Chi-square test of homogeneity
Proportions (counts) · Three or more independent groups · reference distribution: χ²((r−1)(c−1))
When to use it
Compare the distribution of a categorical variable across three or more independent groups.
Null hypothesis
The proportion of each category is the same in every group.
Assumptions
- Independent observations
- Expected frequencies of at least 5
Test statistic
\chi^2 = \sum \dfrac{(O - E)^2}{E}Effect size
Cramér’s V, from 0 to 1.
V = \sqrt{\dfrac{\chi^2}{N\,(\min(r, c) - 1)}}How to report it
χ²(4, N = 300) = 11.2, p = .024, V = .14
In R and Python
R
chisq.test(table)Python
stats.chi2_contingency(table)In Python, stats is scipy.stats and np is numpy.
Variants and alternatives
- Chi-square of independence, which uses the same computation
- After rejecting H₀: Marascuilo
Where it sits in the catalog
Proportions (counts). For categorical answers, such as yes or no, hit or miss: the data are how often each category appears, and the test compares proportions.
Three or more independent groups. Several groups of different individuals, tested at once: one test per pair would inflate the type I error.
In the decision tree
- What do you want to do? Compare groups, or one group with a reference value
- What kind of response did you measure? Categorical: yes or no, or categories
- How many groups or measurements? Three or more independent groups