Chi-Square Tests and Degrees of Freedom

The Chi-Square test, or χ² test indicates the existence of a relationship between two categorical variables. To expatiate, this analysis will mirror concert organizers using chi-square tests to determine whether the genre of music: Afro or Jazz affects the audience attendance. Essentially, the test checks whether or not observed data fits those that would be expected, assuming that there is no association whatsoever. The chi-square test helps in determining if there is a relationship between music genre and attendance.

To compute the chi-square test, the following formula is used:

Where O is the observed value
E is the expected value
If the p-value <=0.05, we reject the null hypothesis, and if p-value > 0.05, we fail to reject the null hypothesis.
The steps to conducting the chi-square test include:

1. Define the hypothesis, both null and alternative hypothesis
2. Gather and organize the data
3. Calculate the expected frequencies
4. Compute the chi-square test
5. Draw the conclusion Degrees of freedom indicate the number of independent observations or variables that can vary in an analysis without breaking any constraints, readily available to estimate a parameter. In chi-square tests, there are three ways for calculating the degrees of freedom:

a). Goodness of Fit Test
In this test, it checks whether the observed distribution of a single categorical variable matches the expected distribution. In this context, we analyze the frequency distribution of how often the audiences choose Afro versus Jazz concerts.
df = k-1 where:
k = number of categories

b). Test of Independence
The test assesses the relationship between two categorical variables, such as music genre (Afro/Jazz) and the attendance level (high/level)
df = (r-1) x (c-1) where:
r is the number of rows,
c is the number of columns in the contingency table

c). Test for Homogeneity
Entails comparison of the distribution of the categorical variable across different populations. In this context, we would compare the two music genres: (Afro and Jazz) and how they vary between different cities where the concerts are held.

Lets assume there are three cities and two music genres, the df = (3-1)*(2-1) = 2

It is worth noting that the shape of the chi-square distribution evolves as the df increases. This is attributed to how the sum of squared differences between the observed and expected frequencies depend on the number of independent comparisons made.
Notably, the df is not always monotonically decreasing, the shape is dependent of the freedom of the data to vary. Employing the concert planning analogy, the more the elements juggled such as venues, audience preferences and genres, it becomes inherently complex, leading to varied potential outcomes.