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In [[statistics]], '''Cochran's C test''',<ref>W.G. Cochran, The distribution of the largest of a set of estimated variances as a fraction of their total, Annals of Human Genetics (London) 11(1), 47–52 (January 1941).</ref> named after [[William Gemmell Cochran|William G. Cochran]], is a [[one-tailed test|one-sided]] upper limit variance [[outlier]] test. The C test is used to decide if a single [[Estimation theory|estimate]] of a [[variance]] (or a [[standard deviation]]) is [[statistical significance|significantly]] larger than a group of variances (or standard deviations) with which the single estimate is supposed to be comparable. The C test is discussed in many text books <ref>D.L. Massart, B.G.M. Vandeginste, L.M.C. Buydens, S. de Jong, P.J. Lewi, J. Smeyers-Verbeke, ''Handbook of Chemometrics and Qualimetrics'': Part A, Elsevier, Amsterdam, The Netherlands, 1997 ISBN 0-444-89724-0.</ref><ref name="QA">P. Konieczka, J. Namieśnik, Quality Assurance and Quality Control in the Analytical Chemical Laboratory – A Practical Approach, CRC Press, Boca Raton, Florida, 2009; ISBN 978-1-4200-8270-8.</ref><ref>J.K. Taylor, Quality Assurance of Chemical Measurements, 4<sup>th</sup> printing, Lewis Publishers, Chelsea, Michigan, 1988; ISBN 0-87371-097-5.</ref> and has been recommended by [[IUPAC]] <ref>[[William Horwitz|W. Horwitz]], Harmonized protocol for the design and interpretation of collaborative studies, Trends in Analytical Chemistry 7(4), 118–120 (April 1988).</ref> and [[International Organization for Standardization|ISO]].<ref name="ISO">[[International Organization for Standardization|ISO]] Standard 5725–2:1994, “[[Accuracy]] (trueness and precision) of measurement methods and results – Part 2: Basic method for the determination of [[repeatability]] and [[reproducibility]] of a standard measurement method”, International Organization for Standardization, Geneva, Switzerland, 1994; | |||
http://www.iso.org/iso/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=11834</ref> Cochran's C test should not be confused with [[Cochran's Q test|Cochran's Q]] test, which applies to the [[analysis]] of two-way [[randomized block design]]s. | |||
The C test assumes a balanced design, i.e. the considered full [[data set]] should consist of individual data series that all have equal size. The C test further assumes that each individual data series is [[normally distributed]]. Although primarily an outlier test, the C test is also in use as a simple alternative for regular [[homoscedasticity]] tests such as [[Bartlett's test|Bartlett's]] test, [[Levene's test|Levene's]] test and the [[Brown–Forsythe test]] to check a [[statistical data]] set for [[homogeneity of variance]]s. An even simpler way to check homoscedasticity is provided by [[Hartley's test|Hartley's F<sub>max</sub> test]],<ref name="QA" /> but Hartley's F<sub>max</sub> test has the disadvantage that it only accounts for the minimum and the maximum of the variance range, while the C test accounts for all variances within the range. | |||
==Description== | |||
The C test detects one exceptionally large variance value at a time. The corresponding data series is then omitted from the full data set. According to ISO standard 5725 <ref name="ISO" /> the C test may be [[iterate]]d until no further exceptionally large variance values are detected, but such practice may lead to excessive rejections if the underlying data series are not normally distributed. | |||
The C test evaluates the [[ratio]]: | |||
::<math>C_j = \frac{S_j^2}{\displaystyle \sum_{i=1}^N S_i^2}</math> | |||
where: | |||
:''C<sub>j</sub>'' = Cochran's C statistic for data series ''j'' | |||
:''S<sub>j</sub>'' = standard deviation of data series ''j'' | |||
:''N'' = number of data series that remain in the data set; ''N'' is decreased in steps of 1 upon each iteration of the C test | |||
:''S<sub>i</sub>'' = standard deviation of data series i (1 ≤ ''i'' ≤ ''N'') | |||
The C test tests the [[null hypothesis]] (H<sub>0</sub>) against the [[alternative hypothesis]] (H<sub>a</sub>): | |||
:H<sub>0</sub>: All variances are equal. | |||
:H<sub>a</sub>: At least one variance value is significantly larger than the other variance values. | |||
==Critical values== | |||
