Chi-Square

Purpose:

Chi-square tests examine categorical data. There are two types:

  • Goodness of Fit: Tests whether the observed frequency distribution of ONE categorical variable matches an expected distribution.
  • Test of Independence: Tests whether TWO categorical variables are associated with each other.

Context Used:

  • Goodness of Fit: ONE categorical variable with 2+ levels; comparing observed frequencies to expected frequencies.
  • Test of Independence: TWO categorical variables; testing whether they are associated.
Do NOT use chi-square if: Your variables are continuous → use correlation or regression | You want to compare means → use t-test or ANOVA | Expected cell frequencies are below 5 in more than 20% of cells → use Fisher's Exact Test

Assumptions:

  • Categorical data: Both variables must be categorical. Chi-square cannot be used with continuous variables.
  • Independence of observations: Each participant contributes to only one cell in the frequency table.
  • Expected cell frequency: Expected frequencies in each cell should be at least 5. If violated, use Fisher's Exact Test (available in Jamovi under the same menu).
Chi-square does not assume normality — that assumption applies to tests with continuous DVs only.

Jamovi Walkthrough:

Goodness of Fit:

  1. Click "Frequencies" → "One Sample Proportion Tests" → "N outcomes: χ²"
  2. Move the categorical variable into the "Variable" box
  3. Under "Expected Proportions," enter the expected proportion for each level
  4. Check "χ² test"

Test of Independence:

  1. Click "Frequencies" → "Independent Samples" → "χ² test of association"
  2. Move one variable into "Rows" and the other into "Columns"
  3. Check "χ²" under "Statistics"
  4. Check "Phi and Cramér's V" for effect size

R Walkthrough:

Goodness of Fit:

observed <- table(df$variable)
expected_prop <- c(0.55, 0.25, 0.15, 0.05)

chisq.test(x = observed, p = expected_prop)

Test of Independence:

tbl <- table(df$variable1, df$variable2)
chisq.test(tbl)

cramerV(tbl)    # Cramer's V — DescTools package

Output Interpretation:

p-value:

  • If p < .05 → reject the null hypothesis. There IS a significant difference or association.
  • If p > .05 → retain the null hypothesis. No significant difference or association.

Effect size — Cramér's V (Test of Independence only): Ranges from 0 to 1. Benchmarks for a 2×2 table (Cohen, 1988):

Cramér's VInterpretation
≈ .10Small effect
≈ .30Medium effect
≈ .50Large effect
Cramér's V benchmarks shift for larger tables (more categories). The values above apply to 2×2 tables. Cramér's V is not applicable for Goodness of Fit tests.

APA Format:

Sample write-ups:

Goodness of Fit:

A chi-square goodness of fit test examined whether attachment style was equally distributed in the sample. Observed frequencies were: secure (n = 48), anxious (n = 27), and avoidant (n = 25). The test was statistically significant, χ²(, N = ) = , p = , indicating that attachment styles were not equally distributed in this sample.

Test of Independence:

A chi-square test of independence examined whether biological sex was associated with anxiety disorder diagnosis. The association was statistically significant, χ²(, N = ) = , p = , Cramér's V = , indicating a [small/medium/large] association between sex and anxiety diagnosis.

Note: For Test of Independence, always report Cramér's V alongside χ², df, N, and p.

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