Cochran's C test explained

Cochran's

C test,^[1] named after William G. Cochran, is a one-sided upper limit variance outlier statistical 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 F_max test, but Hartley's F_max 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 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:

C _{j} = (S ^{2} _{j})/(\displaystyle \sum ^{N} _{i =1} S ^{2} _{i})

where:

C_j = Cochran's C statistic for data series j

S_j = 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_i = standard deviation of data series i (1 ≤ i ≤ N)

The C test tests the null hypothesis (H₀) against the alternative hypothesis (H_a):

H₀: All variances are equal.

H_a: 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_j exceeds the upper limit critical value C_UL. C_UL 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_UL have been tabulated at significance levels α = 0.01,^{[7] [8]} α = 0.025, and α = 0.05.^[8] C_UL can also be calculated from:

C _{UL(\alpha,n,N)} = \left [ 1+ (N -1)/(F _{c(\alpha/N,(n-1),(N-1)(n-1))}) \right ] ^{-1} .

Here:

C_UL = upper limit critical value for one-sided test on a balanced design

α = significance level, e.g., 0.05

n = number of data points per data series

F_c = critical value of Fisher's F ratio; F_c can be obtained from tables of the F distribution^[9] 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 n_j in data series j.^[10]

See also

• Bartlett's test
• Levene's test
• Brown–Forsythe test
• Hartley's test
• F-test of equality of variances

External links

• Critical C values
• Generalized Variance Outlier Test
• Critical F values

Notes and References

1. 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).
2. 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 .
3. P. Konieczka, J. Namieśnik, Quality Assurance and Quality Control in the Analytical Chemical Laboratory – A Practical Approach, CRC Press, Boca Raton, Florida, 2009; .
4. J.K. Taylor, Quality Assurance of Chemical Measurements, 4th printing, Lewis Publishers, Chelsea, Michigan, 1988; .
5. [William Horwitz|W. Horwitz]
6. [International Organization for Standardization|ISO]
7. R. Moore, Mathematics Department, Macquarie University, Sydney, Australia, 1999: http://faculty.washington.edu/heagerty/Books/Biostatistics/TABLES/Cochran.
8. R.U.E. 't Lam, Scrutiny of variance results for outliers: Cochran's test optimized, Analytica Chimica Acta 659, 68–84 (2010);
9. Table of critical values of the F-distribution: NIST
10. R.U.E. 't Lam, Variance Outlier Test, blog: http://rtlam.blogspot.com/