The sample variance of data series ''j'' is considered an outlier at [[significance level]] ''α'' if ''C<sub>j</sub>'' exceeds the upper limit [[critical value]] C<sub>UL</sub>. C<sub>UL</sub> depends on the desired significance level ''α'', the number of considered data series ''N'', and the number of data points (''n'') per data series. Selections of values for C<sub>UL</sub> have been tabulated at significance levels α = 0.01,<ref name="ISO" /><ref name="Moore">R. Moore, Mathematics Department, Macquarie University, Sydney, Australia, 1999: http://faculty.washington.edu/heagerty/Books/Biostatistics/TABLES/Cochran.</ref><ref name="tabulated">R.U.E. 't Lam, Scrutiny of variance results for outliers: Cochran's test optimized, ''Analytica Chimica Acta'' 659, 68–84 (2010); {{doi|10.1016/j.aca.2009.11.032}}</ref> α = 0.025,<ref name="tabulated" /> and α = 0.05.<ref name="ISO" /><ref name="Moore" /><ref name="tabulated">R.U.E. 't Lam, Scrutiny of variance results for outliers: Cochran's test optimized, Analytica Chimica Acta 659, 68–84 (2010); {{doi|10.1016/j.aca.2009.11.032}}</ref> ''C''<sub>UL</sub> can also be calculated from:<ref name="tabulated" /><ref name="exist" /> | |||
:<math>C_\text {UL}(\alpha,n,N) = \left [ 1+ \frac{N-1}{F_\text {c}(\alpha/N,(n-1),(N-1)(n-1))} \right ]^{-1} .</math> | |||
Here: | |||
:''C''<sub>UL</sub> = upper limit critical value for one-sided test on a balanced design | |||
:''α'' = significance level | |||
:''n'' = number of data points per data series | |||
:''F''<sub>c</sub> = critical value of [[F-test of equality of variances|Fisher's F]] ratio; ''F''<sub>c</sub> can be obtained from tables of the [[F distribution]]<ref name="values">Table of critical values of the F-distribution:[http://www.itl.nist.gov/div898/handbook/eda/section3/eda3673.htm NIST]</ref> or using computer software for this function. | |||
==Generalization== | |||
The C test can be generalized to include unbalanced designs, one-sided lower limit tests and [[two-tail|two-sided]] tests at any significance level ''α'', for any number of data series ''N'', and for any number of individual data points ''n<sub>j</sub>'' in data series ''j''.<ref name="tabulated" /><ref name="exist">R.U.E. 't Lam, Variance Outlier Test, blog: http://rtlam.blogspot.com/</ref> | |||
==See also== | |||
*[[Bartlett's test]] | |||
*[[Levene's test]] | |||
*[[Brown–Forsythe test]] | |||
*[[Hartley's test]] | |||
*[[F-test of equality of variances]] | |||
==External links== | |||
*[http://faculty.washington.edu/heagerty/Books/Biostatistics/TABLES/Cochran Critical C values] <ref name="Moore" /> | |||
*[http://rtlam.blogspot.com/ Generalized Variance Outlier Test] <ref name="exist" /> | |||
*[http://www.itl.nist.gov/div898/handbook/eda/section3/eda3673.htm Critical F values] <ref name="values" /> | |||
==References== | |||
{{Reflist}} | |||
{{Statistics}} | |||
[[Category:Statistical tests]] | |||
Latest revision as of 14:04, 9 January 2013
In statistics, Cochran's C test,[1] named after William G. Cochran, is a one-sided upper limit variance outlier test. The C test is used to decide if a single estimate of a variance (or a standard deviation) is significantly larger than a group of variances (or standard deviations) with which the single estimate is supposed to be comparable. The C test is discussed in many text books [2][3][4] and has been recommended by IUPAC [5] and ISO.[6] Cochran's C test should not be confused with Cochran's Q test, which applies to the analysis of two-way randomized block designs.
The C test assumes a balanced design, i.e. the considered full data set should consist of individual data series that all have equal size. The C test further assumes that each individual data series is normally distributed. Although primarily an outlier test, the C test is also in use as a simple alternative for regular homoscedasticity tests such as Bartlett's test, Levene's test and the Brown–Forsythe test to check a statistical data set for homogeneity of variances. An even simpler way to check homoscedasticity is provided by Hartley's Fmax test,[3] but Hartley's Fmax test has the disadvantage that it only accounts for the minimum and the maximum of the variance range, while the C test accounts for all variances within the range.
