Signature
U_TTO = T_y / T_x
| Inputs | Definition | Unit |
|---|---|---|
T_y | Years in full health, followed by death, that the respondent judges equal to T_x years in the health state; at least 0 and at most T_x | years |
T_x | Years the respondent would live in the health state, followed by death, in the time trade-off question; above zero | years |
U_TTO | Time trade-off utility of the health state on the scale with full health at 1 and immediate death at 0 | utility (dimensionless) |
|---|
Function
Health state utility from a time trade-off answer
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.
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Implementations
Excel
Time trade-off utility from named cells
With the two durations in cells named TTOStateYears and TTOFullHealthYears, the formula returns U_TTO in a cell named TTOUtility.
=TTOFullHealthYears/TTOStateYears
Assumptions
Linear QALY model behind the time trade-off rule
The rule holds under the linear QALY model, with the utility of life duration linear in years, full health valued at 1 and immediate death at 0, together with expected utility, the conditions under which Lipman, Brouwer and Attema state that time trade-off and standard gamble weights are often derived. Their study corrects the weights for loss aversion, probability weighting and nonlinear utility of life duration, and the corrected weights are not reproduced here.
Chronic state judged at least as good as immediate death
The state lasts T_x years and is followed by death, and the respondent judges it at least as good as immediate death, so T_y is at least 0 and at most T_x. Torrance describes the use of the techniques for both chronic and temporary states and for states considered better and worse than death, and the rule here covers only the chronic state judged at least as good as immediate death.
One respondent's value and the public valuation in the NICE reference case
The rule gives the value of one respondent for one state. PMG36 section 4.3.4 bases the valuation of health-related quality of life on public preferences from a representative sample of the UK population using a choice-based method, so reference-case utilities summarise many respondents' answers, usually through a value set such as HE-FM-EQ3L-002.
Worked examples
Illustrative time trade-off answer for one chronic state
A respondent judges 7.5 years in full health equal to 10 years in a health state, each followed by death. The time trade-off utility of the state is 0.75. The inputs are illustrative.
T_x = 10; T_y = 7.5; U_TTO = 0.75
Limiting case of a state judged as good as full health
A respondent who gives up no time, judging 10 years in full health equal to 10 years in the state, values the state as full health. This limiting case, with illustrative durations of 10 years, checks an implementation.
T_x = 10; T_y = 10; U_TTO = 1
Limiting case of a state judged as bad as immediate death
A respondent who would give up all 10 years in the state, judging it equal to immediate death, values it at 0, the lower end of the range the rule can return. The inputs are illustrative.
T_x = 10; T_y = 0; U_TTO = 0
Common errors
Mixing standard gamble and time trade-off utilities in one comparison
Morimoto and Fukui report a strong tendency for the standard gamble to give the highest values of the three methods they review. In the review of 32 studies by Arnold and colleagues the mean values were 0.81 across the cases valued by standard gamble and 0.77 across those valued by time trade-off, and PMG36 section 4.3.5 states that results from different methods or instruments cannot always be compared. Valuing the same state at 0.81 for one option and 0.77 for its comparator over 5 years creates a difference of 0.20 QALYs that no treatment produced (illustrative, derived).
Recording the share of time given up as the time trade-off utility
The share of life given up, T_x minus T_y over T_x, is one minus U_TTO. In the first worked example it is 0.25, so recording it as the utility values a state the respondent rated at 0.75 at a third of that value (derived).
Applying the time trade-off ratio to a state judged worse than dead
A respondent who prefers immediate death to T_x years in the state has no T_y of zero or more that makes the two sides equal, so T_y over T_x cannot fall below 0 and the ratio stops at 0 (derived). Values below 0, which the live HE-VR-QALY-001 allows for states worse than dead, need a task framed for states worse than death, whose use Torrance describes, and no formula for that case is given here.
Sources
Lipman, Brouwer and Attema on time trade-off weights under the linear QALY model
Lipman SA, Brouwer WBF, Attema AE. QALYs without bias? Nonparametric correction of time trade-off and standard gamble weights based on prospect theory. Health Economics. 2019;28(7):843-854. doi:10.1002/hec.3895 (full text read in PubMed Central, PMC6618285). Section 2.1, SG and TTO correction for PT, which states that TTO and SG weights are often derived under expected utility and the linear QALY model, with L(T) = T, U(FH) = 1 and V(D) = 0, and that under these assumptions the indifferences between (beta_x, T_x) and (FH, T_y), and between (beta_x, T_x) and the prospect giving (FH, T_x) with probability p and D otherwise, give the TTO weight U(beta_x) = T_y / T_x and the SG weight U(beta_x) = p (unnumbered expressions). The abstract names the corrections for loss aversion, probability weighting and nonlinear utility of life duration.
Morimoto and Fukui on the standard gamble and time trade-off tasks
Morimoto T, Fukui T. Utilities measured by rating scale, time trade-off, and standard gamble: review and reference for health care professionals. Journal of Epidemiology. 2002;12(2):160-178. doi:10.2188/jea.12.160 (full text read in PubMed Central, PMC10468341). Introduction, which states that the SG was developed by von Neumann and Morgenstern, that one strategy gives perfect health with probability p and death with probability 1-p and the other a certain intermediate health status, that p is varied until the respondent registers indifference, at which point the utility value is p, and that the TTO assesses how long a period in perfect health is equivalent to a given period of ill health. The abstract reports a strong tendency for the rating scale to yield the lowest and the SG the highest values.
Torrance on standard gamble and time trade-off for chronic, temporary and worse-than-death states
Torrance GW. Measurement of health state utilities for economic appraisal. Journal of Health Economics. 1986;5(1):1-30. doi:10.1016/0167-6296(86)90020-2. Abstract, which states that the rating scale, standard gamble and time trade-off are described in detail and that their use is described for both chronic and temporary health states and for both health states considered better than death and those considered worse than death.
Arnold and colleagues on mean standard gamble and time trade-off utilities across 32 studies
Arnold D, Girling A, Stevens A, Lilford R. Comparison of direct and indirect methods of estimating health state utilities for resource allocation: review and empirical analysis. BMJ. 2009;339:b2688. doi:10.1136/bmj.b2688 (full text read in PubMed Central, PMC2714630). Abstract, Results, which reports 32 studies yielding 83 instances and mean utility values of 0.81 for the standard gamble and 0.77 for the time trade off.
PMG36 on valuing health states relative to perfect health and death and on method differences
National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). Published 31 January 2022, last updated 13 October 2026. Chapter 4, section 4.3.2 (valuing health-related quality of life according to preferences for a health state relative to other states, usually perfect health and death), section 4.3.4 (valuation of public preferences from a representative sample of the UK population using a choice-based method) and section 4.3.5 (different methods used to measure health-related quality of life produce different utility values, and results from different methods or instruments cannot always be compared).
Canonical Identity
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