Cochran's Q test

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

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

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 significance level α, the asymptotic critical region is

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

where Χ²_{1 − 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 and a second one has been made available by Fahmy and Bellétoile 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.