Test of Independence

Definition

The test of independence is a statistical hypothesis test used to determine whether two categorical variables are independent of each other or if they are associated. It examines the relationship between the variables to see if they are related or if any observed differences are due to chance alone.

5 Must Know Facts For Your Next Test

1. The test of independence is used to determine if two categorical variables are related or independent of each other.
2. The test statistic for the test of independence follows a chi-square distribution, with the degrees of freedom equal to (r-1)(c-1), where r is the number of rows and c is the number of columns in the contingency table.
3. The null hypothesis for the test of independence is that the two variables are independent, while the alternative hypothesis is that they are associated.
4. The p-value for the test of independence represents the probability of observing a test statistic as extreme or more extreme than the calculated value, assuming the null hypothesis is true.
5. The test of independence is a nonparametric test, meaning it does not require any assumptions about the underlying distribution of the variables.

Review Questions

• Explain the purpose and process of the test of independence.
  • The test of independence is used to determine whether two categorical variables are independent or associated. The process involves constructing a contingency table to organize the observed frequencies of the variables, calculating a test statistic (typically chi-square) based on the differences between the observed and expected frequencies, and then comparing the test statistic to a critical value from the chi-square distribution to determine if the null hypothesis of independence can be rejected. The result of the test indicates whether the two variables are related or if any observed differences are due to chance alone.
• Describe the relationship between the test of independence and the chi-square distribution.
  • The test statistic used in the test of independence follows a chi-square distribution. This means that the test statistic can be approximated by the chi-square distribution, which has one parameter: the degrees of freedom. The degrees of freedom for the test of independence are calculated as (r-1)(c-1), where r is the number of rows and c is the number of columns in the contingency table. The chi-square distribution is used to determine the critical value against which the test statistic is compared to assess the statistical significance of the results and make a decision about the null hypothesis of independence.
• Explain how the test of independence is used to compare the chi-square tests discussed in 11.5.
  • The test of independence is one of the chi-square tests covered in the 11.5 Comparison of the Chi-Square Tests topic. This test is used to determine if two categorical variables are independent or associated, which is a common objective in statistical analysis. The test of independence can be compared to other chi-square tests, such as the goodness-of-fit test and the test of homogeneity, in terms of the underlying assumptions, the hypotheses being tested, and the interpretation of the results. Understanding the similarities and differences between these chi-square tests is important for selecting the appropriate statistical method and correctly interpreting the findings in a given research context.

Related terms

Chi-Square Distribution:

The chi-square distribution is a probability distribution used to approximate the distribution of the test statistic in the test of independence. It is a continuous probability distribution with one parameter, the degrees of freedom.

Contingency Table:

A contingency table is a tabular format used to organize and display the frequencies or counts of the observations for two categorical variables. It is the basis for conducting the test of independence.

Null Hypothesis: In the test of independence, the null hypothesis states that the two categorical variables are independent, meaning there is no relationship between them. The alternative hypothesis is that the variables are associated or dependent.