Equity-weighted incremental net health benefit with and without the weight

Converts the incremental cost into the health displaced elsewhere, Delta_C divided by lambda, and subtracts it from the weighted and from the unweighted QALY gain. The displaced health is not weighted, so the two results differ by w minus 1, times Delta_E. A negative unweighted value with a positive weighted value marks a programme that lowers total population health and would be recommended only on the strength of the equity judgement.

Signature

H_d = Delta_C / lambda; NHB_w = w * Delta_E - H_d; NHB_u = Delta_E - H_d
Inputs
InputsDefinitionUnit
Delta_CCost of the programme minus that of the comparatorcurrency per patient
lambdaCost per QALY used to express the incremental cost as health forgone, GBP 25,000 and GBP 35,000 in NICE's presentation, greater than zerocurrency per QALY
wWeight applied to every QALY gained in the group, 1 for no prioritymultiplier, dimensionless
Delta_EQALYs under the programme minus those under the comparatorQALYs per patient
Output
H_dQALYs expected to be lost elsewhere in the health system because of the programme's incremental cost, unweightedQALYs per patient
NHB_wWeighted QALY gain minus the unweighted health displacedweighted QALYs per patient
NHB_uUnweighted QALY gain minus the health displaced, the effect on total population healthQALYs per patient

Function

Equity-weighted QALY gain, ICER, net health benefit and switching weight function

Maps the incremental QALY gain of a programme in each population group, an equity weight for each group, the incremental cost and the value placed on a QALY to the equity-weighted QALY gain, the equity-weighted ICER, the equity-weighted net health benefit and the weight at which the programme would break even. The weights act on the incremental QALYs only, while costs and the health displaced elsewhere stay unweighted, and with every weight equal to 1 the outputs reduce to the unweighted ICER (HE-FM-ICER-001) and incremental net health benefit (HE-FM-CET-002). The weights come from outside the function, for example the NICE severity weight from absolute and proportional shortfall (HE-FM-ASF-001 and HE-FM-ASF-003) or the weight implied by an Atkinson inequality aversion parameter (HE-FM-ATK-004), and neither is redrafted here. The break-even weight differs from the switching weight between two options in the weighted two-group social welfare formula of the Welfare Economics package, because it compares one programme with the health displaced by its cost. Notation follows the Equity-Weighted Analysis article.

Computational function

  • Computational function for weighted and unweighted population net health benefit across several groups

    Takes the number of patients, the equity weight, the incremental QALYs and the incremental cost per patient in each group, and one value of a QALY, and returns the programme's population QALY gain unweighted and weighted, its total incremental cost, the health displaced, the unweighted and weighted net health benefit and the unweighted and weighted ICER. It applies HE-FM-EQWA-001 to the group totals and HE-FM-EQWA-003 to the whole programme, and it is run once at GBP 25,000 and once at GBP 35,000 per QALY to give the four net health benefits PMG36 asks for. The Math / Code line shows two groups, and the R and Python versions take any number.

    Inputs and outputs: N_1 and N_2 are the numbers of patients in each group, required, above zero; w_1 and w_2 are the equity weights, required, above zero, 1 for no priority; dE_1 and dE_2 are the incremental QALYs per patient; dC_1 and dC_2 are the incremental costs per patient in currency; lambda is the value of a QALY in currency per QALY, above zero; E_u and E_w are the unweighted and weighted population QALY gains; C_tot is the total incremental cost; H_d is the health displaced in QALYs; NHB_u and NHB_w are the unweighted and weighted population net health benefits in QALYs; ICER_u and ICER_w are the unweighted and weighted ICERs in currency per QALY.

    Assumption: Each group carries one weight, the weights act on the incremental QALYs only, and the displaced health is unweighted. The weights are set beforehand, for example from the severity band rule HE-FM-ASF-003.

    Worked example (Article's two subgroups at GBP 35,000 per QALY): Subgroup A has 300 patients at a weight of 1.7 gaining 0.6 QALYs each for GBP 30,000, and subgroup B 700 patients at a weight of 1 gaining 0.8 QALYs for GBP 24,000, all illustrative. The total cost of GBP 25.8 million displaces about 737.1 QALYs, so the population net health benefit is about 2.9 QALYs unweighted and 128.9 weighted, the sums of the article's subgroup values. N_1 = 300; w_1 = 1.7; dE_1 = 0.6; dC_1 = 30000; N_2 = 700; w_2 = 1; dE_2 = 0.8; dC_2 = 24000; lambda = 35000; E_u = 740; E_w = 866; C_tot = 25800000; H_d = 737.1429; NHB_u = 2.8571; NHB_w = 128.8571; ICER_u = 34864.86; ICER_w = 29792.15

