Atkinson Index

Atkinson Index in Distributional Cost-Effectiveness Analysis

The Atkinson index turns a stated degree of concern about inequality into one number, the share of average health, or income, that could be given up, if what remained were shared equally, without making society worse off. In health economics it is used to judge how programmes change the spread of health across a population, and it is one of the social welfare indices used in distributional cost-effectiveness analysis. The formula rests on the equally distributed equivalent (EDE) level of health and on the inequality aversion parameter, epsilon ($\varepsilon$), for which values have been elicited from the public in England. An illustrative two-group example follows, and the index can be compared with the Gini coefficient, the concentration index and the Kolm index.

From social welfare to a measure of inequality

Atkinson's 1970 paper argued that every measure of inequality involves judgements about social welfare, so these are better stated openly than left inside a summary statistic. His device was the equally distributed equivalent level of income, the income per head which, if every person received it, would give the same social welfare as the actual distribution. The measure follows directly from it.

$$I = 1 - \frac{y_{EDE}}{\mu}$$

where $I$ is the Atkinson measure of inequality, $y_{EDE}$ is the equally distributed equivalent income and $\mu$ is mean income. The measure lies between 0, complete equality, and 1, complete inequality. Atkinson's own reading was that a value of 0.3 means that, with incomes equally distributed, only 70% of present national income would be needed for the same social welfare.

Health economists apply the same construction to health, and in distributional cost-effectiveness work usually to quality-adjusted life expectancy (QALE), expected lifetime QALYs. EDE health is the level of health which, if everyone had it, would be judged as good as the actual unequal distribution. The gap between mean and EDE health is the health per person that a decision maker with a given inequality aversion would give up for full equality, and because EDE health is in health units it can be compared across programmes as total QALYs are in cost-effectiveness analysis.

The formula and the inequality aversion parameter

Atkinson showed that requiring the measure to be unaffected by scaling every value up or down in the same proportion, a property he called constant relative inequality aversion, restricts the social welfare function to one family. Under that property, a population whose health is a scaled-up copy of another's has the same inequality. For $n$ people the family gives the following.

$$h_{EDE} = \left[ \frac{1}{n} \sum_{i=1}^{n} h_i^{\,1-\varepsilon} \right]^{\frac{1}{1-\varepsilon}} \quad \text{for} \ \varepsilon \neq 1, \qquad h_{EDE} = \left( \prod_{i=1}^{n} h_i \right)^{\frac{1}{n}} \quad \text{for} \ \varepsilon = 1$$

where $h_{EDE}$ is the equally distributed equivalent level of health, $h_i$ is the health of person $i$ (for example, their QALE), $n$ is the number of people and $\varepsilon$ is the inequality aversion parameter, which is zero or positive. The index is then defined as follows.

$$A_{\varepsilon} = 1 - \frac{h_{EDE}}{\bar{h}}$$

where $A_{\varepsilon}$ is the Atkinson index at aversion level $\varepsilon$ and $\bar{h}$ is mean health. At $\varepsilon = 0$ the EDE equals the mean, so only total health counts. As $\varepsilon$ rises, more weight falls on gains at the lower end of the distribution, and as $\varepsilon$ tends to infinity the EDE becomes the health of the worst-off person, the maximin rule. At $\varepsilon = 1$ the EDE is the geometric mean, and at $\varepsilon = 2$ it is the harmonic mean.

The parameter is easiest to read through the weights it implies. Differentiating the EDE shows that the weight on a small gain to person $i$ is proportional to $h_i^{-\varepsilon}$, so the weight on a gain to a less healthy person relative to a healthier one is as follows.

$$w = \left( \frac{h_{high}}{h_{low}} \right)^{\varepsilon}$$

where $w$ is the relative weight, $h_{high}$ is the health of the healthier person or group and $h_{low}$ that of the less healthy one. With the starting QALE of 62 years for the poorest fifth and 74 years for the richest fifth used in an English elicitation study, an $\varepsilon$ of 10.95 gives $(74/62)^{10.95} \approx 6.94$, and 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.

Values of epsilon elicited in England

The size of $\varepsilon$ is a social value judgement, so it has to come from outside the health data. Robson and colleagues analysed online responses from 244 members of the general public in England. Respondents chose between a programme favouring the richest fifth and one favouring the poorest fifth, and the point at which each switched was converted into an aversion parameter.

