Fieller Method

Fieller Method for Confidence Intervals around Cost-Effectiveness Ratios

Fieller's method finds the incremental cost-effectiveness ratios that a trial's cost and effect data cannot reject, by testing each candidate ratio in turn instead of approximating the sampling distribution of the ratio estimate. The test applies the normal approximation to a linear combination of the cost and effect differences, which is far better behaved than their quotient, and the limits are the two roots of a quadratic equation. The method matters in trial-based economic evaluation because the ICER divides by an often small and imprecise effect difference, and Briggs and Gray's 1999 Health Technology Assessment review recommended Fieller's method or the non-parametric bootstrap whenever patient-level cost and effect data exist. Fieller intervals are usually asymmetric, and when the effect difference is not statistically significant they become unbounded, which correctly reports that the data set no upper limit on the ratio.

Why the ICER needs its own interval method

An ICER is the ratio of two estimated differences in means, and both carry sampling error. Franz notes that the simple ratio estimator has neither an expected value nor a variance, so the familiar confidence interval of estimate plus or minus 1.96 standard errors has no firm footing. Briggs, Mooney and Wonderling describe the variance of a ratio estimator as intractable, and in their Monte Carlo study methods that assumed a normal sampling distribution performed poorly, as did combining the separate intervals on costs and effects. The delta method is one such approximation, with an interval symmetric by construction.

Briggs and Gray date Fieller's treatment of ratios, developed for bioassay, to 1932. Willan and O'Brien applied Fieller's theorem to cost-effectiveness data from trials with a binary effectiveness outcome, calculating confidence intervals for the ratio of the between-treatment differences in costs and effectiveness.

The test statistic behind Fieller's interval

The method turns a question about a ratio into a question about a difference. If $R$ were the true ICER, the quantity $\Delta C - R\,\Delta E$ would have mean zero, and under joint normality of the two differences that quantity is normally distributed. Briggs and Gray divide the quantity by its standard deviation to obtain a standard normal statistic.

$$T(R) = \frac{\Delta \hat{C} - R\,\Delta \hat{E}}{\sqrt{v_C + R^2 v_E - 2R\,c_{CE}}}$$

where $R$ is a candidate value of the ICER, $\Delta \hat{C}$ and $\Delta \hat{E}$ are the estimated differences in mean cost and mean effect between the new treatment and its comparator, $v_C$ and $v_E$ are the estimated variances of those differences, and $c_{CE}$ is their estimated covariance.

The 95% confidence set holds every $R$ whose statistic does not exceed the critical value $z$ of 1.96 in absolute value. Setting the squared statistic equal to $z^2$ gives a quadratic in $R$ with these roots.

$$R_{1,2} = \frac{(\Delta \hat{C}\,\Delta \hat{E} - z^2 c_{CE}) \pm \sqrt{(\Delta \hat{C}\,\Delta \hat{E} - z^2 c_{CE})^2 - (\Delta \hat{E}^2 - z^2 v_E)(\Delta \hat{C}^2 - z^2 v_C)}}{\Delta \hat{E}^2 - z^2 v_E}$$

where $R_{1,2}$ are the two roots, the smaller being the lower limit when the interval is bounded, $z$ is the critical value, and the other symbols are as above. The same expression appears in the tutorial by Poltavskiy and colleagues and in Franz's guide to ratios, and Briggs and Gray give an equivalent form in coefficients of variation. Franz notes that the statistic follows a t distribution exactly only under strict conditions, so the normal distribution usually supplies the critical value.

Bounded, exclusive and unbounded confidence sets

The sign of the leading coefficient, $\Delta \hat{E}^2 - z^2 v_E$, decides the shape of the set. Franz shows that the interval is bounded if and only if the effect difference is significantly different from zero, which is when $\Delta \hat{E}^2 / v_E$ exceeds $z^2$. Otherwise a joint statistic for the two differences separates two further cases.

$$Q = \frac{\Delta \hat{C}^2 v_E - 2\,\Delta \hat{C}\,\Delta \hat{E}\,c_{CE} + \Delta \hat{E}^2 v_C}{v_C v_E - c_{CE}^2}$$

where $Q$ is the joint statistic, written in a form algebraically identical to Franz's expression. If $Q$ exceeds $z^2$, the set contains every value except those between the two roots, an exclusive set made of two unbounded rays. If $Q$ falls below $z^2$, the quadratic has no real roots and the set excludes no value at all.

On the cost-effectiveness plane, Franz's geometric reading makes the limits the slopes of the two rays from the origin that touch a confidence ellipse centred on the point estimate. Once the ellipse touches the cost axis, arbitrarily large ratios become compatible with the data. With a significant positive effect difference the ellipse lies right of the cost axis, so a negative limit within a bounded interval means a cost saving with a health gain, whereas Briggs and Gray stress that negative ICERs in general cannot distinguish the quadrants.

