Functions & Formulae

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

Cost-effectiveness frontier from the expected costs and QALYs of mutually exclusive strategies

f(C_j, Q_j, lambda) = (F, ICER_F, j*)

Maps the expected total costs C_j and expected total QALYs Q_j of a set of mutually exclusive strategies, estimated for one population, perspective and time horizon, to the strategies on the cost-effectiveness frontier, the slopes of the segments joining them and, at a value λ placed on one QALY, the strategy chosen. It builds on live records and does not restate them. The dominance screen is HE-FM-DOM-001 and HE-FM-DOM-002, the extended dominance condition and the blend of neighbours are HE-FM-EXD-001 and HE-FM-EXD-002, the ratio between adjacent frontier strategies is the sequential ICER HE-FM-CEA-001 and the maximum net monetary benefit rule is HE-FM-NMB-003. This package adds the range of λ over which each frontier strategy is chosen, the net monetary benefit a strategy gives up against the frontier choice, the gradient of the IQWiG efficiency frontier and two computational functions that build the frontier for any number of strategies. Notation follows the Cost-Effectiveness Frontier article.

  • Threshold range in which a frontier strategy has the highest net monetary benefit

    lambda_in = (C_k - C_i) / (Q_k - Q_i); lambda_out = (C_j - C_k) / (Q_j - Q_k); best = (lambda >= lambda_in) and (lambda <= lambda_out)

    For a frontier strategy k whose neighbours on the frontier are i, the next less effective, and j, the next more effective, k has the highest net monetary benefit for every λ at or above lambda_in, the slope of the segment entering it, and at or below lambda_out, the slope of the segment leaving it. Both slopes are sequential ICERs (HE-FM-CEA-001), so the strategy chosen at λ is the frontier strategy with the largest ICER not above λ, the decision rule stated by Suen and Goldhaber-Fiebert, and it is the same strategy that HE-FM-NMB-003 selects. The slopes are the switching values at which two neighbours tie. For a middle strategy that is not on the frontier the range is empty, because lambda_in exceeds lambda_out exactly when D_M of HE-FM-EXD-001 is positive.

  • Net monetary benefit a strategy gives up against the frontier strategy chosen at a threshold

    NMB_s = lambda * Q_s - C_s; NMB_k = lambda * Q_k - C_k; L_k = NMB_s - NMB_k

    Measures how far a strategy falls below the frontier at a value λ in net monetary benefit terms, as the net monetary benefit of the strategy s chosen at λ, the one with the highest value (HE-FM-NMB-003, the same strategy as HE-FM-CEF-001 picks), less that of strategy k, each from HE-FM-NMB-001. It is the incremental net monetary benefit of HE-FM-NMB-002 with s as the intervention and k as the comparator, so it is zero for the chosen strategy and not negative for any other. For a dominated or extendedly dominated strategy it is positive at every positive λ, since such strategies never have the highest net monetary benefit. When Q_k lies within the QALY range of the frontier, its smallest value arises at the slope of the frontier segment beneath Q_k, where it equals the saving of the blend of the two frontier neighbours in HE-FM-EXD-002, the vertical distance of k above the frontier.

  • Gradient of an IQWiG efficiency frontier segment and the net cost at which a new intervention reaches the extended last segment

    g = (B_j - B_i) / (K_j - K_i); lambda_g = 1 / g; K_max = K_j + (B_new - B_j) / g

    In the IQWiG efficiency frontier benefit B is on the vertical axis and total net cost per patient K on the horizontal axis, the reverse of the usual cost-effectiveness plane, so the gradient g of a segment is the incremental benefit per incremental cost and its reciprocal lambda_g is the sequential ICER of HE-FM-CEA-001. General Methods version 6.1 extended the last segment to judge a new intervention with more benefit and more cost than the comparators, and took the reciprocal of its gradient as the threshold for the net health benefit used to derive a price. K_max is the net cost per patient at which a new intervention with benefit B_new lies on that extension, where its net health benefit is zero. Version 8.0 (section 4.15) states that extending the last segment is no longer necessary, so K_max is a version 6.1 calculation.