Incremental utility-cost ratio in QALYs gained per one million of net expenditure

Signature

IUCR = Delta_Q / Delta_C * 1000000

Function

Incremental utility-cost function as the inverse of an incremental cost-utility ratio

Computational function

• Computational function: incremental cost-utility and utility-cost ratios with quadrant from arm-level costs and QALYs

  Takes the expected cost and expected QALYs of each arm, or matched probabilistic draws of them, and returns both forms PHARMAC asks to be reported: the ICUR of HE-FM-CUA-001 and the IUCR of HE-FM-ICUR-001, with the quadrant of the cost-effectiveness plane. Draws are averaged before any ratio is formed. The inputs differ from the formula's variables: the function builds Delta_C and Delta_Q from arm-level values, and it returns a missing value for a ratio whose denominator is zero.

  Inputs and outputs: C_1: Expected discounted cost per patient with the intervention, or a vector of probabilistic draws; required. Unit: currency units.; C_0: Expected discounted cost per patient with the comparator, or matched draws; required. Unit: currency units.; Q_1: Expected discounted QALYs per patient with the intervention, or matched draws; required. Unit: QALYs.; Q_0: Expected discounted QALYs per patient with the comparator, or matched draws; required. Unit: QALYs.; scale: Spending unit for the IUCR, default 1,000,000. Unit: currency units.; Delta_C, Delta_Q: Mean incremental cost and mean incremental QALYs. Unit: currency units; QALYs.; ICUR: Mean incremental cost over mean incremental QALYs. Unit: currency units per QALY.; IUCR: Mean incremental QALYs per scale units of mean incremental cost. Unit: QALYs per 1,000,000 currency units.; quadrant: north-east, north-west, south-east, south-west or on an axis. Unit: label.

  Assumption: Both arms share the comparator, population, horizon, price year and discount rate, and draws are matched pairs from one probabilistic run. The quadrant label is returned because the ratios alone do not show whether a result is a trade-off or a case of dominance.

  Worked example (New treatment against usual care): The article's arms give GBP 8,400 and 0.40 QALYs, an ICUR of GBP 21,000 per QALY and an IUCR of about 47.619, in the north-east quadrant. C_1 = 24600; C_0 = 16200; Q_1 = 6.42; Q_0 = 6.02; Delta_C = 8400; Delta_Q = 0.40; ICUR = 21000; IUCR = 47.619

  Worked example (Against the dominated older drug): The older drug costs GBP 18,000 and yields 5.90 QALYs, giving increments of GBP 6,600 and 0.52 QALYs, an ICUR of about GBP 12,692.308 and an IUCR of about 78.788. C_1 = 24600; C_0 = 18000; Q_1 = 6.42; Q_0 = 5.90; Delta_C = 6600; Delta_Q = 0.52; ICUR = 12692.308; IUCR = 78.788

  Worked example (QALY loss at extra cost): With 6.47 QALYs for the comparator the intervention loses 0.05 QALYs at an extra GBP 8,400 (illustrative), so both ratios are negative and the quadrant is north-west. C_1 = 24600; C_0 = 16200; Q_1 = 6.42; Q_0 = 6.47; Delta_C = 8400; Delta_Q = -0.05; ICUR = -168000; IUCR = -5.952

  Excel: =LET(dC,AVERAGE(Cost1)-AVERAGE(Cost0),dQ,AVERAGE(QALY1)-AVERAGE(QALY0),HSTACK(IF(dQ=0,NA(),dC/dQ),IF(dC=0,NA(),dQ/dC*1000000),IF(OR(dC=0,dQ=0),"on an axis",IF(dC>0,"north","south")&"-"&IF(dQ>0,"east","west")))) Excel 365; Cost1, Cost0, QALY1 and QALY0 may be single cells or matched columns of draws, and the result spills ICUR, IUCR and the quadrant into three cells.

