Cross-set gap in a utility gain split into constant, dimension and N3 terms

Scores the same move between two EQ-5D-3L states under two national value sets of the constant and N3 form of HE-FM-EQ3L-001 and finds where the difference between the two gains comes from. Under each set the gain is the constant times the change in the any-problem indicator, plus the decrement relieved on each of the five dimensions, plus the N3 decrement times the change in the N3 indicator; this is the difference of two HE-FM-EQ3L-001 scores, written term by term (derived). Subtracting the two gains shows how much of the gap each dimension and each profile-wide term carries. When neither state is full health the constant cancels within each set, as HE-AS-EQ3L-009 notes, so the gap rests on the dimension weights and the N3 term alone.

Signature

g_A = c_A * (I_0 - I_1) + v_A_MO + v_A_SC + v_A_UA + v_A_PD + v_A_AD + gamma_A * (N3_0 - N3_1); g_B = c_B * (I_0 - I_1) + v_B_MO + v_B_SC + v_B_UA + v_B_PD + v_B_AD + gamma_B * (N3_0 - N3_1); dg = g_A - g_B
Inputs
InputsDefinitionUnit
c_ADecrement value set A charges once for any problem, entered as a positive numberutility, no unit
I_0Equal to 1 when the starting state has a problem on any dimension and 0 for 11111indicator, 0 or 1
I_1Equal to 1 when the end state has a problem on any dimension and 0 for 11111indicator, 0 or 1
v_A_MOValue set A decrement at the starting mobility level less its decrement at the end level, zero when the level does not change and negative when it worsensutility, no unit
v_A_SCValue set A decrement at the starting self-care level less its decrement at the end level, zero when the level does not change and negative when it worsensutility, no unit
v_A_UAValue set A decrement at the starting usual activities level less its decrement at the end level, zero when the level does not change and negative when it worsensutility, no unit
v_A_PDValue set A decrement at the starting pain or discomfort level less its decrement at the end level, zero when the level does not change and negative when it worsensutility, no unit
v_A_ADValue set A decrement at the starting anxiety or depression level less its decrement at the end level, zero when the level does not change and negative when it worsensutility, no unit
gamma_ADecrement value set A charges once when any dimension is at level 3, zero for a set without an N3 termutility, no unit
N3_0Equal to 1 when the starting state has a dimension at level 3indicator, 0 or 1
N3_1Equal to 1 when the end state has a dimension at level 3indicator, 0 or 1
c_BDecrement value set B charges once for any problem, entered as a positive numberutility, no unit
v_B_MOValue set B decrement at the starting mobility level less its decrement at the end level, zero when the level does not change and negative when it worsensutility, no unit
v_B_SCValue set B decrement at the starting self-care level less its decrement at the end level, zero when the level does not change and negative when it worsensutility, no unit
v_B_UAValue set B decrement at the starting usual activities level less its decrement at the end level, zero when the level does not change and negative when it worsensutility, no unit
v_B_PDValue set B decrement at the starting pain or discomfort level less its decrement at the end level, zero when the level does not change and negative when it worsensutility, no unit
v_B_ADValue set B decrement at the starting anxiety or depression level less its decrement at the end level, zero when the level does not change and negative when it worsensutility, no unit
gamma_BDecrement value set B charges once when any dimension is at level 3, zero for a set without an N3 termutility, no unit
Output
g_AUtility of the state after the move less the utility of the state before it, scored with value set Autility, no unit
g_BUtility of the state after the move less the utility of the state before it, scored with value set Butility, no unit
dgGain under value set A less the gain under value set B for the same pair of profilesutility, 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.

Computational function

  • Computational function for comparing the UK MVH and Zimbabwe EQ-5D-3L value sets state by state

    Scores one EQ-5D-3L profile under two national value sets of the constant and N3 form, the UK MVH tariff of HE-FM-EQ3L-002 and the Zimbabwe set of Jelsma and colleagues, and returns both utilities and their difference. The R and Python versions score all 243 states under both sets and count the ordered pairs of states for which the two sets disagree on the direction of the change, the cross-set reversal that HE-FM-CSVS-001 splits by term for one pair. Any other pair of constant and N3 sets can be compared by replacing the two coefficient dictionaries.

