# When doing a t-test for the significance of a regression coefficient, why is the number of degrees of freedom $n-p-1$?

I read here that $n-p-1$ was the number of degrees of freedom I should use when doing a t-test for the significance of a regression coefficient, but I don't understand why. My understanding was that t-tests generally had $n-1$ degrees of freedom.

• Typically the number of parameters is p+1 since there are p slopes and 1 intercept. – PatrickT Feb 3 '19 at 10:45

You lose one degree of freedom for each estimated mean parameter. For an ordinary t-test that's 1 (the mean). For regression, each predictor costs you a degree of freedom. The extra one is for the intercept.

More specifically, the degrees of freedom come from the denominator in the t-test, which is based on the residual sum of squares -- there are $n-p-1$ degrees of freedom in the residual sums of squares.

• In fact if you regress some outcome on a constant (only an intercept in the model) you get the mean, p would then be 0, and one would have n-1 degrees of freedom – Repmat Sep 28 '16 at 14:32

This is not true in general. The number of degrees of freedom of t-test depends on a specific model. They're talking about linear regression. So, t-test for an estimator has $n-p-1$ degrees of freedom where $p$ is number of explanatory parameters in the model.

Degrees of freedom is the number of independent values or quantities which can be assigned to a statistical distribution.

So in this case its n−p−1 because:

n is number of training samples. p is number of predictors. 1 is for intercept.