Unbounded sets are an accurate result. Franz cites work by Gleser and Hwang showing that any method unable to produce unbounded confidence sets for a ratio can depart arbitrarily far from its intended confidence level, which affects almost every alternative. Poltavskiy and colleagues add that a positive lower limit with a negative upper limit may need flipping and reading as two disconnected pieces.

Inputs from a trial-based economic evaluation

Besides the critical value, the formula needs the two mean differences, their variances and their covariance. With independent trial arms, Briggs and Gray take the variance of each difference as the sum of the two arms' variances of the mean. By the same variance-of-a-sum rule, the covariance of the differences is the sum of the two arms' covariances of the means. Briggs and Gray show that the width of an ICER interval changes dramatically with the covariance between numerator and denominator.

Joint normality of the two differences is the central assumption. It rests on the central limit theorem applied to mean differences, the subject of asymptotic normality, and Briggs and Gray caution that the assumption may be hard to justify in small samples because healthcare costs are often substantially skewed.

Worked example with a trial-based cost-utility analysis

All numbers in this example are illustrative. A two-arm trial reports an incremental cost of GBP 3,000 per patient with a standard error of GBP 900, and an incremental effect of 0.15 QALYs with a standard error of 0.06. The correlation between the two differences is 0.2, so their covariance is 0.2 × 900 × 0.06 = 10.8.

1. Point estimate. The ICER is 3,000 / 0.15 = GBP 20,000 per QALY. The effect difference is 0.15 / 0.06 = 2.5 standard errors from zero, above 1.96, so the interval will be bounded.

2. Coefficients. With $z^2$ = 1.96² = 3.8416, the leading coefficient is 0.0225 − 0.01382976 = 0.00867024, since 3.8416 × 0.0036 = 0.01382976. The middle term is 450 − 41.48928 = 408.51072, since 3,000 × 0.15 = 450 and 3.8416 × 10.8 = 41.48928. The constant is 9,000,000 − 3,111,696 = 5,888,304, since 3.8416 × 810,000 = 3,111,696.

3. Discriminant. The squared middle term is 408.51072² = 166,881.0 and the product of the outer coefficients is 0.00867024 × 5,888,304 = 51,053.0, so the discriminant is 166,881.0 − 51,053.0 = 115,828.0, with square root 340.335.

4. Limits. For the lower limit, 408.511 − 340.335 = 68.176 and 68.176 / 0.00867024 = 7,863. For the upper limit, 408.511 + 340.335 = 748.846 and 748.846 / 0.00867024 = 86,370. The interval runs from GBP 7,863 to GBP 86,370 per QALY, far further above the point estimate than below it.

5. Delta method. The coefficients of variation are 900 / 3,000 = 0.3 and 0.06 / 0.15 = 0.4, so the delta-method standard error is 20,000 × √(0.09 + 0.16 − 2 × 0.2 × 0.3 × 0.4) = 20,000 × √0.202 = GBP 8,989. Its interval, 20,000 ± 1.96 × 8,989, runs from GBP 2,382 to GBP 37,618, with an upper limit less than half of Fieller's.

6. Net benefit check. At a threshold equal to the lower limit, incremental net benefit is 7,863.17 × 0.15 − 3,000 = −1,820.5. Since 7,863.17 × 7,863.17 × 0.0036 = 222,586 and 2 × 7,863.17 × 10.8 = 169,845, its variance is 222,586 + 810,000 − 169,845 = 862,741, the standard error is 928.8, and the net benefit lies 1,820.5 / 928.8 = 1.96 standard errors below zero.

7. A less precise trial. With a QALY standard error of 0.09, the effect difference is 0.15 / 0.09 = 1.67 standard errors from zero and the covariance becomes 16.2. The leading coefficient turns negative at 0.0225 − 0.03111696 = −0.00861696, the middle term is 450 − 62.23392 = 387.76608 and the discriminant is 201,101.8, with square root 448.444. The roots are GBP 7,042 and minus GBP 97,042, and because $Q$ is 12.15, above 3.84, the set holds every ICER above GBP 7,042 together with every value below minus GBP 97,042. The delta method would still give a tidy minus GBP 4,101 to GBP 44,101.

8. Interpretation. The first trial rules out ICERs below about GBP 7,900 and above about GBP 86,400 per QALY, and the asymmetry reflects the chance that the QALY gain is much smaller than 0.15. The second sets no upper limit, which the delta method hides.

