Functions & Formulae

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

Health inequity measurement function under stated fairness judgements

F(r_gk, w_k, Delta_j, u_j, Delta_res, u_res) = (R_g, I_DU, I_FG, s)

Maps the age-specific rates of ill health in social groups, a standard population, and a decomposition of the standardised gap between groups into parts attributed to named causes, to estimates of health inequity: the part of a measured difference judged avoidable and unfair. Differences due to factors classed as legitimate, such as age, are first removed by direct standardisation. The inequity is then the part of the remaining gap whose causes are classed as illegitimate, with the unexplained remainder treated as acceptable under direct unfairness or as unfair under the fairness gap. The size of a gap is measured with the gaps, slope and relative indices and concentration index on the Health Inequality page (HE-FN-HINQ-001), and indirect standardisation of health care use for need is on the Horizontal Equity page (HE-FM-HEQ-001); neither is repeated here.

  • Directly age-standardised rate for comparing social groups

    R_g = sum_(k=1)^K [w_k * r_gk]

    Weights the age-specific rates of one social group by the age structure of a common standard population, so that groups with different age structures are compared as if they had the same one. In a health inequity analysis this removes the part of a difference that is due to age, which Whitehead places partly under natural, biological variation and so treats as a legitimate source of difference. The weights sum to 1. With a group's own age shares as the weights, the same sum returns the group's crude rate.

  • Direct unfairness measure of health inequity

    I_DU = sum_(j=1)^J [u_j * Delta_j]

    Fleurbaey and Schokkaert define direct unfairness through the variation that remains when every legitimate source of difference is held at a common reference value, written h_i^DU = f(c_i, l_bar) in the article's simplified notation. Applied to a gap between two groups that has been attributed additively to causes, holding the legitimate causes constant leaves only the parts attributed to causes judged illegitimate, so the inequity is the sum of those parts. The unexplained remainder is dropped, which treats it as acceptable. This group-gap formula is this page's simplification: Fleurbaey and Schokkaert define a hypothetical distribution across individuals and then apply an inequality index to it.

  • Fairness gap measure of health inequity

    I_FG = Delta - sum_(j=1)^J [(1 - u_j) * Delta_j]

    Fleurbaey and Schokkaert define the fairness gap as the difference between the actual distribution and a hypothetical one in which every illegitimate source of difference is set to a common reference value, written h_i^FG = h_i minus f(c_bar, l_i) in the article's simplified notation. Applied to a gap between two groups that has been attributed additively to causes, removing the illegitimate causes leaves only the legitimate parts in the hypothetical gap, so the inequity is the whole gap minus those legitimate parts. The unexplained remainder stays in the result, which treats it as unfair. This group-gap formula is this page's simplification: Fleurbaey and Schokkaert define the fairness gap distribution across individuals and then apply an inequality index to it.

  • Inequity share of a decomposed health gap under stated fairness judgements

    Delta = sum_(j=1)^J [Delta_j] + Delta_res; s = (sum_(j=1)^J [u_j * Delta_j] + u_res * Delta_res) / Delta

    Expresses the inequity in a decomposed gap as a share of the whole gap. The gap is first rebuilt from its explained parts and the unexplained remainder, and the inequity counts each explained part judged unfair and the remainder when it is judged unfair. An indicator of 0 for the remainder gives the direct unfairness result of HE-FM-INEQ-002 and an indicator of 1 the fairness gap result of HE-FM-INEQ-003. Asada and colleagues report results in this form for survey data, with about 60% of health inequality inequitable when unexplained variation was treated as acceptable and almost all of it when it was treated as unfair.