Symmetric inverse semigroup: Difference between revisions

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

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