Signature
w = (h_high / h_low)^epsilon
| Inputs | Definition | Unit |
|---|---|---|
h_high | Current health of the healthier person or group, for example QALE of the richest fifth | health units, for example years of QALE |
h_low | Current health of the less healthy person or group, for example QALE of the poorest fifth. Above zero | health units, in the same unit as h_high |
epsilon | Atkinson inequality aversion parameter, zero or above. At 0 every gain has a weight of 1 | none |
w | Weight on a small health gain to the less healthy person or group relative to the same gain to the healthier one | ratio, without unit |
|---|
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 relative weight in one cell
With the two health levels in cells named HealthHigh and HealthLow and the aversion parameter in Epsilon, Excel returns the relative weight.
=(HealthHigh/HealthLow)^Epsilon
Assumptions
Small gains at current health levels for the Atkinson weight
The weight applies to small gains evaluated at the stated health levels. As a programme narrows the gap the ratio of health levels falls, and so does the weight on further gains.
Worked examples
Implied weight at the English median epsilon
With QALE of 62 years for the poorest fifth and 74 years for the richest fifth, the starting point of the English elicitation study, an epsilon of 10.95 gives a weight of about 6.94. Robson and colleagues report an implied weight of 6.95: a QALY gained by the poorest fifth counts almost seven times as much as one gained by the richest fifth.
h_high = 74; h_low = 62; epsilon = 10.95; w = 6.94
Implied weight in the two-group example at epsilon of 2
In the article's two-group example, with baseline QALE of 60 and 72 years, an epsilon of 2 gives a gain to the less healthy group 1.44 times the weight of a gain to the healthier group.
h_high = 72; h_low = 60; epsilon = 2; w = 1.44
Common errors
Inverting the health ratio in the Atkinson weight
Writing the ratio as the lower health level over the higher gives the reciprocal, about 0.144 instead of 6.94 at epsilon of 10.95 with QALE of 62 and 74 years, which reads as if gains to the richest fifth counted more.
Quoting the Atkinson weight without its health levels
The same epsilon implies different weights at different health gaps. At epsilon of 10.95 the weight is about 6.94 between QALE of 62 and 74 years but about 7.36 between 60 and 72 years, so a statement such as seven times describes epsilon only at the stated levels.
Sources
Implied weight on gains to the poorest fifth in the English elicitation
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, with the initial QALE of 62 and 74 years, and Table 1, which reports the median Atkinson epsilon of 10.95 and the implied weight of 6.95 on a marginal gain to the poorest fifth relative to the richest fifth.
Canonical Identity
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