Slope Index of Inequality

Slope Index of Inequality Calculation and Use Across Ranked Socioeconomic Groups

The slope index of inequality (SII) turns a socioeconomic gradient in health into one number in the units of the health measure itself, such as years of life expectancy or percentage points of screening uptake. Every group in the ranking contributes in proportion to its share of the population, so the result reflects the whole distribution of income, education or area deprivation. Health economists meet the index in national monitoring of health inequality and in distributional cost-effectiveness analysis, where it is one of the standard absolute inequality measures. Its relative counterpart comes in two versions that give different numbers from the same fit, and the index is closely tied to the concentration index.

Ranking groups by relative rank

The index starts from a ranking of the population by socioeconomic position, from the most disadvantaged to the most advantaged. Harper and Lynch, in a monograph for the US National Cancer Institute, attribute the index to Preston, Haines and Pamuk. Each group covers a range of the cumulative population distribution and is scored at the midpoint of that range, which is its relative rank.

$$R_j = \sum_{k=1}^{j-1} p_k + \frac{p_j}{2}$$

where $R_j$ is the relative rank of group $j$, $p_k$ is the population share of group $k$, and groups are numbered $j = 1, \dots, J$ in that order.

Mean health in each group is then regressed on its relative rank. The WHO Handbook on health inequality monitoring describes the same steps, calls for an appropriate regression model and compares the predicted values at rank 1 and rank 0.

$$\bar{y}_j = \beta_0 + \beta_1 R_j, \qquad SII = \hat{y}(R = 1) - \hat{y}(R = 0) = \beta_1$$

where $\bar{y}_j$ is mean health in group $j$, $\beta_0$ is the predicted health of a hypothetical person at the very bottom of the ranking, $\beta_1$ is the slope, $\hat{y}(R)$ is the predicted value at rank $R$, and $SII$ is the slope index of inequality. Because rank spans the interval 0 to 1, a one-unit change in rank covers the whole distribution, so the slope itself is the index.

Harper and Lynch state that grouped data are fitted by weighted least squares with weights equal to group population size, while individual-level data, with each person's own relative rank, are self-weighting under ordinary linear regression. In the weighted linear case the slope has a covariance form.

$$\beta_1 = \frac{\sum_{j} p_j (R_j - \bar{R})(\bar{y}_j - \mu)}{\sum_{j} p_j (R_j - \bar{R})^2} = \frac{\text{cov}(y, R)}{\text{var}(R)}$$

where $\bar{R}$ is the population-weighted mean rank, equal to 0.5, $\mu$ is mean health in the whole population, and $\text{cov}(y, R)$ and $\text{var}(R)$ are the population-weighted covariance and variance.

Sign, units and why group sizes matter

The sign depends on the direction of the ranking and on whether the indicator measures good or ill health. The WHO handbook places the most disadvantaged at rank 0 and the most advantaged at rank 1, so a positive value means the indicator is more prevalent among the advantaged and a negative value means it is more prevalent among the disadvantaged, with a flat line giving zero. The handbook notes that the value has the same unit as the health indicator, which makes it easy to explain to non-technical audiences as the difference between the worst-off and best-off individual.

Weighting by group size is the main reason to prefer the index to a simple gap between the extreme groups, since Harper and Lynch note that it lets the index incorporate changes in the distribution of social groups over time. The WHO handbook illustrates this with education-based inequality in modern contraceptive use in the Philippines. Between 1993 and 2008 the difference between the group with secondary or higher education and the group with none rose from 20.8 to 27.1 percentage points, while the slope index fell slightly from 15.7 to 14.3 points and the concentration index fell from 0.08 to 0.04, because the complex measures accounted for the shift of the population between education groups.

Relative index of inequality in its two versions

An absolute index gives the size of a gap in natural units, while a relative index gives it in proportion to the level of health. Harper and Lynch set out two relative versions computed from the same regression.

$$RII = \frac{SII}{\mu}, \qquad RII_{KM} = \frac{\beta_0}{\beta_0 + \beta_1}$$

where $RII$ is the relative index of inequality defined as the slope divided by mean population health, and $RII_{KM}$ is the Kunst and Mackenbach version, the ratio of predicted health at the bottom of the ranking to predicted health at the top. The WHO handbook also uses a ratio of the two predicted values.

