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Cochran's Q Test

Includes: Cochran’s Q, Percent of 1’s per condition.
Purpose: Test whether matched binary outcomes (0/1) have the same proportion of “success” across k ≥ 3 related conditions (an extension of McNemar’s test).

Overview

Cochran’s Q test is the binary-outcome analogue of the Friedman test:

  • Friedman: repeated-measures test for ordinal/continuous outcomes (ranks within each block).
  • Cochran’s Q: repeated-measures test for binary outcomes (0/1), testing whether the success rates differ across conditions.

BESHStatNG reports:

  • the Q test statistic,
  • the two-sided p-value (chi-square approximation, df = k − 1),
  • and the percent of 1’s in each column (condition).

Example dataset

Download the CSV used in the screenshots:

The dataset is in wide format (each column is a condition/time point):

  • Time1, Time2, Time3

Each row is one subject/block with a binary response in each condition.

Screenshots (BESHStatNG)

Input

Cochran's Q – Input

Results

Cochran's Q – Results

When to use it

Use Cochran’s Q when:

  • you have k ≥ 3 related measurements per subject (repeated measures / matched sets),
  • the response is binary (coded as 0/1),
  • you want to test whether the proportion of 1’s differs across the conditions.

Key requirements / considerations:

  • Matched blocks: rows represent subjects/blocks; the comparison is within rows.
  • Independence of blocks: different rows (subjects) are independent.
  • Binary coding: values should be 0/1. (If you have “Yes/No”, recode to 1/0 first.)

If your outcome is not binary, use:

  • Friedman Test for ordinal/continuous repeated measures, or
  • Skillings–Mack Test for incomplete block designs (missing values).

Inputs in Excel

  • Data: Select a rectangular range containing all conditions (columns) and subjects (rows).
    Each column is one condition; each row is one subject/block.

Missing values

Rows with missing / non-numeric cells in any condition are excluded (only complete blocks are analyzed).

Output destination

  • Output range (current sheet)
  • New worksheet
  • New workbook

Steps in the add-in

  1. In Excel ribbon: BESH Stat NG → Analyse → Nonparametric → Cochran's Q Test
  2. Select the Data range (all conditions/columns).
  3. Choose an output location.
  4. Click Compute.

Output and how to read it

BESHStatNG writes:

Cochran’s Q table

  • Q: the Cochran’s Q test statistic
  • Two-sided p-value: chi-square approximation with df = \(k-1\)

Example interpretation (from the screenshot):

  • \(Q = 14.8889\), p = 0.000585 (df = 2)
  • Conclusion: strong evidence that at least one condition has a different success proportion than the others.

Percent of 1’s per condition

For each column, BESHStatNG reports:

\[ \%\;1\text{s} = 100 \cdot \frac{C_j}{n} \]

where \(C_j\) is the column total (number of 1’s) and \(n\) is the number of subjects (rows).

This is an effect-size style summary to help interpret which conditions are higher/lower.

What it does (math and implementation details)

Let:

  • \(n\) = number of subjects/blocks (rows)
  • \(k\) = number of conditions (columns)
  • \(x_{ij} \in \{0,1\}\) = response for subject \(i\) in condition \(j\)

Define:

  • Column totals \(C_j = \sum_{i=1}^{n} x_{ij}\)
  • Row totals \(R_i = \sum_{j=1}^{k} x_{ij}\)
  • Total \(T = \sum_{j=1}^{k} C_j = \sum_{i=1}^{n} R_i\)

Cochran’s Q statistic

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

Under \(H_0\) (all conditions have the same success probability), \(Q\) is approximately chi-square distributed:

\[ Q \;\dot\sim\; \chi^2_{k-1} \qquad\Rightarrow\qquad p = 1 - F_{\chi^2(k-1)}(Q) \]

Relation to other nonparametric tests

  • McNemar’s test (k = 2): Cochran’s Q reduces to McNemar’s matched-pairs test when there are exactly two conditions.
  • Friedman test: Cochran’s Q plays a similar role for binary data as Friedman does for ordinal/continuous repeated measures.
    If your values are not 0/1, Friedman is usually the correct choice.

R code (reference using standard R functions)

Using DescTools (direct wide-format input)

# Cochran's Q example (wide format)
dat <- read.csv("022cochransq.csv")

# Ensure a numeric matrix with 0/1 values
X <- as.matrix(dat)

# DescTools provides CochranQTest()
# install.packages("DescTools")
library(DescTools)

res <- CochranQTest(X)
res

Using rstatix (long-format workflow)

# install.packages("rstatix")
library(rstatix)
library(tidyr)
library(dplyr)

dat <- read.csv("022cochransq.csv")

long <- dat %>%
  mutate(id = row_number()) %>%
  pivot_longer(cols = -id, names_to = "time", values_to = "y")

res <- long %>%
  cochran_qtest(y ~ time | id)
res

Why R results may differ slightly from BESHStatNG

Most implementations (including DescTools::CochranQTest) use the same \(Q\) formula and chi-square approximation, so results typically match exactly.

If you see differences, the most common reasons are:

  • Row exclusion rules: ensure both tools use the same set of complete rows (no missing values).
  • Coding differences: confirm values are coded as 0/1 (not TRUE/FALSE or text).

Notes

  • Cochran’s Q is an omnibus test: a significant result indicates at least one condition differs, but it does not identify which pairs differ.
  • If you need pairwise follow-up comparisons for binary repeated measures, consider:
  • pairwise McNemar tests with multiplicity correction, or
  • a GEE logistic model for a model-based approach (especially with covariates).

See also