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 Cochran^{1}.
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:
- Oon performs a measure of the number of occurrences of the event being searched.
- This measure is redone on these same individuals to compare the results.
- We re-do this measure…
Individual |
Measure 1 |
Measure 2 |
… |
Measure k |
Sum |
1 |
n1 |
||||
2 |
n2 |
||||
… |
… |
||||
Individual n |
nn |
||||
Sum |
S1 |
S2 |
… |
S_{k} |
S_{n} |
Step 1-Assumptions
π is the probability of the occurrence of our event. The test hypotheses are:
H0: π_{1} =… = π_{k} : 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 χ^{2} to K-1 degree of freedom.
Either it is determined by searching directly in the table of χ^{2}, 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 χ^{2}^{ }and is calculated in Excel using the formula:
Chidist (practical value; dof)
Step 5-Interpretation
Result | Statistical conclusion | Practical conclusion |
---|---|---|
Practical value ≥ Critical value | We reject H0 | At least one of our series of values is statistically different from the others at the given level of risk α. |
Practical value < Critical value | We retain H0 | Our series of values are statistically identical or close to the given level of risk α. |
Result | Statistical conclusion | Practical conclusion |
---|---|---|
p-value > α | We retain H0 | Our data series are identical or close to the risk of being wrong with p-value% |
p-Value < α | We reject H0 | At 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