How to Calculate the P-Value from a Chi-Square Statistic in R with Practical Examples

The Chi-Square test is a powerful statistical method used to determine if observed frequencies in categorical data differ significantly from expected frequencies. In R, calculating the Chi-Square statistic and the associated p-value is straightforward. This guide covers two main types of Chi-Square tests: the Goodness of Fit Test and the Test of Independence, with practical examples using real data.

Introduction

The Chi-Square test, a key tool in categorical data analysis, evaluates whether observed frequencies differ significantly from expected frequencies. This test helps in understanding distributions and relationships in categorical data. The two main types of Chi-Square tests are:

• Goodness of Fit Test: Checks if observed data matches a theoretical distribution.
• Test of Independence: Evaluates if two categorical variables are independent.

Overview of the Chi-Square Test

The Chi-Square statistic is calculated as:

\[ \chi^2 = \sum \frac{(O_i – E_i)^2}{E_i} \]

where:

• \( O_i \) is the observed frequency of the \( i \)-th category,
• \( E_i \) is the expected frequency of the \( i \)-th category.

The p-value from the Chi-Square statistic helps determine if the difference between observed and expected frequencies is statistically significant.

Example 1: Goodness of Fit Test

The Goodness of Fit test evaluates if observed frequencies in a single categorical variable match expected frequencies. For example, let’s test if a die is fair, where each side should ideally occur with equal probability.

# Observed frequencies (from 60 rolls of a die)
observed <- c(8, 10, 12, 11, 9, 10)

# Expected frequencies (assuming each side has equal probability)
expected <- rep(10, times = 6)

# Perform the Chi-Square goodness of fit test
chisq_test <- chisq.test(x = observed, p = rep(1/6, 6))

# Display the test result
chisq_test

This code performs a goodness of fit test to assess if the observed frequencies align with a fair die. The result includes the Chi-Square statistic, degrees of freedom, and p-value.

Example 2: Test of Independence

The Test of Independence examines if two categorical variables are related. Suppose we have survey data showing preferences for three types of snacks (Chips, Chocolate, and Fruit) across different age groups (Under 18, 18-35, 35+). We want to test if snack preference is independent of age group.

# Observed frequencies in a contingency table
observed_table <- matrix(c(30, 10, 15, 25, 20, 30, 10, 15, 20),
                         nrow = 3, byrow = TRUE,
                         dimnames = list(AgeGroup = c("Under 18", "18-35", "35+"),
                                         Snack = c("Chips", "Chocolate", "Fruit")))
chisq_test_independence <- chisq.test(observed_table)

# Display the test result
chisq_test_independence

The output shows whether the age group and snack preference variables are independent, based on the p-value from the Chi-Square statistic.

Using pchisq to Calculate P-Values

While chisq.test() provides an easy way to conduct a Chi-Square test in R, you can calculate the p-value directly using the pchisq function. This is useful if you already have the Chi-Square statistic and degrees of freedom.

Syntax for pchisq

The pchisq function calculates the cumulative probability for the Chi-Square distribution. To find the upper-tail probability (the p-value), use lower.tail = FALSE:

p_value <- pchisq(chi_square_stat, df, lower.tail = FALSE)

Here:

• chi_square_stat is the calculated Chi-Square statistic.
• df is the degrees of freedom for the test.
• lower.tail = FALSE specifies that we are interested in the upper tail.

Why the Upper Tail Represents the P-Value

In hypothesis testing, the p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from your sample data, assuming the null hypothesis is true. For the Chi-Square test, this probability is found in the upper tail of the Chi-Square distribution. But why the upper tail?

The Chi-Square distribution is used to measure the total deviation of observed frequencies from expected frequencies. Since it is a sum of squared values, the Chi-Square statistic is always non-negative, and larger values indicate a greater difference between observed and expected frequencies.

Interpreting the Upper Tail

In the Chi-Square distribution, the upper tail represents the area to the right of the calculated Chi-Square statistic. This area gives us the probability of obtaining a Chi-Square statistic at least as large as the observed value. This is important because:

• Extreme values indicate strong deviations: Larger Chi-Square values represent significant deviations between observed and expected values, which could suggest that the observed data does not fit the expected distribution (for a goodness of fit test) or that variables are not independent (for a test of independence).
• P-value as a measure of extremity: By calculating the upper tail area, we obtain the p-value, which tells us how likely it is to observe such a strong deviation by chance. If this p-value is very small (typically < 0.05), it suggests that the observed deviations are unlikely to occur if the null hypothesis is true, leading us to question or reject the null hypothesis.

Using pchisq with lower.tail = FALSE in R allows us to calculate this upper-tail area directly, giving the exact p-value for the Chi-Square statistic. In this way, the p-value provides a quantitative measure of whether the observed data significantly deviates from expectations.

Example 1: Goodness of Fit Test Using pchisq

Assume we have a Chi-Square statistic of 1 with 5 degrees of freedom from the die example. We can calculate the p-value with pchisq:

# Chi-Square statistic and degrees of freedom
chi_square_stat <- 1
df <- 5

# Calculate p-value using pchisq
p_value <- pchisq(chi_square_stat, df, lower.tail = FALSE)
p_value

[1] 0.9626

This p-value of 0.9626 indicates no significant difference, suggesting that the die is likely fair.

Example 2: Test of Independence Using pchisq

For the snack preference example, let’s assume we calculated a Chi-Square statistic of 9.23 with 4 degrees of freedom. Here’s how to use pchisq to calculate the p-value:

# Chi-Square statistic and degrees of freedom
chi_square_stat <- 9.23
df <- 4

# Calculate p-value using pchisq
p_value <- pchisq(chi_square_stat, df, lower.tail = FALSE)
p_value

[1] 0.056

This p-value of 0.056 suggests a potential relationship between age group and snack preference, though it is close to the conventional significance level of 0.05.

Conclusion

This post has covered how to calculate the p-value of a Chi-Square statistic in R for both the Goodness of Fit Test and the Test of Independence. With chisq.test() and pchisq, you can assess the significance of differences in categorical data, gaining insights into distributions and relationships between variables.

Try the Chi-Square to P-Value Calculator

To calculate the p-value from a Chi-Square statistic quickly for your own data, check out our Chi-Square to P-Value Calculator on the Research Scientist Pod.

Have fun and happy researching!