Chi-squared test

Chi-squared test

The Chi-squared test is a statistical test used to determine the independence between two categorical variables. It is named after the Greek letter "chi" (χ), which is used to represent the test statistic. The test is widely used in various fields, including biology, social sciences, and market research.

Background

The Chi-squared test was first introduced by Karl Pearson in the late 19th century. It is based on the principle of comparing observed frequencies with expected frequencies to determine if there is a significant association between two variables. The test is particularly useful when dealing with nominal or ordinal data.

Test Procedure

To perform a Chi-squared test, the following steps are typically followed:

1. Formulate the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis assumes that there is no association between the variables, while the alternative hypothesis assumes that there is an association.

2. Collect data and organize it into a contingency table. A contingency table displays the observed frequencies of each combination of categories for the two variables being tested.

3. Calculate the expected frequencies for each cell in the contingency table. This is done by assuming that the variables are independent and calculating the expected frequencies based on the marginal totals.

4. Calculate the Chi-squared test statistic using the formula:

  χ² = Σ((O - E)² / E)

  where χ² is the test statistic, O is the observed frequency, and E is the expected frequency.

5. Determine the degrees of freedom for the test. This is calculated as (r - 1) * (c - 1), where r is the number of rows and c is the number of columns in the contingency table.

6. Compare the calculated test statistic with the critical value from the Chi-squared distribution table. If the calculated test statistic is greater than the critical value, the null hypothesis is rejected, indicating a significant association between the variables.

Interpretation

The Chi-squared test provides a p-value, which represents the probability of obtaining the observed data under the assumption of independence. A small p-value (typically less than 0.05) indicates that there is strong evidence against the null hypothesis, suggesting that the variables are not independent.

It is important to note that the Chi-squared test only determines if there is an association between variables, but it does not provide information about the strength or direction of the association. To further analyze the relationship between variables, additional statistical tests or measures, such as Cramer's V or Phi coefficient, can be used.

Applications

The Chi-squared test has a wide range of applications in various fields. Some common examples include:

- Biology: Testing the independence between genotype and phenotype in genetic studies. - Social Sciences: Analyzing survey data to determine if there is a relationship between demographic variables. - Market Research: Assessing the association between customer preferences and product features.

See Also

- Categorical Data Analysis: A broader field of statistical methods for analyzing categorical data. - Pearson's Chi-squared test: A specific type of Chi-squared test used for testing goodness of fit.

References

1. Pearson, K. (1900). "On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling". Philosophical Magazine. 50 (302): 157–175.

2. Agresti, A. (2002). "Categorical Data Analysis". Wiley.