The monograph's smoking example shows how far apart the two versions can be. With US adults ranked by income-to-poverty ratio, the fitted line had an intercept of 33.7 and a slope of minus 18.1, and the average smoking rate was 24.6%, so the SII was minus 18.1 percentage points. Dividing by the mean gives minus 0.74, read as a 74% decline in smoking prevalence across the whole income distribution. The Kunst and Mackenbach version divides 33.7 by the predicted rate at the top, 15.6 (33.7 minus 18.1), giving 2.16, so smoking is about 2.2 times as common at the bottom as at the top. A report of a relative index therefore needs to name the version used.

Link to the concentration index

The slope index and the concentration index start from the same ranked population and differ mainly in scale. Wagstaff, Paci and van Doorslaer, in a 1991 appraisal of methods for measuring inequalities in health, concluded that only these two were likely to give an accurate picture of socioeconomic inequalities in health. Harper and Lynch give the exact relationship.

$$C = 2\,\text{var}(R)\, RII, \qquad AC = \mu C = 2\,\text{var}(R)\, SII$$

where $C$ is the relative concentration index, $AC$ is the absolute concentration index, $\text{var}(R)$ is the variance of relative rank, and $RII$ is the slope divided by mean health. Harper and Lynch conclude that the two families give the same rank ordering of health distributions and differ in scale, and they count the associated concentration curve as an extra advantage of the concentration index. The scale factor, the variance of rank, moves a little when group shares change, as the worked example below shows. With individual-level ranks it is close to 1/12, which makes the concentration index about one sixth of the relative index.

Worked example with screening uptake across four income groups

All figures in this example are illustrative. A health authority records uptake of a cancer screening programme in four income groups, with group 1 the lowest income group and group 4 the highest. In year 1 the groups hold 20%, 30%, 30% and 20% of the eligible population, with uptake of 50%, 58%, 64% and 70%.

1. Relative ranks. The cumulative shares are 0.20, 0.50, 0.80 and 1.00, so the midpoints are 0.10, 0.35, 0.65 and 0.90 (for example, 0.20 plus half of 0.30 gives 0.35).

2. Mean uptake. The weighted products 0.2 × 50, 0.3 × 58, 0.3 × 64 and 0.2 × 70 sum to 10 + 17.4 + 19.2 + 14 = 60.6%.

3. Covariance and variance. The rank deviations from 0.5 are minus 0.40, minus 0.15, 0.15 and 0.40, and the uptake deviations from 60.6 are minus 10.6, minus 2.6, 3.4 and 9.4.

$$\text{cov}(y, R) = 0.2(-0.40)(-10.6) + 0.3(-0.15)(-2.6) + 0.3(0.15)(3.4) + 0.2(0.40)(9.4) = 1.870$$

where each term is a population share multiplied by a rank deviation and an uptake deviation, and the four terms are 0.848, 0.117, 0.153 and 0.752.

$$\text{var}(R) = 0.2(0.16) + 0.3(0.0225) + 0.3(0.0225) + 0.2(0.16) = 0.0775$$

where each term is a population share multiplied by a squared rank deviation.

4. Slope index. The slope is 1.870 / 0.0775 = 24.129 percentage points. The intercept is mean uptake less half the slope, 60.6 minus 12.0645, or 48.5355%, so predicted uptake is about 48.54% at rank 0 and 48.5355 + 24.129 = 72.6645%, about 72.66%, at rank 1. The simple gap between groups 4 and 1 is 20 points (70% against 50%). The index is larger because it compares the two ends of the ranking, not two group midpoints at 0.10 and 0.90.

5. Relative indices and concentration index. Dividing by the mean gives 24.129 / 60.6 = 0.398, and the ratio of predicted values at the top and bottom is 72.6645 / 48.5355 = 1.50. The covariance form of the concentration index gives 2 × 1.870 / 60.6 = 0.0617, and the identity above gives the same value, 2 × 0.0775 × 0.398 = 0.0617.

6. A population shift with no change in group uptake. In year 2 the lowest income group shrinks, and the shares become 10%, 30%, 35% and 25% while uptake in each group stays the same. The midpoint ranks become 0.05, 0.25, 0.575 and 0.875, mean uptake rises to 5 + 17.4 + 22.4 + 17.5 = 62.3%, the covariance is 1.6425 and the variance of rank is 0.076125. The slope index falls to 1.6425 / 0.076125 = 21.58 points, while the simple gap stays at 20.

