EQ-5D-3L utility as one minus a constant, level decrements and an N3 term

Signature

U = 1 - alpha * D - (beta_MO + beta_SC + beta_UA + beta_PD + beta_AD) - gamma * N_3

Function

EQ-5D-3L value set scoring function

Computational function

• Computational function for scoring an EQ-5D-3L profile under any constant and N3 coefficient set

  Takes an EQ-5D-3L profile, as a five-digit string such as 11223 or as five level inputs, and a coefficient set of the main-effects form, and returns the utility with the two indicators D and N3. It applies HE-FM-EQ3L-001 with level 2 and level 3 dummies built from the levels, so any value set tabulated with a constant, ten level dummies and N3 can be scored, and with the UK coefficients it reproduces HE-FM-EQ3L-002. The R and Python versions also score all 243 states.

  Inputs and outputs: l_MO, l_SC, l_UA, l_PD and l_AD are the reported levels, required, whole numbers 1, 2 or 3, in profile order; c_0 is the constant and c_N3 the N3 decrement, required, entered as positive decrements; b_MO2, b_MO3, b_SC2, b_SC3, b_UA2, b_UA3, b_PD2, b_PD3, b_AD2 and b_AD3 are the level 2 and level 3 decrements of each dimension, required, positive; D is the any-problem indicator and N_3 the level 3 indicator, each 0 or 1; U is the utility, no unit.

  Assumption: The value set has the constant and N3 form with no other interaction terms. Sets built differently, such as the US D1 model of Shaw and colleagues 2005, need their own scoring rule. Coefficients printed as negative numbers are entered without their signs.

  Worked example (UK tariff, state 11223): The UK coefficients give 0.255, the value worked in the MVH discussion paper. l_MO = 1; l_SC = 1; l_UA = 2; l_PD = 2; l_AD = 3; c_0 = 0.081; b_MO2 = 0.069; b_MO3 = 0.314; b_SC2 = 0.104; b_SC3 = 0.214; b_UA2 = 0.036; b_UA3 = 0.094; b_PD2 = 0.123; b_PD3 = 0.386; b_AD2 = 0.071; b_AD3 = 0.236; c_N3 = 0.269; D = 1; N_3 = 1; U = 0.255

  Worked example (Dutch coefficients, state 33333): The Dutch time trade-off coefficients of Lamers and colleagues 2006, as tabulated in the EuroQol reference volume, give minus 0.329 for the worst state, against minus 0.594 under the UK tariff. l_MO = 3; l_SC = 3; l_UA = 3; l_PD = 3; l_AD = 3; c_0 = 0.071; b_MO2 = 0.036; b_MO3 = 0.161; b_SC2 = 0.082; b_SC3 = 0.152; b_UA2 = 0.032; b_UA3 = 0.057; b_PD2 = 0.086; b_PD3 = 0.329; b_AD2 = 0.124; b_AD3 = 0.325; c_N3 = 0.234; D = 1; N_3 = 1; U = -0.329

  Worked example (Full health under the UK coefficients): State 11111 sets both indicators to zero and scores 1, a limiting case for any coefficient set. l_MO = 1; l_SC = 1; l_UA = 1; l_PD = 1; l_AD = 1; c_0 = 0.081; b_MO2 = 0.069; b_MO3 = 0.314; b_SC2 = 0.104; b_SC3 = 0.214; b_UA2 = 0.036; b_UA3 = 0.094; b_PD2 = 0.123; b_PD3 = 0.386; b_AD2 = 0.071; b_AD3 = 0.236; c_N3 = 0.269; D = 0; N_3 = 0; U = 1

  Excel: =1-Const*(SUM(Levels)>5)-SUMPRODUCT((Levels=2)*Level2Dec+(Levels=3)*Level3Dec)-CoefN3*(MAX(Levels)=3) With the five levels in a row range named Levels, the five level 2 decrements in a range named Level2Dec and the five level 3 decrements in Level3Dec, of the same shape, and the constant and N3 coefficients in Const and CoefN3, the formula returns U.

  R: eq3l_utility <- function(s, cf) { l <- as.integer(strsplit(as.character(s), "")[[1]]); dims <- c("MO", "SC", "UA", "PD", "AD"); dec <- sum(sapply(1:5, function(d) if (l[d] == 1) 0 else cf[[paste0(dims[d], l[d])]])); 1-cf[["c0"]]*any(l > 1)-dec-cf[["N3"]]*any(l == 3) } Takes the profile as a string and a named vector with elements c0, MO2 to AD3 and N3. With uk <- c(c0 = 0.081, MO2 = 0.069, MO3 = 0.314, SC2 = 0.104, SC3 = 0.214, UA2 = 0.036, UA3 = 0.094, PD2 = 0.123, PD3 = 0.386, AD2 = 0.071, AD3 = 0.236, N3 = 0.269), v <- sapply(apply(expand.grid(rep(list(1:3), 5)), 1, paste, collapse = ""), eq3l_utility, cf = uk) scores all 243 states.

