Concept Architecture
Concept
Theoretically, Fieller Method is a statistical procedure for constructing confidence intervals for the ratio of two estimated parameters. It is based on Fieller's theorem, which accounts for the joint sampling variability and covariance of the numerator and denominator rather than treating the denominator as fixed. In health economics, the method is particularly important for estimating confidence intervals for incremental cost-effectiveness ratios (ICERs), where both incremental costs and incremental effects are subject to sampling uncertainty.
Mathematically, the Fieller method derives confidence limits by solving a quadratic inequality based on the joint distribution of two correlated parameter estimates. Unlike standard ratio approximations, it accommodates uncertainty in both the numerator and denominator and may produce finite, infinite or disjoint confidence intervals when the denominator approaches zero. The method assumes approximate multivariate normality of the parameter estimates.
In practice, the Fieller method is implemented using estimated means, variances, covariance and the appropriate critical value from the t-distribution or normal distribution. It is widely applied in health economic evaluations to quantify uncertainty around ICERs and other ratio estimators, particularly when bootstrap methods are not used.
Purpose
Used to construct statistically valid confidence intervals for ratios of correlated estimates, particularly incremental cost-effectiveness ratios in health economic evaluation.
Mathematical Formulae
Primary Formula
For R = X / Y, the Fieller confidence interval is obtained by solving:
(X ? RY)� � t� ? (Var(X) ? 2R Cov(X,Y) + R� Var(Y))
where:
- R = ratio estimate
- X = numerator estimate
- Y = denominator estimate
- Var(X) = variance of X
- Var(Y) = variance of Y
- Cov(X,Y) = covariance between X and Y
- t = critical value from the t-distribution
Supporting Formulae
R? = X / Y
Var(R?) � not used directly under Fieller's theorem because uncertainty in both X and Y is incorporated through the quadratic inequality.
Related Mathematical Methods
- Fieller's Theorem
- Incremental Cost-Effectiveness Ratio
- Delta Method
- Bootstrap Confidence Intervals
- Multivariate Normal Approximation
- Confidence Interval Estimation
Example
A new treatment has an estimated incremental cost of �2,000 and an estimated incremental health gain of 0.08 QALYs. Both estimates have sampling uncertainty and are positively correlated. Rather than constructing a confidence interval using a simple ratio approximation, the Fieller method combines the estimated variances and covariance to calculate confidence limits for the ICER. Because uncertainty in the denominator is explicitly incorporated, the resulting interval accurately reflects the statistical uncertainty surrounding cost-effectiveness.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| T.INV.2T | =T.INV.2T(0.05,198) | Obtain Fieller critical value |
| VAR.S | =VAR.S(B2:B201) | Estimate variance of incremental costs or effects |
| COVARIANCE.S | =COVARIANCE.S(B2:B201,C2:C201) | Estimate covariance between incremental costs and effects |
| SQRT | =SQRT(A2) | Calculate quadratic components of the confidence interval |
VBA (Optional)
Automate Fieller confidence interval calculations for incremental cost-effectiveness ratios using estimated costs, effects, variances and covariance matrices.
Sources
- Fieller EC. Some Problems in Interval Estimation. Journal of the Royal Statistical Society Series B. 1954.
- Briggs AH, Claxton K, Sculpher MJ. Decision Modelling for Health Economic Evaluation.
- O'Brien BJ, Briggs AH. Analysis of Uncertainty in Health Care Cost-Effectiveness Studies.
- Drummond MF, et al. Methods for the Economic Evaluation of Health Care Programmes.
- ISPOR Good Practice Reports.
Related Concepts (2)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What is the Fieller method?
A technique for calculating a confidence interval around the ratio of two normally distributed variables, useful for measures such as the ICER.
Source: Fieller 1954
What does the Fieller method compute for a ratio of estimates?
The Fieller method computes a confidence interval around the ratio of two normally distributed quantities, handling the awkward fact that dividing one uncertain estimate by another produces a ratio whose distribution is not itself normal. Rather than approximating, it works out the interval exactly from the joint uncertainty of numerator and denominator, which makes it well suited to the incremental cost-effectiveness ratio. This can give more reliable intervals than the delta method when the denominator is uncertain. Bounding a ratio of estimates is its purpose. Drummond and colleagues (2015) discuss this.
Source: Drummond et al. 2015
How does the Fieller method work?
The Fieller method works by using the joint distribution of the numerator and denominator, including their variances and covariance, to derive the set of ratio values consistent with the data at the chosen confidence level, solving a quadratic equation that arises from the ratio structure. This yields the confidence limits directly. So the Fieller method works by treating the ratio through the combined uncertainty of its parts rather than approximating it, which correctly handles the fact that a ratio's distribution can be skewed and even unbounded when the denominator is uncertain, producing a confidence interval that reflects these features better than simpler methods.
Source: Fieller 1954
When is the Fieller method used?
The Fieller method is used when a confidence interval is needed for a ratio of two estimated quantities, particularly when the denominator is uncertain and may be near zero, as with the incremental cost-effectiveness ratio in health economics. In such cases, methods that assume the ratio is approximately normal can fail. So the Fieller method is used to obtain valid confidence intervals for ratios where the uncertainty in the denominator matters, which is common for cost-effectiveness ratios, providing a more reliable interval than the delta method when the denominator's variability makes the ratio's distribution awkward.
Source: Fieller 1954
How does the Fieller method differ from the delta method?
The Fieller method derives a confidence interval for a ratio directly from the joint distribution of its components, correctly handling skew and uncertainty in the denominator, while the delta method approximates the ratio's variance through a linear expansion and assumes approximate normality. The Fieller method can give asymmetric or even unbounded intervals when the denominator is uncertain, which the delta method cannot capture. So the two differ in that the Fieller method accounts for the ratio's true behaviour and the delta method linearises it, with the Fieller method generally more reliable for ratios with uncertain denominators, such as cost-effectiveness ratios.
Source: Drummond et al. 2015
What are the limitations of the Fieller method?
The limitations of the Fieller method include that it assumes the numerator and denominator are approximately normally distributed, which may not hold; that it can produce unusual intervals, such as unbounded or disjoint ones, when the denominator is very uncertain, which, though technically correct, are hard to interpret; and that it is more complex than simple approximations. So the Fieller method is used with attention to its normality assumption and to the interpretation of its intervals, since although it handles ratios more validly than simpler methods, its results can be difficult to communicate when the denominator's uncertainty is large, which is why bootstrapping is sometimes used as an alternative for ratios such as the incremental cost-effectiveness ratio.
Source: Fieller 1954
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 16 Dec 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/fieller-method
- Term code
- HE-ES-SA-065
Stable URI · Machine-readable · Resolvable · CC BY 4.0