Signature
h_EDE = (sum_(i=1)^n [h_i^(1-epsilon)] / n)^(1/(1-epsilon))
| Inputs | Definition | Unit |
|---|---|---|
h_i | Health of person i, or of equal-sized group i, for example quality-adjusted life expectancy. Above zero | health units, for example years of QALE |
epsilon | Inequality aversion parameter, zero or above and not equal to 1 in this formula. Higher values give more weight to gains at the lower end of the distribution. It is a social value judgement, not estimated from the health data; the median elicited from the public in England was 10.95 | none |
n | Number of people, or of equal-sized groups, equal to the length of the h_i list | count |
h_EDE | Level of health which, if everyone had it, would be judged as good as the actual distribution at inequality aversion epsilon | health units, for example years of QALE |
|---|
Function
Atkinson inequality index and equally distributed equivalent health function
Maps a distribution of health across people or equal-sized groups, and a chosen inequality aversion parameter epsilon, to the equally distributed equivalent (EDE) level of health and the Atkinson index. The EDE is the level of health which, if everyone had it, would be judged as good as the actual unequal distribution. The index is the share of mean health that could be given up for full equality without lowering social welfare. In distributional cost-effectiveness analysis the EDE ranks programmes and the index describes how far each one leaves health unevenly spread.
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Implementations
Excel
Atkinson EDE for epsilon other than 1 with SUMPRODUCT
With one health value per person or per equal-sized group in a range named Health and the aversion parameter in a cell named Epsilon, not equal to 1, Excel returns the EDE. SUMPRODUCT evaluates the power for each cell without an array entry.
=(SUMPRODUCT(Health^(1-Epsilon))/COUNT(Health))^(1/(1-Epsilon))
Assumptions
Constant relative inequality aversion in the Atkinson EDE
The social welfare function is symmetric and additively separable, with constant relative inequality aversion, so scaling every health value by the same factor scales the EDE by that factor. Atkinson himself described the additively separable form as restrictive.
Equal weight for each person or group in the Atkinson EDE
Each h_i counts once, so each value stands for one person or for a group of the same size as the others. Groups of unequal size are weighted by their population shares, as in the computational function HE-CF-ATK-001.
Strictly positive health values in the Atkinson EDE
With epsilon above 1 the exponent one minus epsilon is negative, so a value of zero cannot be used, and negative values have no meaning in the formula. It applies to ratio-scale health such as QALE, not to net health gains or losses that can fall to zero or below.
Worked examples
Atkinson EDE for programme U at epsilon of 2
Programme U gives QALE of 60.1 and 72.5 years in two equal-sized groups. At epsilon of 2 the EDE is the harmonic mean of the two values, about 65.7202 years, against a mean of 66.30.
h_i = [60.1,72.5]; n = 2; epsilon = 2; h_EDE = 65.7202
Atkinson EDE for programme T at epsilon of 2
Programme T gives QALE of 60.4 and 72.1 years. At epsilon of 2 the EDE is about 65.7334 years, above the 65.7202 for U, so the ranking by mean health, where U leads by 0.05 years, is reversed. In the article the switching value of epsilon is about 1.6.
h_i = [60.4,72.1]; n = 2; epsilon = 2; h_EDE = 65.7334
Atkinson EDE for programme T at the English median epsilon
At the English median epsilon of 10.95 the EDE for programme T is about 63.73 years, against about 63.51 years for programme U computed the same way, so T is preferred at the elicited level of aversion.
h_i = [60.4,72.1]; n = 2; epsilon = 10.95; h_EDE = 63.7344
Common errors
Setting epsilon to exactly 1 in the general Atkinson EDE formula
At epsilon of 1 the outer exponent one over one minus epsilon divides by zero, and Excel returns #DIV/0!. The EDE at epsilon of 1 is the geometric mean, HE-FM-ATK-003, which is the limit of this formula as epsilon tends to 1.
Treating groups of unequal size as equal in the Atkinson EDE
Entering one value per group when the groups differ in size gives each group the same weight. With baseline QALE of 60 and 72 years and the less healthy group making up 20% of the population, the share-weighted EDE at epsilon of 2 is about 69.23 years, whereas entering the two values once each gives about 65.45 years.
Sources
Atkinson's discrete-distribution form of the inequality measure
Atkinson AB. On the measurement of inequality. Journal of Economic Theory. 1970;2(3):244-263. Page 251, equation (5), which derives the welfare function under constant relative inequality aversion with epsilon of zero or above, and page 257, which gives the measure for discrete distributions and the limiting cases of epsilon of 0 and epsilon tending to infinity.
Atkinson social welfare index formula for EDE health
Asaria M, Griffin S, Cookson R. Distributional cost-effectiveness analysis: a tutorial. Medical Decision Making. 2016;36(1):8-19. Section on evaluating social distributions of health, which gives the Atkinson social welfare index for EDE health with inequality aversion parameter epsilon.
Canonical Identity
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