Functions & Formulae

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

Country-specific value set comparison function

V(s_0, s_1; theta_A, theta_B) = (g_A, g_B, dg)

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.

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

    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

    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.

  • Relative change in a QALY gain and in the ICER when an analysis is rescored with another value set

    r_Q = dQ_B / dQ_A - 1; r_I = dQ_A / dQ_B - 1

    Compares the QALY gain of the same treatment scored with two value sets, each the HE-FM-EQ3L-003 gain or any other QALY gain, and expresses the second as a share of the first. r_Q is the share of the QALY gain that changes with the value set, and with the incremental cost unchanged the ICER of HE-FM-CUA-001 changes by r_I, the reciprocal ratio less 1 (derived). The two changes are not mirror images, since (1 + r_Q) times (1 + r_I) equals 1, so a fall of 13% in the QALY gain raises the ICER by more than 13%. The gap in incremental net health benefit (HE-FM-CET-002) between the two sets is dQ_A minus dQ_B, since the cost term is the same under both (derived).

  • Count-based D1 value set form with interaction terms in place of a constant

    D_1 = max(n_2 + n_3 - 1, 0); I_2 = max(n_2 - 1, 0); I_3 = max(n_3 - 1, 0); U = 1 - (beta_MO + beta_SC + beta_UA + beta_PD + beta_AD) - a_D1 * D_1 - a_I22 * I_2^2 - a_I3 * I_3 - a_I32 * I_3^2

    Scores an EQ-5D-3L profile with the D1 form that Shaw, Johnson and Coons used for the US time trade-off set. There is no constant and no N3 term. The level decrements are subtracted as in HE-FM-EQ3L-001, and the interaction effects of problems on several dimensions enter through counts beyond the first, D1 for dimensions with any problem, I2 for dimensions at level 2 (squared) and I3 for dimensions at level 3 (linear and squared). HE-AS-EQ3L-003 records that this form cannot be scored with the constant and N3 formula, and Rand-Hendriksen and colleagues show that it equals a constant model with every level decrement shifted by the D1 coefficient. The coefficients in the examples are illustrative, not the published US values.