Chi-Squared Test of Independence 5×5 Calculator

Understanding the Chi-Square Test of Independence (5x5) and P-Value Calculation

What is the Chi-Square Test of Independence?

The Chi-Square Test of Independence is a statistical test used to determine if there is a significant association between two categorical variables. It compares observed data with the data that would be expected if the variables were independent. This test is often used with larger contingency tables, such as a 5x5 table, to evaluate relationships across multiple groups and categories.

Why Use the Chi-Square Test of Independence?

• Test for Independence: The Chi-Square Test is frequently used in research to assess if there is an association between two categorical variables, such as survey responses across demographic groups.
• Flexible for Larger Tables: While commonly used for 2x2 tables, the Chi-Square Test can also be applied to larger tables, like 5x5, allowing for analysis across multiple categories simultaneously.
• Simple Interpretation: The test provides a chi-square statistic and p-value, allowing researchers to evaluate results against a significance level to determine if the variables are independent.

How the Chi-Square Statistic is Calculated

The chi-square statistic \( \chi^{2} \) is calculated by comparing observed frequencies to expected frequencies in each cell of the contingency table. The formula is:

• \( O \): Observed frequency in each cell of the table
• \( E \): Expected frequency in each cell, calculated as \( E = \frac{\text{row total} \times \text{column total}}{\text{grand total}} \)

This formula sums the squared differences between observed and expected values, weighted by the expected values, for each cell in the table to produce the chi-square statistic.

Calculating the P-Value from the Chi-Square Statistic

To interpret the chi-square statistic, we calculate a p-value, which represents the probability of observing a chi-square statistic at least as extreme as the one calculated, assuming the null hypothesis of independence is true.

The p-value is calculated using the chi-square cumulative distribution function (CDF):

• \( \chi^2 \): The computed chi-square statistic
• \( \text{df} \): Degrees of freedom for the test, calculated as \( (r - 1) \times (c - 1) \), where \( r \) is the number of rows and \( c \) is the number of columns in the table
• \( F_{\chi^2} \): CDF of the chi-square distribution, representing the probability up to a given chi-square value

The resulting p-value represents the area to the right of the chi-square statistic in the chi-square distribution, indicating the likelihood of observing such a result under the null hypothesis of independence.

Interpretation

A small p-value (typically \( p < 0.05 \)) suggests strong evidence against the null hypothesis, indicating a significant association between the variables. A large p-value suggests that the observed differences may be due to chance, supporting the null hypothesis of independence.

Calculating the Chi-Square CDF and P-Value Programmatically

To determine the p-value for a chi-square statistic, calculate the cumulative distribution function (CDF) and then subtract it from 1. This gives the probability that the chi-square random variable will take on a value at least as extreme as the observed statistic.

Python (using SciPy)

In Python, the scipy.stats library provides a function to get the chi-square CDF:

from scipy.stats import chi2

# Parameters
chi_square_statistic = 10.276
degrees_of_freedom = 16

# Calculate p-value (1 - CDF)
p_value = 1 - chi2.cdf(chi_square_statistic, degrees_of_freedom)

This returns the p-value for the chi-square test.

JavaScript (using jStat)

In JavaScript, the jStat library can be used to calculate the chi-square CDF, then subtract it from 1 to get the p-value:

// Define the chi-square statistic and degrees of freedom
const chiSquareStatistic = 10.276;
const degreesOfFreedom = 16;

// Calculate p-value (1 - CDF)
const pValue = 1 - jStat.chisquare.cdf(chiSquareStatistic, degreesOfFreedom);

This returns the p-value based on the chi-square statistic and degrees of freedom.

R

In R, the pchisq function calculates the CDF, which can then be subtracted from 1 to get the p-value:

# Parameters
chi_square_statistic <- 10.276
degrees_of_freedom <- 16

# Calculate p-value (1 - CDF)
p_value <- 1 - pchisq(chi_square_statistic, degrees_of_freedom)

This returns the p-value for the chi-square test in R.

Using these functions, you can calculate the right-tail p-value from the chi-square CDF, helping you assess the statistical significance of your results.

Example: Testing for Association Between Two Variables with a 5x5 Table

Imagine a study that investigates whether five different treatments affect recovery rates across five age groups. The data is organized in a 5x5 contingency table with observed values for each treatment and age group combination.

Step 1: Observed Values (O)

The observed values are the actual counts from the study, representing frequencies across the 5x5 combinations of treatments and age groups.

Step 2: Calculate Row and Column Totals

To find the expected values, we first calculate the totals for each row and column:

• Row Totals: Sum of observed values across each treatment group
• Column Totals: Sum of observed values across each age group
• Grand Total (N): The sum of all observed values in the table

Step 3: Expected Values (E)

Using the formula \( E = \frac{\text{row total} \times \text{column total}}{\text{grand total}} \), we calculate the expected values for each cell in the 5x5 table.

Step 4: Calculate the Chi-Square Statistic

We use the formula \( \chi^{2} = \sum \frac{(O - E)^2}{E} \) to calculate the chi-square statistic for each cell in the 5x5 table. The statistic is the sum of each cell’s contribution to the chi-square value.

Step 5: Determine the P-Value

Using the chi-square cumulative distribution function (CDF) with degrees of freedom equal to \( (r - 1) \times (c - 1) \), we find the p-value for the computed chi-square statistic:

\( p = 1 - F_{\chi^{2}}(\chi^2, \text{df}) \)

Conclusion

With the calculated p-value, we can determine if there is a statistically significant association between treatments and age groups. A p-value below the significance level (e.g., \( p < 0.05 \)) indicates a likely association, while a p-value above suggests independence between the variables.

Further Reading

• Chi-Square Test - Wikipedia
• Explore More Calculators on the Research Scientist Pod