which has looked atwhether individuals’ preferences for road safety are the same as their preferences for health care. Basedon other research conducted on estimating the monetary value of a QALY, the values that have arisenwhen individuals are asked to provide a value for a QALY based on a health context are much lowerthan those that have been inferred from the VPF (Gyrd-Hansen, 2003; Byrne et al., 2005). This is likelyto be because those values that are inferred here from the VPF incorporate a risk of death, whereasempirical studies that have been conducted in the health field have mainly focused on presenting theQALY gain as an improvement in QoL only. This adds to the argument presented above that the valueof a QALY may differ depending on how the QALY gain arises.

6.3. Value of a QALY literature

The suggestion for using the VPF to calculate a WTP per QALY for use in cost effectiveness analyses(CEAs) is not new (Johannesson and Meltzer, 1998), and there have been a small number of studies thathave attempted to do this. The calculation suggested has been, as in approach 1, to divide the VPF by thediscounted QALYs gained for a saved life. For example, using Swedish data, Johannesson and Meltzer(1998) produce an estimate of a WTP per QALY gained of around US $90 000 (approx. £57 000 in 2003prices). Another study in Sweden (using a VPF of SEK16.3m as accepted by the Swedish Government foruse in traffic safety planning) estimated the monetary value of a QALY to be SEK655 000 (approx.£59 000). This value is currently used as a reference by the Swedish pricing and reimbursement boardwhen deciding on reimbursement for pharmaceuticals (Persson and Hjelmgren, 2003).

An examination of studies that have calculated the value of a life was undertaken by Hirth et al.(2000) in an attempt to work out an implicit value of a QALY. Results were categorised by the type ofestimation process used to estimate the value of life e.g. contingent valuation, human capital andrevealed preference. For contingent valuation studies, the median value of a QALY was $161 305 in1997 US dollars (approx. £102 000 in 2003 prices). This estimate was based on eight contingentvaluation studies that asked either WTP for risk reduction or willingness to accept risk questions.

In an Australian study by Abelson (2003), a value of a QALY of A$108 000 (approx. £51 000 in 2003prices) was estimated. This was calculated using a VPF of A$2.5m and the assumptions of 40 years ofremaining life expectancy and a discount rate of 3%.

Papers by Williams (2004) and Rawlins and Culyer (2004) have proposed values of a QALY, which arenot based on the VPF. Based on their experience and observations of the NICE process, Rawlins andCulyer state that interventions with a cost per QALY of over £25 000–£35 000 would need special reasons

Table III. Estimates of the value of a QALY from the prevention of non-fatal injuries

Weighted mean (£) of injuries X, W, S and R Median (£) of injuries X, W, S and R

Mean SG 436 584 328 262Median SG 21 519 6414

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for being accepted, which could be interpreted as the revealed preference of NICE. Williams proposes amuch lower figure of £18 000 based on the premise that in the UK this is the amount that we have in realresources to provide all the needs of the average citizen. Other commentators, too, have hinted that suchlower ‘cut-off’ values, even as low as £15 000 per QALY, are required in the UK (Maynard et al., 2004).

How can these proposed values, based on a wealth of experience, be reconciled with those reported inthis paper? First, Williams implies that, with this amount available to spend on meeting all needs, it isexcessive to spend almost twice that (assuming a £30 000 benchmark) on health care alone andreinforces this by stating that a £30 000 benchmark represents more than 20 times the UK averagehealth-care expenditure per person per annum. However, given that payments to health care are basedon many contributing so that few can benefit, and that relatively few people use the NHS (at least as aninpatient and, thus, for more serious conditions) each year, such a benchmark of £30 000 might beappropriate. Second, the same commentators have pointed out that the NHS has not been good atexpediting the removal of ‘redundant technologies’, which once achieved may also mean that a higherthreshold is indeed affordable (Maynard and Street, 2006). Third, moving to the other end of thespectrum of affordability, the use of a range of values for different QALY types, where valuations ofQoL improvements alone and of life extension are less than those involving live saving, would likelyimply an overall move towards the lower end of the values reported in this paper. It depends on whattypes of QALY are being bought at the margin by the NHS, and more likely these will be of the QoLenhancing and life extension (i.e. low valuation) types.

This final point then raises serious questions about QALYs themselves and, thus, the use of QALY-based CEA as a tool for priority setting. On the other hand, we have merely raised questions aboutdifferential value of QALY types, which require further empirical testing and do not necessarily negatein full the QALY approach.

