Expected loss of each of two options from five PSA draws of incremental net health benefit

With a new option and a comparator whose net health benefit is set to zero, the loss from adopting the new option in a draw is the incremental net health benefit forgone when it is negative, and the loss from keeping the comparator is the gain forgone when it is positive. EL_new and EL_comp average these losses over all five equally likely draws, the two-option case of the expected loss of each strategy in equation 3 of Alarid-Escudero and colleagues, whose expected loss curves give the loss of every option as HE-FM-CEAC-002 gives the probability of every option. For the option chosen on current evidence the loss is the mean opportunity loss of HE-FM-EVPI-001, so the smaller of the two equals the per-person EVPI of HE-FM-EVPI-001 and HE-FM-EVPI-002, and the larger is the figure that HE-ER-EVPI-001 warns must not be reported as EVPI. Their difference D_EL equals the mean incremental net health benefit, because the positive part of a number less the positive part of its negative is the number itself (derived, the two-option case of equation 4 of Alarid-Escudero and colleagues), so the option with the smaller expected loss is the option with the higher expected net health benefit (HE-FM-ENB-002).

Signature

EL_new = (max(-INHB_1, 0) + max(-INHB_2, 0) + max(-INHB_3, 0) + max(-INHB_4, 0) + max(-INHB_5, 0)) / 5; EL_comp = (max(INHB_1, 0) + max(INHB_2, 0) + max(INHB_3, 0) + max(INHB_4, 0) + max(INHB_5, 0)) / 5; D_EL = EL_comp - EL_new
Inputs
InputsDefinitionUnit
INHB_1Incremental QALYs less incremental cost divided by the threshold for the new option against the comparator in the first drawQALYs per person
INHB_2Incremental QALYs less incremental cost divided by the threshold for the new option against the comparator in the second drawQALYs per person
INHB_3Incremental QALYs less incremental cost divided by the threshold for the new option against the comparator in the third drawQALYs per person
INHB_4Incremental QALYs less incremental cost divided by the threshold for the new option against the comparator in the fourth drawQALYs per person
INHB_5Incremental QALYs less incremental cost divided by the threshold for the new option against the comparator in the fifth drawQALYs per person
Output
EL_newMean over the draws of the incremental net health benefit forgone in draws where the new option is worse than the comparator, zero or aboveQALYs per person
EL_compMean over the draws of the incremental net health benefit forgone in draws where the new option is better than the comparator, zero or aboveQALYs per person
D_ELExpected loss from keeping the comparator less expected loss from adopting the new option, equal to the mean incremental net health benefitQALYs per person

Function

Probabilistic sensitivity analysis run that turns joint parameter draws into stored decision outputs

Maps a decision model f, the joint distribution of its uncertain inputs and a cost-effectiveness threshold lambda to the outputs stored for every draw and their summaries. Each of the N draws theta_i is one complete set of inputs, applied to every option so that incremental results keep their dependence, and the model returns each option's cost and QALYs in that draw. The summaries are live records and are not restated here. Expected net monetary and net health benefit from the draws are HE-FM-ENB-001 and HE-FM-ENB-002, the probability that each option is cost-effective and the acceptability frontier are HE-FM-CEAC-001 to HE-FM-CEAC-003, and per-person EVPI from the same draws is HE-FM-VOI-001 with the forms in HE-FM-EVPI-001 to HE-FM-EVPI-004. A probabilistic ICER is the two-option ratio HE-FM-ICER-001 taken on expected values (HE-AS-ICER-003). The Monte Carlo standard error of a mean output over independent draws is the standard deviation across draws divided by the square root of N, the form of HE-FM-ANTV-002 with one observation per draw, and the number of draws for a target standard error is HE-FM-ABM-003 (HE-FM-ABM-004 for a share of draws); antithetic pairs are HE-FN-ANTV-001. Why a run at the mean inputs differs from the mean of the runs is covered by HE-FM-EV-001 to HE-FM-EV-004, and input distributions by HE-FN-BETA-001, HE-CF-BETA-001, HE-FM-BETA-005 and HE-FM-DIRI-001. The probability of a wrong decision and the Monte Carlo error of that probability belong to the Decision Uncertainty page. This package adds the two-parameter Cholesky step that draws a correlated pair of normal inputs, the expected loss of each of two options with the identity that links the two losses to expected net health benefit, and a computational function that runs the whole analysis on a small fixed set of draws. Notation follows the Probabilistic Sensitivity Analysis article.

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Implementations

  • Excel

    Expected loss of each option from a column of incremental net health benefit draws

    With one incremental net health benefit per draw in a range named IncNHB, the first formula returns the expected loss from adopting the new option in a cell named ExpLossNew and the second the expected loss from keeping the comparator in ExpLossComp. SUMPRODUCT needs no array entry, and the range can hold any number of draws.

    =SUMPRODUCT((IncNHB<0)*(0-IncNHB))/COUNT(IncNHB); =SUMPRODUCT((IncNHB>0)*IncNHB)/COUNT(IncNHB)

Assumptions

  • Two options with the comparator as zero reference in the expected loss

    The decision is between one new option and one comparator, and every draw holds the new option's incremental net health benefit at the same threshold, so the comparator's net benefit is zero in every draw. The five-draw form is written out for the calculator; the Excel and R forms take any number of draws, and with three or more options each option's loss is the mean of the best net benefit in each draw less its own, equation 3 of Alarid-Escudero and colleagues.

