Functions & Formulae

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

Expected net benefit of sampling as population sample information less the cost of a study

g(EVSI_pop(n), C(n)) = ENBS(n)

Maps one proposed study design, through its population expected value of sample information and its expected total cost, to the expected net benefit of sampling, the net value of that study. Per-person EVPI and its population form are on the Value of Information page (HE-FM-VOI-001 and HE-FM-VOI-002), and the expected net benefit that fixes the current decision is on the Expected Net Benefit page (HE-FN-ENB-001). EVSI is defined on the EVSI page (HE-FN-EVSI-001), and population EVSI, per-person EVSI scaled by the discounted beneficiary population, is HE-FM-EVSI-004 there. This package adds ENBS for one design, the expected cost of a study with the opportunity loss to participants, the cost of waiting under only in research, the ceiling set by population EVPI and a search for the efficient sample size. Notation follows the Expected Net Benefit of Sampling article.

  • ENBS of one study design from its population EVSI and the study cost

    ENBS = EVSI_pop - C_n

    Gives the net value of one proposed design of sample size n as its population EVSI less the expected total cost of the study, the definition used by the ISPOR task force. Population EVSI comes from HE-FM-EVSI-004, which multiplies the design's per-person EVSI by the discounted number of people whose treatment the result will inform, scaled in the same way as population EVPI in HE-FM-VOI-002. A positive value means the design is expected to repay its cost, and the design with the greatest value among those feasible is the efficient one (HE-CF-ENBS-001).

  • Expected total cost of a two-arm study with the opportunity loss to participants

    C_n = C_F + c_V * n + s_inf * n * abs(INB)

    Adds the three cost components that Fenwick and colleagues list, a fixed cost, a variable cost for each participant and the opportunity cost for participants who receive the option expected to be inferior while the study runs. Each such participant forgoes the absolute value of the expected incremental net benefit per person on current information, so with a share s_inf of the n participants on that arm the loss term is s_inf times n times abs(INB). The result is the C_n entered in HE-FM-ENBS-001.

  • ENBS of a study run under only in research with the cost of waiting

    ENBS_w = EVSI_pop - C_n - N_w * max(INB, 0)

    Subtracts from the population EVSI of a design its expected study cost and the net benefit forgone by people treated outside the study while it runs, when the new treatment is available only in research. When the new treatment is expected to be better, with INB above zero, each of the N_w people waiting forgoes INB. When it is not, standard care outside the study is already the expected best option and waiting costs nothing in this sense. Under approval with research N_w is zero and the result equals the ENBS of HE-FM-ENBS-001, so the same trial can have a different ENBS under the two arrangements.

  • Ceiling on the ENBS of a study design from population EVPI

    ENBS_ub = EVPI_pop - C_n

    No finite study delivers perfect information, so the population EVSI of any design is at most population EVPI (HE-FM-VOI-002) and the ENBS of a design costing C_n cannot exceed population EVPI less that cost. When the ceiling is zero or below for every feasible design, research is not worthwhile and the value of information process stops before any EVSI is computed, the first screen in the sequence set out by the ISPOR task force. A positive ceiling is necessary but not sufficient, so ENBS itself is still required.