How To Calculate The Expected Value For Chi Square

How to Calculate the Expected Value for Chi-Square Tests

The chi-square (χ²) test is a powerful statistical tool used to analyze categorical data and determine if there's a significant association between two or more variables. Understanding how to calculate the expected values is crucial for correctly conducting and interpreting a chi-square test. This article provides a comprehensive guide, walking you through the process step-by-step, with illustrative examples.

Understanding the Chi-Square Test and Expected Values

Before diving into the calculations, let's establish a foundational understanding. The chi-square test compares observed frequencies (the actual counts you gather from your data) with expected frequencies (the counts you would expect if there were no relationship between the variables). A significant difference between observed and expected frequencies suggests a relationship exists.

The expected value for each cell in your contingency table (the table organizing your categorical data) is crucial for the chi-square calculation. It represents the frequency you'd anticipate in that cell if the variables were independent. In simpler terms, it's what you'd expect to see if there's no association.

The formula for calculating the expected value is:

(Row Total * Column Total) / Grand Total

Where:

• Row Total: The sum of observed frequencies in the row.
• Column Total: The sum of observed frequencies in the column.
• Grand Total: The total number of observations in the entire table.

Step-by-Step Calculation of Expected Values

Let's illustrate the process with a concrete example. Imagine a study investigating the relationship between smoking and lung cancer. Here's a hypothetical contingency table showing the observed frequencies:

	Lung Cancer	No Lung Cancer	Row Total
Smoker	80	20	100
Non-Smoker	20	180	200
Column Total	100	200	300

1. Calculate Row Totals and Column Totals:

This step is straightforward; simply sum the observed frequencies for each row and column. The table above already shows these totals.

2. Calculate the Grand Total:

This is the sum of all observed frequencies in the table (or the sum of row totals or column totals). In this case, the grand total is 300.

3. Calculate the Expected Value for Each Cell:

Now, we apply the formula for each cell in the contingency table. Let's break it down:

• Expected Value for Smoker & Lung Cancer: (100 * 100) / 300 = 33.33
• Expected Value for Smoker & No Lung Cancer: (100 * 200) / 300 = 66.67
• Expected Value for Non-Smoker & Lung Cancer: (200 * 100) / 300 = 66.67
• Expected Value for Non-Smoker & No Lung Cancer: (200 * 200) / 300 = 133.33

This results in a new table showing the expected values:

	Lung Cancer	No Lung Cancer	Row Total
Smoker	33.33	66.67	100
Non-Smoker	66.67	133.33	200
Column Total	100	200	300

Interpreting Expected Values

The expected values tell us what the frequencies would be if there were no association between smoking and lung cancer. Notice that the expected frequencies are quite different from the observed frequencies. This difference is crucial for the chi-square test. A large difference suggests a potential relationship.

Performing the Chi-Square Test

Once you've calculated the expected values, you can proceed with the chi-square test itself. The formula for the chi-square statistic is:

χ² = Σ [(Observed - Expected)² / Expected]

This means you need to:

1. Find the difference between each observed and expected value: (Observed - Expected)
2. Square this difference: (Observed - Expected)²
3. Divide the squared difference by the expected value: (Observed - Expected)² / Expected
4. Sum these values across all cells: Σ [(Observed - Expected)² / Expected]

Let's calculate the chi-square statistic for our example:

• Smoker & Lung Cancer: (80 - 33.33)² / 33.33 ≈ 66.7
• Smoker & No Lung Cancer: (20 - 66.67)² / 66.67 ≈ 35.56
• Non-Smoker & Lung Cancer: (20 - 66.67)² / 66.67 ≈ 35.56
• Non-Smoker & No Lung Cancer: (180 - 133.33)² / 133.33 ≈ 17.78

χ² = 66.7 + 35.56 + 35.56 + 17.78 ≈ 155.6

Degrees of Freedom and p-value

To interpret the chi-square statistic, you need the degrees of freedom (df) and the associated p-value. The degrees of freedom for a chi-square test of independence are calculated as:

(Number of rows - 1) * (Number of columns - 1)

In our example: (2 - 1) * (2 - 1) = 1

Using a chi-square distribution table or statistical software (like R, SPSS, or Python with SciPy), you find the p-value corresponding to your chi-square statistic (155.6) and degrees of freedom (1). The p-value will be extremely small, indicating strong evidence to reject the null hypothesis (that smoking and lung cancer are independent). This supports the conclusion that there is a significant association between smoking and lung cancer.

Larger Contingency Tables

The principles remain the same for larger contingency tables with more rows and columns. You simply extend the calculations to include all cells. The degrees of freedom will also increase accordingly.

Assumptions of the Chi-Square Test

It's crucial to remember that the chi-square test rests on several assumptions:

• Independence: Observations must be independent of each other.
• Expected Frequencies: Expected frequencies in each cell should be at least 5. If not, consider techniques like Fisher's exact test or combining categories.
• Categorical Data: The data must be categorical.

Conclusion: Mastering Expected Values for Chi-Square Analysis

Calculating expected values is a fundamental step in performing a chi-square test of independence. By carefully following the steps outlined above, you can accurately determine these values, conduct the test, and draw valid conclusions about the relationships between your categorical variables. Remember to always check the assumptions of the test to ensure the validity of your results. Understanding the underlying concepts and methodology allows you to correctly interpret the statistical significance and draw meaningful insights from your data analysis. Proficiency in chi-square analysis is valuable in various fields, including healthcare, social sciences, and market research, enabling data-driven decision-making across diverse applications.