7. CONCLUSION

The research presented in this paper has set out to establish a WTP-based value of a QALY, whichcan be used in UK policy making. Two approaches have been developed, which present alternativeways of thinking about how people may value life. The first approach follows the work of studiescurrently in the literature (Hirth et al., 2000; Johannesson and Meltzer, 1998; Persson andHjelmgren, 2003) and assumes that the value of life depends only on future QALY gains. Thisapproach was further developed to show how the results can be interpreted as a group aggregatevaluation that is more realistic than the current assumption of linear proportionality, which is currentlyused in the literature.

The second approach presents a new method for modelling a WTP-based value of a QALY taking intoaccount the inverted-U-shaped relationship between the VPF and age. While this approach may be viewedas limited because of the negative values attached to younger people, this approach does represent moreclosely the preference of individuals in their attitudes to risk reductions, which arise from safety.

APPENDIX A: LIFE EXPECTANCY GAINS VERSION OF APPROACH 1

Suppose that a large representative group of individuals is offered small individual reductions, dpi,i5 1,2,y,n, in the risk of death during the coming year. Denoting the ith individual’s MRS of wealthfor the risk of death during the coming year by mi, it follows that the aggregate WTP for the riskreduction will be given by



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V ¼Xn

midpi ðA1Þ

Furthermore, the risk reductions will necessarily result in gains in life expectancy for each of then individuals. More specifically, as shown in Jones-Lee (1976), denoting the ith individual’s remaininglife expectancy by Ei and probability of surviving the coming year by Pi, then

@Ei

@Pi¼

Ei � 1

PiðA2Þ

However, for the typical individual Pi will be very close to 1 and Ei will exceed 30 so that to a reasonableapproximation we can express as follows:

@Ei

@Pi� Ei ðA3Þ

It therefore follows that the individual gains in life expectancy, dEi, resulting from the individual riskreductions, dpi, will be given by

dEi ¼ Eidpi; i ¼ 1; 2; . . . ; n ðA4Þ

Now suppose that the individual risk reductions are all set equal to 1=P

n Ei. It then followsimmediately from Equation (A4) that taken across the whole group of n individuals, the aggregate gainin life expectancy will be given byP

n

dEi ¼Pn

Ei1PnEi

ðA5Þ

¼ 1 ðA6Þ

That is, the aggregate gain in life expectancy will be exactly one year.But from Equation (A1), with dpi ¼ 1=

PEi,

V ¼

PZPZ

Mi

EiðA7Þ

that is,

V ¼M

EðA8Þ

where M and E denote, respectively, the arithmetic means of Mi and Ei so that if the group of nindividuals is indeed representative then the WTP-based value of an aggregate gain of one year of lifeexpectancy is given by the population mean of Mi (which is under the standard WTP definition, theVPF) divided by the population mean of Ei.

The key assumption underpinning the above argument is that all of the affected individuals enjoy thesame risk reduction. Suppose that instead the individual risk reduction is allowed to differ and is in factset so that all n individuals enjoy the same gain in life expectancy. More specifically, suppose that therisk reductions are set so that

dpi ¼1

nEi; i ¼ 1; 2; . . . ; n ðA9Þ

It then follows from (A4) and (A8) that

dEi ¼1

n; i ¼ 1; 2; . . . ; n ðA10Þ

Taken over the whole group of n individuals, the aggregate gain in life expectancy would therefore beexactly one year. In turn, from (A1) and (A9), aggregate WTP would be given by

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V ¼Xn

Mi

nEiðA11Þ

that is,V ¼

1

n

Xn

Mi

EiðA12Þ

In this case it is therefore clear that the Value of a life year (VOLY) will be given by the arithmetic meanof Mi=Ei, as opposed to the arithmetic of Mi divided by the arithmetic mean of Ei as in the previouscase. Notice also that the mean of the ratio Mi=Ei will be equal to the ration of the means of Mi and Ei

only in exceptional circumstances.

APPENDIX B: MATHEMATICAL EXPOSITION OF APPROACH 2B

Mi ¼ aEbi ðB1Þ

To solve Equation (B1) it is first necessary to take logs of both sides to make it into a linear equation.Thus Equation (B1) becomes

ln Mi ¼ ln aþ b ln Ei ðB2Þ

When Ei 5 8

ln aþ b ln 8 ¼ ln 0:45� 1 427 340 ðB3Þ

When Ei 5 39

ln aþ b ln 39 ¼ ln 1:36� 1 427 340 ðB4Þ

Dividing (B4) from (B3) results in

ln1:36

0:45¼ b ln

39

8

b ¼1:10599

1:64317

b ¼ 0:67309 ðB5Þ

Substitute (B5) into (B3) to obtain a:

ln aþ ð0:67309 ln 8Þ ¼ ln 0:45� 1 427 340

ln a ¼ 13:37� 1:3595

a ¼ 164 932 ðB6Þ

Hence, from Equations (B1), (B5) and (B6)

