Chi-Square Tests

What Are Chi-Square Tests?

Chi-square (χ²) tests analyze categorical data (counts/frequencies). They test whether observed frequencies differ from expected frequencies.

🎯 When to Use Chi-Square

Perfect for: - Categorical variables (groups, categories) - Count data (frequencies) - Proportions - Cross-tabulations

NOT for: - Continuous data (use t-test/ANOVA/regression instead) - Small expected frequencies (< 5 per cell)

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📊 Two Types of Chi-Square Tests

1. Goodness of Fit

Question: Does one categorical variable match expected proportions?

Example: Do students prefer online, hybrid, or in-person equally?

2. Test of Independence

Question: Are two categorical variables related?

Example: Is major choice related to gender?

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✅ Chi-Square Goodness of Fit

Purpose

Compare observed frequencies to expected frequencies for one variable.

Example Scenario

You surveyed 100 students about preferred course format: - Online: 45 students - Hybrid: 30 students
- In-person: 25 students

Question: Do students have equal preferences (33.3% each)?

R Code

# Observed frequencies
observed <- c(45, 30, 25)

# Expected frequencies (equal preference = 33.33 each)
# You can specify proportions
expected_prop <- c(1/3, 1/3, 1/3)

# Or just let R assume equal
chisq.test(observed)

# With specific expected proportions
chisq.test(observed, p = expected_prop)

# Visualization
barplot(observed, 
        names.arg = c("Online", "Hybrid", "In-Person"),
        main = "Course Format Preferences",
        col = "steelblue",
        ylab = "Number of Students")

Interpreting Output

• χ² statistic: Measures how much observed differs from expected
• df: Degrees of freedom = (categories - 1)
• p-value: If < .05, preferences differ from expected

Reporting

"A chi-square goodness of fit test showed that course format preferences differed significantly from equal distribution, χ²(2) = [statistic], p = [p-value]. Students preferred online (45%) over hybrid (30%) and in-person (25%) formats."

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🔗 Chi-Square Test of Independence

Purpose

Test whether two categorical variables are related.

Example Scenario

Is there a relationship between major (STEM vs Humanities) and preferred format (Online vs In-Person)?

	Online	In-Person	Total
STEM	35	15	50
Humanities	20	30	50
Total	55	45	100

R Code

# Create contingency table
data_table <- matrix(c(35, 20, 15, 30), 
                     nrow = 2, 
                     byrow = TRUE)

rownames(data_table) <- c("STEM", "Humanities")
colnames(data_table) <- c("Online", "In-Person")

# View table
data_table

# Run chi-square test
result <- chisq.test(data_table)
result

# See expected frequencies
result$expected

# If you have raw data instead:
# chisq.test(table(data$major, data$format))

# Visualization
barplot(data_table, 
        beside = TRUE,
        legend = TRUE,
        col = c("steelblue", "coral"),
        main = "Format Preference by Major",
        xlab = "Format",
        ylab = "Count")

Interpreting Output

• χ² statistic: Measures strength of association
• df: (rows - 1) × (columns - 1)
• p-value: If < .05, variables are related
• Expected frequencies: What you'd expect if no relationship

Check Expected Frequencies

All expected frequencies should be ≥ 5. If not, results may be unreliable.

Effect Size: Cramér's V

# Calculate Cramér's V
library(lsr)
cramersV(data_table)

# Or manually:
n <- sum(data_table)
min_dim <- min(nrow(data_table) - 1, ncol(data_table) - 1)
V <- sqrt(result$statistic / (n * min_dim))
V

Interpretation:

V value	Strength (df=1)	Strength (df=2)
0.10	Small	Small
0.30	Medium	Medium
0.50	Large	Large

Reporting

"A chi-square test of independence showed a significant association between major and format preference, χ²(1) = [statistic], p = [p-value], V = [Cramér's V]. STEM students preferred online format (70%) while Humanities students preferred in-person (60%)."

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📋 Assumptions & Requirements

Critical Requirements

1. Expected Frequencies ≥ 5 - Check: result$expected - If violated: Use Fisher's Exact Test (for 2×2 tables)

2. Independent Observations - Each person counted only once - No repeated measures

3. Categories Mutually Exclusive - Each observation in exactly one category

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🔧 Advanced Options

Fisher's Exact Test

For 2×2 tables with small expected frequencies:

fisher.test(data_table)

Post-Hoc Tests

For larger tables (> 2×2), examine standardized residuals:

result <- chisq.test(data_table)
result$stdres  # Standardized residuals

# Values > |2| suggest that cell contributes to significance

Multiple Comparisons

For multiple chi-square tests, adjust p-values:

# Example: comparing 3 groups
p_values <- c(0.03, 0.04, 0.01)
p.adjust(p_values, method = "bonferroni")

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🎓 Interactive Learning

Practice with Real Data

Try our Chi-Square Interactive Module

Features: - Visual demonstrations - Immediate feedback - Real-world scenarios - Expected frequency calculations

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💡 Common Mistakes

❌ Using chi-square for continuous data
✅ Use t-test, ANOVA, or regression instead

❌ Expected frequencies < 5
✅ Use Fisher's Exact Test or combine categories

❌ Reporting without percentages
✅ Always report both counts and percentages

❌ Testing more than 2 variables
✅ Chi-square works with 2 variables only

❌ Assuming causation
✅ Chi-square shows association, not causation

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📊 Quick Decision Guide

graph TD
    A[Do you have categorical data?] -->|Yes| B[How many variables?]
    A -->|No| C[Use t-test/ANOVA/regression]
    B -->|One variable| D[Goodness of Fit]
    B -->|Two variables| E[Test of Independence]

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📚 Quick Reference

Test Type	Use When	R Function	Effect Size
Goodness of Fit	1 variable vs expected	chisq.test(observed)	-
Test of Independence	2 variables	chisq.test(table)	Cramér's V
Fisher's Exact	2×2, small frequencies	fisher.test(table)	Odds ratio

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🔢 Sample Size Considerations

Minimum Requirements: - Expected frequency ≥ 5 in each cell - For 2×2 tables: At least 20-40 total observations - For larger tables: At least 5 observations per cell expected

If violated: - Combine categories (if logical) - Use Fisher's Exact Test (2×2 only) - Collect more data

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