The median parameter was 10.95 for the Atkinson form and 0.15 for the Kolm form; the median was used because respondents concerned only with the worst-off imply an infinite parameter. Of the weighted responses, 81.51% were willing to give up some total health to reduce health inequality and 2.91% were strict health maximisers. The authors read the result as weighting health gains to the poorest fifth six to seven times as highly as gains to the richest fifth at current QALE.

These estimates depend on how the question is asked. The same paper notes that an earlier English study by Dolan and Tsuchiya implies an equivalent Atkinson $\varepsilon$ of 28.9, and suggests face-to-face administration, the absence of a reflective video and half-year response categories as possible reasons. The values describe stated public preferences, and the DCEA tutorial notes that estimates of the aversion implicit in actual allocation decisions would be a useful complement.

How distributional cost-effectiveness analysis uses Atkinson measures

Distributional cost-effectiveness analysis (DCEA), set out by Cookson, Griffin, Asaria and colleagues at the University of York, has two stages. The first models the social distribution of health under each option, in the tutorial example QALE by subgroups defined by area deprivation, sex and ethnicity, including who bears the health opportunity cost of the spending. The second evaluates those distributions against two objectives, improving total health and reducing unfair health inequality, and this is where the Atkinson index and EDE enter.

The tutorial by Asaria, Griffin and Cookson illustrates the evaluation stage with options for the NHS Bowel Cancer Screening Programme. It reports relative indices, which do not change when all values are scaled, and absolute indices, which do not change when a constant is added. It then applies dominance rules based on the Lorenz curve, which rank some options with minimal restriction on the welfare function, and uses Atkinson and Kolm welfare functions for the rest. The universal reminder produced 1,000 more population QALYs than the targeted reminder, but the targeted reminder was preferred above $\varepsilon = 8$ (Atkinson) or $\alpha = 0.12$ (Kolm). Robson and colleagues note that their elicited values exceed both thresholds, so on those values the targeted reminder would be chosen.

Index values for QALE are small. For the no-screening distribution the Atkinson index was 0.00171 at $\varepsilon = 1$ and 0.06253 at $\varepsilon = 30$. The authors stress that DCEA can also use other forms of equity weighting.

Worked example with two programmes and two socioeconomic groups

The figures below are illustrative. A population is split into two equal-sized income groups with QALE of 60 and 72 years. Two programmes of equal cost are compared, with gains already net of the health opportunity cost of the spending.

1. No inequality aversion. At $\varepsilon = 0$ the EDE equals the mean, so U (66.30) beats T (66.25) by 0.05 years per person, the ranking a conventional analysis of total QALYs gives.

2. Moderate aversion. At $\varepsilon = 1$ the EDE is the geometric mean. For U it is the square root of 60.1 × 72.5 = 4357.25, which is 66.0095. For T it is the square root of 60.4 × 72.1 = 4354.84, which is 65.9912. U still leads, by about 0.018 years.

3. Higher aversion. At $\varepsilon = 2$ the EDE is the harmonic mean, $h_{EDE} = 2 h_1 h_2 / (h_1 + h_2)$, where $h_1$ and $h_2$ are the QALE of the two groups. For programme U it is as follows.

$$h_{EDE,U} = \frac{2 \times 60.1 \times 72.5}{60.1 + 72.5} = \frac{8714.5}{132.6} = 65.7202$$

where $h_{EDE,U}$ is the equally distributed equivalent QALE under programme U. For programme T it is as follows.

$$h_{EDE,T} = \frac{2 \times 60.4 \times 72.1}{60.4 + 72.1} = \frac{8709.68}{132.5} = 65.7334$$

where $h_{EDE,T}$ is the equally distributed equivalent QALE under programme T. T now has the higher EDE, so the ranking has reversed.

4. The Atkinson index. At $\varepsilon = 2$ the index is $A_U = 1 - 65.7202/66.3 = 0.00874$ for U and $A_T = 1 - 65.7334/66.25 = 0.00780$ for T, against $A_{base} = 1 - 65.4545/66 = 0.00826$ at baseline. U widens relative inequality and T narrows it.

5. Switching value and weights. Solving numerically for equal EDEs gives a switching value of $\varepsilon$ of about 1.6. At $\varepsilon = 2$ a gain to the poorer group carries $(72/60)^{2} = 1.44$ times the weight of a gain to the richer group. At the English median of 10.95, the EDEs computed the same way are 63.51 for U and 63.73 for T.