    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; D is the any-problem indicator and N_3 the level 3 indicator, each 0 or 1; U_UK and U_ZW are the utilities under the UK and Zimbabwe sets and dU is U_UK less U_ZW, no unit.

    Assumption: Both sets take the constant and N3 form of HE-FM-EQ3L-001, with the Zimbabwe N3 decrement zero. Full health 11111 is fixed at 1 under both, although Jelsma and colleagues' model predicts 0.90 for it.

    Worked example (Article state 21232): The UK tariff gives 0.088 and the Zimbabwe set 0.453, so the UK value is 0.365 lower. l_MO = 2; l_SC = 1; l_UA = 2; l_PD = 3; l_AD = 2; D = 1; N_3 = 1; U_UK = 0.088; U_ZW = 0.453; dU = -0.365

    Worked example (Jelsma and colleagues' state 22331): The Zimbabwe set gives 0.315, the value worked in the paper, and the UK tariff minus 0.003. l_MO = 2; l_SC = 2; l_UA = 3; l_PD = 3; l_AD = 1; D = 1; N_3 = 1; U_UK = -0.003; U_ZW = 0.315; dU = -0.318

    Worked example (Worst state 33333): The UK tariff gives minus 0.594 and the Zimbabwe set minus 0.145. l_MO = 3; l_SC = 3; l_UA = 3; l_PD = 3; l_AD = 3; D = 1; N_3 = 1; U_UK = -0.594; U_ZW = -0.145; dU = -0.449

    Worked example (Full health 11111): Both sets give 1, a limiting case. l_MO = 1; l_SC = 1; l_UA = 1; l_PD = 1; l_AD = 1; D = 0; N_3 = 0; U_UK = 1; U_ZW = 1; dU = 0

    Excel: =1-UKConst*(SUM(Levels)>5)-SUMPRODUCT((Levels=2)*UKModerate+(Levels=3)*UKExtreme)-UKSevereTerm*(MAX(Levels)=3) fills UKUtility, =1-ZimConst*(SUM(Levels)>5)-SUMPRODUCT((Levels=2)*ZimModerate+(Levels=3)*ZimExtreme) fills ZimUtility and =UKUtility-ZimUtility fills CrossSetGap. The five levels are in a row range named Levels, the level 2 and level 3 decrements of each set in row ranges of the same shape named UKModerate, UKExtreme, ZimModerate and ZimExtreme, and the constants and the UK N3 decrement in UKConst, ZimConst and UKSevereTerm.

    R: 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); zw <- c(c0 = 0.100, MO2 = 0.056, MO3 = 0.204, SC2 = 0.092, SC3 = 0.231, UA2 = 0.043, UA3 = 0.135, PD2 = 0.067, PD3 = 0.302, AD2 = 0.046, AD3 = 0.173, N3 = 0); score <- function(l, cf) { 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) }; g <- as.matrix(expand.grid(rep(list(1:3), 5))); vu <- apply(g, 1, score, cf = uk); vz <- apply(g, 1, score, cf = zw); du <- vu-vz; gu <- outer(vu, vu, "-"); gz <- outer(vz, vz, "-"); rev <- sum(gu*gz < 0) gives the 243 differences in du and the number of ordered pairs whose direction reverses in rev.

    Python: from itertools import product; 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}; zw = {"c0": 0.100, "MO2": 0.056, "MO3": 0.204, "SC2": 0.092, "SC3": 0.231, "UA2": 0.043, "UA3": 0.135, "PD2": 0.067, "PD3": 0.302, "AD2": 0.046, "AD3": 0.173, "N3": 0}; score = lambda l, cf: 1-cf["c0"]*any(x > 1 for x in l)-sum(cf[d + str(x)] for d, x in zip(["MO", "SC", "UA", "PD", "AD"], l) if x > 1)-cf["N3"]*any(x == 3 for x in l); S = list(product((1, 2, 3), repeat=5)); du = [score(s, uk)-score(s, zw) for s in S]; rev = sum(1 for a in S for b in S if (score(b, uk)-score(a, uk))*(score(b, zw)-score(a, zw)) < 0) gives the same differences and reversal count.

