Signature
D_1 = max(n_2 + n_3 - 1, 0); I_2 = max(n_2 - 1, 0); I_3 = max(n_3 - 1, 0); U = 1 - (beta_MO + beta_SC + beta_UA + beta_PD + beta_AD) - a_D1 * D_1 - a_I22 * I_2^2 - a_I3 * I_3 - a_I32 * I_3^2
| Inputs | Definition | Unit |
|---|---|---|
n_2 | Count of the five dimensions reported at level 2 | count, whole number 0 to 5 |
n_3 | Count of the five dimensions reported at level 3 | count, whole number 0 to 5 |
beta_MO | D1 value set decrement for the mobility level reported, zero at level 1 | utility, no unit |
beta_SC | D1 value set decrement for the self-care level reported, zero at level 1 | utility, no unit |
beta_UA | D1 value set decrement for the usual activities level reported, zero at level 1 | utility, no unit |
beta_PD | D1 value set decrement for the pain or discomfort level reported, zero at level 1 | utility, no unit |
beta_AD | D1 value set decrement for the anxiety or depression level reported, zero at level 1 | utility, no unit |
a_D1 | Disutility coefficient on D1, entered with its estimated sign, negative when extra dimensions with problems add less than their level decrements | utility per dimension, no unit |
a_I22 | Disutility coefficient on the square of I2, entered with its estimated sign | utility, no unit |
a_I3 | Disutility coefficient on I3, entered with its estimated sign | utility per dimension, no unit |
a_I32 | Disutility coefficient on the square of I3, entered with its estimated sign | utility, no unit |
D_1 | Number of dimensions above level 1, less 1, and 0 when no dimension has a problem | count |
|---|---|---|
I_2 | Number of dimensions at level 2, less 1, and 0 when none is at level 2 | count |
I_3 | Number of dimensions at level 3, less 1, and 0 when none is at level 3 | count |
U | Utility of the state described by the profile, with full health at 1 and dead at 0 | utility, no unit |
Function
Country-specific value set comparison function
Takes one EQ-5D-3L health profile, or the pair of profiles a treatment moves patients between, and the coefficients of two national value sets, and returns the utility gain under each set, the gap between the two gains and the resulting change in the QALY gain and in the incremental cost-effectiveness ratio. The scoring rule itself is live and is not restated here. The main-effects form with a constant and level decrements, which the article writes as U(s) equal to 1 minus c I(s) minus the sum of the level decrements beta_d(l_d), is HE-FM-EQ3L-001 with the N3 decrement set to zero; HE-FM-EQ3L-002 fixes it at the UK MVH tariff, HE-CF-EQ3L-001 scores any constant and N3 set such as the Dutch one, and HE-FM-EQ3L-003 gives the QALY difference between two states under one set. The records here compare sets. HE-FM-CSVS-001 splits the cross-set gap in a utility gain by term, HE-FM-CSVS-002 turns two QALY gains into relative changes in the gain and the ICER (HE-FM-CUA-001), and HE-FM-CSVS-003 gives the count-based D1 form that the US set uses in place of a constant. Five-level value sets belong to the EQ-5D-5L value set concept, and QALYs from different sets are not added (HE-ER-QALY-003). Notation follows the Country-Specific Value Set article.
Try this function
Implementations
Excel
Count-based D1 value set utility from named levels, decrements and coefficients
With the five reported levels in a row range named Levels, the decrements at those levels in a range named DecrementsAtLevels and the four count coefficients in CoefExtraDims, CoefModerateSq, CoefExtreme and CoefExtremeSq, the first two formulas fill the cells CountModerate and CountExtreme and the third returns the utility held in UtilityDOneForm.
=COUNTIF(Levels,2); =COUNTIF(Levels,3); =1-SUM(DecrementsAtLevels)-CoefExtraDims*MAX(CountModerate+CountExtreme-1,0)-CoefModerateSq*MAX(CountModerate-1,0)^2-CoefExtreme*MAX(CountExtreme-1,0)-CoefExtremeSq*MAX(CountExtreme-1,0)^2
Assumptions
Counts beyond the first dimension and no linear I2 term
D1, I2 and I3 count dimensions beyond the first, so a state with one problem carries none of the count terms and full health 11111 scores 1. Following Rand-Hendriksen and colleagues, the US model was fitted on disutilities, 1 minus the time trade-off value, and the linear I2 term was not statistically significant and was left out of the final model, which is why only its square appears.
Coefficient signs as estimated, entered as disutility coefficients
In the published US tariff as tabulated by Rand-Hendriksen and colleagues, the D1, I3 and I3 squared terms raise the value of a state and the I2 squared term lowers it. In this subtracting form a_D1, a_I3 and a_I32 are therefore entered as negative numbers and a_I22 as a positive one. The published US coefficients are not entered here because the coefficient table of Shaw and colleagues 2005 was not checked at text level; the examples use illustrative values with the same sign pattern.
Equivalence of the D1 form and a constant form
Rand-Hendriksen and colleagues show that the D1 model and a model with a constant give identical values for all 242 states other than full health when the constant equals minus a_D1 and every level decrement is shifted by a_D1, with the I2 squared, I3 and I3 squared coefficients unchanged. The D1 form is a reparameterisation, so the difference from a constant and N3 set lies in the count terms, not in the D1 term.
