Atkinson index from equally distributed equivalent health

Signature

A_epsilon = 1 - h_EDE / h_bar

Function

Atkinson inequality index and equally distributed equivalent health function

Computational function

• Computational function: Atkinson index from subgroup health levels and population shares

  Takes the inputs a distributional cost-effectiveness analysis usually holds, the mean health of each population subgroup and the share of the population in each subgroup, together with a chosen epsilon, and returns mean health, the equally distributed equivalent health and the Atkinson index. It weights each group by its population share, as in Atkinson's measure for discrete distributions, and switches to the share-weighted geometric mean when epsilon equals 1, where the general formula HE-FM-ATK-002 is undefined. The inputs therefore differ from the formula's variables: HE-FM-ATK-002 and HE-FM-ATK-003 need one value per person or per equal-sized group, and the function accepts groups of any size.

  Inputs and outputs: h_g: Mean health of each subgroup, for example QALE by deprivation group; required, every value above zero. Unit: health units, for example years.; s_g: Share of the population in each subgroup, in the same order as h_g; required, zero or above. The R and Python versions divide by the total, so counts can be entered. Unit: proportion.; G: Number of subgroups, the length of h_g and s_g. Unit: count.; epsilon: Inequality aversion parameter; required, zero or above. Unit: none.; h_bar: Share-weighted mean health. Unit: health units.; h_EDE: Equally distributed equivalent health at epsilon. Unit: health units.; A_epsilon: Atkinson index at epsilon. Unit: proportion from 0 to 1.

  Assumption: Everyone in a subgroup is given the subgroup's mean health, so inequality within subgroups is ignored and only variation between the subgroups that define unfair inequality counts. Shares sum to 1 after normalisation.

  Worked example (Programme U, two equal-sized groups, epsilon of 2): With QALE of 60.1 and 72.5 years and equal shares, the function returns a mean of 66.30, an EDE of about 65.7202 and an index of about 0.00874, as in HE-EX-ATK-001. h_g = [60.1,72.5]; s_g = [0.5,0.5]; G = 2; epsilon = 2; h_bar = 66.3; h_EDE = 65.7202; A_epsilon = 0.00874

  Worked example (Groups of unequal size): If the less healthy group with baseline QALE of 60 years makes up 20% of the population and the healthier group with 72 years makes up 80%, the mean is 69.6 years, the EDE at epsilon of 2 about 69.2308 and the index about 0.00531. The figures are illustrative. h_g = [60,72]; s_g = [0.2,0.8]; G = 2; epsilon = 2; h_bar = 69.6; h_EDE = 69.2308; A_epsilon = 0.00531

  Excel: =1-IF(Epsilon=1,EXP(SUMPRODUCT(Share,LN(Health))/SUM(Share)),(SUMPRODUCT(Share,Health^(1-Epsilon))/SUM(Share))^(1/(1-Epsilon)))/(SUMPRODUCT(Share,Health)/SUM(Share)) With subgroup health in a range named Health, population shares or counts in a range named Share and the parameter in Epsilon, the formula returns the Atkinson index; the part inside IF on its own returns the EDE.

  R: atkinson_groups <- function(h, s, eps) { s <- s/sum(s); m <- sum(s*h); ede <- if (eps == 1) exp(sum(s*log(h))) else sum(s*h^(1-eps))^(1/(1-eps)); c(mean = m, ede = ede, index = 1-ede/m) } Returns the mean, the EDE and the index; applying it over a vector of epsilon values with sapply gives the table across a range of aversion.

  Python: def atkinson_groups(h, s, eps): t = sum(s); m = sum(si*hi for si, hi in zip(s, h))/t; ede = math.exp(sum(si*math.log(hi) for si, hi in zip(s, h))/t) if eps == 1 else (sum(si*hi**(1-eps) for si, hi in zip(s, h))/t)**(1/(1-eps)); return m, ede, 1-ede/m Uses the math module, where math.log is the natural logarithm.

