Functions & Formulae

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

Differential discounting of incremental costs and health effects in a cost-effectiveness ratio

ICER = sum_(t=0)^T [dC_t * (1 + r_C)^(-t)] / sum_(t=0)^T [dQ_t * (1 + r_H)^(-t)]; 1 + r_H = (1 + r_C) / (1 + g_v)

Converts the yearly streams of incremental costs and incremental QALYs to present values at two different annual rates, a cost rate and a usually lower health rate, before dividing one by the other. The package covers the ratio for a programme with costs and QALYs in different years, the exact health rate implied by growth in the money value of health, the effective cost rate implied by a growing cost-effectiveness threshold, and the factor by which delay changes the ratio. Separate discounting of two streams in general is HE-FM-DR-003 on the discount rate page and the two-option ratio HE-FM-ICER-001. Notation follows the Differential Discounting article.

  • Incremental cost per QALY of a programme with two cost instalments and one later QALY gain under separate cost and health rates

    C_d = C_0 + C_a / (1 + r_C)^a; Q_d = Q_b / (1 + r_H)^b; ICER = C_d / Q_d

    Discounts a cost paid at the start (year 0) and a second cost in year a at the cost rate, discounts a QALY gain that arrives in year b at the health rate, and divides the discounted cost by the discounted QALY gain. It is the article's prevention programme, whose costs fall early and whose health gain falls late, so the pair of rates can change its ratio a great deal; a treatment with all its cost in year 0 is the special case with no second instalment.

  • Health discount rate from the cost discount rate and growth in the consumption value of health

    r_H = (1 + r_C) / (1 + g_v) - 1; r_H_sub = r_C - g_v

    When the money value of a unit of health grows at g_v a year, discounting the volume of health at r_H gives the same present value as inflating each year's QALYs at g_v and discounting them at the cost rate, provided one plus r_H equals one plus r_C divided by one plus g_v. Gravelle and Smith derive this exact form and summarise it as the cost rate minus the growth rate, the subtraction rule the Dutch guideline uses.

  • Effective cost discount rate and health forgone when the cost-effectiveness threshold grows

    r_C_star = (1 + r_H) * (1 + g_k) - 1; H_PV = C_t / (k_0 * (1 + g_k)^t * (1 + r_H)^t)

    With a fixed budget, a cost in year t displaces health equal to the cost divided by the threshold in that year, and the forgone health is then discounted at the health rate. If the threshold grows at g_k a year, a pound spent later displaces less health, so costs carry an effective rate r_C_star with one plus r_C_star equal to (one plus r_H) times (one plus g_k), approximately the health rate plus the threshold growth.

  • Change in a programme's discounted cost per QALY when the whole programme is delayed under differential rates

    F_s = ((1 + r_H) / (1 + r_C))^s; ICER_s = ICER_0 * F_s

    Moving every cost and QALY of a programme s years later multiplies its discounted cost by (1 + r_C)^(-s) and its discounted QALYs by (1 + r_H)^(-s), so the ratio is multiplied by ((1 + r_H) / (1 + r_C))^s. The factor is below one whenever the health rate is below the cost rate, so the ratio improves with every year of delay, the postponement paradox of Keeler and Cretin; under equal rates it is one.