Chi squared test for contingency tables

Introduction to Chi-Squared Test for Contingency Tables

The Chi-Squared Test for Contingency Tables is a statistical method used to assess the relationship between two categorical variables.
This test works by comparing observed and expected frequencies in a contingency table, which is a way of displaying data that shows the frequencies of different variables and their combinations.
The null hypothesis for this test states that the variables are independent, meaning there is no relationship between them.

Calculating Expected Frequencies

To perform this test, you first have to calculate the expected frequencies for each cell in your contingency table under the null hypothesis.
These expected frequencies are calculated by multiplying the row total and column total for each cell, and dividing by the grand total of all observations.

Carrying Out a Chi-Squared Test

Once you have the expected frequencies, the next step is to calculate a test statistic, which follows a Chi-Squared distribution.
This test statistic is calculated by summing the squared differences between observed and expected frequencies, divided by the expected frequencies, for all cells in the table. The formula for this is Σ [(O - E)^2 / E].
The degrees of freedom for this test is calculated by multiplying the number of rows minus one by the number of columns minus one.

Interpreting the Result of a Chi-Squared Test

The resulting test statistic is checked against the Chi-Squared distribution with the appropriate degrees of freedom to determine the p-value.
If the p-value is lower than your chosen significance level (commonly 0.05), then you reject the null hypothesis and conclude that the variables are dependent.
Importantly, rejecting the null hypothesis only shows that there is a relationship between the variables, it does not reveal the nature or strength of this relationship.

Further Considerations

Before applying the Chi-Squared Test for Contingency tables, it’s important to ensure the data meets its assumptions. These include the requirement that all observations are independent and that the variables are categorical.
The interpretation of the Chi-Squared test is rooted in probability. Obtaining a significant result implies that you would get the observed pattern of results (or more extreme) less than your threshold level of times (typically 5%) by producing random data assuming there is no relationship between the variables.
Remember, the Chi-Squared Test for Contingency Tables is a valuable tool in identifying whether a relationship exists between two categorical variables, but it cannot provide information about the nature or size of this relationship.