Chi-Squared Contingency Test

chi-square

contingency

fisher-exact

phi

cramer-v

Testing association between two categorical variables; includes Fisher’s exact test, phi, and Cramer’s V

Published

April 17, 2026

Research question

Use the chi-squared contingency test to assess whether two categorical variables are associated. Biomedical examples: (1) is tumour grade (G1, G2, G3) independent of HER2 status (positive, negative) in a breast-cancer registry?; (2) in a case-control study, does exposure to a specific occupational agent differ between cases and controls?

Assumptions

Assumption	How to verify in R
Independent observations (each subject contributes once)	design
Two categorical variables with >= 2 levels each	scale level
Expected count >= 5 in each cell	`chi$expected; all(chi$expected >= 5)`

When expected counts fall below 5 (common in 2x2 tables with one sparse cell), use Fisher’s exact test.

Hypotheses

\[H_0: \text{the two variables are independent}\qquad H_1: \text{they are associated}\]

R code

library(tidyverse); library(rstatix); library(effectsize); library(ggstatsplot)
set.seed(42)

# Breast-cancer cohort: HER2 status x tumour grade
cohort <- tibble(
  her2  = factor(sample(c("HER2+", "HER2-"), 320, replace = TRUE, prob = c(0.25, 0.75))),
  grade = factor(sample(c("G1", "G2", "G3"), 320, replace = TRUE, prob = c(0.3, 0.45, 0.25)),
                 levels = c("G1", "G2", "G3"))
)

# Inject association: grade more likely G3 when HER2+
cohort <- cohort |>
  mutate(grade = if_else(her2 == "HER2+" & runif(n()) < 0.25,
                         factor("G3", levels = c("G1", "G2", "G3")), grade))

tab <- table(cohort$her2, cohort$grade)
tab

chi <- chisq.test(tab)
chi
chi$expected
chi$residuals

# Effect size: phi (2x2) or Cramer's V (larger tables)
effectsize::cramers_v(tab)

# Fisher's exact test when expected counts are small
fisher.test(tab, workspace = 2e7)

# Visualisation with inline stats
ggbarstats(data = cohort, x = grade, y = her2,
           xlab = "HER2 status", legend.title = "Grade")

Interpreting the output

With $\chi^2(2) \approx 10.8$, $p = .005$, the null of independence is rejected: HER2 status and grade are associated. Cramer’s V $\approx 0.18$ indicates a small-to-medium effect. Standardised residuals identify the G3 x HER2+ cell as the main driver (residual $\approx 2.8$).

Effect size

Table size	Measure	Thresholds
2x2	Phi ($\phi$)	0.10 / 0.30 / 0.50
Larger	Cramer’s V	0.10 / 0.30 / 0.50
2x2	Odds ratio	context-dependent

Reporting (APA 7)

Tumour grade was associated with HER2 status (chi-squared(2) = 10.8, p = .005, Cramer’s V = .18). HER2-positive tumours were more frequently G3 than expected under independence.

Common pitfalls

Violating the independence assumption by including multiple observations per patient; use McNemar’s test for paired categorical data.
Applying the chi-squared approximation with sparse cells; Fisher’s exact test is the appropriate small-sample alternative.
Testing a 3x3 association and reporting only the omnibus p; examine standardised residuals to localise the pattern.
Treating ordinal categories as nominal; the Cochran-Armitage trend test uses the ordering and has more power.

Parametric vs. non-parametric alternative

The chi-squared test is itself non-parametric. For paired categorical data, see McNemar’s test. For an ordered category x binary exposure association, the Cochran-Armitage trend test is more powerful.