Functions & Formulae

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

Value measurement function for MCDA in HTA from criterion performance, ranges and swing points

m(x_aj, x_worst_j, x_best_j, p_j) = (v_aj, w_j, V_a)

Maps each technology's performance on each criterion, the fixed range of each criterion and the swing points given to each criterion to a partial value score per criterion on a 0 to 100 interval scale, a normalised weight per criterion and an overall value per technology. This is the value measurement approach, which the first ISPOR MCDA task force report describes as by far the most widely used in health care. HE-FM-MCDH-001 gives the linear partial value score, HE-FM-MCDH-002 the swing weights and the additive overall value, and HE-CF-MCDH-001 runs both for three technologies with a ranking and a scan of one weight. The second report notes that additive functions are also used in health-related quality of life instruments, and HE-FM-15D-001, the 15D score as an additive weighted sum of level values with importance weights elicited from population samples, is a live instance of that form; here the weights come from swing points elicited over stated ranges. When one weight varies and the others keep their ratio, each total is linear in that weight, and the weight at which the totals of two technologies are equal, a change of ranking only when it is at least 0 and at most 1, takes the form of the switching weight w_s of HE-FM-WE-004, which that record computes only for the QALYs of two groups, here with the other-criteria blend in place of group A QALYs and the first score less that blend in place of group B QALYs (derived). Dividing a cost by an overall value takes the form of HE-FM-ACER-001, and the second report states that such a cost-benefit ratio needs benefits on a ratio scale, which these totals lack. The partial value score has the algebra of 100 times the blend share p of HE-FM-EXD-002, which that record computes only for the expected effects of three options ordered along a frontier; here the ends are the anchors of a criterion's fixed range and run in decreasing order when lower performance is better. The weights have the algebra of the share in HE-FM-CJA-003, which that record computes for conjoint part-worth ranges, and the overall value equals the points-weighted mean (p_1 v_1 + p_2 v_2 + p_3 v_3) / (p_1 + p_2 + p_3), the form of HE-FM-CRAT-001, which computes it for member counts and expected costs (derived). The swing of HE-FM-OWSA-001 is the width of a tornado bar, not a swing weight. Notation follows the MCDA in HTA article, with v_1, v_2 and v_3 for v_j(x_aj) and V for V(a).

  • Linear partial value score of a technology on one MCDA criterion

    v = 100 * (x - x_worst) / (x_best - x_worst)

    Places a technology's performance on one criterion on a 0 to 100 scale whose ends are the worst and best performance of the criterion's fixed range, so equal changes in performance earn equal changes in score. The same line serves a criterion on which lower performance is better, such as a rate of adverse events, by entering the worse, higher level as x_worst, as the first ISPOR MCDA task force appendix does for its criterion B. Zero marks the bottom of the chosen range, not an absence of value, so differences in score can be compared but ratios cannot. Where equal changes in performance are not equally valued, the score is read from an elicited nonlinear partial value function instead.

  • Swing-weighted additive value of a technology on three MCDA criteria

    w_1 = p_1 / (p_1 + p_2 + p_3); w_2 = p_2 / (p_1 + p_2 + p_3); w_3 = p_3 / (p_1 + p_2 + p_3); V = w_1 * v_1 + w_2 * v_2 + w_3 * v_3

    Divides each criterion's swing points by the total points to give weights that sum to one, then multiplies each partial value score by its criterion's weight and adds the three products, the additive (weighted-sum) model that both ISPOR MCDA task force reports describe as the usual aggregation in health care. With partial values on a 0 to 100 scale the overall value is on the same scale. The weights act as scaling constants for the ranges over which the swings were judged, so a change of range calls for new points. The formula is written for three criteria, the article's health gain, severity and strength of evidence; more criteria add terms in the same way, and HE-CF-MCDH-001 applies it to several technologies.