Fieller limits, net benefit and decisions

Heitjan showed that the Fieller confidence set for the ICER equals the set of threshold values at which the incremental net benefit of the new treatment is not significantly different from zero. Hoch and colleagues show that the 95% limits of incremental net benefit, plotted against willingness to pay, cross zero at the Fieller limits, and that net benefit regression helps when Fieller's theorem yields no interval. Step 6 of the example shows this.

The NICE manual, NICE technology appraisal and highly specialised technologies guidance (PMG36), asks in section 6.3.2, when decision-making modifiers are applied, for net health benefits at values of GBP 25,000 and GBP 35,000 per QALY gained, and section 6.3.5 lists the degree of certainty around the ICER among the factors considered above a most plausible ICER of GBP 25,000. In the first illustrative trial both values lie inside the Fieller interval, where incremental net benefit is only 0.47 and 1.06 standard errors above zero. Briggs and Gray suggest that cost-effectiveness acceptability curves may give decision makers more useful information than an interval for the ratio, partly because an ICER is only appropriate when the cost and effect differences share a sign.

Performance against the delta method and the bootstrap

Monte Carlo comparisons favour Fieller's method over normal approximations. Polsky and colleagues found the miscoverage of bootstrap and Fieller intervals more accurate than that of box and Taylor series intervals, with Taylor series intervals asymmetrically underestimating the upper limit. Severens, De Boo and Konst found both unsatisfactory when the effect difference was close to zero, but Fieller's interval best when the difference was significant, recommending Fieller's method as easier to compute than the bootstrap.

The non-parametric bootstrap resamples patients and makes no normality assumption, which suits skewed costs. Franz notes that standard bootstrap limits cannot be unbounded and so fall under the Gleser and Hwang result, with a variant due to Hwang, which solves the same quadratic, as the exception.

Misreadings and limits

Most errors with Fieller intervals treat the ratio like a mean. Five recur.

1. Treating an unbounded set as a failed calculation. An unbounded or exclusive set is the correct answer when the effect difference is not significant.
2. Reporting the interval only when it is bounded. Franz warns that this rule uses a conditional confidence level, which can be arbitrarily low.
3. Reading a negative limit without the quadrant. A negative ratio can mean savings with a health gain or extra cost with a health loss, so both differences need reporting.
4. Trusting joint normality in small, skewed samples. A few very costly patients can leave a modest trial's differences far from normal.
5. Taking the limits as the whole of the uncertainty. Briggs and Gray point out that statistical analysis only summarises uncertainty due to sampling variation, leaving inputs without sample data, such as unit costs, to sensitivity analysis.

Sources

• Briggs AH, Gray AM. Handling uncertainty when performing economic evaluation of healthcare interventions. Health Technology Assessment. 1999;3(2). https://doi.org/10.3310/hta3020
• Briggs AH, Mooney CZ, Wonderling DE. Constructing confidence intervals for cost-effectiveness ratios: an evaluation of parametric and non-parametric techniques using Monte Carlo simulation. Statistics in Medicine. 1999;18(23):3245-3262. https://pubmed.ncbi.nlm.nih.gov/10602149/
• Franz VH. Ratios: a short guide to confidence limits and proper use. arXiv:0710.2024 [stat.AP]; 2007. https://arxiv.org/abs/0710.2024
• Heitjan DF. Fieller's method and net health benefits. Health Economics. 2000;9(4):327-335. https://pubmed.ncbi.nlm.nih.gov/10862076/
• Hoch JS, Hay A, Isaranuwatchai W, et al. 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. https://doi.org/10.1186/s12885-019-5779-x
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). Published 31 January 2022, last updated 13 October 2026. Sections 6.3.2 and 6.3.5. https://www.nice.org.uk/process/pmg36
• Polsky D, Glick HA, Willke R, Schulman K. Confidence intervals for cost-effectiveness ratios: a comparison of four methods. Health Economics. 1997;6(3):243-252. https://pubmed.ncbi.nlm.nih.gov/9226142/
• Poltavskiy E, Liu D, Chen S, Bang H, Zhao H. Mean cost and cost-effectiveness ratios with censored data: a tutorial and SAS macros. Biostatistics and Epidemiology. 2025;9(1). https://doi.org/10.1080/24709360.2025.2565537
• Severens JL, De Boo TM, Konst EM. Uncertainty of incremental cost-effectiveness ratios: a comparison of Fieller and bootstrap confidence intervals. International Journal of Technology Assessment in Health Care. 1999;15(3):608-614. https://pubmed.ncbi.nlm.nih.gov/10874387/
• Willan AR, O'Brien BJ. Confidence intervals for cost-effectiveness ratios: an application of Fieller's theorem. Health Economics. 1996;5(4):297-305. https://pubmed.ncbi.nlm.nih.gov/8880166/