Functions & Formulae

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

Health forgone for equity, switching weight and implied inequality aversion function

t(s_g, g_g_max, g_g_fair, h_g) = (H_forgone, w_star, epsilon_star)

Takes two options of equal cost, the population share of a worse-off and a better-off group, each group's net health gain per person under each option after the health displaced elsewhere, and each group's baseline health. It returns the total health forgone by choosing the fairer option (HE-FM-EET-001), the weight on the worse-off group's gain at which the two options tie, and the Atkinson inequality aversion that this weight implies at baseline health (HE-FM-EET-003). The weight is the switching weight w_s of the live record HE-FM-WE-004 (function HE-FN-WE-001), with each group's QALYs read as its population share times its net gain per person, and it is cited there rather than restated. It exceeds 1 by the health forgone divided by the share-weighted extra gain to the worse-off group, the link that HE-VT-EET-003 checks, and it differs from the equity switching weight HE-FM-EQWA-004, which sets one programme against the health displaced by its cost. The implied aversion inverts the live Atkinson relative weight HE-FM-ATK-004, whose test HE-VT-ATK-005 recovers epsilon from a weight. Equally distributed equivalent health itself is live as HE-FM-ATK-002 and HE-FM-ATK-003 (function HE-FN-ATK-001, computational function HE-CF-ATK-001) and is not restated, and the exact switching value of the aversion parameter, found by solving the EDE comparison numerically, belongs to the Distributional Cost-Effectiveness package. Notation follows the Equity-Efficiency Trade-off article.

  • Total health forgone for equity between a health-maximising and a fairer option

    H_max = s_worse * g_worse_max + s_better * g_better_max; H_fair = s_worse * g_worse_fair + s_better * g_better_fair; H_forgone = H_max - H_fair; f_forgone = H_forgone / H_max

    Gives the mean net health gain per person of each option as the population-share weighted sum of the two groups' net gains, and the health forgone for equity as the gain of the health-maximising option less the gain of the fairer option. Asaria and colleagues call this difference in total health the health opportunity cost of choosing the option that reduces inequality, and Cookson and colleagues describe counting it as equity constraint analysis. The share f_forgone expresses the same loss as a fraction of the health-maximising option's net gain. A positive H_forgone means a trade-off exists, and a value of zero or below means the fairer option costs no health.

  • Implied inequality aversion from the switching weight by inverting the Atkinson relative weight

    epsilon_star = log(w_star) / log(h_better / h_worse)

    Inverts the live Atkinson relative weight HE-FM-ATK-004, in which a small gain to the worse-off group counts (h_better / h_worse) to the power epsilon times a gain to the better-off group. Setting that weight equal to the switching weight w_star, the w_s of HE-FM-WE-004 for the two options, and taking natural logarithms gives the approximate inequality aversion at which the two options tie. Cookson and colleagues note that an equity parameter implies a set of weights once combined with the existing distribution of health, and this formula reads the relation backwards. It is an approximation because the weights are taken at baseline health, while the exact switch solves the comparison of equally distributed equivalent health. The function log is the natural logarithm.

Equity-Efficiency Trade-off — Functions & Formulae | HealthEconomics.wiki