Functions & Formulae

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

Expected value of sample information function for a proposed study design

e(NB(d, theta), p(theta), p(X | theta)) = EVSI

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.

  • EVSI of a study with a positive or negative result for two options

    P_pos = p * q + (1 - p) * (1 - q); pF_pos = p * q / P_pos; pF_neg = p * (1 - q) / (1 - P_pos); B_pos = pF_pos * NB_BF + (1 - pF_pos) * NB_BU; B_neg = pF_neg * NB_BF + (1 - pF_neg) * NB_BU; V_S = P_pos * max(NB_A, B_pos) + (1 - P_pos) * max(NB_A, B_neg); V_0 = max(NB_A, p * NB_BF + (1 - p) * NB_BU); EVSI = V_S - V_0

    Evaluates the EVSI expectation exactly when the proposed study can give only two results and the choice is between standard care A and a new option B whose net benefit depends on whether B works (state F) or not (state U). Bayes theorem turns the prior probability p of state F and the chance q that the result points the right way into the probability of each result and the posterior probability of F after it. The better option is chosen after each result, and the expected value of choosing after the study, V_S, is compared with the value of choosing now, V_0. A result adds value only when it changes the choice, so EVSI is zero when every result leaves the current choice in place and equals EVPI when q is 1.

  • Preposterior variance of incremental net benefit for a study of size n

    n0 = sigma2 / v0; nu = n / (n + n0); s2 = nu * v0

    Gives the variance, across the results a proposed study could produce, of the posterior mean of incremental net benefit, the preposterior variance, when the prior and the study estimate are both normal. The prior variance v0 is written as the per-patient sampling variance sigma2 divided by a prior effective sample size n0, and a study of n patients then resolves the fraction nu, n divided by n plus n0, of the prior variance. The preposterior variance is zero when n is zero, rises with n and tends to the prior variance as n grows without limit. It also equals the prior variance less the posterior variance, the known reduction in variance on which the analytic EVSI depends.

  • Closed-form EVSI for two options with normal incremental net benefit

    s = sqrt(s2); z = abs(m0) / s; phi_z = exp(-z * z / 2) / sqrt(2 * 3.14159265); EVSI = s * (phi_z - z * Q_z)

    Applies the unit normal loss integral to the preposterior distribution of incremental net benefit, which is normal with the prior mean m0 and the preposterior variance s2 of HE-FM-EVSI-002. EVSI is the expected gain from switching away from the option preferred now in the results that reverse the sign of the posterior mean, s times phi_z less the absolute value of m0 times Q_z, where z is the absolute prior mean in preposterior standard deviations. phi_z is the standard normal density written out with exp and sqrt, and Q_z is the standard normal upper-tail probability at z, entered from Excel's 1-NORM.S.DIST(z,TRUE) because the calculator has no normal distribution function. With s2 equal to the prior variance the same expression gives per-person EVPI under normality, the bound in HE-FM-VOI-001.

  • Population EVSI for a constant annual beneficiary population over the decision's lifetime

    D = (1 + r) * (1 - (1 + r)^(-Y)) / r; EVSI_pop = EVSI * N * D

    Scales per-person EVSI to everyone whose treatment the study result would inform, N people a year for Y years, with later cohorts discounted at rate r, the population value of research in equation 15 of the second ISPOR value of information report. The factor D is the present value of one person a year in years 0 to Y minus 1, so the first cohort is not discounted, as in population EVPI (HE-FM-VOI-002), which allows a different N_t in each period. Population EVSI is the figure set against the cost of the study, and their difference, the expected net benefit of sampling, has its own package.