Mi ¼ 164 932E0:67309i ðB7Þ


dMi

dEi¼ ð0:67309� 164 932ÞE�0:3269i ðB8Þ¼ 111 013E�0:3269i ðB9Þ

As in approach 2a it is assumed the probability density function takes the form

rðEiÞ ¼ gE1=2i ðB10Þ

which entails that



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Z 39

8

gE1=2i dEi ¼ 1 ðB11Þ

so that

g½23sE

3=2i �

398 ¼ 1 ðB12Þ

Hence,

147:3g ¼ 1 ðB13Þ


g ¼ 0:00678 ðB14Þ

From Equations (B9), (B11) and (B14) we then have

V2b ¼R 398 111 013� E�0:3269i � 0:00678� E

1=2i dEi ðB15Þ

¼R 398747:86E0:1731

i dEi ðB16Þ

¼�637:52E1:17309

i

�398

ðB17Þ

¼ 40 029 ðB18Þ

Hence,

V2b ¼ 40 029 ðB19Þ

APPENDIX C: ESTIMATING THE VALUE OF A QALY FROM THE ELICITED VALUES OFPREVENTING SERIOUS INJURIES

Using the UK DfT’s VSIs, a value of a QALY can be estimated, which incorporates only changes inQoL (Department for Transport, 2004). The VSI is based on the valuation of eight different severities ofserious injuries, which range from injuries that will last only a few days and require no hospitaltreatment through to permanent paralysis and brain damage (Jones-Lee et al., 1995).

The EQ-5D was used to calculate the QoL loss for each of the eight injuries. Most of the injuries aredescribed over three time periods: in-hospital effects, aftereffects within the first year and aftereffectsover remaining life expectancy. Thus, for each injury, we first calculated a utility value using the EQ-5Dtariff for each of these three time periods. For each time period the QALY loss was calculated asfollows:

QoL loss5QoL weight for UK population�EQ-5D tariff value for injury andQALY loss5QoL loss�Duration (years)

The duration of ‘effects over remaining life expectancy’ was taken to be weighted UK remaining lifeexpectancy minus the duration of time periods one and two. The total QALY loss for each injury is thenthe sum of the QALY losses across each time period.

The original estimation of the VSI was based on a population sample survey in which respondentsvalued each injury against death. The SG technique was used to obtain this value, which was then usedto calculate the MRS of the injury against death. The value of preventing this injury is then the fractionof the VPF that is given by the MRS, i.e. if the MRS is 0.09, the value of preventing this injury will be9% of the VPF. Thus, the value of a QALY for each injury, i5 1–8,

¼VPF�MRSi

QALY lossiðC1Þ

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and

V ¼X8i¼1

ðValue of a QALYÞiPi ðC2Þ

where Pi is the probability of occurrence of injury i.The value of a QALY was estimated using both the mean and median best SG results presented by

Jones-Lee et al. (1995). As it turned out, only four injuries (X, W, S and R) were used in the calculationof the value of a QALY as these were the only injuries for which SG values were directly derived. Asillustrated above, each injury is weighted by its probability of occurrence so that a weighted mean valueof a QALY is estimated. Mean and median values of a QALY were calculated from the individual valueof a QALY estimate for each of the four injuries, leading to the results in Table III.

The results estimated using the mean SG values are significantly higher than those of any of the otherapproaches. The mean SG values for each of the health states are positive values greater than zero,while the median SG values for the less severe injuries W and X are equal to zero. The use of the medianSG results would appear to be more appropriate as 75% of the sample returned a zero SG valuation forinjury X and 81% of the sample gave a zero valuation for injury W.

Using the median SG values the results (in the range £6400–21 500) are more in line with what wouldbe expected with the value of a QALY estimated on the basis of changes in QoL only being lower thanthose estimated using approaches 1 and 2 in the main text.

ACKNOWLEDGEMENTS

This work was supported by National Co-ordinating Centre for Research Methodology Grant NumberRM03/JH13. We would like to acknowledge the Social Value of a QALY team and Andy Marshall forcomments on earlier versions of this paper. Helen Mason is supported by an ESRC post doctoralfellowship. Cam Donaldson holds the Health Foundation chair in Health Economics at NewcastleUniversity and is an NIHR senior Investigator.’

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modelling the monetary value of a qaly: a new approach based on uk data

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