    Test (State 33333 gap): With Levels at 33333 the cell CrossSetGap is minus 0.449, the UK minimum minus 0.594 less the Zimbabwe value minus 0.145. Expected result TRUE. FALSE shows the UK N3 decrement charged to the Zimbabwe set as well, which gives minus 0.180. Excel check =ABS(CrossSetGap-(-0.594-(-0.145)))<0.0005

    Test (Jelsma and colleagues' state 22331): With Levels at 22331 the cell ZimUtility rounds to 0.315, the value worked in the paper. Expected result TRUE. FALSE shows the Zimbabwe constant left out, which gives 0.415. Excel check =ROUND(ZimUtility,3)=0.315

    Common error (Reading the mean gap as a uniform shift): Over all 243 states the UK value is lower in 235, higher in 7 and equal only at 11111, with a mean difference of minus 0.314, a minimum of minus 0.519 at 32133 and a maximum of plus 0.026 at 11211. Of the 58,806 ordered pairs of distinct states, 4,116, or 7.0%, change direction between the two sets (computed here with the code above), so a single offset between two national sets cannot stand in for rescoring.

    Source: Jelsma J, Hansen K, De Weerdt W, De Cock P, Kind P. How do Zimbabweans value health states? Population Health Metrics. 2003;1(1):11. Results, Valuations subsection, Table 5 and the worked value for 22331. The UK coefficients are those of HE-FM-EQ3L-002, from 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_UK = 1 - 0.081 * D - 0.069 * (min(l_MO - 1, 1) - max(l_MO - 2, 0)) - 0.314 * max(l_MO - 2, 0) - 0.104 * (min(l_SC - 1, 1) - max(l_SC - 2, 0)) - 0.214 * max(l_SC - 2, 0) - 0.036 * (min(l_UA - 1, 1) - max(l_UA - 2, 0)) - 0.094 * max(l_UA - 2, 0) - 0.123 * (min(l_PD - 1, 1) - max(l_PD - 2, 0)) - 0.386 * max(l_PD - 2, 0) - 0.071 * (min(l_AD - 1, 1) - max(l_AD - 2, 0)) - 0.236 * max(l_AD - 2, 0) - 0.269 * N_3; U_ZW = 1 - 0.1 * D - 0.056 * (min(l_MO - 1, 1) - max(l_MO - 2, 0)) - 0.204 * max(l_MO - 2, 0) - 0.092 * (min(l_SC - 1, 1) - max(l_SC - 2, 0)) - 0.231 * max(l_SC - 2, 0) - 0.043 * (min(l_UA - 1, 1) - max(l_UA - 2, 0)) - 0.135 * max(l_UA - 2, 0) - 0.067 * (min(l_PD - 1, 1) - max(l_PD - 2, 0)) - 0.302 * max(l_PD - 2, 0) - 0.046 * (min(l_AD - 1, 1) - max(l_AD - 2, 0)) - 0.173 * max(l_AD - 2, 0); dU = U_UK - U_ZW

Try this function

Implementations

  • Excel

    Cross-set gap in a utility gain from named constant, relieved decrement and N3 cells

    With the two constants in ConstSetA and ConstSetB, the any-problem indicators of the two states in AnyProblemBefore and AnyProblemAfter, the five relieved decrements of each set in the ranges RelievedSetA and RelievedSetB, the N3 decrements in SevereTermSetA and SevereTermSetB and the N3 indicators in SevereBefore and SevereAfter, the first two formulas return the gains held in GainSetA and GainSetB and the third the gap held in GainGapAcrossSets.

