Functions & Formulae

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

Atkinson inequality index and equally distributed equivalent health function

f(h_i, epsilon) = (h_EDE, A_epsilon)

Maps a distribution of health across people or equal-sized groups, and a chosen inequality aversion parameter epsilon, to the equally distributed equivalent (EDE) level of health and the Atkinson index. The EDE is the level of health which, if everyone had it, would be judged as good as the actual unequal distribution. The index is the share of mean health that could be given up for full equality without lowering social welfare. In distributional cost-effectiveness analysis the EDE ranks programmes and the index describes how far each one leaves health unevenly spread.

  • Atkinson index from equally distributed equivalent health

    A_epsilon = 1 - h_EDE / h_bar

    Compares the equally distributed equivalent health at a chosen epsilon, from HE-FM-ATK-002 or HE-FM-ATK-003, with mean health. The index is one minus their ratio, so it lies between 0, complete equality, and 1. Atkinson's own reading is that a value of 0.3 means an equal distribution would need only 70% of present national income to give the same social welfare; in health, an index of 0.04 means an equal distribution at 96% of current mean health would be judged as good.

  • Atkinson equally distributed equivalent health for epsilon other than 1

    h_EDE = (sum_(i=1)^n [h_i^(1-epsilon)] / n)^(1/(1-epsilon))

    Gives the equally distributed equivalent health as the power mean of the n health values with exponent one minus epsilon. At epsilon of 0 it equals the arithmetic mean, so only total health counts; at epsilon of 2 it is the harmonic mean; and as epsilon rises it moves towards the health of the worst-off person, the maximin rule. Each h_i is the health of one person, or of one of n groups of equal size.

  • Atkinson equally distributed equivalent health at epsilon of 1

    h_EDE = exp(sum_(i=1)^n [log(h_i)] / n)

    Gives the equally distributed equivalent health when epsilon equals 1, where the general formula HE-FM-ATK-002 is undefined. The EDE is then the geometric mean of the n health values, the nth root of their product, written here as the exponential of the mean natural logarithm. The function log is the natural logarithm. It corresponds to Atkinson's logarithmic welfare function and is the limit of the general formula as epsilon tends to 1.

  • Relative weight on a health gain implied by Atkinson inequality aversion

    w = (h_high / h_low)^epsilon

    Gives the weight on a small health gain to a less healthy person or group relative to the same gain to a healthier one, implied by an Atkinson epsilon. It follows from the EDE formula, in which the weight on a small gain to person i is proportional to h_i raised to the power minus epsilon. It turns epsilon into a statement a decision committee can read: how many times a QALY to the less healthy group counts.