[Statistics] Chi-Squared Test

This post covers chi-squared test.

1. Introduction

In the realm of statistics, the chi-squared test stands as a powerful tool for analyzing categorical data and determining the presence of associations or dependencies between variables. In this blog post, we'll delve into the fascinating origins and history of the chi-squared test, explore its underlying principles, provide a practical usage example, and shed light on the mathematical expression and calculation involved.

2. Origin and History

The chi-squared test owes its existence to the pioneering work of Karl Pearson, a renowned British statistician, who developed it around the turn of the 20th century. Pearson's contributions to statistical theory and methodology laid the foundation for modern statistical analysis, and the chi-squared test stands as one of his significant legacies.

3. Principle of the Chi-Squared Test

The chi-squared test assesses whether there is a statistically significant association or dependence between two categorical variables. It compares the observed frequencies in a contingency table with the expected frequencies that would occur under the assumption of independence. The test calculates a chi-squared statistic, which is then compared to a critical value or p-value to determine the significance of the association.

4. Mathematical Expression and Calculation

The chi-squared test involves calculating the chi-squared statistic using the formula

χ² = Σ((O - E)² / E),

where:

- χ² represents the chi-squared statistic,

- O refers to the observed frequency in each cell of the contingency table, and

- E represents the expected frequency in each cell under the assumption of independence.

To calculate the expected frequency for each cell, the formula is:

E = (row total × column total) / grand total.

After obtaining the chi-squared statistic, its value is compared to the critical value from the chi-squared distribution with degrees of freedom determined by the contingency table dimensions.

Degrees of freedom can be calculated as df = (the number of rows - 1) × (the number of columns -1)

Alternatively, the p-value can be calculated, indicating the probability of obtaining a chi-squared statistic as extreme as the observed one, assuming no association between variables.

5. Usage Example

Suppose a researcher wants to investigate whether there is a relationship between gender (male or female) and the preference for different music genres (rock, pop, hip-hop). The researcher collects data from a sample of 200 individuals and records their gender and music genre preferences. The data is then organized into a contingency table as follows:

	Rock	Pop	Hip-hop	Row Total
Male	40	60	20	120
Female	30	50	40	120
Column Total	70	60	110	240

To perform the chi-squared test, we need to calculate the expected frequencies and then the chi-squared statistic.

Step 1: Calculate Expected Frequencies:

To calculate the expected frequencies, we use the formula:

E = (row total × column total) / grand total.

For example, the expected frequency for the cell "Male-Rock" can be calculated as follows:

E(Male-Rock) = (row total for Male × column total for Rock) / grand total = (120 × 70) / 240 = 35.

Using the same formula, we can calculate the expected frequencies for all the other cells in the contingency table.

	Rock	Pop	Hip-hop	Row Total
Male	35	55	30	120
Female	35	55	30	120
Column Total	70	60	110	240

Step 2: Calculate the Chi-Squared Statistic:

Next, we calculate the chi-squared statistic using the formula:

χ² = Σ((O - E)² / E),

where Σ represents the summation sign, O is the observed frequency, and E is the expected frequency.

Let's calculate the chi-squared statistic for the example:

	Rock	Pop	Hip-hop
Male	[(40 - 35)² / 35]	[(60 - 55)² / 55]	[(20 - 30)² / 30]
Female	[(40 - 35)² / 35]	[(60 - 55)² / 55]	[(20 - 30)² / 30]

χ² = [(40 - 35)² / 35] + [(60 - 55)² / 55] + [(20 - 30)² / 30] + [(30 - 35)² / 35] + [(50 - 55)² / 55] + [(40 - 30)² / 30]

= (25 / 35) + (25 / 55) + (100 / 30) + (25 / 35) + (25 / 55) + (100 / 30)

≈ 0.71 + 0.45 + 3.33 + 0.71 + 0.45 + 3.33

≈ 9.004

Step 3: Determine the Critical Value or P-value:

To interpret the chi-squared statistic, we compare it to the critical value from the chi-squared distribution with degrees of freedom determined by the contingency table dimensions. Alternatively, we can calculate the p-value, indicating the probability of obtaining a chi-squared statistic as extreme as the observed one, assuming no association between variables.

