How To Find Expected Value Chi Square

Let's explore the concept of expected value in the context of the Chi-Square test, a crucial element in statistical analysis. Understanding how to calculate expected values is fundamental to interpreting the results of Chi-Square tests, which are widely used to determine if there's a statistically significant association between categorical variables.

Chi-Square Test: An Overview

The Chi-Square test is a statistical hypothesis test used to determine if there is a significant association between two categorical variables. It works by comparing the observed frequencies (the actual data collected) with the expected frequencies (the frequencies you would expect if there were no association between the variables). There are a couple of Chi-Square tests, namely:

• Chi-Square Test of Independence: Used to determine if there is a significant association between two categorical variables.
• Chi-Square Goodness of Fit Test: Used to determine if the observed sample data matches an expected distribution.

The core of the Chi-Square test lies in comparing what you actually observed in your data with what you expected to see if there were no relationship between the variables you are investigating. The difference between these observed and expected values forms the basis for calculating the Chi-Square statistic.

Expected Value: The Theoretical Foundation

The expected value represents the theoretical frequency of each cell in a contingency table, assuming there is no association between the variables being studied. In simpler terms, it's the value you would anticipate seeing in each category if the two variables were completely independent. The Chi-Square test leverages these expected values to quantify the difference between what was observed and what was predicted under the null hypothesis of independence.

The Formula for Calculating Expected Value

Calculating the expected value is straightforward. For each cell in a contingency table, the expected value is calculated using the following formula:

Expected Value = (Row Total * Column Total) / Grand Total

Where:

• Row Total is the total number of observations in the row containing the cell.
• Column Total is the total number of observations in the column containing the cell.
• Grand Total is the total number of observations in the entire table.

This formula essentially distributes the overall sample size proportionally across the cells based on the marginal totals (row and column totals).

Step-by-Step Guide to Finding Expected Value Chi-Square

Let's break down the process of finding the expected value for a Chi-Square test with a detailed, step-by-step guide. We'll use an example to illustrate each step.

Example Scenario:

Suppose we want to investigate if there's an association between smoking habits and the development of lung cancer. We collect data from a sample of individuals and categorize them based on whether they are smokers or non-smokers and whether they have been diagnosed with lung cancer or not.

Step 1: Create a Contingency Table

The first step is to organize the data into a contingency table (also known as a cross-tabulation). This table will show the observed frequencies for each combination of categories.

Step 2: Calculate Row Totals, Column Totals, and Grand Total

As shown in the table above, we need to calculate the row totals, column totals, and the grand total of all observations.

• Row Totals:
  • Smoker: 65 + 35 = 100
  • Non-Smoker: 15 + 85 = 100
• Column Totals:
  • Lung Cancer: 65 + 15 = 80
  • No Lung Cancer: 35 + 85 = 120
• Grand Total: 100 + 100 = 80 + 120 = 200

Step 3: Apply the Expected Value Formula to Each Cell

Now, we apply the expected value formula to each cell in the contingency table:

• Expected Value (Smoker, Lung Cancer):

  (Row Total for Smoker * Column Total for Lung Cancer) / Grand Total

  (100 * 80) / 200 = 40

• Expected Value (Smoker, No Lung Cancer):

  (Row Total for Smoker * Column Total for No Lung Cancer) / Grand Total

  (100 * 120) / 200 = 60

• Expected Value (Non-Smoker, Lung Cancer):

  (Row Total for Non-Smoker * Column Total for Lung Cancer) / Grand Total

  (100 * 80) / 200 = 40

• Expected Value (Non-Smoker, No Lung Cancer):

  (Row Total for Non-Smoker * Column Total for No Lung Cancer) / Grand Total

  (100 * 120) / 200 = 60

Step 4: Create a Table of Expected Values

Organize the calculated expected values into a table that mirrors the original contingency table:

Step 5: Calculate the Chi-Square Statistic

Now that we have both the observed values and the expected values, we can calculate the Chi-Square statistic. The formula for the Chi-Square statistic is:

Χ² = Σ [(Observed Value - Expected Value)² / Expected Value]

Where:

• Χ² is the Chi-Square statistic.
• Σ means "sum of".

Applying this formula to our example:

Χ² = [(65-40)² / 40] + [(35-60)² / 60] + [(15-40)² / 40] + [(85-60)² / 60]

Χ² = [625 / 40] + [625 / 60] + [625 / 40] + [625 / 60]

Χ² = 15.625 + 10.417 + 15.625 + 10.417

Χ² = 52.084

Step 6: Determine the Degrees of Freedom

The degrees of freedom (df) are needed to determine the p-value. For a Chi-Square test of independence, the degrees of freedom are calculated as:

df = (Number of Rows - 1) * (Number of Columns - 1)

In our example:

df = (2 - 1) * (2 - 1) = 1 * 1 = 1

Step 7: Determine the P-value

Using the calculated Chi-Square statistic (52.084) and the degrees of freedom (1), we can find the p-value using a Chi-Square distribution table or statistical software. The p-value represents the probability of observing a Chi-Square statistic as extreme as, or more extreme than, the one calculated, assuming there is no association between the variables.

In this case, the p-value is extremely small (close to 0).

Step 8: Interpret the Results

Finally, we compare the p-value to a significance level (alpha), typically set at 0.05.