Worked examples
Illustrative D1 value set score for EQ-5D-3L state 21111 with one problem
Illustrative coefficients, a level 2 decrement of 0.15 and a level 3 decrement of 0.50 on every dimension, a_D1 of minus 0.12, a_I22 of 0.01, a_I3 of minus 0.10 and a_I32 of minus 0.02, are used; they are not the published US values. With one problem every count is zero, so the utility is 1 less the mobility decrement, 0.85.
beta_MO = 0.15; beta_SC = 0; beta_UA = 0; beta_PD = 0; beta_AD = 0; n_2 = 1; n_3 = 0; a_D1 = -0.12; a_I22 = 0.01; a_I3 = -0.10; a_I32 = -0.02; D_1 = 0; I_2 = 0; I_3 = 0; U = 0.85
Illustrative D1 value set score for EQ-5D-3L state 22222
With the same illustrative coefficients, five moderate problems cost 0.75 in level decrements, D1 of 4 gives back 0.48 and I2 of 4 costs 0.16 through its square, so the utility is 0.57.
beta_MO = 0.15; beta_SC = 0.15; beta_UA = 0.15; beta_PD = 0.15; beta_AD = 0.15; n_2 = 5; n_3 = 0; a_D1 = -0.12; a_I22 = 0.01; a_I3 = -0.10; a_I32 = -0.02; D_1 = 4; I_2 = 4; I_3 = 0; U = 0.57
Illustrative D1 value set score for EQ-5D-3L state 22331 with mixed levels
With the same illustrative coefficients, two moderate and two extreme problems cost 1.30 in level decrements. D1 is 3, I2 and I3 are each 1, and the count terms add 0.36 minus 0.01 plus 0.10 plus 0.02, giving 0.17.
beta_MO = 0.15; beta_SC = 0.15; beta_UA = 0.50; beta_PD = 0.50; beta_AD = 0; n_2 = 2; n_3 = 2; a_D1 = -0.12; a_I22 = 0.01; a_I3 = -0.10; a_I32 = -0.02; D_1 = 3; I_2 = 1; I_3 = 1; U = 0.17
Illustrative D1 value set score for the worst EQ-5D-3L state 33333
With the same illustrative coefficients, five extreme problems cost 2.50 in level decrements, and D1 of 4 and I3 of 4 give back 0.48, 0.40 and 0.32, so the utility is minus 0.30, a state below dead.
beta_MO = 0.50; beta_SC = 0.50; beta_UA = 0.50; beta_PD = 0.50; beta_AD = 0.50; n_2 = 0; n_3 = 5; a_D1 = -0.12; a_I22 = 0.01; a_I3 = -0.10; a_I32 = -0.02; D_1 = 4; I_2 = 0; I_3 = 4; U = -0.30
Common errors
Counting every dimension with a problem in D1
D1 counts dimensions beyond the first. Counting all of them gives a single-problem state a D1 credit it should not have, so with the illustrative coefficients 21111 scores 0.97 in place of 0.85 and 22222 scores 0.69 in place of 0.57 (computed here).
Scoring a D1 value set with the constant and N3 formula
Entering a D1 set's level decrements into HE-FM-EQ3L-001 with a zero constant and N3 drops all the count terms. With the illustrative coefficients 22222 then scores 0.25 in place of 0.57 and 33333 minus 1.50 in place of minus 0.30 (computed here), so a D1 set needs its own scoring rule, as HE-AS-EQ3L-003 states.
Sources
Shaw, Johnson and Coons' D1 valuation model for the US EQ-5D-3L set
Shaw JW, Johnson JA, Coons SJ. US valuation of the EQ-5D health states. Development and testing of the D1 valuation model. Medical Care. 2005;43(3):203-220. Abstract, which reports that respondents from a US probability sample valued 13 of 243 states by time trade-off, that methods were investigated to account for interaction effects caused by problems on several dimensions, and that the D1 model included ordinal terms for departures from perfect health as well as interaction effects, and that the D1 model, in a random effects specification, gave a good fit to the observed time trade-off values. Only the abstract was checked; the coefficient table was not seen.
D1 terms, the dropped I2 term and the equivalent constant model
Rand-Hendriksen K, Augestad LA, Dahl FA. A critical re-evaluation of the regression model specification in the US D1 EQ-5D value function. Population Health Metrics. 2012;10(1):2. Methods, subsection The D1 variable, model, and tariff values, which defines the ten level dummies, I2 and I3 as the numbers of dimensions at levels 2 and 3 beyond the first, their squares, and D1 as the number of dimensions beyond the first not at level 1, states that I2 was excluded from the final model and that the model was fitted on disutilities, with Table 1 listing the published US D1 tariff; Results and Table 2, which state that the constant of the equivalent model is the negative of the D1 coefficient, that using D1 adds the constant to each of the ten dummy coefficients, and that the two forms give identical values for all 242 predicted states. Open access.
EuroQol reference volume definitions of the D1, I2 and I3 terms
Janssen B, Szende A, Ramos-Goñi JM. Data and methods. In Szende A, Janssen B, Cabases J, editors. Self-reported population health. An international perspective based on EQ-5D. Dordrecht, Springer, 2014, chapter 2. Notes to Table 2.2, which define D1 as the additional number of dimensions at either level 2 or level 3, I2 as the number of dimensions at level 2 beyond the first and I3 as the number of dimensions at level 3 beyond the first, terms used by the United States set of Shaw 2005 in the table. Open access.
Canonical Identity
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