Cochran's Q test


In statistics, in the analysis of two-way randomized block designs where the response variable can take only two possible outcomes, Cochran's Q test is a non-parametric statistical test to verify whether k treatments have identical effects. 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 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.

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 1Treatment 2Treatment k
Block 1X11X12X1k
Block 2X21X22X2k
Block 3X31X32X3k
Block bXb1Xb2Xbk

Description

Cochran's Q test is
The Cochran's Q test statistic is
where

Critical region

For significance level α, the asymptotic critical region is
where Χ21 − α,k − 1 is the -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, 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 in a way that is common to all treatments within each block.

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