Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Health inequality measures across ordered population groups

M(h_j, f_j) = (AG, RG, SII, RII, C)

Maps the mean health of population groups, ordered from the most to the least disadvantaged, and the share of the population in each group to summary measures of socioeconomic-related health inequality: the absolute and relative gaps between the extreme groups, the slope index of inequality, the relative index of inequality in two forms and the concentration index. The measures describe how health varies with socioeconomic position and do not say whether that variation is unfair, which is the separate judgement of health inequity. The Atkinson index, which ignores the ranking of groups, has its own records on the Atkinson index page (HE-FN-ATK-001) and is not repeated here.

  • Absolute and relative health gap between the extreme groups

    AG = h_J - h_1; RG = h_J / h_1

    Compares mean health in the most advantaged group J with mean health in the most disadvantaged group 1, as a difference in the units of the health variable and as a ratio without units. The WHO handbook calls these the difference and the ratio; Harper and colleagues call them the rate difference and the rate ratio.

  • Slope index of inequality for grouped health data

    SII = cov_hr / var_r

    Gives the slope of the line fitted by population-weighted least squares to group mean health against the midpoint rank of each group, with the population ranked from the most disadvantaged (rank 0) to the most advantaged (rank 1). The slope equals predicted health at rank 1 minus predicted health at rank 0, so the index is in the units of the health variable. From group values, cov_hr is the sum over groups of f_j times (h_j minus mu) times (r_j minus 0.5), and var_r is the sum of f_j times (r_j minus 0.5) squared, because the population-weighted mean rank is 0.5. The computational function HE-CF-HINQ-001 carries out the full procedure from group means and population shares.

  • Relative index of inequality as a ratio of predicted health

    alpha = mu - SII / 2; RII_ratio = (alpha + SII) / alpha

    Divides predicted health at rank 1 by predicted health at rank 0 from the same fitted line as the slope index. The line passes through mean health at the mean rank of 0.5, so the intercept alpha, the predicted value at rank 0, is mean health minus half the slope index, and the predicted value at rank 1 is alpha plus the slope index. A value above 1 shows higher health in the advantaged groups.

  • Relative index of inequality as the slope index over mean health

    RII_mean = SII / mu

    Divides the slope index of inequality by population-weighted mean health. The relative index values reported in the distributional cost-effectiveness tutorial by Asaria and colleagues equal their slope index divided by mean QALE, so this form appears in that literature alongside the ratio form of the WHO handbook. Both forms rise with relative inequality but take different values, so the form used is stated with the result.

  • Concentration index of health for grouped data

    C = 2 * cov_hr / mu

    Gives the concentration index from the covariance between health and fractional rank in the socioeconomic distribution. For grouped data the fractional rank of each group is the cumulative population share up to the midpoint of the group, and the mean and covariance are weighted by population share, so cov_hr is the same covariance as in the slope index. The index is positive when health is concentrated among the better-off and negative when it is concentrated among the worse-off.