Functions & Formulae

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

Health state utility from a time trade-off answer

v(T_x, T_y) = U_TTO

Maps one respondent's answer to a time trade-off question, the years T_y in full health judged equal to T_x years in a health state, to the utility of that state on the scale with full health at 1 and immediate death at 0. The standard gamble needs no separate formula, since the expected value of a gamble giving full health, valued at 1, with probability p and immediate death, valued at 0, otherwise is HE-FM-CHN-001 with two branches (the general form is HE-FN-EV-001) and equals p, so the indifference probability is itself the utility. The time trade-off rule HE-FM-UTIL-001 is the equality of the QALYs of the two sides of the question, each valued with HE-FM-QALY-001, solved for the utility. Lipman, Brouwer and Attema print it, and no live record states it. The utilities enter QALYs through HE-FM-QALY-001 and cost per QALY through HE-FM-ICER-001 and HE-FM-CUA-001 (HE-CF-ICUR-001 from arm-level costs and QALYs), and the article's worked example, including its toxicity period valued at minus 0.20, is a re-application of HE-FM-QALY-001, HE-FM-DISU-001 and HE-FM-ICER-001 and is not restated. Utilities of EQ-5D profiles come from value sets, HE-FN-EQ3L-001 and the UK time trade-off tariff HE-FM-EQ3L-002, and utility decrements from HE-FN-DISU-001 and HE-CF-DISU-001. Notation follows the Utility article, with the time trade-off durations of Lipman, Brouwer and Attema.

  • Time trade-off utility of a chronic state under the linear QALY model

    U_TTO = T_y / T_x

    The time trade-off utility is the ratio of the years in full health T_y to the years in the state T_x that the respondent judges equal. The rule follows from the indifference once full health is valued at 1, immediate death at 0 and utility of life duration is linear in years, as Lipman, Brouwer and Attema set out, and it gives one respondent's value for one state judged at least as good as immediate death. The standard gamble asked about the same state gives the indifference probability itself as the utility (HE-FN-UTIL-001), and the two methods tend to give different values (HE-ER-UTIL-001).