    Worked example (Same subgroups at GBP 25,000 per QALY): The displaced health rises to 1,032 QALYs, and the population net health benefit falls to minus 292 QALYs unweighted and minus 166 weighted, so the programme lowers health on both measures. N_1 = 300; w_1 = 1.7; dE_1 = 0.6; dC_1 = 30000; N_2 = 700; w_2 = 1; dE_2 = 0.8; dC_2 = 24000; lambda = 25000; E_u = 740; E_w = 866; C_tot = 25800000; H_d = 1032; NHB_u = -292; NHB_w = -166; ICER_u = 34864.86; ICER_w = 29792.15

    Worked example (Both weights set to 1): With no priority weight the weighted and unweighted results are identical, about 2.9 QALYs at GBP 35,000 per QALY, the limiting case that checks the weighting step. N_1 = 300; w_1 = 1; dE_1 = 0.6; dC_1 = 30000; N_2 = 700; w_2 = 1; dE_2 = 0.8; dC_2 = 24000; lambda = 35000; E_u = 740; E_w = 740; C_tot = 25800000; H_d = 737.1429; NHB_u = 2.8571; NHB_w = 2.8571; ICER_u = 34864.86; ICER_w = 34864.86

    Excel: =SUMPRODUCT(GroupSize,EquityWeight,QALYGainPerPatient)-SUMPRODUCT(GroupSize,CostPerPatient)/QALYValue With one row per group in ranges named GroupSize, EquityWeight, QALYGainPerPatient and CostPerPatient, and the value of a QALY in QALYValue, the formula returns the weighted population net health benefit. =SUMPRODUCT(GroupSize,QALYGainPerPatient)-SUMPRODUCT(GroupSize,CostPerPatient)/QALYValue gives the unweighted value, and =SUMPRODUCT(GroupSize,CostPerPatient)/SUMPRODUCT(GroupSize,EquityWeight,QALYGainPerPatient) the weighted ICER.

    R: eq_nhb <- function(N, w, dE, dC, lambda) { Eu <- sum(N*dE); Ew <- sum(N*w*dE); C <- sum(N*dC); Hd <- C/lambda; c(E_u = Eu, E_w = Ew, C_tot = C, H_d = Hd, NHB_u = Eu-Hd, NHB_w = Ew-Hd, ICER_u = C/Eu, ICER_w = C/Ew) } Vectorised over N, w, dE and dC, with one element per group.

    Python: def eq_nhb(N, w, dE, dC, lam): Eu = sum(n*e for n, e in zip(N, dE)); Ew = sum(n*v*e for n, v, e in zip(N, w, dE)); C = sum(n*c for n, c in zip(N, dC)); Hd = C/lam; return {"E_u": Eu, "E_w": Ew, "C_tot": C, "H_d": Hd, "NHB_u": Eu-Hd, "NHB_w": Ew-Hd, "ICER_u": C/Eu, "ICER_w": C/Ew} Takes lists with one element per group and the value of a QALY, and returns a dictionary of the outputs.

    Test (Weighted minus unweighted net health benefit equals the extra weighted gain): The difference is the sum over groups of patients times weight minus 1 times QALY gain, 300 × 0.7 × 0.6 = 126 QALYs in the article's example, at any value of a QALY. Expected result TRUE. Excel check =ABS((SUMPRODUCT(GroupSize,EquityWeight,QALYGainPerPatient)-SUMPRODUCT(GroupSize,QALYGainPerPatient))-SUMPRODUCT(GroupSize,EquityWeight-1,QALYGainPerPatient))<0.000001

    Test (Article's weighted population net health benefit reproduced): The two subgroups give 128.9 weighted QALYs at GBP 35,000 per QALY when rounded to one decimal place. Expected result TRUE. Excel check =ROUND(SUMPRODUCT({300;700},{1.7;1},{0.6;0.8})-SUMPRODUCT({300;700},{30000;24000})/35000,1)=128.9

    Common error (Reading a positive pooled result as a gain in every group): At GBP 35,000 per QALY the pooled unweighted net health benefit is about 2.9 QALYs, yet subgroup A alone loses about 77.1 QALYs and subgroup B gains 80.0. PMG36 asks for cost-effectiveness estimates separately for each relevant subgroup (section 4.9.1), so pooled figures are reported beside the subgroup values, not in place of them.

    Source: 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 6 Committee recommendations, section 6.3.2, which asks for net health benefit at GBP 25,000 and GBP 35,000 per QALY, both with and without the QALY weighting applied, and chapter 4 Economic evaluation, section 4.9.1, which asks for clinical and cost-effectiveness estimates separately for each relevant subgroup of patients. Cookson R, Mirelman AJ, Griffin S, Asaria M, Dawkins B, Norheim OF, Verguet S, Culyer AJ. Using cost-effectiveness analysis to address health equity concerns. Value in Health. 2017;20(2):206-212. Section on equity-weighting analysis, on weights applied to the health benefits of people with different characteristics.

