On the Power of the <i>F</i>-test for Hypotheses in a Linear Model

We improve students’ understanding of the <i>F</i>-test for linear hypotheses in a linear model by explaining elements that affect the power of the test. Including true restrictions in a joint null hypothesis affects test power in a way that is not generally known. Asking a student whether including the true restrictions in the null hypothesis will increase or decrease power, the student is likely to say: “I don’t know.” The student’s answer is not bad because the power depends on the noncentrality parameter and the degrees of freedom. We show that adding true restrictions to a linear hypothesis cannot decrease the noncentrality parameter of the <i>F</i>-statistic, a result many will find counterintuitive. Adding true restrictions can increase or decrease <i>F</i>-test power depending on the offsetting negative effect of reducing the numerator degrees of freedom. We provide illustrative examples of these results and prove them for the general case.