Functions & Formulae

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

Net benefit approach to a two-option cost-effectiveness decision and its sampling uncertainty

b(lambda, Delta_E, Delta_C) = INB

Maps a threshold lambda and the differences in mean effect and mean cost between a new option and its comparator, or the per-arm trial data behind them, to incremental net benefit, whose sign gives the adoption decision, and to the sampling uncertainty around it. Option-level and incremental net monetary benefit, the maximum net monetary benefit rule and expected net benefit from simulations are on the Net Monetary Benefit and Expected Net Benefit pages (HE-FM-NMB-001 to HE-FM-NMB-003, HE-FM-ENB-001 and HE-FM-ENB-002). Incremental net health benefit at a threshold is HE-FM-CET-002, the ICER is HE-FM-ICER-001 and dominance is HE-FM-DOM-001. A normal interval for incremental net benefit built from the standard errors of the two differences is HE-FM-AN-004 with HE-CF-AN-001, its bootstrap variance is HE-FM-BSE-003, the acceptability curve and frontier are HE-FM-CEAC-001 to HE-FM-CEAC-003 and EVPI is HE-FM-VOI-001. This package adds Stinnett and Mullahy's interval for incremental net health benefit from two independent trial arms, the identity that links the sign of incremental net benefit to the ICER in each quadrant of the cost-effectiveness plane, and net-benefit regression. Notation follows the Net Benefit Approach article.

  • Stinnett and Mullahy confidence interval for incremental net health benefit from two trial arms

    INHB = E_1 - E_0 - (C_1 - C_0) / lambda; Var_INHB = (s_E1^2 + s_C1^2 / lambda^2 - 2 * r_1 * s_E1 * s_C1 / lambda) / n_1 + (s_E0^2 + s_C0^2 / lambda^2 - 2 * r_0 * s_E0 * s_C0 / lambda) / n_0; SE_INHB = sqrt(Var_INHB); L_INHB = INHB - z * SE_INHB; U_INHB = INHB + z * SE_INHB; z_INHB = INHB / SE_INHB

    Estimates incremental net health benefit with the analogy estimator, the difference in arm mean effects less the difference in arm mean costs divided by lambda, and estimates its variance from each arm's sample variances of effect and cost, their within-arm correlation and the arm size (Stinnett and Mullahy, equations 6 and 7). The estimator is linear in the four sample means, so its variance is a direct sum of sample terms with no Taylor series approximation, and the interval is a normal (Wald) interval as in HE-FM-AN-001 (their equation 8). The statistic z_INHB, the estimate divided by its standard error, converts to the probability that net health benefit is positive through the standard normal cumulative distribution function, which the calculator lacks, so Excel NORM.S.DIST or R pnorm does that step. Multiplying every term by lambda gives the money-scale result of HE-FM-AN-004.

  • Incremental net monetary benefit written through the ICER in each quadrant of the cost-effectiveness plane

    R = Delta_C / Delta_E; INMB = Delta_E * (lambda - R)

    Rewrites incremental net monetary benefit (HE-FM-NMB-002) as the effect difference times the gap between the threshold and the ICER (HE-FM-ICER-001), which holds whenever Delta_E is not zero. The sign of INMB is the sign of Delta_E times the sign of lambda minus R. With a health gain (north-east and south-east) a positive INMB means R below lambda, and with a health loss (south-west and north-west) it means R above lambda, so the same ratio can call for opposite decisions. This is the ambiguity between quadrants that Stinnett and Mullahy describe. Plotted against lambda, the expression is the line that Hoch and colleagues describe, with intercept minus Delta_C, slope Delta_E and a zero crossing at the ICER.

  • Net-benefit regression coefficient on a treatment indicator

    NB_1 = lambda * E_1 - C_1; NB_0 = lambda * E_0 - C_0; alpha = NB_0; beta = NB_1 - NB_0

    Each patient's net benefit at threshold lambda, lambda times the patient's effect minus the patient's cost, is regressed on a treatment indicator T_i, 1 for the new option and 0 for the comparator, in the model NB_i = alpha + beta T_i + epsilon_i. With no other regressors the fitted intercept is the comparator arm's mean net benefit and the coefficient is the difference in arm means, which equals incremental net monetary benefit (HE-FM-NMB-002), so the confidence interval for beta is the interval for incremental net benefit. The calculator works from the arm means. Covariates added to the regression give an adjusted estimate of beta, which Hoch, Briggs and Willan proposed for imperfect randomisation and for subgroups.

Net Benefit Approach — Functions & Formulae | HealthEconomics.wiki