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Cochran’s Q is a generalization of the McNemar test and allows to process more than 2 sets of matched data.

Introduction

William Gemmel Cochran (born 1909) is a Scottish statistician. It was by working on the yield of farmland that he developed the test of the Q of Cochran1.

Cochran’s Q is a generalization of the test of McNemar For processing more than 2 sets of matched data. This is the case for example when we put in place plans of experiments on the same samples.

 

The principle

One wants to compare the occurrence of an event at several different times on the same population of n individuals:

  1. Oon performs a measure of the number of occurrences of the event being searched.
  2. This measure is redone on these same individuals to compare the results.
  3. We re-do this measure…

Individual

Measure 1

Measure 2

Measure k

Sum

1

n1

2

n2

Individual n

nn

Sum

S1

S2

Sk

Sn

Step 1-Assumptions

π is the probability of the occurrence of our event. The test hypotheses are:

H0: π1 =… = πk : The probabilities of the event are identical on all measurements

H1: At least one measure differs from other

Step 2-Practical value

The test statistic is to measure the level of variability between the results of the different tests. Thus, at the most this variability will be great, the more we can conclude that the test is significant. The statistics are as follows:

Step 3-Critical value

The practical value is compared to the critical value that we are referring to the distribution law of χ2 to K-1 degree of freedom.

Either it is determined by searching directly in the table of χ2, or via the EXCEL spreadsheet with the function: CHIINV (risk α; dof).

In view of the fact that we are comparing more than 2 samples, there is no point in doing a bilateral test.

Step 4 – The p-value

The p-value of the test allows to conclude definitively on the model. It follows a law of χ2 and is calculated in Excel using the formula:

Chidist (practical value; dof)

Step 5-Interpretation

ResultStatistical conclusionPractical conclusion
Practical value ≥ Critical valueWe reject H0At least one of our series of values is statistically different from the others at the given level of risk α.
Practical value < Critical valueWe retain H0Our series of values are statistically identical or close to the given level of risk α.
ResultStatistical
conclusion
Practical conclusion
p-value > αWe retain H0Our data series are identical or close to the risk of being wrong with p-value%
p-Value < αWe reject H0At least 1 of our 2 series of data is statistically different from the others at the risk of being wrong with p-value%

Source

1 – F. Yates (1982) – Obtituary: William Gemmell Cochran, 1909, 1980

R. Rafiq (2008) – Population comparison

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