In statistics, in the analysis of twoway randomized block designs where the response variable can take only two possible outcomes (coded as 0 and 1), Cochran's Q test is a nonparametric statistical test to verify whether k treatments have identical effects.^{[1]}^{[2]}^{[3]} It is named for William Gemmell Cochran. Cochran's Q test should not be confused with Cochran's C test, which is a variance outlier test. Put in less technical terms, requires that there only be a binary response (success/failure or 1/0) and that there be 2 or more matched groups (groups of the same size). The test assesses whether the proportion of successes is the same between groups. Often used to assess if different observers of the same phenomenon have consistent results amongst themselves (interobserver variability).
Contents

Background 1

Description 2

Critical region 3

Assumptions 4

Related tests 5

References 6
Background
Cochran's Q test assumes that there are k > 2 experimental treatments and that the observations are arranged in b blocks; that is,

Treatment 1

Treatment 2

\cdots

Treatment k

Block 1

X_{11}

X_{12}

\cdots

X_{1k}

Block 2

X_{21}

X_{22}

\cdots

X_{2k}

Block 3

X_{31}

X_{32}

\cdots

X_{3k}

\vdots

\vdots

\vdots

\ddots

\vdots

Block b

X_{b1}

X_{b2}

\cdots

X_{bk}

Description
Cochran's Q test is

H_{0}: The treatments are equally effective.

H_{a}: There is a difference in effectiveness among treatments.
The Cochran's Q test statistic is

T = k\left(k1\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(kX_{i\bullet}\right)}
where

k is the number of treatments

X_{• j} is the column total for the j^{th} treatment

b is the number of blocks

X_{i •} is the row total for the i^{th} block

N is the grand total
Critical region
For significance level α, the critical region is

T > \chi^2_{1\alpha,k1}
where Χ^{2}_{1 − α,k − 1} is the (1 − α)quantile of the chisquared 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..
Assumptions
Cochran's Q test is based on the following assumptions:

A large sample approximation; in particular, it assumes that b is "large".

The blocks were randomly selected from the population of all possible blocks.

The outcomes of the treatments can be coded as binary responses (i.e., a "0" or "1") in a way that is common to all treatments within each block.
Related tests

When using this kind of design for a response that is not binary but rather ordinal or continuous, one instead uses the Friedman test or Durbin tests.

The case where there are exactly two treatments is equivalent to McNemar's test, which is itself equivalent to a twotailed sign test.
References

^

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^ National Institute of Standards and Technology. Cochran Test
This article incorporates public domain material from websites or documents of the National Institute of Standards and Technology.
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