Functions & Formulae

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

Equity-weighted QALY gain, ICER, net health benefit and switching weight function

e(w_g, Delta_E_g, Delta_C, lambda) = (Delta_E_w, ICER_w, NHB_w, w_star)

Maps the incremental QALY gain of a programme in each population group, an equity weight for each group, the incremental cost and the value placed on a QALY to the equity-weighted QALY gain, the equity-weighted ICER, the equity-weighted net health benefit and the weight at which the programme would break even. The weights act on the incremental QALYs only, while costs and the health displaced elsewhere stay unweighted, and with every weight equal to 1 the outputs reduce to the unweighted ICER (HE-FM-ICER-001) and incremental net health benefit (HE-FM-CET-002). The weights come from outside the function, for example the NICE severity weight from absolute and proportional shortfall (HE-FM-ASF-001 and HE-FM-ASF-003) or the weight implied by an Atkinson inequality aversion parameter (HE-FM-ATK-004), and neither is redrafted here. The break-even weight differs from the switching weight between two options in the weighted two-group social welfare formula of the Welfare Economics package, because it compares one programme with the health displaced by its cost. Notation follows the Equity-Weighted Analysis article.

  • Equity-weighted QALY gain summed over population groups

    V_1 = w_1 * Delta_E_1; V_2 = w_2 * Delta_E_2; Delta_E_w = V_1 + V_2

    Multiplies the incremental QALY gain in each group by that group's equity weight to give the group's equity-weighted value, and adds these values over the groups that the programme affects. The calculator form is written for two groups, V_1 and V_2, and each further group adds one more term w_g times Delta_E_g to the sum. A weight above 1 counts each QALY gained in a priority group as more than one QALY, and a weight of 1 leaves a group's gain unchanged. The result replaces the unweighted QALY gain in the ICER and in net health benefit, so the comparison with costs follows the usual rules once the weights are fixed.

  • Equity-weighted ICER for one group and the equivalent threshold

    ICER_w = Delta_C / (w * Delta_E); lambda_w = w * lambda

    Divides the incremental cost by the weighted QALY gain of a group that carries a single weight w. Because the weight is the same for everyone in the group, the weighted ICER equals the unweighted ICER divided by w, and judging it against a threshold lambda gives the same decision as judging the unweighted ICER against w times lambda. The second part of the formula returns that equivalent threshold, the route the Norwegian committee proposed in 2014 through differential threshold ranges by severity.

  • Equity-weighted incremental net health benefit with and without the weight

    H_d = Delta_C / lambda; NHB_w = w * Delta_E - H_d; NHB_u = Delta_E - H_d

    Converts the incremental cost into the health displaced elsewhere, Delta_C divided by lambda, and subtracts it from the weighted and from the unweighted QALY gain. The displaced health is not weighted, so the two results differ by w minus 1, times Delta_E. A negative unweighted value with a positive weighted value marks a programme that lowers total population health and would be recommended only on the strength of the equity judgement.

  • Equity switching weight at which a programme breaks even

    w_star = Delta_C / (lambda * Delta_E)

    Sets the weighted net health benefit to zero and solves for the weight, giving the incremental cost divided by lambda times the QALY gain, which is the unweighted ICER divided by lambda. A weight at or above w_star makes the programme acceptable at lambda, so the value states how much concern for the group is needed to justify a programme that is not cost-effective without weights. A value below 1 means the programme is cost-effective unweighted.