Comment on “On the Power of the F-test for Hypotheses in a Linear Model,” by Griffiths and Hill (2022)

The authors establish and illustrate some relationships among the noncentrality parameters identified with three F tests T 1 , T 2 , and T 3 applicable to a setting where the data consist of the realized values of the elements of an N × 1 random vector y that is distributed as N ( X β , σ 2 I ) (MVN with mean vector X β and variance-covariance matrix σ 2 I ). In what follows, it is shown that these relationships can be established in a relatively simple way that uses only readily available results, that provides insights into the underlying rationale, and that lends itself to some potentially useful extensions. The three F -tests can be regarded as pertaining to a J 1 × 1 vector τ 1 = R 1 β and a J 2 × 1 vector τ 2 = R 2 β formed from J = J 1 + J 2 linearly independent estimable linear combinations of the elements of β : for a J 1 × 1 vector of constants r 1 and a J 2 × 1 vector of constants r 2 , T 1 is a test of the null hypothesis τ = r where τ = ( τ (cid:2) 1 , τ (cid:2) 2 ) (cid:2) and r = ( r (cid:2) 1 , r (cid:2) 2 ) (cid:2) , T 2 is a test of the null hypothesis τ 1 = r 1 (when β is unrestricted), and T 3 is a test of the null hypothesis τ 1 = r 1 when β is subject to the restriction τ 2 = r 2 . The noncentrality parameters identified with T 1 , T 2 , and T 3 are