Functions & Formulae

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

Utility decrements, their QALY losses and the rules for combining them on a stated baseline

d = U_0 - U_1; Q_loss = d * dur

Expresses a harm such as an adverse event, a second chronic condition or an unpleasant treatment as a fall in utility from a stated baseline, turns it into lost QALYs over the harm's duration, and estimates the utility of people with two conditions from single-condition data by the additive, multiplicative or minimum rule of NICE DSU TSD 12. QALYs from periods of constant utility are HE-FM-QALY-001 and the expected QALY loss from adverse events HE-FM-AER-006. Notation follows the Disutility article.

  • Utility of two conditions combined by the additive, multiplicative and minimum rules on a shared baseline

    U_add = U_0 - (U_0 - U_A) - (U_0 - U_B); U_mult = U_0 * (U_A / U_0) * (U_B / U_0); U_min = min(U_0, U_A, U_B)

    When utilities exist only for people with one condition each, TSD 12 sets out three conventional estimates for people with both: the additive rule subtracts both absolute decrements, the multiplicative rule applies both proportional effects, and the minimum rule assumes no further loss beyond the worse single condition. With a shared baseline U_0 the multiplicative value equals U_0 minus d_A minus d_B plus d_A d_B / U_0, so with two positive decrements it is higher than the additive value by d_A d_B / U_0. TSD 12 advises the multiplicative rule when combined data are unavailable.

  • QALY gain and cost per QALY from avoiding a second condition under a chosen combination rule

    G = (U_A - U_AB) * dur; CPQ = C_case / G

    A person who avoids condition B stays at the utility of condition A alone instead of the combined utility, so the QALY gain per case avoided is the difference times the years B would have lasted, and the cost per QALY gained divides the cost per case avoided by that gain. The combined utility comes from HE-FM-DISU-002 under the rule chosen.

  • Ara and Brazier general population EQ-5D baseline by age and sex with a proportional decrement

    U_gp = 0.9508566 + 0.0212126 * male - 0.0002587 * age - 0.0000332 * age^2; Loss = (1 - m) * U_gp

    Ara and Brazier's regression of EQ-5D index scores from the 2003 and 2006 Health Survey for England gives mean general population utility as a quadratic in age with a shift for men. Applying a proportional decrement m to this baseline, as the multiplicative rule does, makes the absolute loss shrink as the baseline falls with age, whereas a fixed additive decrement costs the same at every age. The equation is shown for illustration; PMG36 asks for a recent source, and the norms used should match the instrument and value set of the other utilities.