  R: icur_iucr <- function(C_1, C_0, Q_1, Q_0, scale = 1e6) { dC <- mean(C_1)-mean(C_0); dQ <- mean(Q_1)-mean(Q_0); quadrant <- if (dC == 0 || dQ == 0) "on an axis" else paste0(if (dC > 0) "north" else "south", "-", if (dQ > 0) "east" else "west"); list(Delta_C = dC, Delta_Q = dQ, ICUR = if (dQ != 0) dC/dQ else NA, IUCR = if (dC != 0) dQ/dC*scale else NA, quadrant = quadrant) } Base R; icur_iucr(24600, 16200, 6.42, 6.02) returns an ICUR of 21000 and an IUCR of 47.619.

  Python: def icur_iucr(C_1, C_0, Q_1, Q_0, scale=1e6): dC = float(np.mean(C_1))-float(np.mean(C_0)); dQ = float(np.mean(Q_1))-float(np.mean(Q_0)); quadrant = "on an axis" if dC == 0 or dQ == 0 else ("north" if dC > 0 else "south")+"-"+("east" if dQ > 0 else "west"); return {"Delta_C": dC, "Delta_Q": dQ, "ICUR": dC/dQ if dQ != 0 else None, "IUCR": dQ/dC*scale if dC != 0 else None, "quadrant": quadrant} Needs import numpy as np; returns the same values as the R function.

  Test (Ratio of means from two draws): Matched draws C_1 = (24600, 24600), C_0 = (16200, 16200), Q_1 = (6.42, 6.02) and Q_0 = (6.02, 6.00), giving incremental costs of GBP 8,400 and QALY gains of 0.40 and 0.02 as in the article, give an ICUR of GBP 40,000 per QALY, not the mean of the draw-level ratios of GBP 220,500, and an IUCR of 25.000. Expected result: TRUE. Excel check: =ROUND((AVERAGE(24600,24600)-AVERAGE(16200,16200))/(AVERAGE(6.42,6.02)-AVERAGE(6.02,6.00)),0)=40000

  Common error (Taking ratios before averaging the draws): Computing an ICUR or IUCR for each draw and then averaging returns GBP 220,500 per QALY in the two-draw case above, more than five times the ratio of means. The mean of draw-level ratios is not the ratio of expected cost to expected QALYs.

  Source: Pharmaceutical Management Agency (PHARMAC). Prescription for Pharmacoeconomic Analysis: Methods for Cost-Utility Analysis. Version 2.2. Wellington: PHARMAC; 2015. Sections 9.1, 9.2 and 9.4; Paulden M. Why it's time to abandon the ICER. PharmacoEconomics. 2020;38(8):781-784. doi:10.1007/s40273-020-00915-5. Section 3, decision rules by quadrant.

  Delta_C = C_1 - C_0; Delta_Q = Q_1 - Q_0; ICUR = Delta_C / Delta_Q; IUCR = Delta_Q / Delta_C * 1000000

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Implementations

• Excel

  Incremental utility-cost ratio from named cells in Excel

  With the incremental QALYs in a cell named DeltaQ and the incremental cost in DeltaC, Excel returns the QALYs gained per one million units of spending, or #N/A when the incremental cost is zero.

  =IF(DeltaC=0,NA(),DeltaQ/DeltaC*1000000)

Assumptions

• Both increments of the utility-cost ratio from one comparison

  Delta_Q and Delta_C come from the same comparator, population and time horizon and are both discounted, with incremental defined as the value for the proposed treatment minus the value for the comparator. Each is an expected value: across probabilistic draws or subgroups the costs and QALYs are averaged first and the ratio is formed last, as PHARMAC requires when aggregating subgroups.

• Quadrant fixes how an incremental utility-cost ratio is read

  The ratio is undefined when Delta_C is zero and carries no direction of its own. PHARMAC notes that in the two trade-off cases, where the ratio is positive, its value can inform a choice. When the intervention gains QALYs at extra cost, a higher IUCR is better; when it saves money but loses QALYs, a positive IUCR counts QALYs lost per one million saved and a lower value is better. A negative IUCR marks dominance one way or the other, so the signs of Delta_Q and Delta_C are reported with it.

