Signature
INB = lambda * Delta_E - Delta_C; SE_INB = sqrt(lambda^2 * SE_E^2 + SE_C^2 - 2 * lambda * Cov_CE); INB_L = INB - z * SE_INB; INB_U = INB + z * SE_INB; z_INB = INB / SE_INB
| Inputs | Definition | Unit |
|---|---|---|
lambda | Cost-effectiveness threshold, the value placed on one unit of effect | currency per unit of effect, for example pounds per QALY |
Delta_E | Estimated difference in mean effect, intervention minus comparator | effect |
Delta_C | Estimated difference in mean cost, intervention minus comparator | currency |
SE_E | Standard error of Delta_E | effect |
SE_C | Standard error of Delta_C | currency |
Cov_CE | Estimated covariance between Delta_C and Delta_E | currency times effect |
z | Standard normal critical value, 1.96 for a two-sided 95% interval | none |
INB | Estimated incremental net monetary benefit at threshold lambda | currency |
|---|---|---|
SE_INB | Standard error of INB at threshold lambda | currency |
INB_L | Lower confidence limit for INB | currency |
INB_U | Upper confidence limit for INB | currency |
z_INB | INB divided by its standard error; the standard normal cumulative distribution function at z_INB is the probability that net benefit is positive | none |
Function
Large-sample normal confidence interval function
Maps an estimate and its estimated sampling variance to an approximate confidence interval by treating the estimator as normally distributed, which asymptotic normality justifies in large samples even when patient-level data are skewed. In trial-based cost-effectiveness analysis the same operation is applied to a difference in mean cost or effect, to the ICER through the delta method, to the pair of differences through Fieller's method, and to incremental net benefit, which is linear in the two differences.
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Implementations
Excel
Net benefit interval and probability in five cells
With named cells Threshold, DeltaEffect, DeltaCost, SeEffect, SeCost, CovCostEffect and ZValue, and the first two results named IncNB and SeINB, the formulas return INB, its standard error, the two limits and the probability that INB is positive.
=Threshold*DeltaEffect-DeltaCost; =SQRT(Threshold^2*SeEffect^2+SeCost^2-2*Threshold*CovCostEffect); =IncNB-ZValue*SeINB; =IncNB+ZValue*SeINB; =NORM.S.DIST(IncNB/SeINB,TRUE)
Assumptions
Normal approximation for a linear combination of mean differences
INB is a linear combination of two differences in means, so the central limit theorem applies to it directly and the approximation does not break down when Delta_E is close to zero. The probability describes sampling uncertainty in one dataset only.
Fixed, stated threshold for the net benefit interval
lambda is fixed by the analyst, and the interval and probability are recomputed at each threshold. Across a range of thresholds the probabilities trace the acceptability curve.
Worked examples
Net benefit interval at £30,000 per QALY
At an illustrative threshold of £30,000 per QALY, incremental net benefit is £1,000 with a variance of 1,800,000 and a standard error of about £1,342. The 95% interval runs from about -£1,630 to £3,630, which includes zero, and z_INB of about 0.745 gives a probability of about 0.77 that net benefit is positive.
lambda = 30000; Delta_E = 0.10; Delta_C = 2000; SE_E = 0.04; SE_C = 600; Cov_CE = 0; z = 1.96; INB = 1000; SE_INB = 1341.6408; INB_L = -1629.62; INB_U = 3629.62; z_INB = 0.7454
Threshold equal to the ICER gives zero net benefit
At a threshold equal to the ICER of £20,000 per QALY, incremental net benefit is zero, so z_INB is zero and the probability that net benefit is positive is one half, a limiting case that checks the implementation.
lambda = 20000; Delta_E = 0.10; Delta_C = 2000; SE_E = 0.04; SE_C = 600; Cov_CE = 0; z = 1.96; INB = 0; SE_INB = 1000; INB_L = -1960; INB_U = 1960; z_INB = 0
Common errors
Using one minus the two-sided p-value as the probability of positive net benefit
At £30,000 per QALY in the article's example the two-sided p-value for INB is about 0.456, and one minus it, about 0.544, understates the probability that net benefit is positive, which is about 0.772. The two-sided p-value has to be converted to a one-sided value first.
Omitting the covariance term from the net benefit variance
When the cost and effect differences are correlated, dropping the covariance term misstates the standard error: with positively correlated differences the variance is overstated, and with negatively correlated differences it is understated, which shifts both the interval and the probability.
Sources
Central limit theorem standard errors for incremental net benefit
Nixon RM, Wonderling D, Grieve RD. Non-parametric methods for cost-effectiveness analysis: the central limit theorem and the bootstrap compared. Health Economics. 2010;19(3):316-333. Abstract: trial-based analyses estimate incremental net benefit with 95% confidence intervals and compute acceptability curves; central limit theorem standard errors were accurate for moderate to large samples (more than 50) even with highly skewed costs.
Net benefit regression and one-sided p-values for the acceptability curve
Hoch JS, Hay A, Isaranuwatchai W, Thavorn K, Leighl NB, Tu D, Trenaman L, Dewa CS, O'Callaghan C, Pater J, Jonker D, Chen BE, Mittmann N. Advantages of the net benefit regression framework for trial-based economic evaluations of cancer treatments: an example from the Canadian Cancer Trials Group CO.17 trial. BMC Cancer. 2019;19:552. Methods section and Discussion: INB equals willingness to pay times the effect difference minus the cost difference. Note to Table 1 in the Results: two-sided p-values for INB are converted to one-sided values to draw the acceptability curve.
Net health benefit as the alternative to ratio intervals
Stinnett AA, Mullahy J. Net health benefits: a new framework for the analysis of uncertainty in cost-effectiveness analysis. Medical Decision Making. 1998;18(2 Suppl):S68-S80. Abstract: confidence intervals for cost-effectiveness ratios have theoretical limitations that make them inappropriate in many situations, and the net health benefit framework is proposed as the alternative.
Canonical Identity
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