7. Interpretation. In year 1, a move up the whole income ranking is associated with about 24 points higher uptake. The fall in year 2 comes entirely from the changed composition of the population, since no group's uptake moved, and the differences between groups 2, 3 and 4 are now spread over longer stretches of the rank scale. The simple gap cannot register the shift, but the example also shows that the index can change without any change in outcomes within groups, so group shares should be reported alongside it.

The index in UK monitoring and distributional cost-effectiveness analysis

The Office for National Statistics uses the slope index of inequality to summarise inequality in life expectancy and healthy life expectancy by area deprivation. Its bulletin released on 15 April 2026 groups small areas in England into deciles of the Index of Multiple Deprivation (IMD 2025 for 2022 to 2024) and explains that the index takes every group into account and weights each by its population size. For 2022 to 2024 the ONS estimated the slope index for life expectancy at birth in England at 10.4 years for males and 8.0 years for females, and for healthy life expectancy at birth at 19.3 and 20.1 years. The bulletin warns that the analyses for England and Wales use different deprivation indices and are not comparable.

In economic evaluation the index describes how an option changes the distribution of health. In the distributional cost-effectiveness tutorial by Asaria, Griffin and Cookson, the slope index appears among the absolute (translation invariant) measures calculated for the quality-adjusted life expectancy distributions of four strategies for the NHS Bowel Cancer Screening Programme, one of them no screening. It was lowest with no screening, at 12.887 years, and rose slightly under each screening strategy, to 12.941 years with standard screening and 12.944 years with a universal reminder. The authors suggest calculating a range of inequality measures when there is no clear choice between them.

Assumptions and choices that move the index

The index needs groups with a natural order. The WHO handbook states that the slope index and the concentration index are not appropriate for non-ordered subgroups, where measures such as the absolute mean difference or the Theil index are used instead.

A straight line through ranked groups also assumes a roughly linear gradient. Harper and Lynch caution that when middle groups carry a disproportionate share of ill health, gradient measures may show no disparity where one could be argued to exist, so the group data should be inspected first. Sergeant and Firth note that the standard estimator of the relative index applies only to linear variation in rates and propose a spline-based approach with a parametric bootstrap for confidence intervals, which also handles a standardising covariate such as age.

For binary outcomes the model matters. A weighted linear fit reads directly in percentage points, whereas a model such as logistic regression gives predicted values at the two extremes that are then subtracted, and the two need not agree. The WHO handbook reports standard errors alongside the slope index in its benchmarking tables, and the ONS bulletin describes some recent changes as uncertain and refers readers to the confidence intervals in its dataset.

Common misreadings of the slope index

A common error is to read the index as an observed difference between two real groups. It is a predicted difference between hypothetical individuals at ranks 0 and 1. A fall in the index is not by itself evidence that disadvantaged groups have gained, since it can follow from a population shift between groups or from worsening health among the advantaged, so it belongs alongside group-level results and a relative measure. Nor does the index say whether a gradient is unfair, which is a question for health equity.

Sources

• Asaria M, Griffin S, Cookson R. Distributional cost-effectiveness analysis: a tutorial. Medical Decision Making. 2016;36(1):8-19. https://doi.org/10.1177/0272989X15583266
• Harper S, Lynch J. Methods for Measuring Cancer Disparities: Using Data Relevant to Healthy People 2010 Cancer-Related Objectives. NCI Cancer Surveillance Monograph Series, Number 6. NIH Publication No. 05-5777. Bethesda, MD: National Cancer Institute; 2005. https://drum.lib.umd.edu/items/4f9ca2ad-c0fa-4aa2-a4ff-acc1d27b5502
• Office for National Statistics. Healthy life expectancy by national area deprivation, England and Wales: between 2013 to 2015 and 2022 to 2024. Statistical bulletin, released 15 April 2026. https://www.ons.gov.uk/peoplepopulationandcommunity/healthandsocialcare/healthinequalities/bulletins/healthylifeexpectancybynationalareadeprivationenglandandwales/between2013to2015and2022to2024
• Sergeant JC, Firth D. Relative index of inequality: definition, estimation, and inference. Biostatistics. 2006;7(2):213-224. https://doi.org/10.1093/biostatistics/kxj002
• Wagstaff A, Paci P, van Doorslaer E. On the measurement of inequalities in health. Social Science and Medicine. 1991;33(5):545-557. https://doi.org/10.1016/0277-9536(91)90212-U
• World Health Organization. Handbook on health inequality monitoring: with a special focus on low- and middle-income countries. Geneva: World Health Organization; 2013. https://iris.who.int/handle/10665/85345