Utility of two conditions combined by the additive, multiplicative and minimum rules on a shared baseline

Signature

U_add = U_0 - (U_0 - U_A) - (U_0 - U_B); U_mult = U_0 * (U_A / U_0) * (U_B / U_0); U_min = min(U_0, U_A, U_B)

Function

Utility decrements, their QALY losses and the rules for combining them on a stated baseline

Computational function

• Computational function: combined utility, QALY gain and cost per QALY under three combination rules for any number of conditions

  Combines any number of single-condition utilities on a shared baseline under the three TSD 12 rules, and returns for each rule the combined utility, the utility without the last condition, the QALY gain from avoiding the last condition over a stated duration and the cost per QALY gained. The inputs differ from the formula's: a vector of single-condition utilities of any length, a duration and a cost per case avoided. TSD 12 and the article give the rules for two conditions; applying the same rule to more conditions (sum of decrements, product of ratios, lowest value) is the convention stated here; TSD 12 section 3.2.1 notes that very little research has explored the accuracy of the methods for more than two simultaneous conditions.

  Inputs and outputs: U0: Utility of people with none of the conditions. Unit: utility.; U: Utilities of people with each single condition, the condition avoided last. Unit: utility.; dur: Years the last condition would have lasted. Unit: years.; cost: Cost per case of the last condition avoided. Unit: pounds.; combined: Combined utility with all conditions under each rule (HE-FM-DISU-002). Unit: utility.; without_last: Combined utility without the last condition. Unit: utility.; qaly_gain: Undiscounted QALY gain per case avoided (HE-FM-DISU-003). Unit: QALYs.; cost_per_qaly: Cost per QALY gained, missing when the gain is zero. Unit: pounds per QALY.

  Assumption: One shared baseline, no measured utilities for combined states, a constant decrement over dur and no deaths, as in the article.

  Worked example (Conditions A and B, article example): With 0.80, 0.70 and 0.60, five years and 20,000 pounds per case of B avoided, the combined utilities are 0.50, 0.525 and 0.60, the gains 1.00, 0.875 and 0.50 QALYs and the costs per QALY 20,000, 22,857.14 and 40,000 pounds, as in the article. U0 = 0.8; U = 0.7, 0.6; dur = 5; cost = 20000; cost_per_qaly = 20000, 22857.14, 40000

  Worked example (A third condition with utility 0.75 avoided): Adding a third condition with single-condition utility 0.75 and avoiding it instead gives combined utilities of 0.45, 0.4922 and 0.60, gains of 0.25, 0.1641 and 0 QALYs and costs per QALY of 80,000 and 121,904.76 pounds, with no gain under the minimum rule because the third condition is not the worst (computed here for illustration). U0 = 0.8; U = 0.7, 0.6, 0.75; dur = 5; qaly_gain = 0.25, 0.1641, 0

  Excel: With NeitherUtil named and the single-condition utilities in the range SingleUtils, =NeitherUtil-SUMPRODUCT(NeitherUtil-SingleUtils) returns the additive value, =NeitherUtil*PRODUCT(SingleUtils/NeitherUtil) the multiplicative value (entered as an array formula in older versions) and =MIN(NeitherUtil,SingleUtils) the minimum, held in UAddAll, UMultAll and UMinAll; the same formulas on the range without the last condition give the values without it, and HE-FM-DISU-003 gives the gains.

  R: combine_util <- function(U0, U, dur, cost) { k <- length(U); f <- function(u) c(additive = U0-sum(U0-u), multiplicative = U0*prod(u/U0), minimum = min(c(U0, u))); w <- f(U); wo <- f(U[-k]); g <- (wo-w)*dur; data.frame(rule = names(w), combined = w, without_last = wo, qaly_gain = g, cost_per_qaly = ifelse(g > 0, cost/g, NA), row.names = NULL) } Base R only; combine_util(0.80, c(0.70, 0.60), 5, 20000) returns the article example.

  Python: def combine_util(U0, U, dur, cost): f = lambda u: {"additive": U0-sum(U0-x for x in u), "multiplicative": U0*math.prod(x/U0 for x in u), "minimum": min([U0]+list(u))}; w = f(U); wo = f(U[:-1]); g = {k: (wo[k]-w[k])*dur for k in w}; return {k: {"combined": w[k], "without_last": wo[k], "qaly_gain": g[k], "cost_per_qaly": cost/g[k] if g[k] > 0 else None} for k in w} Needs import math (Python 3.8 or later for math.prod); returns the same values as the R function.

