Cochran's test is a non-parametricstatistical test to verify whether k treatments have identical effects in the analysis of two-way randomized block designs where the response variable is binary.[1][2][3] It is named after William Gemmell Cochran. Cochran's Q test should not be confused with Cochran's C test, which is a variance outlier test. Put in simple technical terms, Cochran's Q test requires that there only be a binary response (e.g. success/failure or 1/0) and that there be more than 2 groups of the same size. The test assesses whether the proportion of successes is the same between groups. Often it is used to assess if different observers of the same phenomenon have consistent results (interobserver variability).[4]
Cochran's Q test assumes that there are k > 2 experimental treatments and that the observations are arranged in bblocks; that is,
Treatment 1
Treatment 2
Treatment k
Block 1
X11
X12
X1k
Block 2
X21
X22
X2k
Block 3
X31
X32
X3k
Block b
Xb1
Xb2
Xbk
The "blocks" here might be individual people or other organisms.[5] For example, if b respondents in a survey had each been asked k Yes/No questions, the Q test could be used to test the null hypothesis that all questions were equally likely to elicit the answer "Yes".
where Χ21 − α,k − 1 is the (1 − α)-quantile of the chi-squared distribution with k − 1 degrees of freedom. The null hypothesis is rejected if the test statistic is in the critical region. If the Cochran test rejects the null hypothesis of equally effective treatments, pairwise multiple comparisons can be made by applying Cochran's Q test on the two treatments of interest.
The exact distribution of the T statistic may be computed for small samples. This allows obtaining an exact critical region. A first algorithm had been suggested in 1975 by Patil[6] and a second one has been made available by Fahmy and Bellétoile[7] in 2017.
^William G. Cochran (December 1950). "The Comparison of Percentages in Matched Samples". Biometrika. 37 (3/4): 256–266. doi:10.1093/biomet/37.3-4.256. JSTOR2332378.
^Conover, William Jay (1999). Practical Nonparametric Statistics (Third ed.). Wiley, New York, NY USA. pp. 388–395. ISBN9780471160687.
^National Institute of Standards and Technology. Cochran Test
^Robert R. Sokal & F. James Rohlf (1969). Biometry (3rd ed.). New York: W. H. Freeman. pp. 786–787. ISBN9780716724117.
^Kashinath D. Patil (March 1975). "Cochran's Q test: Exact distribution". Journal of the American Statistical Association. 70 (349): 186–189. doi:10.1080/01621459.1975.10480285. JSTOR2285400.
^Fahmy T.; Bellétoile A. (October 2017). "Algorithm 983: Fast Computation of the Non-Asymptotic Cochran's Q Statistic for Heterogeneity Detection". ACM Transactions on Mathematical Software. 44 (2): 1–20. doi:10.1145/3095076.