In multiple regression, what is the formula for calculating degrees of freedom (df)?

In multiple regression, the formula for calculating degrees of freedom (df) is based on the total number of observations and the number of parameters being estimated in the model. The correct formula is expressed as N - p - 1, where N represents the total number of observations (data points) and p symbolizes the number of independent predictors in the regression model.

The reason this formula is appropriate lies in the statistical reasoning underpinning degrees of freedom. When fitting a regression model, each predictor adds an additional constraint or parameter that must be estimated, thus reducing the available degrees of freedom. The subtraction of 1 accounts for the overall mean of the dependent variable being estimated as well, which is another parameter.

Overall, the degrees of freedom indicate the number of values in the final calculation that are free to vary. In the context of multiple regression, it is important to have this correct calculation to accurately assess the reliability of the regression model and its associated statistical tests, such as F-tests for overall significance. Understanding these degrees of freedom is essential for interpreting model results and ensuring valid statistical inferences.