Fieller's theorem explained
In statistics, Fieller's theorem allows the calculation of a confidence interval for the ratio of two means.
Approximate confidence interval
Variables a and b may be measured in different units, so there is no way to directly combine the standard errors as they may also be in different units. The most complete discussion of this is given by Fieller (1954).[1]
and
, and variances
and
and covariance
, and if
are all known, then a (1 -
α) confidence interval (
mL,
mU) for
is given by
(mL,mU)=
-
\mp
\sqrt{\nu11-2
\nu12+
\nu22-g\left(\nu11-
\right)}\right]
where
Here
is an
unbiased estimator of
based on r degrees of freedom, and
is the
-level deviate from the
Student's t-distribution based on
r degrees of freedom.
Three features of this formula are important in this context:
a) The expression inside the square root has to be positive, or else the resulting interval will be imaginary.
b) When g is very close to 1, the confidence interval is infinite.
c) When g is greater than 1, the overall divisor outside the square brackets is negative and the confidence interval is exclusive.
Other methods
One problem is that, when g is not small, the confidence interval can blow up when using Fieller's theorem. Andy Grieve has provided a Bayesian solution where the CIs are still sensible, albeit wide.[2] Bootstrapping provides another alternative that does not require the assumption of normality.[3]
History
Edgar C. Fieller (1907 - 1960) first started working on this problem while in Karl Pearson's group at University College London, where he was employed for five years after graduating in Mathematics from King's College, Cambridge. He then worked for the Boots Pure Drug Company as a statistician and operational researcher before becoming deputy head of operational research at RAF Fighter Command during the Second World War, after which he was appointed the first head of the Statistics Section at the National Physical Laboratory.[4]
See also
- Gaussian ratio distribution
Further reading
- Pigeot . Iris. Iris Pigeot . Schäfer . Juliane . Röhmel . Joachim . Hauschke . Dieter . 2003 . Assessing non-inferiority of a new treatment in a three-arm clinical trial including a placebo . . 22 . 6. 883–899 . 10.1002/sim.1450 . 12627407 . 21180003 .
- Fieller . EC . 1932 . The distribution of the index in a bivariate Normal distribution . . 24 . 3–4. 428–440 . 10.1093/biomet/24.3-4.428 .
- Fieller, EC. (1940) "The biological standardisation of insulin". Journal of the Royal Statistical Society (Supplement). 1:1 - 54.
- Fieller . EC . 1944 . A fundamental formula in the statistics of biological assay, and some applications . Quarterly Journal of Pharmacy and Pharmacology . 17 . 117–123 .
- Motulsky, Harvey (1995) Intuitive Biostatistics. Oxford University Press.
- Senn, Steven (2007) Statistical Issues in Drug Development. Second Edition. Wiley.
- 64 . 234–241 . 2010 . 10.1198/tast.2010.08130 . Hirschberg . . J.. A Geometric Comparison of the Delta and Fieller Confidence Intervals. Lye . J. . 3. 122922413 .
Notes and References
- Fieller, EC. . Some problems in interval estimation. . . 16 . 2. 175–185 . 1954 . 2984043 .
- O'Hagan A, Stevens JW, Montmartin J . Inference for the cost-effectiveness acceptability curve and cost-effectiveness ratio. . . 17. 4 . 339–49 . 2000 . 10.2165/00019053-200017040-00004 . 10947489. 35930223 .
- Campbell. M. K.. Torgerson, D. J.. . 1999. 92. 3. 177–182. 10.1093/qjmed/92.3.177. Bootstrapping: estimating confidence intervals for cost-effectiveness ratios. 10326078 . free.
- Irwin, J. O. . Rest, E. D. Van . Edgar Charles Fieller, 1907-1960 . Journal of the Royal Statistical Society, Series A . 124 . 2 . 275–277 . 1961 . 2984155 . Blackwell Publishing.