  Python: def eq3l_utility(s, cf): l = [int(c) for c in str(s)]; dec = sum(cf[d + str(x)] for d, x in zip(["MO", "SC", "UA", "PD", "AD"], l) if x > 1); return 1-cf["c0"]*any(x > 1 for x in l)-dec-cf["N3"]*any(x == 3 for x in l) Takes the profile as a string or integer and a dictionary with keys c0, MO2 to AD3 and N3. With uk = {"c0": 0.081, "MO2": 0.069, "MO3": 0.314, "SC2": 0.104, "SC3": 0.214, "UA2": 0.036, "UA3": 0.094, "PD2": 0.123, "PD3": 0.386, "AD2": 0.071, "AD3": 0.236, "N3": 0.269}, from itertools import product; v = [eq3l_utility("".join(map(str, p)), uk) for p in product((1, 2, 3), repeat=5)] scores all 243 states.

  Test (UK minimum and states below zero): Scoring all 243 states with the UK coefficients gives a minimum of minus 0.594 at 33333 and 84 states below zero, about a third, matching the negative values printed in the tariff appendix. Expected result TRUE. Excel check =AND(ROUND(1-(0.081+0.314+0.214+0.094+0.386+0.236+0.269),3)=-0.594,ROUND(1-(0.081+0.036+0.123+0.236+0.269),3)=0.255)

  Test (N3 charged once): For 33333 the UK utility equals 1 less the constant, the five level 3 decrements and one N3 decrement, and differs from the five-fold N3 version by four times 0.269, 1.076. Expected result TRUE. Excel check =ROUND((1-1.325-0.269)-(1-1.325-5*0.269),3)=1.076

  Common error (Printed negative signs kept in a subtracting formula): Entering the EuroQol reference volume's negative coefficients into a formula that subtracts them turns every decrement into a gain, so 11223 scores 1.745 instead of 0.255. The signs are dropped, or the coefficients are added to 1 as printed.

  Source: 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. Table 2.2, which tabulates national time trade-off coefficients as a constant, level 2 and level 3 dummies for each dimension and N3, including the UK and Dutch rows used here. The UK coefficients agree with Table 1 of Dolan, Gudex, Kind and Williams 1995.

  D = min(l_MO + l_SC + l_UA + l_PD + l_AD - 5, 1); N_3 = min(max(l_MO - 2, 0) + max(l_SC - 2, 0) + max(l_UA - 2, 0) + max(l_PD - 2, 0) + max(l_AD - 2, 0), 1); U = 1 - c_0 * D - b_MO2 * (min(l_MO - 1, 1) - max(l_MO - 2, 0)) - b_MO3 * max(l_MO - 2, 0) - b_SC2 * (min(l_SC - 1, 1) - max(l_SC - 2, 0)) - b_SC3 * max(l_SC - 2, 0) - b_UA2 * (min(l_UA - 1, 1) - max(l_UA - 2, 0)) - b_UA3 * max(l_UA - 2, 0) - b_PD2 * (min(l_PD - 1, 1) - max(l_PD - 2, 0)) - b_PD3 * max(l_PD - 2, 0) - b_AD2 * (min(l_AD - 1, 1) - max(l_AD - 2, 0)) - b_AD3 * max(l_AD - 2, 0) - c_N3 * N_3

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Implementations

• Excel

  EQ-5D-3L main-effects utility from named level and decrement ranges

  With the five reported levels in a range named Levels, the decrements at those levels in a range named Decrements and the constant and N3 coefficients in cells named Alpha and Gamma, the formula returns U. SUM(Levels)>5 is TRUE for any problem and MAX(Levels)=3 for any level 3, and Excel treats TRUE as 1 when it multiplies.

  =1-Alpha*(SUM(Levels)>5)-SUM(Decrements)-Gamma*(MAX(Levels)=3)

Assumptions

• Additive main effects with no interactions beyond the N3 term

  Each dimension contributes its own decrement whatever the levels on the other dimensions, and the only terms that depend on the whole profile are the constant and N3. Dolan and colleagues reported that none of the models with interactions between dimensions improved the fit significantly and that many introduced inconsistencies into the estimated values, and that without N3 the model underestimated the values of less severe states and overestimated those of more severe ones.

• Level 3 decrement at least as large as the level 2 decrement on each dimension

  On every dimension the decrement at level 3 is no smaller than the one at level 2, and both are zero or positive, so a state that is worse on one dimension and the same on the others never scores higher. A coefficient set that breaks this ordering is not logically consistent.

• Not every national EQ-5D-3L value set takes the constant and N3 form

  The form applies to value sets estimated with a constant, level 2 and level 3 dummies and N3, as the UK tariff was, and to sets reported without the constant or without N3, which are scored by entering zero for the missing coefficient. Sets with any other term cannot be scored this way. The US time trade-off set of Shaw, Johnson and Coons 2005 is a D1 model with terms for the number of dimensions with problems beyond the first and for the numbers at level 2 and at level 3 beyond the first, some of them squared, and no constant or N3 term, so it cannot be scored with this formula and is not formalised here.