Description
The C test detects one exceptionally large variance value at a time. The corresponding data series is then omitted from the full data set. According to ISO standard 5725 [6] the C test may be iterated until no further exceptionally large variance values are detected, but such practice may lead to excessive rejections if the underlying data series are not normally distributed. The C test evaluates the ratio:
where:
- Cj = Cochran's C statistic for data series j
- Sj = standard deviation of data series j
- N = number of data series that remain in the data set; N is decreased in steps of 1 upon each iteration of the C test
- Si = standard deviation of data series i (1 ≤ i ≤ N)
The C test tests the null hypothesis (H0) against the alternative hypothesis (Ha):
- H0: All variances are equal.
- Ha: At least one variance value is significantly larger than the other variance values.
Critical values
The sample variance of data series j is considered an outlier at significance level α if Cj exceeds the upper limit critical value CUL. CUL depends on the desired significance level α, the number of considered data series N, and the number of data points (n) per data series. Selections of values for CUL have been tabulated at significance levels α = 0.01,[6][7][8] α = 0.025,[8] and α = 0.05.[6][7][8] CUL can also be calculated from:[8][9]
![C_{{\text{UL}}}(\alpha ,n,N)=\left[1+{\frac {N-1}{F_{{\text{c}}}(\alpha /N,(n-1),(N-1)(n-1))}}\right]^{{-1}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/96755e5948b974323114069e58afa1cae4212e53)
Here:
- CUL = upper limit critical value for one-sided test on a balanced design
- α = significance level
- n = number of data points per data series
- Fc = critical value of Fisher's F ratio; Fc can be obtained from tables of the F distribution[10] or using computer software for this function.
Generalization
The C test can be generalized to include unbalanced designs, one-sided lower limit tests and two-sided tests at any significance level α, for any number of data series N, and for any number of individual data points nj in data series j.[8][9]
See also
External links
References
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- ↑ W.G. Cochran, The distribution of the largest of a set of estimated variances as a fraction of their total, Annals of Human Genetics (London) 11(1), 47–52 (January 1941).
- ↑ D.L. Massart, B.G.M. Vandeginste, L.M.C. Buydens, S. de Jong, P.J. Lewi, J. Smeyers-Verbeke, Handbook of Chemometrics and Qualimetrics: Part A, Elsevier, Amsterdam, The Netherlands, 1997 ISBN 0-444-89724-0.
- ↑ 3.0 3.1 P. Konieczka, J. Namieśnik, Quality Assurance and Quality Control in the Analytical Chemical Laboratory – A Practical Approach, CRC Press, Boca Raton, Florida, 2009; ISBN 978-1-4200-8270-8.
- ↑ J.K. Taylor, Quality Assurance of Chemical Measurements, 4th printing, Lewis Publishers, Chelsea, Michigan, 1988; ISBN 0-87371-097-5.
- ↑ W. Horwitz, Harmonized protocol for the design and interpretation of collaborative studies, Trends in Analytical Chemistry 7(4), 118–120 (April 1988).
- ↑ 6.0 6.1 6.2 6.3 ISO Standard 5725–2:1994, “Accuracy (trueness and precision) of measurement methods and results – Part 2: Basic method for the determination of repeatability and reproducibility of a standard measurement method”, International Organization for Standardization, Geneva, Switzerland, 1994; http://www.iso.org/iso/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=11834
- ↑ 7.0 7.1 7.2 R. Moore, Mathematics Department, Macquarie University, Sydney, Australia, 1999: http://faculty.washington.edu/heagerty/Books/Biostatistics/TABLES/Cochran.
- ↑ 8.0 8.1 8.2 8.3 8.4 R.U.E. 't Lam, Scrutiny of variance results for outliers: Cochran's test optimized, Analytica Chimica Acta 659, 68–84 (2010); 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. Cite error: Invalid
<ref>tag; name "tabulated" defined multiple times with different content - ↑ 9.0 9.1 9.2 R.U.E. 't Lam, Variance Outlier Test, blog: http://rtlam.blogspot.com/
- ↑ 10.0 10.1 Table of critical values of the F-distribution:NIST