Signature
n0 = sigma2 / v0; nu = n / (n + n0); s2 = nu * v0
| Inputs | Definition | Unit |
|---|---|---|
sigma2 | Variance contributed by one patient, or one patient per arm, to the study's estimate of incremental net benefit, so that the estimate from n has variance sigma2 divided by n | (currency per patient) squared |
v0 | Variance of incremental net benefit per patient on current evidence, for example the variance of its probabilistic analysis draws | (currency per patient) squared |
n | Number of patients in the proposed study, counted in the units in which sigma2 is defined, for example patients per arm | patients |
n0 | Number of patients whose data would carry the same information about incremental net benefit as current evidence | patients |
|---|---|---|
nu | Share of the prior variance of incremental net benefit resolved by the study | dimensionless |
s2 | Variance across possible study results of the posterior mean of incremental net benefit per patient | (currency per patient) squared |
Function
Expected value of sample information function for a proposed study design
Maps the current uncertainty about a decision model's parameters, the sampling distribution of the data a proposed study would collect and the net benefit of each option to the expected gain from choosing after the study reports rather than now. EVSI equals the expectation, over the datasets X the study could produce, of the largest posterior expected net benefit among the options, less the largest expected net benefit on current evidence, the form given as equation 10 of the second ISPOR value of information report and equation 2 of Kunst and colleagues. It replaces the perfect knowledge of a parameter subset in EVPPI (HE-FM-VOI-003) with the data of a study of stated design and size, so it is never negative and does not exceed EVPPI for the parameters the study informs or per-person EVPI (HE-FM-VOI-001). This package gives the exact form for a study with two possible results, the normal preposterior variance, closed-form EVSI for two options and population EVSI over the decision's lifetime on the basis used for population EVPI (HE-FM-VOI-002). The expected net benefit of sampling, population EVSI less the cost of the study, has its own package. Notation follows the Expected Value of Sample Information article.
Try this function
Implementations
Excel
Preposterior variance of incremental net benefit from named cells
With the prior variance in a cell named PriorVarINB, the per-patient sampling variance in SamplingVar and the study size in StudySize, the formula returns s2. Its square root is the preposterior standard deviation used by HE-FM-EVSI-003.
=PriorVarINB*StudySize/(StudySize+SamplingVar/PriorVarINB)
Assumptions
Normal prior and normal study estimate of incremental net benefit
Incremental net benefit has a normal prior with variance v0, and the study estimate is normal around the true value with known variance sigma2 divided by n, the conjugate pair for which Jalal and Alarid-Escudero derive the variance fraction (equations 7 to 12). With skewed priors or study estimates the expression is an approximation.
Study that informs incremental net benefit directly
The study estimates incremental net benefit itself, so enough data would resolve all of its prior uncertainty. When a study informs only some parameters, the reducible variance is the part explained by those parameters, and EVSI is bounded by their EVPPI (HE-FM-VOI-003) rather than by EVPI.
Sampling variance of a two-arm trial with equal arms
For a two-arm trial with n patients in each arm, the estimate of incremental net benefit is a difference of two independent means, so sigma2 is the sum of the per-patient variances of net benefit in the two arms and n is the number per arm.
Worked examples
Preposterior variance for 100 patients per arm against a prior effective size of 200
Illustrative inputs with a prior standard deviation of incremental net benefit of GBP 1,000 per patient, a prior variance of 1,000,000, and per-patient standard deviations of net benefit of GBP 10,000 in each arm give a sampling variance of 200,000,000 and a prior effective sample size of 200 per arm. A trial of 100 per arm resolves a third of the prior variance, a preposterior variance of about 333,333.
v0 = 1000000; sigma2 = 200000000; n = 100; n0 = 200; nu = 0.333333; s2 = 333333.33
Preposterior variance with no new patients
With n equal to zero the posterior mean equals the prior mean whatever happens, so the preposterior variance is zero and the study has no value, the lower limiting case.
v0 = 1000000; sigma2 = 200000000; n = 0; n0 = 200; nu = 0; s2 = 0
Common errors
Posterior variance used in place of the preposterior variance
After 100 per arm the posterior variance is about 666,667, the uncertainty that remains, and the preposterior variance about 333,333, the uncertainty the study resolves. Entering the posterior variance in HE-FM-EVSI-003 gives an EVSI of about GBP 134.97 per patient instead of GBP 61.68 for the illustrative prior, more than twice the correct value.
Variance of one arm used for a two-arm comparison
Setting sigma2 to the variance of one arm, 100,000,000, halves n0 to 100 and raises the fraction resolved by 100 per arm to a half instead of a third, which overstates EVSI at about GBP 99.82 per patient instead of GBP 61.68.
Sources
Jalal and Alarid-Escudero on the variance of the preposterior distribution
Jalal H, Alarid-Escudero F. A Gaussian approximation approach for value of information analysis. Medical Decision Making. 2018;38(2):174-188. Equations 7 and 8 give a normal prior for the parameter with variance sigma squared over n0 and a normal sample mean with variance sigma squared over n, equation 9 gives the conjugate normal posterior with variance sigma squared over n plus n0, and equation 12 gives the variance of the posterior mean over the possible datasets, the preposterior distribution, as n over n0 plus n times the prior variance. The text states that this variance tends to zero as n falls to zero and to the prior variance as n grows.
Jalal and colleagues on the variance fraction of preposterior incremental net benefit
Jalal H, Goldhaber-Fiebert JD, Kuntz KM. Computing expected value of partial sample information from probabilistic sensitivity analysis using linear regression metamodeling. Medical Decision Making. 2015;35(5):584-595. The EVSI section and Box 2 compute the preposterior variance of incremental net benefit as the prior variance times the variance fraction n over n plus n0, and state that with no additional sample EVSI is zero and with an infinite sample the preposterior variance equals the prior variance and EVSI equals EVPI.
Canonical Identity
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