• Coefficients entered as positive decrements

  The constant, the level decrements and the N3 term are entered as positive amounts that are subtracted from 1. Tables that print them as negative coefficients added to 1, as the EuroQol reference volume does, describe the same model, and their signs are dropped before entry.

Worked examples

• Illustrative EQ-5D-3L state 21121 with two level 2 problems

  With the article's illustrative coefficients, which are not the published UK values, state 21121 has moderate mobility problems and moderate pain. The constant 0.08, the mobility decrement 0.07 and the pain decrement 0.12 give a utility of 0.73.

  alpha = 0.08; beta_MO = 0.07; beta_SC = 0; beta_UA = 0; beta_PD = 0.12; beta_AD = 0; gamma = 0.27; D = 1; N_3 = 0; U = 0.73

• Illustrative EQ-5D-3L state 22221 with four level 2 problems

  Adding level 2 problems on self-care, with an illustrative decrement of 0.10, and usual activities, 0.04, lowers the illustrative utility by 0.14 to 0.59. No dimension is at level 3, so N3 is not charged.

  alpha = 0.08; beta_MO = 0.07; beta_SC = 0.10; beta_UA = 0.04; beta_PD = 0.12; beta_AD = 0; gamma = 0.27; D = 1; N_3 = 0; U = 0.59

• Illustrative EQ-5D-3L state 21131 crossing the N3 threshold

  Moving pain alone in 21121 to level 3 replaces the illustrative pain decrement 0.12 with 0.38 and brings in the N3 decrement 0.27, so the utility falls by 0.53 to 0.20. Half of the fall comes from the N3 term.

  alpha = 0.08; beta_MO = 0.07; beta_SC = 0; beta_UA = 0; beta_PD = 0.38; beta_AD = 0; gamma = 0.27; D = 1; N_3 = 1; U = 0.20

• Illustrative EQ-5D-3L state 31131 with N3 charged once

  Moving mobility in 21131 to level 3 as well, with an illustrative decrement of 0.30, gives a utility of minus 0.03, a state valued as worse than dead. The second dimension at level 3 does not add a second N3 decrement.

  alpha = 0.08; beta_MO = 0.30; beta_SC = 0; beta_UA = 0; beta_PD = 0.38; beta_AD = 0; gamma = 0.27; D = 1; N_3 = 1; U = -0.03

• EQ-5D-3L full health 11111 scores one under any coefficient set

  With every dimension at level 1 all five level decrements are zero and both indicators are zero, so the utility is 1 whatever the constant and N3 coefficients. This limiting case checks that the constant is not charged for full health.

  alpha = 0.08; beta_MO = 0; beta_SC = 0; beta_UA = 0; beta_PD = 0; beta_AD = 0; gamma = 0.27; D = 0; N_3 = 0; U = 1

Common errors

• Charging the EQ-5D-3L N3 decrement once for each dimension at level 3

  N3 is a single indicator. Subtracting the illustrative 0.27 for both level 3 dimensions of 31131 gives minus 0.30 instead of minus 0.03, and under the UK tariff charging it five times for 33333 gives minus 1.670 instead of minus 0.594, so the most severe states are understated.

• Reading the minus 0.594 floor as a property of the EQ-5D-3L instrument

  The lowest value belongs to the value set and not to the questionnaire. Scoring 33333 with the Dutch time trade-off coefficients of Lamers and colleagues 2006 tabulated in the EuroQol reference volume gives minus 0.329, against minus 0.594 with the UK tariff, so a negative mean utility partly reflects the set used.

Sources

• Dolan's model for valuing all EQ-5D-3L states from a subset

  Dolan P. Modeling valuations for EuroQol health states. Medical Care. 1997;35(11):1095-1108. Abstract, which reports that valuations of 42 states elicited by time trade-off from a representative UK sample were modelled in terms of the level of severity on each dimension, an intercept for any move away from full health and a term for whether any dimension is at its most severe level, to interpolate values for all 243 states.

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• EuroQol reference volume table of EQ-5D-3L time trade-off coefficients

  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. Table 2.2, which gives the time trade-off value set coefficients of 13 countries as a constant, dummies for level 2 and level 3 on each dimension and N3, defined as 1 if any dimension is at level 3, and shows the US set of Shaw 2005 with D1 and count terms in place of the constant and N3. Open access.

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• MVH discussion paper on interaction terms and the EQ-5D-3L N3 term

  Dolan P, Gudex C, Kind P, Williams A. A social tariff for EuroQol. Results from a UK general population survey. Centre for Health Economics Discussion Paper 138. York, University of York, 1995. Pages 10 and 11, which report that none of the models with interactions between dimensions improved the model significantly and many introduced inconsistencies, that without the N3 intercept for any dimension at level 3 the model underestimated less severe and overestimated more severe states, and that TTO scores are explained by an intercept for any move away from full health, two variables for each dimension and N3. Scanned text, read from the page images.

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