How To Calculate Expected Value For Chi Square

How to Calculate Expected Value for Chi-Square Tests

The chi-square (χ²) test is a powerful statistical tool used to analyze categorical data. It assesses the difference between observed frequencies and expected frequencies within different categories of a variable. Understanding how to calculate the expected values is crucial for correctly performing and interpreting a chi-square test. This in-depth guide will walk you through the process, explaining the underlying concepts and offering practical examples.

Understanding Expected Values in Chi-Square Tests

Before diving into the calculations, let's clarify what expected values represent in a chi-square context. The expected value for a cell in a contingency table is the frequency you would expect to observe if there were no association between the variables being studied. It's a theoretical frequency based on the marginal totals (row and column sums) of your data. The difference between the observed and expected values is what drives the chi-square statistic. A large difference suggests a statistically significant association between the variables, while a small difference implies no significant relationship.

Types of Chi-Square Tests and Expected Value Calculations

There are several types of chi-square tests, each with slightly different approaches to calculating expected values. The most common are:

• Goodness-of-fit test: This test compares the observed distribution of a single categorical variable to an expected distribution.
• Test of independence: This test examines the association between two categorical variables.

Calculating Expected Values for the Goodness-of-Fit Test

In a goodness-of-fit test, the expected value for each category is calculated based on the overall sample size and the hypothesized proportions for each category.

Formula:

Expected value (Eᵢ) = (Total sample size) * (Hypothesized proportion for category i)

Example:

Let's say you're testing whether a die is fair. You roll the die 60 times and observe the following frequencies:

Face	Observed Frequency (Oᵢ)
1	8
2	12
3	9
4	10
5	11
6	10

The hypothesized proportion for each face of a fair die is 1/6. The total sample size is 60. Therefore, the expected value for each face is:

Eᵢ = 60 * (1/6) = 10

In this case, the expected frequency for each face is 10.

Calculating Expected Values for the Test of Independence

The test of independence assesses whether two categorical variables are independent. The expected value for each cell in the contingency table is calculated using the marginal totals.

Formula:

Expected value (Eᵢⱼ) = (Row total for row i) * (Column total for column j) / (Total sample size)

where:

• Eᵢⱼ is the expected value for the cell in row i and column j.
• Row total for row i is the sum of observed frequencies in row i.
• Column total for column j is the sum of observed frequencies in column j.
• Total sample size is the sum of all observed frequencies.

Example:

Let's consider a study investigating the relationship between smoking status and lung cancer. The observed data is shown below:

	Lung Cancer	No Lung Cancer	Total
Smoker	40	60	100
Non-smoker	10	90	100
Total	50	150	200

Let's calculate the expected value for the cell representing "Smoker" and "Lung Cancer":

E₁₁ = (100) * (50) / 200 = 25

Similarly, we can calculate the expected values for other cells:

• E₁₂ = (100) * (150) / 200 = 75
• E₂₁ = (100) * (50) / 200 = 25
• E₂₂ = (100) * (150) / 200 = 75

The complete table with observed and expected values would look like this:

	Lung Cancer (Observed/Expected)	No Lung Cancer (Observed/Expected)	Total
Smoker	40/25	60/75	100
Non-smoker	10/25	90/75	100
Total	50	150	200

Interpreting Expected Values and the Chi-Square Statistic

The expected values themselves don't directly indicate statistical significance. Instead, they are used to calculate the chi-square statistic (χ²):

Formula:

χ² = Σ [(Oᵢⱼ - Eᵢⱼ)² / Eᵢⱼ]

where:

• Oᵢⱼ is the observed frequency for cell i,j.
• Eᵢⱼ is the expected frequency for cell i,j.
• The summation is across all cells in the contingency table.

A higher chi-square value suggests a greater discrepancy between observed and expected frequencies, indicating a stronger association (or departure from the expected distribution in the goodness-of-fit test). You then compare the calculated chi-square value to a critical value from the chi-square distribution, based on your chosen significance level (alpha) and degrees of freedom.

Degrees of Freedom and the Chi-Square Distribution

The degrees of freedom (df) are crucial for determining the critical value for the chi-square test. The df represent the number of independent pieces of information used to estimate the parameters of the distribution.

• Goodness-of-fit test: df = k - 1, where k is the number of categories.
• Test of independence: df = (number of rows - 1) * (number of columns - 1)

The chi-square distribution is a probability distribution that depends on the degrees of freedom. If the calculated chi-square value exceeds the critical value (obtained from a chi-square table or statistical software), you reject the null hypothesis. The null hypothesis is that there is no association between the variables (or that the observed distribution follows the expected distribution).

Practical Applications and Considerations

Chi-square tests are widely used in various fields, including:

• Market research: Analyzing customer preferences and demographics.
• Healthcare: Evaluating treatment effectiveness and disease prevalence.
• Social sciences: Investigating relationships between social variables.
• Biology: Studying genetic inheritance patterns.

Important Considerations:

• Sample size: Chi-square tests are more reliable with larger sample sizes. Small sample sizes can lead to inaccurate results.
• Expected cell frequencies: Generally, expected cell frequencies should not be too small (often a rule of thumb is at least 5). If expected cell frequencies are too low, alternative statistical methods may be more appropriate.
• Independence of observations: The observations in your data set should be independent of one another.
• Statistical software: Statistical software packages (like R, SPSS, SAS, or Python with libraries like SciPy) simplify chi-square calculations and provide p-values directly.

Conclusion: Mastering Expected Value Calculations for Chi-Square Tests

Accurately calculating expected values is fundamental to conducting a valid chi-square test. By understanding the formulas and applying them correctly, you can effectively analyze categorical data and draw meaningful conclusions about the relationships between variables. Remember to consider the limitations of the chi-square test and use appropriate statistical software to perform the calculations and obtain p-values for interpreting your results. Mastering this technique empowers you to explore various datasets and make data-driven decisions. Further exploration of advanced statistical methods can enhance your analytical skills, allowing you to handle more complex research questions and gain deeper insights from your data.