Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Expected value of perfect information function for a single adoption decision

evpi(NB_ds, lambda) = EVPI

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.

  • Per-person EVPI as the mean opportunity loss of the current decision

    L_s = NBmax_s - NBcur_s; EVPI = sum_(s=1)^S [L_s] / S

    In each draw s the opportunity loss L_s is the highest net benefit available in that draw less the net benefit, in the same draw, of the option d* chosen on current evidence. EVPI is the mean of L_s over the S equally weighted draws. Because d* is the option with the highest mean net benefit, the result equals the mean of the maxima less the maximum of the means in HE-FM-VOI-001, and it also shows which draws carry the loss. L_s is zero in every draw where d* is also the best option.

  • Two-option EVPI in QALYs from incremental net health benefit draws

    INHB_s = dQ_s - dC_s / lambda; EVPI_H = sum_(s=1)^S [max(INHB_s, 0)] / S - max(sum_(s=1)^S [INHB_s] / S, 0); EVPI_M = lambda * EVPI_H

    With one new option and one comparator, the comparator's net benefit can be set to zero in every draw and the new option's to its incremental net health benefit INHB_s, its incremental QALYs less its incremental cost divided by the threshold. EVPI in QALYs is then the mean of the positive part of INHB_s less the positive part of its mean. Multiplying by the threshold gives the same EVPI in money, because net monetary benefit is the threshold times net health benefit in every draw.

  • EVPI as the probability of a wrong current decision times the mean loss when wrong

    EVPI = P_err * L_wrong

    Splits the mean opportunity loss into how often the option chosen on current evidence is not the best option, P_err, and the mean loss in the draws where it is not, L_wrong. The product equals EVPI because the draws where the chosen option is best add no loss. P_err is one minus the acceptability of the chosen option (HE-FM-CEAC-003), so the formula links the acceptability curve to EVPI and shows why a small chance of a costly error can give a larger EVPI than a large chance of a small one.

  • Two-option EVPI for a normally distributed incremental net benefit

    z = abs(mu_INB) / sigma_INB; pdf_z = exp(-z^2 / 2) / 2.5066283; EVPI_N = sigma_INB * (pdf_z - z * (1 - cdf_z))

    When the incremental net benefit of the new option over its comparator is normal with mean mu_INB and standard deviation sigma_INB, EVPI has a closed form, sigma_INB times the unit normal loss integral at z, the absolute mean divided by the standard deviation. pdf_z is the standard normal density at z, computed in the formula, and cdf_z is the standard normal cumulative probability at z, which the site calculator cannot compute and so takes as an input from NORM.S.DIST or a table. The form needs no draws and has no Monte Carlo error. For a given sigma_INB the loss term is largest at z equal to zero, where EVPI is sigma_INB divided by the square root of 2 pi, about 0.3989 sigma_INB.