Multiple regression degrees of freedom $f$-test.

Question

I'm finding conflicting information from college textbooks on calculating the degrees of freedom for a a global $F$-test on a multiple regression. To be absolutely clear, assume there are 50 observations and 3 independent variables. Can you please tell me the df for the numerator and denominator? I have found 2 sets of numbers in college texts. One indicating the numerator is equal to $P$, in this case 3, and alternatively $P-1$. For the denominator I am finding $n-p$,which in this case would be 47, and alternatively, $n-p-1$. Perhaps I am misunderstanding the material and there are circumstances when one vs. the other formula applies. I've not done any regression analysis in more than 25 years and now find I'm stuck on a Christmas vacation project I wanted to do with my son. So any help that would explain, in a gentle way, (I can't get through the quadratic explanation, or something that will bury me in calculus) how to determine the df would be appreciated. Concrete examples would be very beneficial. Also, if there is a good practical walk through of multiple regression/Anova that will show some examples and explain concepts (but please do not recommend Regression for Dummies) I'd appreciate a referral to that as well. Thanks for your help.

Answer 1

The correct approach is to use $p - 1$ in the numerator (degrees of freedom of the model) and $n - p$ in the denominator (degrees of freedom of the error), where $p$ is the number of predictors and $n$ is the number of observations. The sum of these two numbers gives the total degrees of freedom, i.e. $n - 1$. Note that $p$ is the number of predictors, which is equal to the number of independent variables $+1$: this is because the intercept has also to be counted as a predictor.

In your example, the number of predictors is 4 (3 independent variables plus the intercept), so that the numerator is $4 - 1 = 3$. The denominator is $50 - 4 = 46$, and the total variance has $50 - 1 = 49$ degrees of freedom.

It is likely that the discrepancies found between textbooks about the presence of "$p - 1$" or "$p$" in the numerator, and about "$n - p$" or "$n - p - 1$" in the denominator simply depend on the meaning we give to $p$. The letter "$p$" in this context is commonly used to describe the number of predictors: however, if it is (inappropriately) used to describe the number of independent variables, then the numerator can become equal to $p$ and the denominator can become equal to $n - p - 1$.

Lastly, there is another reason for confusion. Sometimes regression models are built by dropping the intercept. This approach, which forces the regression line to go through the origin, is rarely used because of a number of potential pitfalls. However, when this approach is chosen, the number of predictors and that of independent variables are equal. In this case, it is correct to set $p$ as the numerator and $n - p - 1$ as the denominator.