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{{expert-subject|Statistics|date=November 2010}}


In [[statistics]], '''Fieller's theorem''' allows the calculation of a [[confidence interval]] for the ratio of two [[arithmetic mean|means]].


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==Approximate confidence interval==
Variables ''a'' and ''b'' may be measured in different units, so there is no way to directly combine the [[standard error]]s as they may also be in different units. The most complete discussion of this is given by Fieller (1954).<ref>{{cite journal |author=Fieller, EC. |title=Some problems in interval estimation. |journal=[[Journal of the Royal Statistical Society, Series B]] |volume=16 |issue= 2|pages=175–185 |year=1954 |jstor=2984043 }}</ref>
 
Fieller showed that if ''a'' and ''b'' are (possibly [[correlation|correlated]]) [[sample mean|means of two samples]] with [[expected value|expectations]] <math>\mu_a</math> and <math>\mu_b</math>, and variances <math>\nu_{11}\sigma^2</math> and <math>\nu_{22}\sigma^2</math> and covariance <math>\nu_{12}\sigma^2</math>, and if <math>\nu_{11}, \nu_{12}, \nu_{22} </math> are all known, then a (1&nbsp;&minus;&nbsp;''α'') confidence interval (''m''<sub>L</sub>,&nbsp;''m''<sub>U</sub>) for <math>a/b</math> is given by
 
: <math>(m_L, m_{U}) = \frac{1}{(1-g)} \left[\frac{a}{b} - \frac{g\nu_{12}}{\nu_{22}} \mp \frac{t_{r,\alpha}s}{b} \sqrt{\nu_{11} - 2\frac{a}{b}\nu_{12} + \frac{a^2}{b^2} \nu_{22} - g\left(\nu_{11} - \frac{\nu_{12}^2}{\nu_{22}}\right)} \right]</math>
 
where
:<math>g=\frac{t^{2}_{r,\alpha}s^2\nu_{22}}{b^2}.</math>
Here <math>s^2</math> is an [[bias of an estimator|unbiased estimator]] of <math>\sigma^2</math> based on r degrees of freedom, and <math>t_{r,\alpha}</math> is the <math>\alpha</math>-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.
 
== 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 that ''a'' and ''b'' are [[multivariate normal distribution|jointly]] [[normal distribution|normally distributed]], and that ''b'' is not too near zero (i.e. more specifically, that the standard error of ''b'' is small compared to ''b''),
 
: <math>\operatorname{Var} \left( \frac{a}{b} \right) = \left( \frac{a}{b} \right)^{2} \left( \frac{\operatorname{Var}(a)}{a^2} + \frac{\operatorname{Var}(b)}{b^2}\right).</math>
 
From this a 95% confidence interval can be constructed in the usual way (degrees of freedom for ''t''<sup>&nbsp;*</sup> 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'',&nbsp;''y''):
 
: <math>\operatorname{Var}(f) = \left(\frac{\partial f}{\partial x}\right)^2 \operatorname{Var}(x) + \left(\frac{\partial f}{\partial y}\right)^2 \operatorname{Var}(y) + 2\frac{\partial f}{\partial x}.\frac{\partial f}{\partial y} \operatorname{Cov}(x,y)</math>
 
to obtain either,
 
: <math>\operatorname{Var}\left(\log_e \frac{a}{b}\right) = \frac{\operatorname{Var}(a)}{a^2} + \frac{\operatorname{Var}(b)}{b^2}</math>
 
or
 
: <math>\operatorname{Var}\left(\log_{10} \frac{a}{b}\right) = \left(\log_{10}e\right)^2 \left( \frac{\operatorname{Var}(a)}{a^2} + \frac{\operatorname{Var}(b)}{b^2}\right).</math>
 
===Case 2===
Assume that ''a'' and ''b'' are jointly [[multivariate normal distribution|normally distributed]], and that b is near zero (i.e. SE(''b'') is ''not'' small compared to ''b'').
 
First, calculate the intermediate quantity:
 
: <math>g=\left(\frac{t^{*}}{b}\right)^{2}\operatorname{Var}(b).</math>
 
You cannot calculate the confidence interval of the quotient if <math>g\ge 1</math>, as the CI for the denominator ''μ''<sub>''b''</sub> will include zero.
 
