Fieller's theorem

In statistics, Fieller's theorem allows the calculation of a confidence interval for the ratio of two means.

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 in Fieller (1954).

Fieller showed that if "a" and "b" are (possibly correlated) means of two samples with expectations $mu\_a$ and $mu\_b$, and variances $u\_\{11\}sigma^2$ and $u\_\{22\}sigma^2$ and covariance $u\_\{12\}sigma^2$, then a (1 − "α") confidence interval ("m"_L, "m"_U) for $mu\_a/mu\_b$ is given by

: $(m\_L,\; m\_\{U\})\; =\; frac\{1\}\{(1-g)\}\; left\; [frac\{a\}\{b\}\; -\; frac\{g\; u\_\{12\{\; u\_\{22\; mp\; frac\{t\_\{r,alpha\}s\}\{b\}\; sqrt\{\; u\_\{11\}\; -\; 2frac\{a\}\{b\}\; u\_\{12\}\; +\; frac\{a^2\}\{b^2\}\; u\_\{22\}\; -\; gleft(\; u\_\{11\}\; -\; frac\{\; u\_\{12\}^2\}\{\; u\_\{22\; ight)\}\; ight]$

where $g=frac\{t^\{2\}\_\{r,alpha\}s^\{2\}\; u\_\{22\{b^2\}$, $s^2$ is an unbiased estimator of $sigma^2$ based on r degrees of freedom, and $t\_\{r,alpha\}$ is the $alpha$-level "t"-deviate 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.

Approximate formulae

These equations approximation to the full formula, and are obtained via a Taylor series expansion of a function of two variables and then taking the variance (i.e. a generalisation to two variables of the formula for the approximate standard error for a function of an estimate).

Case 1

Assume "a" and "b" are jointly normally distributed. "b" is not too near zero (i.e. more specifically, if the standard error of "b" is small compared to "b"),

: $operatorname\{Var\}\; left(\; frac\{a\}\{b\}\; ight)\; =\; left(\; frac\{a\}\{b\}\; ight)^\{2\}\; left(\; frac\{Var(a)\}\{a^2\}\; +\; frac\{Var(b)\}\{b^2\}\; ight).$

From this a 95% confidence interval can be constructed in the usual way (degrees of freedom for "t" ^* is equal to the total number of values in the numerator and denominator minus 2).

This can be expressed in a more useful form for when (as is usually the case) logged data is used, using the following relation for a function of "x" and "y", say ƒ("x", "y"):

: $operatorname\{Var\}(f)\; =\; left(frac\{partial\; f\}\{partial\; x\}\; ight)^2\; operatorname\{Var\}(x)\; +\; left(frac\{partial\; f\}\{partial\; y\}\; ight)^2\; operatorname\{Var\}(y)\; +\; 2frac\{partial\; f\}\{partial\; x\}.frac\{partial\; f\}\{partial\; y\}\; operatorname\{Cov\}(x,y)$

to obtain either,

: $operatorname\{Var\}left(log\_e\; frac\{a\}\{b\}\; ight)\; =\; frac\{operatorname\{Var\}(a)\}\{a^2\}\; +\; frac\{operatorname\{Var\}(b)\}\{b^2\}$

or

: $operatorname\{Var\}left(log\_\{10\}\; frac\{a\}\{b\}\; ight)\; =\; left(log\_\{10\}e\; ight)^2\; left(\; frac\{operatorname\{Var\}(a)\}\{a^2\}\; +\; frac\{operatorname\{Var\}(b)\}\{b^2\}\; ight).$

Case 2

Assume "a" and "b" are jointly normally distributed. b is near zero (i.e. SE("b") is "not" small compared to "b").

First, calculate the intermediate quantity:

: $g=left(frac\{t^\{*\{b\}\; ight)^\{2\}operatorname\{Var\}(b).$

You cannot calculate the confidence interval of the quotient if $gge\; 1$, as the CI for the denominator "μ"_"b" will include zero.

However if then we can obtain

: $operatorname\{Var\}\; left(\; frac\{a\}\{b\}\; ight)\; =\; left(\; frac\{a\}\{b(1-g)\}\; ight)^\{2\}\; left((1-g)frac\{operatorname\{Var\}(a)\}\{a^2\}\; +\; frac\{operatorname\{Var\}(b)\}\{b^2\}\; ight).$

Other

[http://pharmacoeconomics.adisonline.com/pt/re/phe/fulltext.00019053-200017040-00004.htm;jsessionid=L11JGT4n7nyzPW0GxJQ6jX7SXB1nvHBfCV62YCdlnZp3p2fpNj9K!-406629960!181195629!8091!-1#P92] The extension is that when g is not small, CLs can blow up when using Fieller's theorem. Andy Grieve has provided a Bayesian solution where the CLs are still sensible, albeit wide.

History

[http://www.jstor.org/pss/2984155] Edgar C. Fieller (1907–1960) was a statistician employed in the pharmaceutical industy (Boots).

References

[http://en.wikipedia.org/wiki/Ratio_distribution#Gaussian_ratio_distribution] Also see the Wikipedia entry on Gaussian ratio distribution.

Notes

* [http://biomet.oxfordjournals.org/cgi/content/citation/24/3-4/428] Fieller, EC. (1932) The distribution of the index in a bivariate Normal distribution. Biometrika 24(3–4):428–440.
* [http://www.jstor.org/pss/2983630] Fieller, EC. (1940) The biological standardisation of insulin. Journal of the Royal Statistical Society (Supplement). 1:1–54.
* [http://ul-newton.lib.cam.ac.uk/cgi-bin/Pwebrecon.cgi?v1=2&ti=1,2&cnt=25&SL=None&search%5Farg=fieller%20Quarterly%20Journal%20of%20Pharmacology&search%5Fcode=FT%2A&PID=YFBP6wpiuFa1Dwg0RrHeFjI97Y1z-&SEQ=20080717133422&SID=1] Fieller, EC. (1944) A fundamental formula in the statistics of biological assay, and some applications. Quarterly Journal of Pharmacy and Pharmacology. 17: 117-123.
* [http://www.jstor.org/pss/2984043] Fieller, EC. (1954). Some problems in interval estimation. Journal of the Royal Statistical Society B, 16:175–185.
* [http://www.us.oup.com/us/catalog/general/subject/LifeSciences/?view=usa&ci=9780195086072] Motulsky, Harvey (1995) Intuitive Biostatistics. Oxford University Press.
* [http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470018771.html] Senn, Steven (2007) Statistical Issues in Drug Development. Second Edition. Wiley.