    =ConstSetA*(AnyProblemBefore-AnyProblemAfter)+SUM(RelievedSetA)+SevereTermSetA*(SevereBefore-SevereAfter); =ConstSetB*(AnyProblemBefore-AnyProblemAfter)+SUM(RelievedSetB)+SevereTermSetB*(SevereBefore-SevereAfter); =GainSetA-GainSetB

Assumptions

  • Both national value sets of the constant and N3 main-effects form

    Each set is scored by HE-FM-EQ3L-001, a constant for any problem, a decrement for each level above 1 and at most one N3 decrement, so a set reported without the constant or without N3 enters zero for it. The Zimbabwe set of Jelsma and colleagues has a constant and ten level decrements and no N3 term, since their model with N3 valued moderate problems on several dimensions below extreme problems and was discarded. A set with count terms, such as the D1 form of HE-FM-CSVS-003, is not split this way.

  • Same profile pair, levels and coefficient signs under both value sets

    Both gains are for the same starting and end profiles, read in the order mobility, self-care, usual activities, pain or discomfort and anxiety or depression. Every coefficient is a positive decrement (HE-AS-EQ3L-004), so a relieved decrement is positive for a dimension that improves. Full health 11111 is fixed at 1 through I = 0 under both sets, the convention of HE-FM-EQ3L-001 and the treatment Rand-Hendriksen and colleagues describe for the US D1 tariff, in which the value of 11111 is given and not predicted. Jelsma and colleagues do not state this for the Zimbabwe set, whose model predicts 0.90 for 11111 (their Table 6), so a Zimbabwe gain into 11111 rests on this convention (HE-ER-CSVS-002).

Worked examples

  • Article's illustrative sets A and B for the move out of 21232 into 11121

    With the article's illustrative decrements, which are not taken from any published set, relieving moderate mobility and usual activities problems, moving pain down one level to level 2 and relieving moderate anxiety gains 0.45 under set A and 0.39 under set B. The constant cancels and neither set has an N3 term. Of the gap of 0.06, pain contributes 0.09 (0.27 against 0.18), anxiety 0.01, while mobility (minus 0.03) and usual activities (minus 0.01) pull the other way, so pain alone carries more than the whole gap (computed here), in line with the article's reading that set B gives severe pain a smaller decrement.

    c_A = 0.08; c_B = 0.05; I_0 = 1; I_1 = 1; v_A_MO = 0.07; v_A_SC = 0; v_A_UA = 0.04; v_A_PD = 0.27; v_A_AD = 0.07; v_B_MO = 0.10; v_B_SC = 0; v_B_UA = 0.05; v_B_PD = 0.18; v_B_AD = 0.06; gamma_A = 0; gamma_B = 0; N3_0 = 1; N3_1 = 0; g_A = 0.45; g_B = 0.39; dg = 0.06

  • UK MVH and Zimbabwe EQ-5D-3L sets for the move out of 21232 into 11121

    Scoring the article's two states with the UK tariff of HE-FM-EQ3L-002 (set A) gives 0.088 and 0.796, a gain of 0.708, and with the Zimbabwe coefficients of Jelsma and colleagues (set B) gives 0.453 and 0.833, a gain of 0.380. Of the gap of 0.328, the UK N3 decrement of 0.269 carries about 82%, and the four dimension differences together only 0.059, so the two sets differ on this move mainly because one of them charges extra for any level 3 problem (computed here).

    c_A = 0.081; c_B = 0.100; I_0 = 1; I_1 = 1; v_A_MO = 0.069; v_A_SC = 0; v_A_UA = 0.036; v_A_PD = 0.263; v_A_AD = 0.071; v_B_MO = 0.056; v_B_SC = 0; v_B_UA = 0.043; v_B_PD = 0.235; v_B_AD = 0.046; gamma_A = 0.269; gamma_B = 0; N3_0 = 1; N3_1 = 0; g_A = 0.708; g_B = 0.380; dg = 0.328