Choosing T costs 0.05 years of QALE per person, and T is preferred under any Atkinson welfare function with $\varepsilon$ above about 1.6. Reporting the switching value shows a committee how much inequality aversion its choice implies.

Atkinson, Gini, concentration and Kolm measures compared

Inequality measures differ in whether they rank people by a social variable and in the value judgements they build in. Atkinson showed that the Gini coefficient, for typical distributions, attaches more weight to transfers in the centre of the distribution than at the tails, a weighting he doubted would accord with social values. Wagstaff made the parallel point in health, arguing that the Gini coefficient and the concentration index embody one particular set of value judgements, and generalising them allows others to be reflected.

The concentration index measures socioeconomic-related inequality. People are ranked by living standards, and the index is twice the area between the concentration curve and the line of equality, bounded between minus one and one for a non-negative variable. It is unchanged when the health variable is multiplied by a constant but not when a constant is added. The Atkinson index depends only on the distribution of health levels, with no social ranking; it reflects socioeconomic inequality only when, as in DCEA, the distribution is built from the health of socioeconomic subgroups.

The Kolm index is the absolute counterpart of the Atkinson index. Its EDE is

$$h_{EDE}^{Kolm} = -\frac{1}{\alpha} \ln \left( \frac{1}{n} \sum_{i=1}^{n} e^{-\alpha h_i} \right)$$

where $h_{EDE}^{Kolm}$ is the Kolm equally distributed equivalent health, $\alpha$ is the absolute inequality aversion parameter and the other terms are as defined above. Adding the same number of years to everyone leaves the Kolm gap between mean and EDE unchanged, whereas the Atkinson index falls. Because $\alpha$ multiplies health, its value depends on the unit of measurement.

Assumptions and choices that change the result

Several choices made before any calculation can change which option comes out ahead. The first is relative against absolute inequality. An equal proportional gain leaves the Atkinson index unchanged, an equal gain in years leaves Kolm inequality unchanged, and the two can rank programmes differently. The DCEA tutorial treats this as a value judgement and suggests calculating several measures when there is no clear choice.

The formula also needs positive values on a ratio scale. With $\varepsilon$ of 1 or more a single zero sets the EDE to zero, negative values cannot be used, and an index of inequality in ill health is not the mirror image of one in health. DCEA further asks which social variables define unfair inequality, and variation unrelated to them is ignored. How the opportunity cost falls matters too, and the tutorial shared it equally across subgroups in the base case and tested allocating it all to the least healthy or the healthiest. Finally, the welfare function is symmetric and additively separable, which Atkinson himself called restrictive.

Reading Atkinson results in economic evaluation

An index value describes a welfare loss, not a percentage gap between groups. An index of 0.04 means that an equal distribution at 96% of current mean health would be judged as good as the current one, at the chosen $\varepsilon$. Values at different $\varepsilon$ cannot be compared with one another.

The index alone does not choose between options. The option with more inequality can still have the higher EDE health, as programme U does at $\varepsilon = 1$ in the worked example, so decisions rest on the EDE, or an inequality-adjusted net health benefit, not on the index. The median of 10.95 is not an agreed standard; it is one survey estimate of public preferences in one country, and another English study with the same basic questionnaire implied a much higher value. Showing results across a range of $\varepsilon$, with the switching point, is more informative.

Sources

• Asaria M, Griffin S, Cookson R. Distributional cost-effectiveness analysis: a tutorial. Medical Decision Making. 2016;36(1):8-19. doi:10.1177/0272989X15583266
• Asaria M, Griffin S, Cookson R, Whyte S, Tappenden P. Distributional cost-effectiveness analysis of health care programmes: a methodological case study of the UK Bowel Cancer Screening Programme. Health Economics. 2015;24(6):742-754. doi:10.1002/hec.3058
• Atkinson AB. On the measurement of inequality. Journal of Economic Theory. 1970;2(3):244-263. doi:10.1016/0022-0531(70)90039-6
• O'Donnell O, van Doorslaer E, Wagstaff A, Lindelow M. Analyzing Health Equity Using Household Survey Data: A Guide to Techniques and Their Implementation. Washington, DC: World Bank; 2008. Chapter 8, The concentration index.
• 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. doi:10.1002/hec.3430
• Wagstaff A. Inequality aversion, health inequalities and health achievement. Journal of Health Economics. 2002;21(4):627-641. doi:10.1016/s0167-6296(02)00006-1