Cochren's Q-test

Cochran’s Q test is a non - parametric statistical test used to test whether two or more effects have the same effect on Russian groups . In this case, the response of the group can take only 2 possible values (denoted as 0 and 1) ^[1] ^[2] ^[3] ^[4] . The criterion was named after William Cochren . The Cochren Q-test should not be confused with the Cochren G-test . When using the Q-criterion, it is assumed that the result of the impact is described by only two types (for example, success / failure, 1/0) and there are more than 2 groups of the same size. The criterion determines whether the success rate is the same in different groups. Often it is used to determine whether different observers of the same phenomenon receive a similar result (variability of subjective expert assessment) ^[5] .

Experimental conditions

It is assumed that k > 2 experimental influences take place and that the observations are grouped into b blocks in

	Impact 1	Impact 2	$\cdots$ ${\ displaystyle \ cdots}$	Impact k
Block 1	X ₁₁	X ₁₂	$\cdots$ ${\ displaystyle \ cdots}$	X _{1 k}
Block 2	X ₂₁	X ₂₂	$\cdots$ ${\ displaystyle \ cdots}$	X _{2 k}
Block 3	X ₃₁	X ₃₂	$\cdots$ ${\ displaystyle \ cdots}$	X _{3 k}
$\vdots$ ${\ displaystyle \ vdots}$	$\vdots$ ${\ displaystyle \ vdots}$	$\vdots$ ${\ displaystyle \ vdots}$	$\ddots$ ${\ displaystyle \ ddots}$	$\vdots$ ${\ displaystyle \ vdots}$
Group b	X _{b 1}	X _{b 2}	$\cdots$ ${\ displaystyle \ cdots}$	X _{b k}

Description

Cochren's Q-test:

Null hypothesis (H ₀ ): the effects have the same effect.

Alternative hypothesis (H _a ): there is a difference in the effectiveness of the various effects.

Cochren's Q-test statistics:

T=k\left(k-1\right){\frac {\sum \limits _{j=1}^{k}\left(X_{\bullet j}-{\frac {N}{k}}\right)^{2}}{\sum \limits _{i=1}^{b}X_{i\bullet }\left(k-X_{i\bullet }\right)}}

{\ displaystyle T = k \ left (k-1 \ right) {\ frac {\ sum \ limits _ {j = 1} ^ {k} \ left (X _ {\ bullet j} - {\ frac {N} { k}} \ right) ^ {2}} {\ sum \ limits _ {i = 1} ^ {b} X_ {i \ bullet} \ left (k-X_ {i \ bullet} \ right)}}}

where

k is the number of effects

X _{• j} is the sum in the column for the j- th impact,

b is the number of groups

X _{i •} - the amount of the line for the i- th group,

N is the total amount.

Critical Area

For significance level α, the critical region:

T>\chi _{1-\alpha ,k-1}^{2}

{\ displaystyle T> \ chi _ {1- \ alpha, k-1} ^ {2}}

where Χ ² _{1 - α, k - 1} - (1 - α) is the quantile of the chi-square distribution with k - 1 degrees of freedom. The null hypothesis is rejected if the statistics are in a critical region. If the null hypothesis about the same effect of effects is rejected by the Q-criterion, pairwise multiple comparisons can be made using the Cochren Q-criterion to evaluate two effects of interest.

An approximate distribution of statistics T can be calculated for a small number of studied objects. This allows you to roughly estimate the critical area. The first algorithm was proposed in 1975 by Patil ^[6] , the second by Fami and Beletual ^[7] in 2017.

Assumptions

Cochren's Q-test is applicable when applying the following assumptions:

a large number of objects must be examined, b must be large .
groups should be randomly selected from the entire possible set of groups.
impact on groups can be described by a dichotomous variable that takes only 2 possible values (for example, “0” or “1”)

Related criteria

Friedman's criterion or Darbin's criterion can be used in cases where the response takes not 2 values, but several possible ones or any value in a certain interval.
When it comes to exactly two influences, the Cochren Q-test is equivalent to the McNemar criterion , which, in turn, is equivalent to the criterion of signs .

Links

“Cochren’s Q Test” on the Knowledge Portal

Notes

↑ Q Cochren test
↑ William G. Cochran. The Comparison of Percentages in Matched Samples (Eng.) // Biometrika : journal. - 1950 .-- December ( vol. 37 , no. 3/4 ). - P. 256-266 . - DOI : 10.1093 / biomet / 37.3-4.256 . (eng.)
↑ Conover, William Jay. Practical Nonparametric Statistics. - Third. - Wiley, New York, NY USA, 1999. - P. 388–395. (eng.)
↑ National Institute of Standards and Technology. Cochran Test
↑ Mohamed M. Shoukri. Measures of interobserver agreement . - Boca Raton: Chapman & Hall / CRC, 2004 .-- ISBN 9780203502594 . (eng.)
↑ Kashinath D. Patil. Cochran's Q test: Exact distribution (Eng.) // Journal of the American Statistical Association : journal. - 1975 .-- March ( vol. 70 , no. 349 ). - P. 186-189 . - DOI : 10.1080 / 01621459.1975.10480285 . (eng.)
↑ Fahmy T .; Bellétoile A. Algorithm 983: Fast Computation of the Non-Asymptotic Cochran's Q Statistic for Heterogeneity Detection (English) // ACM Transactions on Mathematical Software (TOMS): journal. - 2017 .-- October ( vol. 44 , no. 2 ). - P. 1-20 . - DOI : 10.1145 / 3095076 . (eng.)

[1] Q Cochren test

[2] William G. Cochran. The Comparison of Percentages in Matched Samples (Eng.) // Biometrika : journal. - 1950 .-- December ( vol. 37 , no. 3/4 ). - P. 256-266 . - DOI : 10.1093 / biomet / 37.3-4.256 . (eng.)

[3] Conover, William Jay. Practical Nonparametric Statistics. - Third. - Wiley, New York, NY USA, 1999. - P. 388–395. (eng.)

[4] National Institute of Standards and Technology. Cochran Test

[5] Mohamed M. Shoukri. Measures of interobserver agreement . - Boca Raton: Chapman & Hall / CRC, 2004 .-- ISBN 9780203502594 . (eng.)

[6] Kashinath D. Patil. Cochran's Q test: Exact distribution (Eng.) // Journal of the American Statistical Association : journal. - 1975 .-- March ( vol. 70 , no. 349 ). - P. 186-189 . - DOI : 10.1080 / 01621459.1975.10480285 . (eng.)

[7] Fahmy T .; Bellétoile A. Algorithm 983: Fast Computation of the Non-Asymptotic Cochran's Q Statistic for Heterogeneity Detection (English) // ACM Transactions on Mathematical Software (TOMS): journal. - 2017 .-- October ( vol. 44 , no. 2 ). - P. 1-20 . - DOI : 10.1145 / 3095076 . (eng.)