  • UK MVH and Zimbabwe EQ-5D-3L sets for a return to full health out of 21121

    When the end state is 11111 the constant no longer cancels, and 11111 is fixed at 1 under both sets (HE-AS-CSVS-002), not at the 0.90 that the Zimbabwe model predicts. Under the UK tariff the gain is 0.081 plus 0.069 plus 0.123, or 0.273, and under the Zimbabwe set 0.100 plus 0.056 plus 0.067, or 0.223. The gap of 0.050 is the sum of a constant term of minus 0.019, a mobility term of 0.013 and a pain term of 0.056 (computed here).

    c_A = 0.081; c_B = 0.100; I_0 = 1; I_1 = 0; v_A_MO = 0.069; v_A_SC = 0; v_A_UA = 0; v_A_PD = 0.123; v_A_AD = 0; v_B_MO = 0.056; v_B_SC = 0; v_B_UA = 0; v_B_PD = 0.067; v_B_AD = 0; gamma_A = 0.269; gamma_B = 0; N3_0 = 0; N3_1 = 0; g_A = 0.273; g_B = 0.223; dg = 0.050

Common errors

  • Reading a cross-set gain gap as a difference in dimension weights

    Two sets can agree closely on every dimension and still give very different gains because of a profile-wide term. For the move out of 21232 into 11121 the four dimension differences between the UK and Zimbabwe sets add up to only 0.059, while the UK N3 decrement adds 0.269, so 82% of the gap of 0.328 comes from a term the Zimbabwe set does not have (computed here). Splitting the gap by term shows this before the gap is put down to national preferences about pain or mobility.

  • Charging the Zimbabwe constant to full health in a gain to 11111

    Jelsma and colleagues' Table 6 lists a predicted value of 0.90 for 11111, which is their constant 0.900 with no level effect subtracted (derived). Using that value for the end state of a return to full health out of 21121 gives a gain of 0.123 in place of 0.223 under the Zimbabwe set and inflates the UK and Zimbabwe gap to 0.150 in place of 0.050 (computed here). Fixing 11111 at 1, through I = 0, keeps every set on the scale anchored at full health.

Sources

  • Zimbabwe EQ-5D-3L time trade-off coefficients in Jelsma and colleagues' Table 5

    Jelsma J, Hansen K, De Weerdt W, De Cock P, Kind P. How do Zimbabweans value health states? Population Health Metrics. 2003;1(1):11. Results, Valuations subsection, Table 5 (main effects of the residual maximum likelihood linear mixed model on 12,663 observations), which gives the constant 0.900 and the level 2 and level 3 effects for mobility (minus 0.056, minus 0.204), self-care (minus 0.092, minus 0.231), usual activities (minus 0.043, minus 0.135), pain or discomfort (minus 0.067, minus 0.302) and anxiety or depression (minus 0.046, minus 0.173), the worked value of 0.315 for state 22331, 0.90 less 0.056, 0.092, 0.135 and 0.302, the text on the discarded N3 model and Table 6 with a predicted mean of 0.90 for 11111. Open access.

    View source →

  • EuroQol reference volume table of national EQ-5D-3L time trade-off value set forms

    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 the time trade-off value sets of 13 countries as a constant, level 2 and level 3 dummies for each dimension and, for some sets, N3 or other terms; the Zimbabwe row (Jelsma 2003, random effects) prints the constant minus 0.100 and the same ten level coefficients as Jelsma and colleagues' Table 5, with no N3 entry. Open access.

    View source →

  • Fixed value of 1 for EQ-5D state 11111 outside the predicted states

    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 Handling the constant and state 11111, which states that state 11111 was given a fixed value of 1 and death a fixed value of 0, and that the authors treat the value of 11111 as axiomatic and outside the scope of the tariff prediction, with the intercept a constant for the 242 other states. The paper concerns the US D1 tariff; applying the same fixed value to the Zimbabwe set is this package's convention, not a statement of Jelsma and colleagues. Open access.

    View source →

Canonical Identity

Persistent URI
https://healtheconomics.wiki/formula/cross-set-gap-in-a-utility-gain-split-into-constant-dimension-and-n3-terms

Stable URI · Machine-readable · Resolvable · CC BY 4.0