Cochran's Q test

Cochran's ${\displaystyle Q}$ test is a

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	${\displaystyle \cdots }$	Treatment k
Block 1	X11	X12	${\displaystyle \cdots }$	X1k
Block 2	X21	X22	${\displaystyle \cdots }$	X2k
Block 3	X31	X32	${\displaystyle \cdots }$	X3k
${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \ddots }$	${\displaystyle \vdots }$
Block b	Xb1	Xb2	${\displaystyle \cdots }$	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".

Description

Cochran's Q test is

Null hypothesis (H₀): the treatments are equally effective.

Alternative hypothesis (H_a): there is a difference in effectiveness between treatments.

The Cochran's Q test statistic is

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

where

k is the number of 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

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

where Χ²_{1 − α,k − 1} is the (1 − α)-

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.

Assumptions

Cochran's Q test is based on the following assumptions:

1. If the large sample approximation is used (and not the exact distribution), b is required to be "large".
2. The blocks were randomly selected from the population of all possible blocks.
3. 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

• The Friedman test or Durbin test can be used when the response is not binary but ordinal or continuous.
• When there are exactly two treatments the Cochran Q test is equivalent to McNemar's test, which is itself equivalent to a two-tailed sign test.

References