Chi-square test: A comprehensive guide for use in statistics
The chi-square test (χ² test) is one of the basic statistical methods for analyzing whether there is a relationship between two categorical variables. This test is particularly useful when you are working with frequencies or contingency tables and want to check whether observed differences between groups are random or indicate a true association. In this blog post, we will explain the basics of the chi-square test, discuss its different applications and illustrate it with an example in R.
What is the chi-square test?
The chi-square test checks whether the observed frequencies in categories differ significantly from the expected frequencies. There are two main types of chi-square test:
- Chi-square goodness-of-fit test: This test checks whether the distribution of a single categorical variable corresponds to an expected distribution.
- Chi-square independence test (Test of Independence): This test checks whether there is an association or a correlation between two categorical variables.
1. chi-square goodness-of-fit test
The goodness-of-fit test is used to test whether the observed frequencies of a single categorical variable match an expected frequency distribution. For example, one could examine whether a die is fair by comparing the frequency of each result after multiple rolls with the theoretical, uniform distribution.
Example:
Suppose you roll a die 60 times and get the following results:
| Number of points | Frequency |
|---|---|
| 1 | 8 |
| 2 | 10 |
| 3 | 12 |
| 4 | 9 |
| 5 | 11 |
| 6 | 10 |
The expected frequencies would be 10 in each case, since with a fair die each result should occur with a probability of ( \frac{1}{6} ).
The adjustment test will now check whether the differences between the observed and expected frequencies are random or significant.
2. chi-square independence test
The independence test is used to test whether two categorical variables are independent of each other. A typical example would be the question of whether gender (male/female) and the presence of a disease (yes/no) are related.
Example:
Let's imagine a survey in which 100 people are asked about their gender and their opinion of a new product (like/dislike). The following contingency table shows the results:
| Like | Do not like | Total | |
|---|---|---|---|
| Male | 30 | 20 | 50 |
| Female | 10 | 40 | 50 |
| Total | 40 | 60 | 100 |
The chi-square independence test will now check whether gender and opinion of the product are statistically independent of each other.
Performing the chi-square test in R
The chi-square test can be easily performed in R. Let's take the second example (independence test) and carry out the test in R.
# Daten in einer Kontingenztabelle
data <- matrix(c(30, 20, 10, 40), nrow = 2, byrow = TRUE)
colnames(data) <- c("Gefällt", "Gefällt nicht")
rownames(data) <- c("Männlich", "Weiblich")
data
# Durchführung des Chi-Quadrat-Tests
chisq.test(data)The output provides the chi-square value, the degrees of freedom and the p-value. If the p-value is less than the significance level (e.g. 0.05), we can reject the null hypothesis and assume that there is a correlation between the variables.
Interpretation of the chi-square test
The null hypothesis of the chi-square test states that there is no difference between the observed and expected frequencies (adjustment test) or that the variables are independent of each other (independence test).
- If the p-value is less than the specified significance level (e.g. 0.05), we reject the null hypothesis. This means that there is a statistically significant difference between the groups or that the variables are not independent of each other.
- If the p-value is greater than the significance level, we cannot reject the null hypothesis, which means that the differences are probably random and there is no relationship between the variables.
Assumptions of the chi-square test
There are some important assumptions that must be taken into account in the chi-square test:
- Categorical data: The test is applied to nominal or ordinal (categorical) data.
- Expected frequencies: The expected frequencies in each cell of the contingency table should ideally be greater than 5. If the expected frequencies are too small, the test may be biased.
- Independence of the observations: The observations in the different groups should be independent of each other.
Conclusion
The chi-square test is a simple yet powerful tool for examining relationships between categorical variables. It provides a quick and effective method for deciding whether differences in frequencies between groups are random or indicate a genuine relationship. By using and interpreting it correctly, you can gain valuable insights from your data.
Although the chi-square test is useful in many situations, you should make sure that the assumptions are met and consider alternative methods such as the Fisher test if the conditions for the chi-square test are not met.
