What is Cochran's Q test for?

Cochran's Q is used to test whether the proportion of 'success' differs across three or more related treatments or time points, when the outcome is binary and the same subjects are measured every time.

What are the assumptions of Cochran's Q?

The data must be paired (matched subjects), the outcome must be binary, each subject must have a score for every treatment, and the groups being compared must be k ≥ 3.

How is the p-value calculated?

The Q statistic is compared to a chi-square distribution with k – 1 degrees of freedom. This calculator computes the chi-square tail probability.

Cochran's Q Test Calculator

Test whether the success proportion is the same across 3 or more related treatments (or time points), when the outcome is binary (0/1, yes/no) and measured on the same subjects. This is the natural extension of the McNemar test to k > 2 conditions.

Number of subjects

each row = one subject

Number of treatments k ≥ 3

each column = treatment

Q statistic

—

Higher = more evidence of difference

—

df = k − 1

p-value

—

Chi-square approximation

How Cochran's Q test works

Suppose you test the same subjects under k different conditions, and for each condition the outcome is success (1) or failure (0). You want to know if the success rate is the same for all k conditions.

Notation:

k = number of treatments/conditions
n = number of subjects (matched)
C_j = column total for treatment j (sum over subjects)
R_i = row total for subject i (how many successes that subject had across all treatments)
T = total number of successes over all cells = ΣC_j

Test statistic:

\( Q = \frac{(k-1)\left[k \sum_{j=1}^{k} C_j^2 - T^2 \right]}{kT - \sum_{i=1}^{n} R_i^2} \)

Under the null hypothesis (all proportions equal), Q ~ χ² with k − 1 degrees of freedom.

Assumptions

k ≥ 3 related groups (matched, same subjects)
Binary outcome (0/1)
No missing cells (every subject measured in every condition)

Post-hoc?

Cochran's Q only tells you that at least two conditions differ. To see which ones differ, you can run pairwise McNemar tests with a multiple-comparisons adjustment (Bonferroni, Holm).

Audit: Complete

Formula (LaTeX) + variables + units

This section shows the formulas used by the calculator engine, plus variable definitions and units.

Formula (extracted LaTeX)

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Formula (extracted text)

Notation: k = number of treatments/conditions n = number of subjects (matched) Cj = column total for treatment j (sum over subjects) Ri = row total for subject i (how many successes that subject had across all treatments) T = total number of successes over all cells = ΣCj Test statistic: \( Q = \frac{(k-1)\left[k \sum_{j=1}^{k} C_j^2 - T^2 \right]}{kT - \sum_{i=1}^{n} R_i^2} \) Under the null hypothesis (all proportions equal), Q ~ χ² with k − 1 degrees of freedom.

Variables and units

No variables provided in audit spec.

Sources (authoritative):

NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures
FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/

Changelog

Version: 0.1.0-draft
Last code update: 2026-01-19

0.1.0-draft · 2026-01-19

Initial audit spec draft generated from HTML extraction (review required).
Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
Confirm sources are authoritative and relevant to the calculator methodology.

Verified by Ugo Candido on 2026-01-19
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