If a categorical variable has k levels, how many dummy variables are needed?

When working with categorical variables in regression analysis, it is essential to accurately represent the information these variables convey. A categorical variable that has k levels requires the use of dummy variables to include it meaningfully in a regression model.

To fully capture the information from k levels without redundancy, you only need k-1 dummy variables. Each dummy variable represents one level of the categorical variable, but including a dummy variable for each of the k levels would lead to perfect multicollinearity, which can distort the regression model. This redundancy occurs because the information from the k levels can be fully reconstructed from the k-1 dummy variables. Essentially, one of the levels acts as a reference group and is represented by the absence of all dummy variables.

By using k-1 dummy variables, the model can adequately account for the effects associated with the k levels while maintaining proper statistical integrity. This approach ensures that the regression coefficients for the k-1 dummy variables indicate how each of these levels compares to the reference group, providing clear interpretability.