Cochran's Q Test
Also known as: Cochran Q Testi, Cochran's Q, Q test for related proportions
Cochran's Q test is a nonparametric hypothesis test introduced by William G. Cochran in 1950 for comparing proportions across three or more related binary measurements. It extends McNemar's test to the multiple-condition case and is the method of choice when every participant is observed under each condition and the outcome is recorded as a simple success/failure (1/0).
Key highlights
- Makes no distributional assumptions about the underlying binary outcomes.
- Naturally removes between-subject variability, increasing sensitivity to true condition effects.
- Directly extends the familiar McNemar logic to the multi-condition case, keeping interpretation straightforward.
- Widely available and recognised in social science, health, and educational research.
Intuition
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How it works
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When to use it
Use Cochran's Q test when you have a block (repeated-measures or matched) design with three or more conditions and a binary outcome variable. Two assumptions must hold: the data must form a genuine block structure in which every participant provides an observation under every condition, and the outcome must be genuinely dichotomous (not collapsed from a continuous variable). The minimum recommended sample size is roughly 10 subjects. With only two conditions, McNemar's test is the appropriate choice. When the outcome is ordinal or continuous rather than binary, Friedman's test is the correct nonparametric alternative.
Strengths & limitations
- Makes no distributional assumptions about the underlying binary outcomes.
- Naturally removes between-subject variability, increasing sensitivity to true condition effects.
- Directly extends the familiar McNemar logic to the multi-condition case, keeping interpretation straightforward.
- Widely available and recognised in social science, health, and educational research.
- Restricted to binary (0/1) outcomes — continuous, ordinal, or nominal outcomes with more than two categories require a different test.
- Requires a complete block design; missing data in any condition for any subject complicate the analysis.
- The chi-squared approximation may be unreliable when the number of subjects is small or when success rates are very close to 0 or 1.
- A significant overall Q does not reveal which specific pairs of conditions differ; pairwise follow-up tests with multiplicity correction are needed.
Common pitfalls
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Applications
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Frequently asked
How is Cochran's Q different from McNemar's test?
McNemar's test compares proportions across exactly two related binary measurements. Cochran's Q is the extension to three or more conditions; with k = 2, the two tests yield identical results.
What should I do after a significant Q?
Conduct pairwise McNemar tests for every pair of conditions and apply a multiplicity correction — Bonferroni (divide α by the number of pairs) or the more powerful Holm step-down procedure — to control the family-wise error rate.
What if my outcome is not truly binary?
If the outcome is ordinal or continuous, use Friedman's test instead. Artificially collapsing a continuous variable into 0/1 to apply Cochran's Q discards information and is not recommended.
How large a sample do I need?
The chi-squared approximation generally holds adequately with at least 10 subjects. Very small samples or extreme success rates (near 0 % or 100 %) in any condition can make the approximation unreliable; an exact permutation version of the test is preferable in those cases.
Sources
- 1.Cochran, W. G. (1950). The comparison of percentages in matched samples. Biometrika, 37(3–4), 256–266.
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ScholarGate. (2026, June 1). Cochran Q Test. ScholarGate. https://scholargate.app/statistics/cochran-q-test