Signature
r = (y / 20)^(1 / (s - 2)) - 1
| Inputs | Definition | Unit |
|---|---|---|
y | Largest number of days of ill health in year s that still makes the delaying treatment worthwhile | days, above zero |
s | Year in which the delayed illness would fall, counted from now, so the delay is s minus 2 years | years, above 2 |
r | Constant annual discount rate for health implied by the answer, negative when the respondent would rather be ill sooner | a proportion per year |
|---|
Function
Social time preference rate, stepped long-term discount factors and implied individual rates for health
Maps the components of social time preference to the HM Treasury social time preference rate and its health discount rate, maps a stepped long-term schedule of rates to the discount factor for any year of an appraisal, and maps one answer to a stated choice between ill health at two dates to the constant annual rate it implies. The rates it returns feed the single-amount factor of the Discount Rate page (HE-FM-DR-001) and the stream totals of the Discounting page (HE-FN-DISC-001, HE-FM-DISC-001 to HE-FM-DISC-004 and HE-CF-DISC-001). A health rate derived from growth in the consumption value of health is HE-FM-DDIS-002 under HE-FN-DDIS-001, and the quasi-hyperbolic weight D(k) = beta lambda^k of the article is HE-FM-HYPD-004 under HE-FN-HYPD-001, so neither is repeated here. Notation follows the Time Preference article.
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Implementations
Excel
Implied annual rate from a TEMPUS open-ended answer in named cells
With the days accepted in DaysLater and the year of the delayed illness in YearLater, the first formula returns the implied rate; name its cell ImpliedRateOut. The second uses RRI, available from Excel 2013, which returns the same rate as the growth rate that takes 20 to DaysLater over YearLater minus 2 periods.
=(DaysLater/20)^(1/(YearLater-2))-1; =RRI(YearLater-2,20,DaysLater)
Assumptions
Constant annual rate over the delay between the two illnesses
The equation applies the same factor b to every year between year 2 and year s, so r is the one constant rate that fits a single answer. Answers at several delays give one rate each, and TEMPUS reports that stationarity was violated and that hyperbolic models fitted its data better than the discounted utility model.
Days of ill health weighed by their number in the TEMPUS equation
The equation sets 20 days in year 2 against y days in year s by count alone, so y days of the illness are worth y times one day of it and the illness is the same at both dates. An answer that reflects a change in severity or circumstances over time is read as time preference.
Individual implied rate used for description and not as the social rate
The rate describes one person's trade-off. TEMPUS warns that its implied rates should not be over-emphasised because the sample was unrepresentative, Attema and colleagues report average rates as high as 45% across studies, and the reference-case rate is set by guidance (PMG36 section 4.5.1).
Worked examples
TEMPUS-style answer of 24 days in year 4
The article's illustrative respondent accepts up to 24 days in year 4, a delay of 2 years, which implies about 9.54% a year.
y = 24; s = 4; r = 0.0954
TEMPUS-style answer of 30 days in year 9
Accepting up to 30 days in year 9, a delay of 7 years, implies about 5.96% a year, as in the article's table (illustrative answer).
y = 30; s = 9; r = 0.0596
TEMPUS-style answer of 36 days in year 13
Accepting up to 36 days in year 13, a delay of 11 years, implies about 5.49% a year. Across the three answers the rate falls with delay, the pattern the article reads as a violation of stationarity (illustrative answers).
y = 36; s = 13; r = 0.0549
TEMPUS-style answer of exactly 20 days gives zero time preference
A respondent who accepts no more than the original 20 days at any later date is indifferent to timing, so the implied rate is zero whatever the delay, a limiting case that checks an implementation.
y = 20; s = 7; r = 0
TEMPUS-style answer below 20 days gives negative time preference
Accepting at most 18 days in year 4 means the respondent would rather have the illness sooner, which implies a rate of about minus 5.13% a year (illustrative answer, computed here for illustration). Van der Pol and Cairns report that a substantial number of respondents had negative or zero rates.
y = 18; s = 4; r = -0.0513
Common errors
TEMPUS delay counted from now in place of from year 2
Using s in the exponent instead of s minus 2 spreads the trade-off over years that are not part of the delay. For 24 days in year 4 the rate becomes 4.66% instead of 9.54% (computed here for illustration), about half the true value.
TEMPUS ratio inverted to 20 over y
Writing (20/y) in place of (y/20) turns positive time preference into negative. For 24 days in year 4 the rate becomes minus 8.71% instead of plus 9.54% (computed here for illustration), so the respondent would be misclassified as preferring ill health sooner.
Individual implied rate used as the social discount rate
Applying the respondent's long-delay rate of 5.49% would leave a weight of 0.343 on a QALY in year 20 against 0.503 at the 3.5% of the NICE reference case, as in step 3 of the article's worked example, so prevention with distant gains would be valued about a third lower.
Sources
TEMPUS equation for the implied discount rate from an open-ended answer
Cairns JA, van der Pol MM. The estimation of marginal time preference in a UK-wide sample (TEMPUS) project. Health Technology Assessment. 2000;4(1) i-iv, 1-83. Chapter 5, open-ended method, section on estimating implied discount rates, page 20, equation (1), r = (y/20)^(1/(s minus 2)) minus 1 with b = 1/(1 + r), for y days of ill health s years ahead judged equivalent to 20 days 2 years ahead; the same chapter classifies responses into positive, negative and zero rates. The executive summary reports that stationarity was violated, that hyperbolic models fitted better and that the rates should not be over-emphasised because the sample was unrepresentative.
Negative and zero time preference for health in stated choices
van der Pol MM, Cairns JA. Negative and zero time preference for health. Health Economics. 2000;9(2) 171-175. Abstract, which reports that the majority of respondents have positive discount rates but a substantial number have negative or zero rates.
Range of elicited time preference rates for health across studies
Attema AE, Brouwer WBF, Claxton K. Discounting in economic evaluations. PharmacoEconomics. 2018;36(7) 745-758. Section on the height of discount rates, which reports that studies measuring health-related time preferences found average discount rates up to 45%, with much heterogeneity across methods and samples.
Canonical Identity
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