# Thread: Multiple linear regression: partial F-test

1. ## Multiple linear regression: partial F-test

"Suppose that in a MULTIPLE linear regression analysis, it is of interest to compare a model with 3 independent variables to a model with the same response varaible and these same 3 independent variables plus 2 additional independent variables.
As more predictors are added to the model, the coefficient of multiple determination (R^2) will increase, so the model with 5 predicator variables will have a higher R^2.
The partial F-test for the coefficients of the 2 additional predictor variables (H_o: β_4=β_5=0) is equivalent to testing that the increase in R^2 is statistically signifcant."

I don't understand the bolded sentence. Why are they equivalent?

Thanks for explaining!

[also under discussion in Talk Stats forum]

2. Originally Posted by kingwinner
"Suppose that in a MULTIPLE linear regression analysis, it is of interest to compare a model with 3 independent variables to a model with the same response varaible and these same 3 independent variables plus 2 additional independent variables.
As more predictors are added to the model, the coefficient of multiple determination (R^2) will increase, so the model with 5 predicator variables will have a higher R^2.
The partial F-test for the coefficients of the 2 additional predictor variables (H_o: β_4=β_5=0) is equivalent to testing that the increase in R^2 is statistically signifcant."

I don't understand the bolded sentence. Why are they equivalent?

Thanks for explaining!

[also under discussion in Talk Stats forum]

It's a test to see if the variables associated with $\beta_4$ and $\beta_5$ are of any value.
By that we mean, do these x's help in explaining y any better than the first three terms in your model.
Note that $R^2$ increases as we introduce more terms, but the mean squares may not.
The mean squares, in particular MSE is SSE divided by n minus the number of parameters in the model.
SSE will decrease as you add terms, but so will the denominator of MSE.