Worked examples

• Utility-cost ratio for the new treatment against usual care

  The article's illustrative new treatment adds 0.40 QALYs at an extra GBP 8,400 per patient against usual care, an ICUR of GBP 21,000 per QALY. The IUCR is about 47.619 QALYs gained per GBP 1 million of extra spending, the article's figure of about 47.6.

  Delta_Q = 0.40; Delta_C = 8400; IUCR = 47.619

• Utility-cost ratio against a dominated comparator

  Against the article's dominated older drug the new treatment adds 0.52 QALYs at an extra GBP 6,600, an ICUR of about GBP 12,692 per QALY. The IUCR rises to about 78.788 QALYs per GBP 1 million, so the wrong comparator flatters the result in both forms.

  Delta_Q = 0.52; Delta_C = 6600; IUCR = 78.788

• Utility-cost ratio with a QALY gain of 0.05

  Holding the incremental cost at GBP 8,400 and cutting the QALY gain to 0.05, as in the article's table, gives an ICUR of GBP 168,000 per QALY and an IUCR of about 5.952 QALYs per GBP 1 million. The ICUR is eight times its base value and the IUCR one eighth of it.

  Delta_Q = 0.05; Delta_C = 8400; IUCR = 5.952

• Utility-cost ratio for a QALY loss at extra cost

  With a QALY change of minus 0.05 at the same extra cost, the north-west case in the article's table, the ICUR is minus GBP 168,000 per QALY and the IUCR about minus 5.952. The negative value records that the intervention is dominated, not that it buys QALYs cheaply.

  Delta_Q = -0.05; Delta_C = 8400; IUCR = -5.952

Common errors

• Reading a utility-cost ratio as if lower were better

  The IUCR runs in the opposite direction to the ICUR. In the article's table, at an incremental cost of GBP 8,400, an IUCR of 47.619 is the GBP 21,000 per QALY case and an IUCR of 5.952 the GBP 168,000 per QALY case, so reading the smaller IUCR as the better result inverts the conclusion. The direction of a reported figure has to be checked before figures from PHARMAC and from other agencies are compared.

• Averaging subgroup utility-cost ratios directly

  PHARMAC states that directly weighting the estimates for each subgroup is not appropriate. In an illustrative case with two equal subgroups, one with increments of 0.40 QALYs and GBP 8,400 (IUCR 47.619) and one with 0.10 QALYs and GBP 20,000 (IUCR 5.000), the mean of the two ratios is 26.310. Averaging first gives 0.25 QALYs and GBP 14,200, an IUCR of 17.606, which is the correct aggregate.

• Ignoring the quadrant of a positive utility-cost ratio

  An illustrative intervention that saves GBP 8,400 and loses 0.40 QALYs has the same IUCR of 47.619 as the article's example, but it counts QALYs lost per GBP 1 million saved. At GBP 25,000 per QALY its incremental net monetary benefit is minus GBP 1,600, so the same figure that supports the north-east case rejects the south-west one, as Paulden describes for the ICER.

Sources

• PHARMAC on incremental utility-cost ratios in cost-utility results

  Pharmaceutical Management Agency (PHARMAC). Prescription for Pharmacoeconomic Analysis: Methods for Cost-Utility Analysis. Version 2.2. Wellington: PHARMAC; 2015. Chapter 9 Results of Cost-Utility Analysis: section 9.1 (IUCR as incremental QALYs per incremental NZD 1 million net expenditure, discounted, with the calculation), section 9.2 (aggregate across subgroups by weighting average net cost and net QALY gain, then dividing), section 9.3 (four categories of result) and section 9.4 (ICURs in effect the inverse of IUCRs, reported alongside them).

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• Paulden on quadrant rules behind the incremental utility-cost ratio

  Paulden M. Why it's time to abandon the ICER. PharmacoEconomics. 2020;38(8):781-784. doi:10.1007/s40273-020-00915-5. Section 3: a positive ratio below the threshold is cost effective in the north-east quadrant but not in the south-west quadrant, and a negative ratio is cost effective in the south-east quadrant but not in the north-west quadrant.

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Canonical Identity