Signature
epsilon_star = log(w_star) / log(h_better / h_worse)
| Inputs | Definition | Unit |
|---|---|---|
w_star | Switching weight on the worse-off group's gain, the w_s of HE-FM-WE-004 with each group's QALYs taken as its population share times its net gain per person, above zero | multiplier, dimensionless |
h_better | Baseline health per person of the better-off group, for example quality-adjusted life expectancy | years |
h_worse | Baseline health per person of the worse-off group, above zero and below h_better | years, in the same unit as h_better |
epsilon_star | Approximate value of the Atkinson inequality aversion parameter at which the two options tie, zero or above when w_star is at least 1 | none |
|---|
Function
Health forgone for equity, switching weight and implied inequality aversion function
Takes two options of equal cost, the population share of a worse-off and a better-off group, each group's net health gain per person under each option after the health displaced elsewhere, and each group's baseline health. It returns the total health forgone by choosing the fairer option (HE-FM-EET-001), the weight on the worse-off group's gain at which the two options tie, and the Atkinson inequality aversion that this weight implies at baseline health (HE-FM-EET-003). The weight is the switching weight w_s of the live record HE-FM-WE-004 (function HE-FN-WE-001), with each group's QALYs read as its population share times its net gain per person, and it is cited there rather than restated. It exceeds 1 by the health forgone divided by the share-weighted extra gain to the worse-off group, the link that HE-VT-EET-003 checks, and it differs from the equity switching weight HE-FM-EQWA-004, which sets one programme against the health displaced by its cost. The implied aversion inverts the live Atkinson relative weight HE-FM-ATK-004, whose test HE-VT-ATK-005 recovers epsilon from a weight. Equally distributed equivalent health itself is live as HE-FM-ATK-002 and HE-FM-ATK-003 (function HE-FN-ATK-001, computational function HE-CF-ATK-001) and is not restated, and the exact switching value of the aversion parameter, found by solving the EDE comparison numerically, belongs to the Distributional Cost-Effectiveness package. Notation follows the Equity-Efficiency Trade-off article.
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Implementations
Excel
Implied inequality aversion from named cells
With the switching weight in SwitchWeight and baseline health in HealthBetter and HealthWorse, the formula returns the implied aversion, held in a cell named ImpliedEpsilon. Excel LN is the natural logarithm, although any base gives the same ratio.
=LN(SwitchWeight)/LN(HealthBetter/HealthWorse)
Assumptions
Weights taken at baseline health for the implied aversion
The switching weight of HE-FM-WE-004 is one linear weight on every QALY to the worse-off group (HE-AS-WE-008), while the Atkinson weight applies to small gains at the stated health levels (HE-AS-ATK-009). Using baseline health gives the approximation, while solving the equally distributed equivalent health of the two options for the tie gives the exact value, about 2.35 in the article's example against 2.38 from this formula. Asaria and colleagues locate such a switch by comparing EDE health across levels of inequality aversion.
Ordered positive health levels and a weight of at least 1
h_better is above h_worse and both are above zero, so the logarithm of the ratio is positive. The switching weight exceeds 1 when the fairer option gives less total health and more to the worse-off group, and a weight of at least 1 then gives an aversion of zero or above, the range of the Atkinson parameter (HE-VR-ATK-014).
Relative inequality concept of the implied aversion
The Atkinson parameter measures aversion to relative inequality (HE-AS-ATK-003). An absolute index such as the Kolm index, which Asaria and colleagues report beside it, gives a different parameter for the same break-even weight, and this formula does not convert between them.
Worked examples
Implied inequality aversion for programmes E and F
In the article's illustrative example the switching weight of HE-FM-WE-004, with the share-weighted net gains, is 0.21 divided by 0.16, or 1.3125. With baseline quality-adjusted life expectancy of 74 and 66 years it gives 0.2719 divided by 0.1144, or about 2.38. The weight of an Atkinson function with aversion above this value favours F at baseline health.
w_star = 1.3125; h_better = 74; h_worse = 66; epsilon_star = 2.3768
Inverting the implied weight of the English elicitation
Robson and colleagues report an implied weight of 6.95 at initial quality-adjusted life expectancy of 62 years for the poorest fifth and 74 for the richest fifth. Inverting it returns about 10.96, against their median Atkinson parameter of 10.95, with the gap due to the weight being rounded to two decimals.
w_star = 6.95; h_better = 74; h_worse = 62; epsilon_star = 10.96
Zero implied aversion when the switching weight is 1
When the two options tie under equal weights, so that no health is forgone and the switching weight is 1, the implied aversion is zero, the health-maximising rule.
w_star = 1; h_better = 74; h_worse = 66; epsilon_star = 0
Common errors
Reading the approximate aversion as the exact switching value
The formula gives about 2.38 for the article's example, but the EDE comparison switches at about 2.35. At an aversion of 2.36 the EDE of F already exceeds that of E by about 0.0002 years, although 2.36 lies below the approximate value.
Using mean health in place of the better-off group's health
Replacing 74 years with the population mean of 70.8 in the ratio gives about 3.87 instead of 2.38, because the weight in HE-FM-ATK-004 compares the two groups, not one group with the mean.
Reporting a negative aversion when no trade-off exists
A switching weight below 1 means the fairer option has at least as much total health. An illustrative weight of 0.8 with 74 and 66 years returns about minus 1.95, which is not a valid Atkinson parameter, and the case is reported as having no trade-off.
Switching weight taken from per-person gains without population shares
Entering the per-person group gains in HE-FM-WE-004, whose group QALYs are totals, gives a switching weight of 0.35 divided by 0.40, or 0.875, in place of 0.21 divided by 0.16, or 1.3125. F then appears to be preferred even with equal weights, and the implied aversion comes out at about minus 1.17 instead of 2.38.
Sources
Cookson and colleagues on weights implied by an equity parameter
Cookson R, Mirelman AJ, Griffin S, Asaria M, Dawkins B, Norheim OF, Verguet S, Culyer AJ. Using cost-effectiveness analysis to address health equity concerns. Value in Health. 2017;20(2):206-212. Section Equity-Weighting Analysis (Valuing the Importance of Equity), which states that an equity parameter indirectly implies a specific set of equity weights when combined with the existing distribution of health, and that these implied weights change with that distribution.
Robson and colleagues on the implied weight at initial health
Robson M, Asaria M, Cookson R, Tsuchiya A, Ali S. Eliciting the level of health inequality aversion in England. Health Economics. 2017;26(10):1328-1334. Methods, which give the initial distribution of quality-adjusted life expectancy as 62 and 74 years, and Table 1 with footnote b, which reports the median Atkinson parameter of 10.95 and the implied weight of 6.95 on a marginal gain to the poorest fifth relative to the richest fifth at initial health.
Asaria and colleagues on locating the switching level of inequality aversion
Asaria M, Griffin S, Cookson R. Distributional cost-effectiveness analysis: a tutorial. Medical Decision Making. 2016;36(1):8-19. Section Analyzing tradeoffs between total health and health inequality using social welfare indices, with Figure 4, which show the comparison of EDE health between two strategies across levels of inequality aversion and state that the targeted strategy would be preferred above an Atkinson parameter of 8.
Canonical Identity
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