Distributional cost-effectiveness comparison of options by social group function
d(h0_g, dQ_g, o_g, H_d, N_g, epsilon) = (h1_g, d_EDE, d_A, quadrant, epsilon_star)
Maps each social group's baseline health, its QALY gain from a programme, its share of the health opportunity cost and its size, together with a chosen inequality aversion, to the group's health under the programme, the change in equally distributed equivalent (EDE) health and in the Atkinson index against the comparator, the programme's quadrant on the equity-efficiency impact plane, and the inequality aversion at which two options give equal EDE health. The health opportunity cost of the incremental cost is the live record HE-FM-CET-002, and the EDE and Atkinson index of each distribution are the live records HE-FM-ATK-002, HE-FM-ATK-003, HE-CF-ATK-001 and HE-FM-ATK-001, which are cited here and not redrafted. Other inequality indices for grouped data, such as the absolute gap and the concentration index, are in HE-FN-HINQ-001, and direct equity weights on QALY gains are in HE-FN-EQWA-001. The two-group closed approximation to the switching aversion belongs to the equity-efficiency trade-off package, while the computational function here solves the EDE comparison itself for any number of groups. Notation follows the Distributional Cost-Effectiveness article.
Group health after a programme net of the group's share of the health opportunity cost
NHB_1 = dQ_1 - o_1 * H_d; NHB_2 = dQ_2 - o_2 * H_d; h1_1 = h0_1 + NHB_1 / N_1; h1_2 = h0_2 + NHB_2 / N_2; NHB_tot = NHB_1 + NHB_2
Subtracts each group's share of the health opportunity cost from the QALYs the group gains, giving the group's net health benefit, and divides that by the number of people in the group to move the group's baseline health per person to its health under the programme. The health opportunity cost H_d is the incremental cost divided by the value of a QALY, the live record HE-FM-CET-002, and the shares o_1 and o_2 say who bears it. Because the shares add to 1, the total of the group net health benefits equals the total QALY gain less H_d, so the distribution of the opportunity cost changes who gains while total health stays the same. The line shows two groups, the more deprived group 1 and the less deprived group 2, and each further group adds a pair of terms of the same form.
Change in mean health, EDE health and Atkinson index of a programme against its comparator
d_mu = mu_1 - mu_0; d_EDE = EDE_1 - EDE_0; d_P = d_mu - d_EDE; dEDE_pop = N_pop * d_EDE; d_A = (1 - EDE_1 / mu_1) - (1 - EDE_0 / mu_0)
Takes the mean health and the equally distributed equivalent health of the programme's distribution and of the comparator's, both at the same inequality aversion epsilon from the live records HE-FM-ATK-002 or HE-FM-ATK-003, or HE-CF-ATK-001 for groups of unequal size, and returns the change in mean health, the change in EDE health per person and across the population, the change in the inequality cost and the change in the Atkinson index. The inequality cost is mean health less EDE health, the health per person that a decision maker with that aversion would give up for full equality, so the change in EDE equals the change in mean health less the change in the inequality cost. A positive change in EDE health means the programme is preferred to the comparator at that epsilon. Each Atkinson index is HE-FM-ATK-001, and only the difference between the two is new here.
Equity-efficiency impact plane quadrant of a programme against its comparator
WW = (d_H >= 0) and (d_I <= 0) and (d_H > 0 or d_I < 0); LL = (d_H <= 0) and (d_I >= 0) and (d_H < 0 or d_I > 0); WL = (d_H > 0) and (d_I > 0); LW = (d_H < 0) and (d_I < 0)
Places a programme on the equity-efficiency impact plane from the sign of its change in total health, net of the health opportunity cost, and the sign of its change in an inequality index, with the comparator at the origin. A win-win programme raises total health and lowers inequality and a lose-lose programme does the reverse, while win-lose and lose-win programmes need a trade-off analysis such as the change in EDE health of HE-FM-DCEA-002. The sources name the four quadrants without classifying a point on an axis, so the formula applies the tie rule of the live dominance record HE-FM-DOM-001, with less inequality taking the place of lower cost. No change on one axis with an improvement on the other counts as win-win, the mirror case counts as lose-lose, and no change on either axis gives no quadrant.