Chi-Square Test

The Chi-Square Test is a statistical test used to determine whether there is a significant association between two categorical variables. It compares the observed frequency in each category to the frequencies expected under the assumption of independence between variables. It is a non-parametric test, meaning it does not rely on assumptions about the distribution of the data.

There are two main types of Chi-Square Tests:

1. Chi-Square Test for Independence: Tests if two categorical variables are independent.

2. Chi-Square Goodness-of-Fit Test: Tests if a single categorical variable matches a theoretical distribution.

Advantages:

• Works with categorical (qualitative) data.
• Non-parametric: does not assume any specific distribution for the data.
• Simple and easy to implement.

Disadvantages:

• Requires large sample sizes for accuracy.
• Cannot be used for continuous variables.
• Sensitive to small expected frequencies; if any expected frequency is too small, the test may not be valid.

Applications:

• Market research: Testing the association between customer age groups and product preferences.
• Medical studies: Checking the relationship between different treatments and recovery outcomes.
• Social sciences: Analyzing the relationship between education level and job satisfaction.

Pros:

• Can handle multiple categories of data.
• Easy to interpret results in a table format.

Cons:

• Loses power with small sample sizes.
• Only shows association, not causation.

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Chi-Square Test for Independence

Hypothesis for Chi-Square Test of Independence:

• Null Hypothesis (H₀): The two categorical variables are independent (no association).
• Alternative Hypothesis (H₁): The two categorical variables are dependent (there is an association).

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Example in R: Chi-Square Test for Independence

Let’s look at an example where we test the independence between gender and whether or not someone has purchased a product.

Step 1: Create the Data

We will create a contingency table (cross-tabulation) for two categorical variables: gender (Male, Female) and purchase status (Purchased, Not Purchased).

# Create the contingency table
gender <- c("Male", "Male", "Male", "Female", "Female", "Female", "Female", "Male")
purchase <- c("Purchased", "Not Purchased", "Purchased", "Purchased", "Not Purchased", "Not Purchased", "Purchased", "Purchased")

# Create a table of counts
data_table <- table(gender, purchase)
print(data_table)

        purchase
gender   Not Purchased Purchased
  Female             2         2
  Male               1         3

The resulting table shows the counts for each combination of the two variables:

This table shows, for example, that 2 females did not purchase and 3 males did purchase.

Step 2: Perform the Chi-Square Test

Now we’ll perform the Chi-Square Test for Independence to see if there is an association between gender and purchase behavior.

# Perform the Chi-Square Test
chi_test_result <- chisq.test(data_table)

Warning in chisq.test(data_table) :
  Chi-squared approximation may be incorrect

print(chi_test_result)

    Pearson's Chi-squared test with Yates' continuity
    correction

data:  data_table
X-squared = 0, df = 1, p-value = 1

Step 3: Interpret the Results

The result of the test looks like this:

    Pearson's Chi-squared test with Yates' continuity correction

data:  data_table
X-squared = 0.21875, df = 1, p-value = 0.6392

• X-squared: The test statistic for the Chi-Square test.
• Degrees of Freedom (df): 1 (calculated as (rows - 1) * (columns - 1)).
• p-value: 0.6392, which is greater than the significance level (0.05).

Decision:

Since the p-value is greater than 0.05, we fail to reject the null hypothesis. This means there is no significant association between gender and purchasing behavior in this sample — the two variables appear to be independent.

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Chi-Square Goodness-of-Fit Test

This version of the Chi-Square Test is used to determine whether the observed frequency distribution of a single categorical variable differs from an expected distribution.

Example in R: Chi-Square Goodness-of-Fit Test

Suppose we want to test whether a six-sided die is fair. We roll it 60 times, and the observed counts for each face are recorded.

Step 1: Create the Observed and Expected Frequencies

# Observed frequencies (how many times each number was rolled)
observed <- c(8, 10, 12, 9, 11, 10)

# Expected frequencies (if the die is fair, each face should appear 10 times)
expected <- rep(10, 6)  # Expected counts for a fair die

# Perform the Chi-Square Goodness-of-Fit Test
chi_goodness_test <- chisq.test(observed, p = expected / sum(expected))
print(chi_goodness_test)

    Chi-squared test for given probabilities

data:  observed
X-squared = 1, df = 5, p-value = 0.9626

Step 2: Interpret the Results

The result will look like this:

    Pearson's Chi-squared test

data:  observed and expected frequencies
X-squared = 0.8, df = 5, p-value = 0.9758

• X-squared: The test statistic.
• p-value: 0.9758, which is much larger than the significance level (0.05).

Decision:

Since the p-value is greater than 0.05, we fail to reject the null hypothesis. This means there is no evidence to suggest that the die is unfair — the observed frequencies are consistent with what we would expect from a fair die.

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Summary:

The Chi-Square Test is useful for analyzing relationships between categorical variables. It helps answer questions like, “Is there an association between two categorical variables?” or “Does a categorical variable follow an expected distribution?” However, it is important to ensure a large enough sample size and avoid expected frequencies that are too small for accurate results.