In this example, with a contingency table of 2 rows and 3 columns, the degrees of freedom are (2 - 1) × (3 - 1) = 2. Assuming a significance level of 0.05, we can consult the chi-squared distribution table or use statistical software to find the critical value or calculate the p-value associated with the chi-squared statistic.

Here's how you can do it using Excel:

Open Excel and click on a blank cell. Enter the formula "=CHIINV(0.05, 2)". Press Enter. The result will be the critical value of the chi-squared distribution with df = 2 and a significance level of 0.05, which should be approximately 5.991.

The chi-squared statistic was calculated to be approximately 9.004, which is larger than the critical value of 5.991. In conclusion, the relationship between these variables was stastically significant, χ²(1, N = 240) = 9.00, p = .011. Male were more likely than women to prefer to Rock and Pop, whereas female were more likel than male to prefer to Hip-hop.

7. Caveats

While the chi-squared test is a valuable statistical tool, it is important to be aware of its caveats and limitations. Understanding these caveats will help researchers interpret the results accurately and make informed decisions. Here are some key caveats of the chi-squared test:

1. Sample Size: The chi-squared test assumes that the sample size is sufficiently large for the test to be valid. If the sample size is small, the test may not provide reliable results, and alternative methods like Fisher's exact test might be more appropriate.

2. Independence Assumption: The chi-squared test assumes that the observations within each cell of the contingency table are independent of each other. Violating this assumption can lead to inaccurate results. For example, if data is collected from related individuals or matched pairs, the independence assumption may not hold.

3. Cell Frequencies: The chi-squared test can produce unreliable results if the expected cell frequencies are too low. It is generally recommended to have an expected frequency of at least 5 in each cell. If expected frequencies are lower, the Fisher's exact test or other specialized tests may be more suitable.

4. Large Sample Sizes: While the chi-squared test performs well with large sample sizes, it can detect even small departures from independence. As a result, a statistically significant result may not necessarily indicate a practically meaningful or substantial association. Researchers should consider the effect size and practical significance in addition to the p-value.

5. Categorical Variables Only: The chi-squared test is specifically designed for analyzing relationships between categorical variables. It is not suitable for continuous or ordinal variables. For analyzing relationships involving continuous variables, alternative statistical methods such as regression analysis might be more appropriate.

6. Assumptions of the Test: The chi-squared test assumes that the data is derived from a random sample and that the expected frequencies are not too small. Violations of these assumptions can affect the validity of the test results.

7. Interpretation of Results: While the chi-squared test can determine whether an association exists between variables, it does not provide information about the direction or strength of the relationship. Additional analyses or measures, such as Cramer's V or Phi coefficient, may be needed to assess the effect size and interpret the practical significance of the association.

It is crucial to consider these caveats and select appropriate statistical tests based on the characteristics of the data and the research question at hand. Consulting with a statistician or expert in statistical analysis can be helpful in ensuring the proper application and interpretation of the chi-squared test.

8. Conclusion

The chi-squared test has become a fundamental statistical tool for analyzing categorical data and detecting associations between variables. Developed by Karl Pearson, this test has a rich history and continues to be widely used in various fields, including social sciences, healthcare research, and market analysis. By comparing observed and expected frequencies, calculating the chi-squared statistic, and interpreting the results, researchers gain insights into the presence or absence of associations in their data.

Understanding the origins, principles, and calculations behind the chi-squared test equips researchers with a powerful statistical tool to explore relationships, make informed decisions, and advance knowledge in their respective domains. As we delve deeper into the world of statistics, embracing the chi-squared test allows us to unlock the potential hidden within categorical data and uncover meaningful insights.