However if <math>g<1</math> then we can obtain
 
: <math> \operatorname{Var} \left( \frac{a}{b} \right) = \left( \frac{a}{b(1-g)} \right)^{2} \left((1-g)\frac{\operatorname{Var}(a)}{a^2} + \frac{\operatorname{Var}(b)}{b^2} \right).</math>
 
=== Other ===
 
One problem is that, when g is not small, CIs can blow up when using Fieller's theorem. Andy Grieve has provided a Bayesian solution where the CIs are still sensible, albeit wide.<ref>{{cite journal |author=O'Hagan, A., Stevens, JW. and Montmartin, J. | title=Inference for the cost-effectiveness acceptability curve and cost-effectiveness ratio. |journal=[[PharmacoEconomics (journal)|Pharmacoeconomics]] |volume=17|issue=4 |pages=339–49 |year=2000 |doi=10.2165/00019053-200017040-00004 |pmid=10947489}}</ref> [[Bootstrapping]] provides another alternative that that does not require the assumption of normality.<ref>{{cite journal|last=Campbell|first=M. K.|coauthors=Torgerson, D. J.|journal=[[QJM: an International Journal of Medicine]]|year=1999|volume=92|issue=3|pages=177–182|doi=10.1093/qjmed/92.3.177|title=Bootstrapping: estimating confidence intervals for cost-effectiveness ratios}}</ref>
 
== History ==
Edgar C. Fieller (1907&ndash;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 UK|Boots Pure Drug Company]] as a statistician and [[operational research]]er 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 (United Kingdom)|National Physical Laboratory]].<ref>{{cite journal |author=Irwin, J. O. and Rest, E. D. Van | title=Edgar Charles Fieller, 1907-1960 |journal=Journal of the Royal Statistical Society, Series A |volume=124 |issue=2 |pages=275–277 |year=1961 | jstor=2984155 |publisher=Blackwell Publishing}}</ref>
 
==See also==
*[[Ratio_distribution#Gaussian_ratio_distribution|Gaussian ratio distribution]]
 
== Notes ==
{{reflist}}
 
== Further reading ==
* Iris Pigeot, Juliane Schafer, Joachim Rohmel and Dieter Hauschke (2003) "Assessing non-inferiority of a new treatment in a three-arm clinical trial including a placebo". ''[[Statistics in Medicine (journal)|Statistics in Medicine]]'', 22:883–899, {{doi| 10.1002/sim.1450}}
* Fieller, EC. (1932) "The distribution of the index in a bivariate Normal distribution". ''[[Biometrika]]'', 24(3&ndash;4):428&ndash;440. {{doi| 10.1093/biomet/24.3-4.428}}
* Fieller, EC. (1940) "The biological standardisation of insulin". ''[[Journal of the Royal Statistical Society]] (Supplement)''. 1:1&ndash;54. {{jstor|2983630}}
* 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. ISBN 0-19-508607-4
* Senn, Steven (2007) ''Statistical Issues in Drug Development''. Second Edition. Wiley. ISBN 0-471-97488-9
*{{cite doi|10.1198/tast.2010.08130}}
 
{{DEFAULTSORT:Fieller's Theorem}}
[[Category:Statistical theorems]]
[[Category:Statistical approximations]]
[[Category:Normal distribution]]

Revision as of 01:47, 22 November 2012

Template:Expert-subject

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]

Fieller showed that if a and b are (possibly correlated) means of two samples with expectations μa and μb, and variances ν11σ2 and ν22σ2 and covariance ν12σ2, and if ν11,ν12,ν22 are all known, then a (1 − α) confidence interval (mLmU) for a/b is given by

(mL,mU)=1(1g)[abgν12ν22tr,αsbν112abν12+a2b2ν22g(ν11ν122ν22)]

where

g=tr,α2s2ν22b2.

Here s2 is an unbiased estimator of σ2 based on r degrees of freedom, and tr,α is the α-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.

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

Var(ab)=(ab)2(Var(a)a2+Var(b)b2).

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(xy):

Var(f)=(fx)2Var(x)+(fy)2Var(y)+2fx.fyCov(x,y)

to obtain either,

Var(logeab)=Var(a)a2+Var(b)b2

or

Var(log10ab)=(log10e)2(Var(a)a2+Var(b)b2).

Case 2

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

First, calculate the intermediate quantity:

g=(t*b)2Var(b).

You cannot calculate the confidence interval of the quotient if g1, as the CI for the denominator μb will include zero.

However if g<1 then we can obtain

Var(ab)=(ab(1g))2((1g)Var(a)a2+Var(b)b2).

Other

One problem is that, when g is not small, CIs 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 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

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Further reading

  • Iris Pigeot, Juliane Schafer, Joachim Rohmel and Dieter Hauschke (2003) "Assessing non-inferiority of a new treatment in a three-arm clinical trial including a placebo". Statistics in Medicine, 22:883–899, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park.
  • Fieller, EC. (1932) "The distribution of the index in a bivariate Normal distribution". Biometrika, 24(3–4):428–440. 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park.
  • Fieller, EC. (1940) "The biological standardisation of insulin". Journal of the Royal Statistical Society (Supplement). 1:1–54. Template:Jstor
  • 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. ISBN 0-19-508607-4
  • Senn, Steven (2007) Statistical Issues in Drug Development. Second Edition. Wiley. ISBN 0-471-97488-9
  • Template:Cite doi
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    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang