Functions & Formulae

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

Estimation of the expected value of partial perfect information from a probabilistic analysis

e(NB_d, theta_i, theta_c) = EVPPI_hat

Maps a decision model, or the probabilistic sample it has produced, and a chosen group of parameters of interest theta_i to an estimate of the expected value of partial perfect information (EVPPI) per person. Each estimator approximates the expected net benefit of every option given theta_i, takes the best option for each value of theta_i, averages, and subtracts the value of the best option on current information. The definition and its order of expectation and maximisation, with a discrete worked case, are on the Value of Information page (HE-FM-VOI-003), the per-person EVPI that bounds every EVPPI from the same sample is HE-FM-VOI-001, and population scaling is HE-FM-VOI-002. This package adds the estimators set out by the ISPOR Value of Information Task Force, with checks that each estimate is at least zero and no larger than the EVPI. Notation follows the Expected Value of Partial Perfect Information article.

  • Two-level nested Monte Carlo EVPPI estimator from incremental net benefit

    ibar_1 = (INB_11 + INB_12) / J; ibar_2 = (INB_21 + INB_22) / J; EVPPI_2L = (max(ibar_1, 0) + max(ibar_2, 0)) / K - max((ibar_1 + ibar_2) / K, 0)

    Draws K values of the parameters of interest theta_i in an outer loop and, for each, J values of the remaining parameters theta_c in an inner loop, averages the incremental net benefit of the new option over the inner draws and keeps the larger of that mean and zero, the value of standard care. The average of the K kept values, less the larger of zero and the mean of the inner means, estimates EVPPI for theta_i, the double-loop scheme of Brennan, Kharroubi, O'Hagan and Chilcott and of Algorithm 2 in the second ISPOR report. The calculator form is written for K = 2 outer and J = 2 inner draws, with INB_kj the run at outer draw k and inner draw j; larger loops add terms in the same pattern, as the Excel and the Computational Function do. With more than two options the maximum is taken over the inner means of every option, and Strong, Oakley and Brennan note that the value is unchanged when net benefits are expressed relative to a baseline option. When theta_i holds every parameter the inner runs at each outer draw are identical and the estimator becomes the per-person EVPI of HE-FM-VOI-001.

  • Single-loop EVPPI shortcut for a model linear in the remaining parameters

    ibar_k = alpha_k + beta_k * mu_c; EVPPI_1L = sum_(k=1)^K [max(ibar_k, 0)] / K - max(sum_(k=1)^K [ibar_k] / K, 0)

    Replaces the inner loop by one model run for each draw of theta_i with the remaining parameters theta_c set at their mean. The second ISPOR report allows this when the model is linear or multilinear in theta_c, with no correlation between complementary parameters that are multiplied together, and prefers single-loop methods where they apply because they reduce Monte Carlo error. Here the incremental net benefit at draw k is alpha_k plus beta_k times theta_c, with alpha_k and beta_k depending on theta_i only, so its expectation over theta_c is that line evaluated at the mean mu_c. Brennan and colleagues show that the shortcut can be substantially inaccurate when even relatively small correlation or nonlinearity is present.

  • Regression-based EVPPI estimator from fitted incremental net benefits

    EVPPI_R = sum_(n=1)^N [max(0, g_2n, g_3n)] / N - max(0, sum_(n=1)^N [g_2n] / N, sum_(n=1)^N [g_3n] / N)

    Regresses the incremental net benefit of each option against a baseline on the parameters of interest across the N runs of one probabilistic sample and treats the fitted values as the conditional expected net benefits, so no inner loop or extra model runs are needed. The estimate is the mean over runs of the largest fitted value, with zero for the baseline, less the largest mean fitted value. This is Equation 9 of Strong, Oakley and Brennan written for three options as incremental net benefits, which they note leaves the result unchanged and needs one regression fewer. Using the fitted values in both terms exploits the positive correlation between them and improves precision. Strong, Oakley and Brennan fit a generalised additive model or a Gaussian process, and Algorithm 4 of the second ISPOR report sets out the same scheme.