Signature
S = L^K; P = K * (L - 1); f_v = n_v / S
| Inputs | Definition | Unit |
|---|---|---|
L | Number of response levels on each dimension, 5 for the EQ-5D-5L | levels |
K | Number of dimensions in the descriptive system, 5 for the EQ-5D | dimensions |
n_v | Distinct health states in the valuation study's time trade-off design | states |
S | Count of distinct profiles, one per combination of levels across the dimensions | states |
|---|---|---|
P | Count of dimension and level decrements when level 1 carries none, before any constant or interaction term | parameters |
f_v | Proportion of the S states that appear in the time trade-off design | proportion |
Function
EQ-5D-5L value set scoring function
Maps an EQ-5D-5L profile, the level 1 to 5 reported on each of mobility, self-care, usual activities, pain or discomfort and anxiety or depression, to a single utility under the coefficients of a chosen five-level value set. The five-level instrument describes 3,125 states, and a main-effects value set gives each dimension its own decrement at each of levels 2 to 5, 20 decrements in all, plus a constant where the set reports one. Once the decrement for the reported level of each dimension has been looked up (HE-FM-EQ5L-001), the profile score is the main-effects form of HE-FM-EQ3L-001 with the N3 decrement set to zero, and the three-level tariff itself is HE-FM-EQ3L-002; the 15D counterpart is HE-FM-15D-001. The utilities it returns weight time in QALYs (HE-FM-QALY-001), and a QALY difference between two scored states is HE-FM-EQ3L-003, which applies only when both states are scored with the same set. Scoring one profile under the value sets of two countries belongs to the Country-Specific Value Set page. Notation follows the EQ-5D-5L Value Set article.
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Implementations
Excel
EQ-5D state count from named cells
With the levels per dimension in a cell named LevelsPerDimension and the number of dimensions in DimensionCount, the formula returns S in a cell named StateCount.
=LevelsPerDimension^DimensionCount
Excel
EQ-5D main-effects decrement count from named cells
With the same two cells, the formula returns P in a cell named DecrementCount.
=DimensionCount*(LevelsPerDimension-1)
Excel
Share of EQ-5D states valued from named cells
With the number of states in the time trade-off design in a cell named StatesValued, the formula returns f_v in a cell named ShareValued.
=StatesValued/StateCount
Assumptions
Every combination of EQ-5D-5L levels counted as a state
Each dimension is answered independently, so every combination of levels is a possible profile, and the count leaves out dead, which anchors the scale and is not a profile. Devlin and colleagues give the count as 3,125, 5 to the power 5, against 243, 3 to the power 5, for the three-level version.
Share of EQ-5D-5L states valued counts the time trade-off design only
n_v counts the distinct states in the time trade-off design. Choice tasks in a hybrid design cover other states, which are not counted here, so f_v is the share of states with a directly observed value on the utility scale.
Worked examples
EQ-5D-5L state count and the England time trade-off design
Five dimensions with five levels describe 3,125 states and need 20 decrements in a main-effects set. The England study valued 86 states by time trade-off, a share of 0.0275, under 3% of the states.
L = 5; K = 5; n_v = 86; S = 3125; P = 20; f_v = 0.0275
EQ-5D-5L state count and the UK time trade-off design
The UK valuation study of Rowen and colleagues 2026 used composite time trade-off for 102 states, a share of 0.0326 of the 3,125.
L = 5; K = 5; n_v = 102; S = 3125; P = 20; f_v = 0.0326
EQ-5D-3L state count and the UK MVH design for comparison
Three levels on five dimensions give 243 states and 10 level decrements. The UK three-level study valued 42 states by time trade-off (HE-SR-EQ3L-001), a share of 0.1728, about six times the England five-level share.
L = 3; K = 5; n_v = 42; S = 243; P = 10; f_v = 0.1728
Common errors
Counting EQ-5D-5L states as levels times dimensions
Multiplying 5 levels by 5 dimensions gives 25, the number of answer options on the questionnaire, not the number of states. The states are every combination of the five answers, 5 to the power 5, 3,125.
Reading every EQ-5D-5L value as directly observed
In the England study 86 states were valued by time trade-off, so the values of the other 3,039 states are model predictions. The choice of model therefore shapes most of a five-level value set, and a dispute about the model is a dispute about most of the values.
Sources
England EQ-5D-5L design and parameter count
Devlin NJ, Shah KK, Feng Y, Mulhern B, van Hout B. Valuing health-related quality of life: an EQ-5D-5L value set for England. Health Economics. 2018;27(1):7-22. Section 1, introduction, which states that the five dimensions and five levels of the EQ-5D-5L describe 3,125 (5 to the power 5) unique health states, against 243 (3 to the power 5) for the three-level EQ-5D; section 2.2, study design, which states that the time trade-off design included 86 states, the worst state 55555 and the five mildest states selected in advance plus 80 states from the design, spread over 10 blocks; section 2.5, which counts 20 parameters (4 levels times 5 dimensions) in the least restrictive model; section 3.4, which refers to the 3,125 health states described by the EQ-5D-5L.
Quality assurance of the England EQ-5D-5L value set on coverage of states
Hernández Alava M, Pudney S, Wailoo A. The EQ-5D-5L value set for England: findings of a quality assurance program. Value in Health. 2020;23(5):642-648. Abstract, which states that the time trade-off experiments covered less than 3% of the possible 3125 five-level health states and the discrete choice experiments 12.5% of possible states, and that significant limitations were identified in the valuation data and the statistical analysis behind the English value set.
UK EQ-5D-5L valuation study design
Rowen D, Mukuria C, Bray N, Carlton J, Longworth L, Meads D, Oluboyede Y, O'Neill C, Yang Y. A United Kingdom value set for the EQ-5D-5L. Value in Health. 2026;29(5):858-869. Abstract, which reports composite time trade-off interviews for 102 health states with a representative sample of UK adults, 1200 interviews in all, and a value set generated with a random effects Tobit model. Cited at abstract level only, and its coefficients are not used here.
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