  Test (Splitting a group leaves the result unchanged): Dividing the less healthy group into two halves with the same health gives the same EDE as the undivided group, because only population shares matter. Expected result: TRUE. Excel check: =ABS(SUMPRODUCT({0.25,0.25,0.5},{60,60,72}^(1-2))^(1/(1-2))-SUMPRODUCT({0.5,0.5},{60,72}^(1-2))^(1/(1-2)))<1E-9

  Test (Epsilon of 1 gives the geometric mean): With equal shares and epsilon of 1 the share-weighted logarithm branch returns the geometric mean of the baseline QALE of 60 and 72 years, about 65.7267, and an index of about 0.00414. Expected result: TRUE. Excel check: =ABS(EXP(SUMPRODUCT({0.5,0.5},LN({60,72})))-SQRT(60*72))<1E-9

  Common error (Weighting the mean but not the EDE by population share): Using share-weighted mean health with an EDE that counts each group once mixes two distributions. In the unequal-shares example it gives an index of about 0.0596 instead of 0.00531, overstating inequality more than tenfold.

  Source: Atkinson AB. On the measurement of inequality. Journal of Economic Theory. 1970;2(3):244-263. Page 257, which gives the measure for discrete distributions with each income level weighted by its frequency in the distribution.

  h_bar = sum_(g=1)^G [s_g * h_g]; h_EDE = (sum_(g=1)^G [s_g * h_g^(1-epsilon)])^(1/(1-epsilon)); A_epsilon = 1 - h_EDE / h_bar

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Implementations

• Excel

  Atkinson index from EDE and mean health in one cell

  With the EDE in a cell named EDE and mean health in a cell named MeanHealth, both computed at the same epsilon and in the same unit, Excel returns the Atkinson index.

  =1-EDE/MeanHealth

Assumptions

• EDE and Atkinson index computed at the same epsilon

  The EDE is computed at the epsilon for which the index is reported. Index values at different epsilon rest on different value judgements and are not compared with each other, so results are shown across a range of epsilon.

• Positive ratio-scale health in the Atkinson index

  Health is measured on a ratio scale with every value above zero, such as QALE in years. Negative values cannot be used, and an index of inequality in ill health is not the mirror image of an index of inequality in health.

• Relative inequality concept of the Atkinson index

  The index measures relative inequality: multiplying every health value by the same factor leaves it unchanged, while adding the same number of years to everyone lowers it. The choice between relative and absolute measures, such as the Kolm index, is a value judgement, and the DCEA tutorial suggests calculating several measures when there is no clear choice.

Worked examples

• Atkinson index for programme U at epsilon of 2

  In the article's illustrative two-group example, programme U gives QALE of 60.1 and 72.5 years in two equal-sized groups, a mean of 66.30 years and an EDE at epsilon of 2 of about 65.7202 years. The Atkinson index is about 0.00874, above the baseline value of about 0.00826, so U widens relative inequality.

  h_EDE = 65.7202; h_bar = 66.3; A_epsilon = 0.008745

• Atkinson index for programme T at epsilon of 2

  Programme T gives QALE of 60.4 and 72.1 years, a mean of 66.25 years and an EDE at epsilon of 2 of about 65.7334 years. The Atkinson index is about 0.00780, below the baseline value of about 0.00826, so T narrows relative inequality.

  h_EDE = 65.7334; h_bar = 66.25; A_epsilon = 0.00780

Common errors

• Choosing the programme with the lower Atkinson index

  The index measures inequality, not social welfare, so the option with the lower index is not necessarily preferred. In the article's example at epsilon of 1, programme U has the higher index, about 0.00438 against 0.00391 for T, yet also the higher EDE, about 66.0095 against 65.9912 years, so U is still ranked first at that epsilon. Decisions rest on the EDE or an inequality-adjusted net health benefit.

• Reading the Atkinson index as a gap between groups

  An index of 0.04 does not mean that the less healthy group has 4% less health than the mean. It means that an equal distribution at 96% of current mean health would be judged as good as the current one at the chosen epsilon, and the same distribution gives a different index at another epsilon.

Sources

• Atkinson's definition of the inequality measure from equally distributed equivalent income

  Atkinson AB. On the measurement of inequality. Journal of Economic Theory. 1970;2(3):244-263. Page 250, which defines the equally distributed equivalent income and the measure I as one minus its ratio to mean income, lying between 0 and 1, with the reading of a value of 0.3.

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• Atkinson index and EDE health in distributional cost-effectiveness analysis

  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 defines EDE health, interprets the gap between mean and EDE health, and reports Atkinson indices for QALE distributions in Table 8.

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