Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

EQ-5D-5L value set scoring function

u5(l_MO, l_SC, l_UA, l_PD, l_AD; theta) = U

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.

  • EQ-5D-5L dimension decrement at the reported level from four level coefficients

    beta_d = b_2 * (min(l_d - 1, 1) - min(max(l_d - 2, 0), 1)) + b_3 * (min(max(l_d - 2, 0), 1) - min(max(l_d - 3, 0), 1)) + b_4 * (min(max(l_d - 3, 0), 1) - max(l_d - 4, 0)) + b_5 * max(l_d - 4, 0)

    Returns the decrement that a five-level value set attaches to the level reported on one dimension. Level 1 carries no decrement, and each of levels 2 to 5 has its own coefficient, which is the full decrement for that level and not a step to be added to the lower ones. The level indicators are written with min and max so that the site calculator picks exactly one coefficient for a whole-number level. Five such lookups, one per dimension, give the decrements beta_d of the article's additive form U(s) = 1 minus their sum, scored for a whole profile in HE-CF-EQ5L-001.

  • England EQ-5D-5L value set of Devlin and colleagues from five reported levels

    U = 1 - (0.058 * (min(l_MO - 1, 1) - min(max(l_MO - 2, 0), 1)) + 0.076 * (min(max(l_MO - 2, 0), 1) - min(max(l_MO - 3, 0), 1)) + 0.207 * (min(max(l_MO - 3, 0), 1) - max(l_MO - 4, 0)) + 0.274 * max(l_MO - 4, 0)) - (0.050 * (min(l_SC - 1, 1) - min(max(l_SC - 2, 0), 1)) + 0.080 * (min(max(l_SC - 2, 0), 1) - min(max(l_SC - 3, 0), 1)) + 0.164 * (min(max(l_SC - 3, 0), 1) - max(l_SC - 4, 0)) + 0.203 * max(l_SC - 4, 0)) - (0.050 * (min(l_UA - 1, 1) - min(max(l_UA - 2, 0), 1)) + 0.063 * (min(max(l_UA - 2, 0), 1) - min(max(l_UA - 3, 0), 1)) + 0.162 * (min(max(l_UA - 3, 0), 1) - max(l_UA - 4, 0)) + 0.184 * max(l_UA - 4, 0)) - (0.063 * (min(l_PD - 1, 1) - min(max(l_PD - 2, 0), 1)) + 0.084 * (min(max(l_PD - 2, 0), 1) - min(max(l_PD - 3, 0), 1)) + 0.276 * (min(max(l_PD - 3, 0), 1) - max(l_PD - 4, 0)) + 0.335 * max(l_PD - 4, 0)) - (0.078 * (min(l_AD - 1, 1) - min(max(l_AD - 2, 0), 1)) + 0.104 * (min(max(l_AD - 2, 0), 1) - min(max(l_AD - 3, 0), 1)) + 0.285 * (min(max(l_AD - 3, 0), 1) - max(l_AD - 4, 0)) + 0.289 * max(l_AD - 4, 0))

    Scores a profile with the EQ-5D-5L value set for England published by Devlin and colleagues in 2018, 1 minus the Table 2 decrement for the level reported on each dimension, with no separate constant decrement. Each bracket is the lookup of HE-FM-EQ5L-001 with that dimension's England coefficients. The lowest value is minus 0.285 for 55555, the highest below full health is 0.950, and 159 of the 3,125 states score below zero. NICE does not use this set for reference-case analyses, and the set NICE now names is not formalised here.

  • Number of EQ-5D-5L health states and main-effects decrements, and the share valued directly

    S = L^K; P = K * (L - 1); f_v = n_v / S

    Counts the states that a descriptive system with K dimensions of L levels describes, the number of decrements a main-effects value set needs when level 1 carries none, and the share of states that respondents valued directly. For the EQ-5D-5L the count is 5 to the power 5, 3,125 states, and 5 dimensions times 4 levels above level 1 give 20 decrements, the size of the England model in HE-FM-EQ5L-002. The same counts give 243 states and 10 level decrements for the EQ-5D-3L, whose UK tariff in HE-FM-EQ3L-002 adds a constant and N3 to them. The share valued shows how much of a value set rests on model prediction.

EQ-5D-5L Value Set — Functions & Formulae | HealthEconomics.wiki