    E_u = N_1 * dE_1 + N_2 * dE_2; E_w = N_1 * w_1 * dE_1 + N_2 * w_2 * dE_2; C_tot = N_1 * dC_1 + N_2 * dC_2; H_d = C_tot / lambda; NHB_u = E_u - H_d; NHB_w = E_w - H_d; ICER_u = C_tot / E_u; ICER_w = C_tot / E_w

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Implementations

  • Excel

    Weighted and unweighted net health benefit from named cells

    With the cells named as for the weighted ICER, the first formula returns the weighted and the second the unweighted net health benefit per patient. Both are run with QALYValue set to 25000 and to 35000.

    =QALYWeight*IncrementalQALYs-IncrementalCost/QALYValue; =IncrementalQALYs-IncrementalCost/QALYValue

Assumptions

  • Displaced health is not equity weighted

    The health lost elsewhere is counted in unweighted QALYs, as in the NICE presentation of net health benefit with and without the weighting (PMG36, section 6.3.2), which is why the weighted value can be positive while total population health falls. Cookson and colleagues note that the distribution of the opportunity cost also matters, which this formula does not capture.

  • Net health benefit reported at two values of a QALY, weighted and unweighted

    PMG36 section 6.3.2 asks for net health benefit at GBP 25,000 and GBP 35,000 per QALY, both with and without the QALY weighting, so four values are normally reported for each weighted programme or subgroup.

  • Per-patient net health benefit scaled by group size

    Delta_E and Delta_C are per patient, so NHB_w and NHB_u are per patient. Population values multiply each by the number of patients in the group, and several groups are added in HE-CF-EQWA-001.

Worked examples

  • Severe subgroup net health benefit at GBP 35,000 per QALY

    In the article's illustrative subgroup A, GBP 30,000 displaces about 0.8571 QALYs at GBP 35,000 per QALY. The weighted net health benefit is about 0.1629 QALYs per patient and the unweighted value about minus 0.2571, so the weight turns a health loss into a weighted gain.

    Delta_C = 30000; lambda = 35000; w = 1.7; Delta_E = 0.6; H_d = 0.8571; NHB_w = 0.1629; NHB_u = -0.2571

  • Severe subgroup net health benefit at GBP 25,000 per QALY

    At GBP 25,000 per QALY the same GBP 30,000 displaces 1.2 QALYs, and net health benefit is minus 0.18 QALYs per patient weighted and minus 0.6 unweighted, both negative.

    Delta_C = 30000; lambda = 25000; w = 1.7; Delta_E = 0.6; H_d = 1.2; NHB_w = -0.18; NHB_u = -0.6

  • Subgroup with a weight of 1 where both net health benefits agree

    Subgroup B gains 0.8 QALYs per patient at an extra GBP 24,000 and carries a weight of 1. At GBP 35,000 per QALY about 0.6857 QALYs are displaced and both net health benefits are about 0.1143 QALYs per patient.

    Delta_C = 24000; lambda = 35000; w = 1; Delta_E = 0.8; H_d = 0.6857; NHB_w = 0.1143; NHB_u = 0.1143

Common errors

  • Dividing the displaced health by the equity weight

    Weighting the displaced health as well, by dividing the GBP 30,000 by 1.7 times GBP 35,000, gives about 0.5158 QALYs per patient for subgroup A instead of 0.1629. Only the QALY gain carries the weight.

  • Reporting only the weighted net health benefit

    For the 300 patients of subgroup A at GBP 35,000 per QALY, the weighted net health benefit is about 48.9 QALYs, while the unweighted value is about minus 77.1 QALYs, a fall in total population health. PMG36 section 6.3.2 asks for both so that this loss stays visible.

Sources

  • NICE presentation of net health benefit with and without QALY weighting

    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 6 Committee recommendations, section 6.3.2 (net health benefit presented at GBP 25,000 and GBP 35,000 per QALY, both with and without the QALY weighting; negative values mean overall health loss from care not funded elsewhere; technologies with negative unweighted net health benefit may still be recommended when modifiers apply).

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  • Net health benefit framework for equity-weighted comparisons

    Stinnett AA, Mullahy J. Net health benefits: a new framework for the analysis of uncertainty in cost-effectiveness analysis. Medical Decision Making. 1998;18(2 Suppl):S68-S80. Abstract, which presents the net health benefits approach as an alternative to cost-effectiveness ratios. The unweighted measure, the QALY gain less the cost divided by the value of a QALY, is the live record HE-FM-CET-002.

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Canonical Identity

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