Signature
z = abs(mu_INB) / sigma_INB; pdf_z = exp(-z^2 / 2) / 2.5066283; EVPI_N = sigma_INB * (pdf_z - z * (1 - cdf_z))
| Inputs | Definition | Unit |
|---|---|---|
mu_INB | Expected incremental net benefit of the new option over its comparator at the stated threshold | QALYs or currency per person |
sigma_INB | Standard deviation of the uncertainty about incremental net benefit, such as the spread across probabilistic draws, above zero | QALYs or currency per person |
cdf_z | Probability that a standard normal variable is at most z, entered from NORM.S.DIST(z,TRUE) or a table | proportion |
z | Absolute mean incremental net benefit divided by its standard deviation | dimensionless |
|---|---|---|
pdf_z | Value of the standard normal density at z, computed as exp(-z^2/2) divided by the square root of 2 pi | dimensionless |
EVPI_N | Expected value of perfect information for one person when incremental net benefit is normal | QALYs or currency per person |
Function
Expected value of perfect information function for a single adoption decision
Maps the net benefit of every option in every probabilistic draw, at a stated cost-effectiveness threshold, to the expected net benefit forgone per person by choosing on current evidence rather than with all uncertainty removed. Its inputs are the draw by option table of net benefit NB_ds and the threshold lambda, and its output is EVPI, in QALYs when net health benefit is used and in currency when net monetary benefit is used. The mean of the maxima less the maximum of the means is live in the Value of Information package as HE-FM-VOI-001, with population scaling in HE-FM-VOI-002, the partial form in HE-FM-VOI-003 and the error probability in HE-FM-CEAC-003. This package adds the opportunity-loss form, the two-option form in health units, the split of EVPI into error probability and loss when wrong, and the closed form for a normally distributed incremental net benefit. Notation follows the Expected Value of Perfect Information article.
Try this function
Implementations
Excel
Normal two-option EVPI with NORM.S.DIST from named cells
With the mean incremental net benefit in a cell named MuINB and its standard deviation in SigmaINB, NORM.S.DIST with FALSE returns the density and with TRUE the cumulative probability at z.
=SigmaINB*(NORM.S.DIST(ABS(MuINB)/SigmaINB,FALSE)-ABS(MuINB)/SigmaINB*(1-NORM.S.DIST(ABS(MuINB)/SigmaINB,TRUE)))
Assumptions
Normal distribution of incremental net benefit for closed-form EVPI
The incremental net benefit is normally distributed. For a mean estimated from trial data this rests on the central limit theorem, while a model's net benefit across draws can be skewed, in which case the closed form is an approximation to EVPI from the draws.
Exactly two options for the one-dimensional normal EVPI
The closed form applies to one incremental net benefit, so to two options. Three options need the two-dimensional loss integral of Lee and colleagues or the draws themselves (HE-FM-EVPI-001).
Mean and standard deviation of incremental net benefit at one threshold
mu_INB and sigma_INB are in the same units and at the same threshold. Both change with the threshold, so the closed form is evaluated again for each threshold, and the threshold at which mu_INB is zero need not be where EVPI peaks if sigma_INB also changes.
Worked examples
Normal approximation to the article's five draws at GBP 25,000 per QALY
A normal distribution with the mean and sample standard deviation of the article's five illustrative INHB values, 0.07 and about 0.2168 QALYs, gives z of about 0.3229 and an EVPI of about 0.0560 QALYs per patient, against 0.06 from the draws themselves. The gap reflects the five draws not being normal.
mu_INB = 0.07; sigma_INB = 0.2168; cdf_z = 0.62661; z = 0.3229; pdf_z = 0.3787; EVPI_N = 0.0560
Normal EVPI when the mean incremental net benefit is zero
With mu_INB equal to zero and the same standard deviation of 0.2168 QALYs, z is zero, cdf_z is 0.5 and EVPI is about 0.3989 times 0.2168, or about 0.0865 QALYs per patient. This limiting case is the largest normal EVPI for that standard deviation.
mu_INB = 0; sigma_INB = 0.2168; cdf_z = 0.5; z = 0; pdf_z = 0.398942; EVPI_N = 0.0865
Normal EVPI for an illustrative GBP 1,000 mean and GBP 2,000 spread
An illustrative incremental net monetary benefit with a mean of GBP 1,000 and a standard deviation of GBP 2,000 gives z of 0.5, a density of about 0.3521 and a cumulative probability of about 0.6915, so EVPI is about GBP 395.59 per person.
mu_INB = 1000; sigma_INB = 2000; cdf_z = 0.69146; z = 0.5; pdf_z = 0.352065; EVPI_N = 395.59
Common errors
Using the signed mean in the normal EVPI loss term
For a mean of minus GBP 1,000 and a standard deviation of GBP 2,000, using z equal to minus 0.5 in the loss term gives about GBP 1,395.59 instead of GBP 395.59, overstating EVPI by exactly the absolute mean. The loss term needs the absolute mean.
Using the Monte Carlo standard error as the spread in normal EVPI
Dividing the standard deviation of the five illustrative draws, about 0.2168 QALYs, by the square root of 5 gives the standard error of their mean, about 0.0970, and an EVPI of about 0.0133 QALYs instead of 0.0560. sigma_INB measures the uncertainty about incremental net benefit, not the precision of the simulation.
Applying the normal EVPI to three or more options
Running the closed form on each pairwise comparison and adding or taking the largest result does not give the EVPI of a choice among three options, because the best option in each state depends on all of them together.
Sources
Lee, Gustafson and Sadatsafavi on the unit normal loss integral for two-option EVPI
Lee TY, Gustafson P, Sadatsafavi M. Closed-form solution of the unit normal loss integral in 2 dimensions, with application in value-of-information analysis. Medical Decision Making. 2023;43(5):621-626. Introduction, which defines the unit normal loss integral for a normal Y with mean mu and variance sigma squared as E(max(Y, 0)) = mu[1-Phi(-mu/sigma)] + sigma phi(-mu/sigma), attributes it to Raiffa and Schlaifer, and states that for two strategies with a normal incremental net benefit EVPI has a closed form using this integral, applicable to model-based and data-driven evaluations. The loss term with the absolute mean follows by subtracting the positive part of mu.
ISPOR methods report EVPI definition used to subtract the current-decision value
Rothery C, Strong M, Koffijberg HE, Basu A, Ghabri S, Knies S, Murray JF, Sanders Schmidler GD, Steuten L, Fenwick E. Value of information analytical methods. Report 2 of the ISPOR Value of Information Analysis Emerging Good Practices Task Force. Value in Health. 2020;23(3):277-286. Section on the expected value of perfect information, which defines EVPI as the expected maximum utility over options less the maximum of the expected utilities. With the comparator at zero, the second term is the positive part of mu_INB.
Canonical Identity
Stable URI · Machine-readable · Resolvable · CC BY 4.0