  • Risk-neutral choice by the smaller expected loss

    The decision maker is risk neutral, so the option chosen on current evidence is the one with the highest expected net benefit, which Alarid-Escudero and colleagues show is the one with the lowest expected loss. Both losses are conditional on the uncertainties inside the probabilistic analysis, as HEMA Report 2 notes for its own example.

  • Draws with zero incremental net health benefit add no expected loss

    A draw in which the incremental net health benefit is exactly zero adds no loss to either option, since neither choice forgoes anything. The probability that the new option is cost-effective counts such a draw as not cost-effective, because HE-FM-CEAC-001 uses a strict inequality, so ties change that probability but not either expected loss.

Worked examples

  • Article's five draws at GBP 25,000 per QALY as expected losses

    In the article's illustrative example the incremental net health benefit in the five draws is 2.4, minus 0.1, minus 0.1, minus 0.6 and minus 0.6 QALYs. Adopting the new treatment loses 1.4 / 5 = 0.28 QALYs per patient on average and keeping usual care loses 2.4 / 5 = 0.48, so the difference of 0.2 QALYs is the expected net health benefit, as the article states. The smaller loss, 0.28, is the article's per-patient EVPI.

    INHB_1 = 2.4; INHB_2 = -0.1; INHB_3 = -0.1; INHB_4 = -0.6; INHB_5 = -0.6; EL_new = 0.28; EL_comp = 0.48; D_EL = 0.2

  • Article's five draws at GBP 35,000 per QALY as expected losses

    At GBP 35,000 per QALY, the higher value in PMG36 section 4.2.16, the fixed cost of GBP 15,000 is worth about 0.4285714 QALYs, so the same survival draws give incremental net health benefits of about 2.5714286, 0.0714286, 0.0714286, minus 0.4285714 and minus 0.4285714 QALYs. The expected loss from adopting falls to about 0.1714286 and that from keeping usual care rises to about 0.5428572 QALYs per patient. These figures are computed here from the article's draws.

    INHB_1 = 2.5714286; INHB_2 = 0.0714286; INHB_3 = 0.0714286; INHB_4 = -0.4285714; INHB_5 = -0.4285714; EL_new = 0.1714286; EL_comp = 0.5428572; D_EL = 0.3714286

  • No expected loss from adopting when every draw favours the new option

    In five illustrative draws with incremental net health benefits of 0.1, 0.2, 0.3, 0.1 and 0.3 QALYs the new option is better in every draw, so the expected loss from adopting it is zero and the expected loss from keeping the comparator equals the mean of 0.2 QALYs. This limiting case has no decision uncertainty and an EVPI of zero.

    INHB_1 = 0.1; INHB_2 = 0.2; INHB_3 = 0.3; INHB_4 = 0.1; INHB_5 = 0.3; EL_new = 0; EL_comp = 0.2; D_EL = 0.2

Common errors

  • Averaging an expected loss over the losing draws only

    Dividing the adoption losses in the article's example, 1.4 QALYs in total, by the 4 draws in which adoption loses gives 0.35 QALYs, the loss when wrong of HE-FM-EVPI-003, not the expected loss of 0.28, the slip that HE-ER-EVPI-005 describes for EVPI. An expected loss is averaged over all draws, including those with no loss.

  • Choosing the option most likely to be cost-effective instead of the smaller expected loss

    In the article's example usual care is the better option in four draws out of five, but keeping it has an expected loss of 0.48 QALYs per patient against 0.28 for adopting the new treatment. Choosing by the probability of 0.8 gives up 0.2 QALYs per patient of expected net health benefit, the HEMA point that the probability alone can point to the option with the lower expected net benefit, and the trap HE-ER-ENB-001 shows in money terms.

Sources

  • Alarid-Escudero and colleagues on the expected loss of each strategy

    Alarid-Escudero F, Enns EA, Kuntz KM, Michaud TL, Jalal H. "Time traveling is just too dangerous" but some methods are worth revisiting, the advantages of expected loss curves over cost-effectiveness acceptability curves and frontier. Value in Health. 2019;22(5):611-618. Section Definition and mathematical notation of expected losses, equations 3 and 4, which define the expected loss of strategy d as the mean over the N PSA samples of the net benefit of the best strategy in each sample less that of d, and state that for a risk-neutral decision maker the optimal strategy, the one with the highest expected benefit, is the one with the lowest expected loss, and that the expected loss of the optimal strategy equals the EVPI. The two-option form with the comparator at zero follows by setting the best net benefit in a draw to the positive part of the incremental net benefit.

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  • HEMA Report 2 on reporting the expected loss of each option with the CEAC

    Steuten L, Thavorn K, Cipriano LE, Guertin JR, McQueen B, Schmidler G, Sullivan S, Wailoo A, Sculpher M. Methods to analyze and present uncertainty in economic evaluation to support HTA decisions. HEMA Report 2. Health Economics Methods Advisory, 2026. Section 6.4, Standard outputs and their limitations (page 52), which states that expected loss shows the magnitude of the potential loss associated with decision uncertainty and should accompany the CEAC, and in its illustrative example gives an expected loss of 0.14 QALYs per patient from choosing the new medicine against 0.24 from choosing standard care, with the difference equal to the expected incremental net health benefit of 0.10 QALYs, while the probability alone would point to the option with the lower expected net benefit.

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Canonical Identity

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