Cochran's Q Test Calculator

Cochran's Q Test is a non-parametric statistical test used to detect differences in proportions across three or more related groups when the dependent variable is binary (e.g., success/failure, yes/no). It is the extension of McNemar's Test to more than two conditions and a special case of Friedman's Test applied to binary data.

Use Cochran's Q Test when your outcome variable is binary, you have three or more related conditions or treatments, and the same subjects are observed under each condition (repeated measures design). Common applications include clinical trials, psychology experiments, and educational research where subjects are exposed to multiple treatments.

The null hypothesis (H₀) states that the proportions of successes are equal across all k conditions — in other words, the treatments are not significantly different. The alternative hypothesis (Hₐ) states that at least one condition has a different proportion of successes from the others.

The Q statistic is calculated as Q = (k−1) × [k × ΣTj² − T..²] / (k × T.. − ΣRi²), where k is the number of conditions, Tj is the column total (successes in condition j), T.. is the grand total of all successes, and Ri is the row total (successes for subject i). Under H₀, Q follows a chi-square distribution with k−1 degrees of freedom.

Enter your data as a matrix in the text area, with one row per subject. Within each row, separate the condition values (0 or 1) with commas. For example, if subject 1 succeeded in conditions A and C but not B, enter '1,0,1'. Ensure every row has exactly k values corresponding to your number of conditions.

If the p-value is less than or equal to your chosen significance level (α), you reject the null hypothesis and conclude that at least one condition has a significantly different proportion of successes. If the p-value is greater than α, you fail to reject the null hypothesis, meaning the data do not provide sufficient evidence of a difference.

Cochran's Q Test only tells you that at least one condition differs — it does not identify which specific conditions are different. For that, post-hoc pairwise McNemar tests with a Bonferroni correction are recommended. The test also assumes that subjects are independent of each other and that the binary classification is consistent across conditions.

If Cochran's Q Test is significant, follow up with pairwise McNemar's Tests between each pair of conditions to identify where the differences lie. Apply a Bonferroni correction (divide α by the number of comparisons) to control the family-wise error rate. Some researchers also use Dunn's test or other post-hoc methods adapted for binary repeated measures.