Signature
h_EDE = exp(sum_(i=1)^n [log(h_i)] / n)
| Inputs | Definition | Unit |
|---|---|---|
h_i | Health of person i, or of equal-sized group i. Above zero | health units, for example years of QALE |
n | Number of people, or of equal-sized groups, equal to the length of the h_i list | count |
h_EDE | Equally distributed equivalent health at inequality aversion epsilon of 1 | 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.
Implementations
Excel
Geometric-mean EDE with Excel GEOMEAN
With one health value per person or per equal-sized group in a range named Health, Excel GEOMEAN returns the EDE at epsilon of 1. The equivalent form =EXP(AVERAGE(LN(Health))) gives the same result in Excel versions with dynamic arrays.
=GEOMEAN(Health)
Assumptions
Health above zero for the geometric-mean EDE
Every health value is above zero, because the logarithm of zero is undefined. A single zero sets the EDE to zero and the Atkinson index to 1 whatever the other values.
Equal weight per person or group in the geometric-mean EDE
Each h_i counts once. Groups of unequal size are weighted by their population shares inside the mean of the logarithms, as in HE-CF-ATK-001.
Worked examples
Geometric-mean EDE for programme U
For programme U the EDE at epsilon of 1 is the square root of 60.1 × 72.5 = 4357.25, which is about 66.0095 years.
h_i = [60.1,72.5]; n = 2; h_EDE = 66.0095
Geometric-mean EDE for programme T
For programme T the EDE at epsilon of 1 is the square root of 60.4 × 72.1 = 4354.84, which is about 65.9912 years. U still leads, by about 0.018 years, so the ranking does not reverse until epsilon reaches about 1.6.
h_i = [60.4,72.1]; n = 2; h_EDE = 65.9912
Common errors
Zero or negative values in the geometric-mean EDE
Applying the geometric mean to a health measure that can be zero or negative, such as QALYs gained or net health benefit, fails: Excel GEOMEAN returns #NUM!, and a single zero computed by hand gives an EDE of zero and an index of 1. The formula needs levels of health, such as QALE, not changes.
Sources
Atkinson's logarithmic welfare function at epsilon of 1
Atkinson AB. On the measurement of inequality. Journal of Economic Theory. 1970;2(3):244-263. Page 251, equation (5), which gives the welfare function as the natural logarithm of income when epsilon equals 1.
Atkinson index at epsilon of 1 for QALE distributions
Asaria M, Griffin S, Cookson R. Distributional cost-effectiveness analysis: a tutorial. Medical Decision Making. 2016;36(1):8-19. Table 8, which reports the Atkinson index at epsilon of 1 for the QALE distributions of the bowel cancer screening strategies, 0.00171 for no screening.
Canonical Identity
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