  Test (Rules in rising order for any number of conditions): When no single-condition utility is above NeitherUtil or below zero, the additive value is no higher than the multiplicative value and the multiplicative no higher than the minimum, because a product of ratios no greater than 1 is at least 1 minus the sum of the proportional decrements and no greater than any one ratio. Expected result: TRUE. FALSE shows the multiplicative rule applied to the raw utilities of the three-condition example (0.315), below the additive 0.45. Excel check: =AND(UAddAll<=UMultAll+1E-12,UMultAll<=UMinAll+1E-12)

  Common error (Reporting one rule's cost per QALY as if the rule did not matter): In the article the decision against an illustrative threshold of 25,000 pounds per QALY changes with the combination rule alone, so all three results belong in the scenario analysis.

  Source: Ara R, Wailoo A. NICE DSU Technical Support Document 12: The use of health state utility values in decision models. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; July 2011 (full text read). Appendix, equations 1, 3 and 5; sections 3.2.1 and 5.2.1.

  U_add = U0 - sum(U0 - U_k); U_mult = U0 * prod(U_k / U0); U_min = min(U0, U_k); qaly_gain = (U_rule without last - U_rule with all) * dur; cost_per_qaly = cost / qaly_gain

Try this function

Implementations

• Excel

  Combined utility under three rules from named cells

  With NeitherUtil, UtilA and UtilB named, the formulas return the additive, multiplicative and minimum estimates, held in UAdd, UMult and UMin.

  =NeitherUtil-(NeitherUtil-UtilA)-(NeitherUtil-UtilB); =NeitherUtil*(UtilA/NeitherUtil)*(UtilB/NeitherUtil); =MIN(NeitherUtil,UtilA,UtilB)

Assumptions

• Shared baseline for both single-condition decrements

  U_0 is the utility of people with neither condition from the same dataset and age group, so the TSD 12 baselines for each condition coincide; TSD 12 states that assuming perfect health as the baseline for people without a condition is inappropriate.

• No utilities measured in people with both conditions

  The rules estimate a combined state that has not been measured; when utilities from people with both conditions exist they are used directly.

Worked examples

• Conditions A and B from a baseline of 0.80

  With 0.80 for neither condition, 0.70 for A and 0.60 for B, the additive rule gives 0.50, the multiplicative 0.80 x 0.875 x 0.75 = 0.525 and the minimum 0.60, as in the article.

  U_0 = 0.8; U_A = 0.7; U_B = 0.6; U_add = 0.5; U_mult = 0.525; U_min = 0.6

• Same conditions combined as if from full health

  Combining the raw utilities on a baseline of 1 gives a multiplicative value of 0.70 x 0.60 = 0.42, as in the article's step 5, and an additive value of 0.30 (computed here for illustration).

  U_0 = 1; U_A = 0.7; U_B = 0.6; U_add = 0.3; U_mult = 0.42; U_min = 0.6

Common errors

• Combining raw utilities as if the baseline were full health

  In the article the multiplicative estimate falls from 0.525 to 0.42 and the QALY gain per case of B avoided rises from 0.875 to 1.40 when the baseline of 0.80 is replaced by 1; TSD 12 states that assuming perfect health as the baseline is inappropriate.

• Choosing the combination rule without testing the others

  TSD 12 reports that no method is unequivocally supported and that all conventional rules produced some large errors, so the rule used should be stated and the others tested in scenario analysis; in the article the cost per QALY moves from 20,000 to 40,000 pounds with the rule alone.

Sources

• TSD 12 additive, multiplicative and minimum methods for combined conditions

  Ara R, Wailoo A. NICE DSU Technical Support Document 12: The use of health state utility values in decision models. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; July 2011 (full text read). Appendix, equations 1, 3 and 5: the additive method assumes a constant absolute decrement relative to the baseline, the multiplicative method a constant proportional decrement relative to the baseline, and the minimum method a decrement equal to the maximum decrement of the single conditions; equations 2 and 4 give the forms with a baseline of perfect health.

  View source →

• TSD 12 evidence on the accuracy of the combination methods and the advice to use the multiplicative method

  Ara R, Wailoo A. NICE DSU Technical Support Document 12: The use of health state utility values in decision models. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; July 2011 (full text read). Section 3.2.1, pp. 13-14: there is no unequivocal evidence supporting one method; although all estimated some values with substantial errors, the multiplicative appears the most accurate overall. Section 5.2.1: when utilities from cohorts with combined conditions are not available, the multiplicative method should be used, with the multiplier estimated from age-adjusted data as a minimum. Executive summary: it is inappropriate to assume the baseline is perfect health.

  View source →

• PMG36 preference for a multiplicative adjustment of utilities

  National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; 2022, last updated 31 March 2026 (full text of chapter 4 read). Section 4.3.7: adjustments to utility values may be needed, for example for age or comorbidities; a multiplicative approach is generally preferred, and the methods used should be clearly documented.

  View source →

Canonical Identity