Signature
v = 100 * (x - x_worst) / (x_best - x_worst)
| Inputs | Definition | Unit |
|---|---|---|
x | Performance of the technology on the criterion in the criterion's natural unit, for example 0.6 QALYs per patient for health gain | the criterion's unit |
x_worst | Performance level scored 0, the worst realistic performance of the range; it is the higher level when lower performance is better | the criterion's unit |
x_best | Performance level scored 100, the best realistic performance of the range, different from x_worst | the criterion's unit |
v | Score of the technology on the criterion, 0 at the worst and 100 at the best performance of the range | points on a 0 to 100 interval scale |
|---|
Function
Value measurement function for MCDA in HTA from criterion performance, ranges and swing points
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).
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Implementations
Excel
Excel linear partial value score for an MCDA criterion
With the performance in a cell named PerformanceLevel and the range ends in cells named WorstPerformance and BestPerformance, the formula returns the score in a cell named PartialValueScore. For the article's technology A the cells hold 0.6, 0 and 0.8 and the formula returns 75. For a lower-is-better criterion the higher level goes in WorstPerformance and the formula is unchanged. A range with equal ends, or two blank end cells, returns #DIV/0!.
=100*(PerformanceLevel-WorstPerformance)/(BestPerformance-WorstPerformance)
Assumptions
Linear partial value function on the MCDA criterion range
Equal changes in performance anywhere in the range are equally valued, the linear case that the first ISPOR task force appendix assumes for its criteria A and B. For its criterion C, where a change near the bottom of the range is valued more than the same change near the top, the appendix uses a nonlinear partial value function, which it says could be developed by the bisection method, and this formula does not apply.
Performance inside the fixed range of the MCDA criterion
x lies in the range whose ends are x_worst and x_best, so v is at least 0 and at most 100; a level outside the range returns a score below 0 or above 100 (derived). For criteria reused across decisions, as in HTA, the second ISPOR report asks for a range covering the best and worst performance that could realistically occur, a global or fixed scale, rather than the range of the technologies in hand.
Interval meaning of a partial value score in MCDA
The score is an interval scale whose zero is the bottom of the range. For a range of 6 (scored 0) to 10 (scored 100) with a linear partial value function, the second ISPOR report states that going from 6 to 8 is as good as going from 8 to 10, but not that 10 is twice as good as 8.
Worked examples
Health gain score of technology A in the MCDA in HTA example
In the article's illustrative example, health gain runs over a fixed range of 0 to 0.8 QALYs per patient with a linear value function. Technology A gains 0.6 QALYs and scores 75; technology B, with 0.4 QALYs, scores 50 in the same way.
x = 0.6; x_worst = 0; x_best = 0.8; v = 75
Lower-is-better criterion score in the ISPOR MCDA appendix
The first ISPOR task force appendix scores its criterion B, on which lower performance is better, over a range of 5 to 10 bb, so 10 bb is the worst level and 5 bb the best. Alternative 1, at 8 bb, scores 40 and alternative 2, at 6.5 bb, scores 70, as in the appendix's Table A2, and in its scenario analysis 6 bb scores 80. Its criterion A, on which higher is better, runs over 65 to 90 aa and gives 80 for 85 aa and 32 for 73 aa.
x = 8; x_worst = 10; x_best = 5; v = 40
Health gain score at the bottom of the range for technology C
Technology C in the article has no health gain, the bottom of the 0 to 0.8 QALY range, and scores 0, the limiting case. A score of 0 marks the worst end of the chosen range, not the absence of value on the criterion.
x = 0; x_worst = 0; x_best = 0.8; v = 0
Common errors
Local range of the appraised technologies in place of a fixed MCDA scale
Scoring health gain on the range of the three technologies in hand, 0 to 0.6 QALYs, in place of the fixed 0 to 0.8 range gives A 100 and B 66.67 in place of 75 and 50, and with the article's weights A's total becomes 85 and B's 73.33 in place of 72.5 and 65 (computed here for illustration). The swing weight of 0.50 was judged for a swing of 0.8 QALYs, so it no longer matches the scores, and the scale moves whenever a new technology widens the range. The second ISPOR report keeps local scales for one-off decisions and asks for global scales where criteria are reused, as in HTA.
Worst and best ends swapped on a lower-is-better MCDA criterion
Entering 5 bb as x_worst and 10 bb as x_best for the ISPOR appendix's criterion B, on which lower is better, returns 100 less the correct score, 60 in place of 40 for alternative 1 and 30 in place of 70 for alternative 2, so the alternative with the lower, better level is scored as the worse one.
Reading MCDA partial value scores as ratios
In the article's example severity scores 50 for a moderate and 100 for a severe condition, which does not make treating a severe condition twice as valuable, because zero marks a mild condition, the bottom of the chosen range. The second ISPOR report makes the same point with a range of 6 to 10, on which 10 scores 100 and 8 scores 50 but 10 is not twice as good as 8.
Sources
ISPOR MCDA task force report 1 appendix on linear partial value functions
Thokala P, Devlin N, Marsh K, Baltussen R, Boysen M, Kalo Z, et al. Multiple criteria decision analysis for health care decision making: an introduction. Report 1 of the ISPOR MCDA Emerging Good Practices Task Force. Value in Health. 2016;19(1):1-13. Appendix, MCDA Modeling Example, Scoring, develops partial value functions over the feasible range of each criterion, with the score often defined on a scale of 0 to 100 points, and assumes linear functions for criterion A (65 to 90 aa, higher better) and criterion B (5 to 10 bb, lower better) in Figures A1 and A2; Table A2 gives the scores 80 and 32 on criterion A and 40 and 70 on criterion B, and the Uncertainty Analysis paragraph scores 6 bb on criterion B at 80. Criterion C has a nonlinear function (Figure A3), which the appendix says could be developed by the bisection method.
ISPOR MCDA task force report 2 on scoring scales and global ranges
Marsh K, IJzerman M, Thokala P, Baltussen R, Boysen M, Kaló Z, et al. Multiple criteria decision analysis for health care decision making: emerging good practices. Report 2 of the ISPOR MCDA Emerging Good Practices Task Force. Value in Health. 2016;19(2):125-137. Step 2b, Report and justify the criteria definitions, states that the use of criteria over multiple decisions, as in HTA, requires a range covering the best and worst performance that could realistically occur (global or fixed scales), while a one-off decision can use the best and worst performance of the alternatives evaluated (local or relative scales). Step 4a, Report and justify the methods used for scoring, notes that a 0 to 100 scale is often used and that, for a range of 6 (scored 0) to 10 (scored 100) with a linear partial value function, going from 6 to 8 is as good as going from 8 to 10, but 10 is not twice as good as 8.
Canonical Identity
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