• If the p-value is less than or equal to alpha, we reject the null hypothesis and conclude that there is a significant association between the variables.
• If the p-value is greater than alpha, we fail to reject the null hypothesis and conclude that there is no significant association between the variables.

In our example, since the p-value is close to 0 and therefore less than 0.05, we reject the null hypothesis. This means we have evidence to suggest that there is a significant association between smoking habits and the development of lung cancer.

Why Expected Values Matter

The expected values are more than just numbers in a formula; they represent the baseline against which we compare our observed data. They provide a crucial reference point for understanding whether the patterns we see in our data are likely due to chance or reflect a real relationship between the variables.

• Assessing Independence: Expected values allow us to assess the independence of categorical variables. If the observed values are substantially different from the expected values, it suggests that the variables are not independent.
• Quantifying Discrepancies: The Chi-Square test uses the difference between observed and expected values to quantify the discrepancies between the observed data and the null hypothesis of independence.
• Informing Decisions: The results of the Chi-Square test, based on the expected values, help us make informed decisions about whether to reject or fail to reject the null hypothesis, leading to meaningful conclusions about the relationships between categorical variables.

Common Pitfalls and How to Avoid Them

While calculating expected values is relatively straightforward, there are some common pitfalls to be aware of:

• Small Sample Sizes: The Chi-Square test is sensitive to small sample sizes. If the expected values in any cell are too small (typically less than 5), the test results may be unreliable. To address this, consider combining categories or using alternative statistical tests.
• Incorrect Calculations: Ensure that you are accurately calculating row totals, column totals, grand totals, and expected values. Double-check your calculations to avoid errors that could lead to incorrect conclusions.
• Misinterpreting Results: Remember that the Chi-Square test only indicates whether there is an association between variables, not the nature or strength of the association. It does not prove causation.
• Forgetting Degrees of Freedom: Failing to correctly calculate the degrees of freedom can lead to an incorrect p-value and erroneous conclusions.
• Applying to Non-Categorical Data: The Chi-Square test is designed for categorical data. Do not apply it to continuous data.

Real-World Applications

The Chi-Square test, and therefore the calculation of expected values, has numerous applications across various fields:

• Healthcare: Analyzing the relationship between risk factors (e.g., smoking, diet) and disease prevalence.
• Marketing: Assessing the effectiveness of different marketing campaigns on customer behavior.
• Education: Evaluating the association between teaching methods and student performance.
• Social Sciences: Investigating the relationship between demographic variables and attitudes or opinions.
• Genetics: Determining if observed genetic ratios deviate significantly from expected Mendelian ratios.

Example 2: Color Preference vs. Gender

Let’s consider another example to further solidify the concept. A researcher wants to investigate whether there is an association between gender and preference for colors (Red, Blue, Green). They collect data from 300 individuals and organize it as follows:

Step 1: Calculate Row Totals, Column Totals, and Grand Total

• Row Totals:
  • Male: 40 + 30 + 20 = 90
  • Female: 60 + 80 + 70 = 210
• Column Totals:
  • Red: 40 + 60 = 100
  • Blue: 30 + 80 = 110
  • Green: 20 + 70 = 90
• Grand Total: 90 + 210 = 100 + 110 + 90 = 300

Step 2: Calculate the Expected Values

• Expected Value (Male, Red) = (90 * 100) / 300 = 30
• Expected Value (Male, Blue) = (90 * 110) / 300 = 33
• Expected Value (Male, Green) = (90 * 90) / 300 = 27
• Expected Value (Female, Red) = (210 * 100) / 300 = 70
• Expected Value (Female, Blue) = (210 * 110) / 300 = 77
• Expected Value (Female, Green) = (210 * 90) / 300 = 63

Step 3: Create a Table of Expected Values

Step 4: Calculate the Chi-Square Statistic

Χ² = Σ [(Observed Value - Expected Value)² / Expected Value]

Χ² = [(40-30)² / 30] + [(30-33)² / 33] + [(20-27)² / 27] + [(60-70)² / 70] + [(80-77)² / 77] + [(70-63)² / 63]

Χ² = [100 / 30] + [9 / 33] + [49 / 27] + [100 / 70] + [9 / 77] + [49 / 63]

Χ² ≈ 3.33 + 0.27 + 1.81 + 1.43 + 0.12 + 0.78

Χ² ≈ 7.74

Step 5: Determine the Degrees of Freedom

df = (Number of Rows - 1) * (Number of Columns - 1)

df = (2 - 1) * (3 - 1) = 1 * 2 = 2

Step 6: Determine the P-value

Using a Chi-Square distribution table or statistical software with Χ² = 7.74 and df = 2, we find that the p-value is approximately 0.021.

Step 7: Interpret the Results

Since the p-value (0.021) is less than the significance level (0.05), we reject the null hypothesis. We conclude that there is a statistically significant association between gender and color preference in this sample.

Conclusion

Calculating expected values is a critical step in performing a Chi-Square test. These values provide the theoretical foundation for comparing observed data against what would be expected if there were no association between the categorical variables being studied. By understanding the formula, following the step-by-step guide, and avoiding common pitfalls, you can confidently calculate expected values and interpret the results of Chi-Square tests, enabling you to draw meaningful conclusions from your data. Whether in healthcare, marketing, education, or other fields, the Chi-Square test, with its reliance on expected values, remains a powerful tool for analyzing categorical data and uncovering significant relationships.