Categorical Data Analysis in R

Chi-Squared Goodness of Fit and Independence Tests

Introduction to Chi-Squared Tests

The Chi-Squared test is a fundamental statistical tool for analyzing categorical data. In this lesson, we explore two primary applications:

1. Goodness of Fit: Does the observed proportion of a single variable (e.g., flower size) match an expected distribution (e.g., equal proportions)?
2. Test of Independence: Is there a significant relationship between two categorical variables (e.g., Species and Size)?

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1. Environment Setup and Data Exploration

We will use the classic iris dataset and transform the continuous Sepal.Length variable into categorical bins.

# Load libraries
library(tidyverse)

# Quick look at the data
head(iris)

  Sepal.Length Sepal.Width Petal.Length Petal.Width Species
1          5.1         3.5          1.4         0.2  setosa
2          4.9         3.0          1.4         0.2  setosa
3          4.7         3.2          1.3         0.2  setosa
4          4.6         3.1          1.5         0.2  setosa
5          5.0         3.6          1.4         0.2  setosa
6          5.4         3.9          1.7         0.4  setosa

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2. Data Preparation

We use the cut() function to discretize Sepal.Length into three distinct categories: Small, Medium, and Large.

# Converting numeric variables into categorical variables
flower <- iris %>% 
  mutate(size = cut(Sepal.Length,
                    breaks = 3,
                    labels = c("Small", "Medium", "Large"))) %>%
  select(Species, size)  

# View the formatted table
table(flower)

            size
Species      Small Medium Large
  setosa        47      3     0
  versicolor    11     36     3
  virginica      1     32    17

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3. Chi-Squared Goodness of Fit Test

This test evaluates one variable. We want to know if the proportions of “Small,” “Medium,” and “Large” flowers are significantly different from each other.

Hypotheses

• : The proportions of Small, Medium, and Large flowers are equal ( each).
• : The proportions are unequal.

# Visualizing the distribution
flower %>%
  ggplot(aes(x = size)) +
  geom_bar(fill = "#a139ca", alpha = 0.7) +
  labs(title = "Goodness of Fit: Observed Frequencies",
       subtitle = "Proportion of flowers by size",
       x = "Size Category",
       y = "Frequency") +
  theme_test(base_size = 15)

# Performing the test
goodness_fit <- flower %>% 
  select(size) %>% 
  table() %>% 
  chisq.test()

goodness_fit

    Chi-squared test for given probabilities

data:  .
X-squared = 28.44, df = 2, p-value = 6.673e-07

Interpretation: The p-value is approximately . Since , we reject the null hypothesis. The proportions of flower sizes in this dataset are statistically unequal.

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4. Chi-Squared Test of Independence

This test evaluates the inter-relationship between two variables: Species and Size.

Hypotheses

• : Species and Size are independent (no relationship).
• : Species and Size are dependent (there is an association).

# Visualizing the relationship
flower %>%
  ggplot(aes(size, fill = Species)) +
  geom_bar(alpha = 0.7) +
  labs(title = "Test of Independence: Species vs. Size",
       subtitle = "Stacked bar chart showing categorical overlap",
       x = "Size Category",
       y = "Count") +
  theme_test(base_size = 15) +
  scale_fill_manual(values = c("setosa" = "#d48ce1",
                               "versicolor" = "#a139ca",
                               "virginica" = "#6d2683"))

Performing the Test

# Run the independence test
independence_test <- table(flower) %>% chisq.test()

independence_test

    Pearson's Chi-squared test

data:  .
X-squared = 111.63, df = 4, p-value < 2.2e-16

Interpretation: The p-value is . Since , we reject the null hypothesis. This suggests that Species and Size are highly dependent—knowing the species helps predict the likely size of the flower.

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5. Checking Assumptions (Expected Values)

A Chi-squared test may be unreliable if more than 20% of the expected frequencies are less than 5. In such cases, Fisher’s Exact Test is preferred.

# View expected frequencies
independence_test$expected

            size
Species         Small   Medium    Large
  setosa     19.66667 23.66667 6.666667
  versicolor 19.66667 23.66667 6.666667
  virginica  19.66667 23.66667 6.666667

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Systematic Checklist (Cheat Sheet)

• Discretizing Data: cut(variable, breaks = n, labels = c(...))
• Contingency Table: table(var1, var2)
• Statistical Test: chisq.test(table_object)
• Checking Validity: test_object$expected
• Alternative for Small Samples: fisher.test(table_object)

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Summary: You have successfully mastered both types of Chi-squared tests! You can now determine if a category distribution is balanced and if two categorical variables influence one another.

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