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 – 3+ Related Proportions (Binary)

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.

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).