Is a partial F-Test on a model reduced by only one variable valid?

Question

For a recent project, I used multiple linear regression to model data. I attempted to choose between my initial full model and a reduced model by performing a partial F-test. The models used were the following:

Full model: $\hat{Y}$ = $B_0$ + $B_1$$x_1$ + $B_2$$x_2$ + $B_3$$x_3$

Reduced Model: $\hat{Y}$ = $B_0$ + $B_2$$x_2$ + $B_3$$x_3$

The reduced model has only one covariate removed - $x_1$. The partial F-Test returned a F-value of 2.162 and a P-value greater than 0.1, so I did not reject the null hypothesis and concluded that the addition of $x_1$ in the model may not make any significant linear contribution to the prediction of $Y$.

I was later told by my teacher that I should not run a partial F-Test on a reduced model of only one less variable. The teacher said that not only should I not do such a thing, but also that doing so would be wrong. I was told to instead just compare the two models using each model's $R^2$ values, F-values, and T-values.

I can see how the partial F-test in this case might not provide helpful information, since the null hypothesis it tests is that $B_1 = 0$, which is the same thing that the T-test on the parameter $x_1$ tests (in the full model). [Note: I'm not sure what the exact relationship between this T-value and the partial F-test's F-value is.] Nonetheless, why would it would be considered wrong to use and interpret the results from the partial F-test?

Answer 1

I don't know what course you were taking and what qualifications your teacher has but it sounds like you are doing a likelihood ratio test on nested models and you wind up rejecting the higher order model in favor of the reduced model. This is valid as long as the reduced model is nested within the full model. It doesn't matter if the full model has 1 additional variable or 5. It is also appropriate to take the full model and test whether or not B1 is 0. This would be a t test. Both tests should give consistent results.

If your teacher said what you claim then the teacher is wrong.

Answer 2

It is also worth noting here that a partial F-test on a single variable is equivalent to a T-test on that variable. These two tests impose the same evidential ordering and have the same p-value function, so they are the same test. (This result is proved in various advanced regression textbooks --- also see here.) You can easily verify this in your particular case by comparing the p-value of your F-test with the p-value of the T-test for the corresponding term under the full model --- they are the same. Thus, if your teacher is suggesting the T-test as a substitute for what you are doing, then that is merely suggesting an equivalent test.