Functions & Formulae

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

Bound corrections of the concentration index for binary and bounded health variables

g(C, mu, a, b) = (W, E)

Maps the standard concentration index of a health variable, its mean and the bounds of its measurement scale to an index whose feasible range does not depend on the mean. For a binary variable the standard index can only lie between mu minus 1 and 1 minus mu in large samples, so Wagstaff divides it by 1 minus the mean, while Erreygers multiplies it by 4 times the mean over the range of the scale, which equals 8 times the covariance of health and fractional rank over that range. The standard index itself, C = 2 cov_hr / mu, is HE-FM-HINQ-005, computed from grouped data in HE-CF-HINQ-001; the curve form by trapezoids is HE-FM-VEQ-002, the horizontal inequity index C_M minus C_N is HE-FM-HEQ-003 and the rule of 75 is HE-FM-HEQ-004. Notation follows the Concentration Index article.

  • Wagstaff normalised concentration index for a binary health variable

    C_min = mu - 1; C_max = 1 - mu; W = C / (1 - mu)

    For a variable that takes only the values 0 and 1, such as being ill, immunised or dead, the concentration index is bounded below by the mean minus 1 and above by 1 minus the mean in large samples, so its feasible range shrinks as the mean rises (Wagstaff 2005). Dividing the index by 1 minus the mean restores a range of minus 1 to 1, so indices for binary outcomes with different means can be compared. The sign is unchanged: a negative value still means that the variable is concentrated among the poor.

  • Erreygers corrected concentration index for a bounded health variable

    E = 4 * mu * C / (b - a)

    Multiplies the standard concentration index by 4 times the mean and divides by the range of the measurement scale, b minus a. Because C equals 2 cov_hr / mu, the corrected index also equals 8 times the covariance of health and fractional rank divided by b minus a, a scaled version of the generalised concentration index mu C. Erreygers designed it to satisfy four requirements: transfer, level independence, cardinal invariance and mirror. For a binary variable a is 0 and b is 1.