Signature
a = Delta_E^2 - z^2 * SE_E^2; b = -2 * (Delta_C * Delta_E - z^2 * Cov_CE); c = Delta_C^2 - z^2 * SE_C^2; R_L = (-b - sqrt(b^2 - 4 * a * c)) / (2 * a); R_U = (-b + sqrt(b^2 - 4 * a * c)) / (2 * a)
| Inputs | Definition | Unit |
|---|---|---|
Delta_E | Estimated difference in mean effect, intervention minus comparator | effect, for example QALYs |
z | Standard normal critical value, 1.96 for a two-sided 95% interval | none |
SE_E | Standard error of Delta_E | effect |
Delta_C | Estimated difference in mean cost, intervention minus comparator | currency |
Cov_CE | Estimated covariance between Delta_C and Delta_E | currency times effect |
SE_C | Standard error of Delta_C | currency |
a | Coefficient of R squared: the squared effect difference less z squared times the squared standard error of the effect difference. Positive only when the effect difference is significant at the chosen level | effect squared |
|---|---|---|
b | Coefficient of R: twice z squared times the covariance, less twice the product of the two differences | currency times effect |
c | Constant term: the squared cost difference less z squared times the squared standard error of the cost difference | currency squared |
R_L | Lower confidence limit for the ICER when a is positive | currency per unit of effect |
R_U | Upper confidence limit for the ICER when a is positive | currency per unit of effect |
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
Fieller limits from three helper cells
The first three formulas fill helper cells named QuadA, QuadB and QuadC; the last two return the lower and upper limits, which bound an interval only when QuadA is positive.
=DeltaEffect^2-ZValue^2*SeEffect^2; =-2*(DeltaCost*DeltaEffect-ZValue^2*CovCostEffect); =DeltaCost^2-ZValue^2*SeCost^2; =(-QuadB-SQRT(QuadB^2-4*QuadA*QuadC))/(2*QuadA); =(-QuadB+SQRT(QuadB^2-4*QuadA*QuadC))/(2*QuadA)
Assumptions
Jointly normal cost and effect differences
The two differences are approximately jointly normal, which the central limit theorem supports because each is a difference of averages. No normal approximation is applied to the ratio itself.
Bounded Fieller interval only when a is positive
The roots bound an interval only when a is positive, that is when the effect difference divided by its standard error exceeds z. When a is negative and the discriminant positive, the confidence set is every value below the smaller root together with every value above the larger root; when the discriminant is also negative, the set is the whole real line.
Worked examples
Fieller interval for the article's trial
With the article's figures the quadratic has a of about 0.0038534, b of -400 and c of 2,617,024, and its roots give a 95% interval of about £7,017 to £96,786 per QALY. The interval is asymmetric about the ICER of £20,000 per QALY.
Delta_C = 2000; Delta_E = 0.10; SE_C = 600; SE_E = 0.04; Cov_CE = 0; z = 1.96; a = 0.0038534; b = -400; c = 2617024; R_L = 7016.89; R_U = 96786.47
No sampling uncertainty collapses the Fieller interval
With both standard errors set to zero the discriminant is zero and both limits equal the ICER, here £2,000 divided by 0.5, a limiting case that checks the implementation.
Delta_C = 2000; Delta_E = 0.5; SE_C = 0; SE_E = 0; Cov_CE = 0; z = 1.96; a = 0.25; b = -2000; c = 4000000; R_L = 4000; R_U = 4000
Common errors
Reading the two roots as an interval when a is negative
With a QALY standard error of 0.06, a is about -0.00383 and the roots are about £6,177 and -£110,622 per QALY. Reporting the span between them as an interval is wrong: the confidence set is every value below about -£110,622 together with every value above about £6,177, so the data place no upper limit on the ICER.
Sources
Fieller's theorem applied to cost-effectiveness ratios
Willan AR, O'Brien BJ. Confidence intervals for cost-effectiveness ratios: an application of Fieller's theorem. Health Economics. 1996;5(4):297-305. Abstract: Fieller's theorem is used to calculate confidence intervals for the ratio of the between-treatment differences in cost and effectiveness, the incremental cost-effectiveness ratio.
Fieller limits as zero crossings of the net benefit confidence limits
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: in a graph of incremental net benefit against willingness to pay, the x-intercepts of the 95% confidence limits give the lower and upper Fieller limits for the ICER.
Canonical Identity
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