A Nearly Similar Powerful Test for Mediation

We derive a new powerful, simple test for mediation. Testing for mediation is empirically important and the no-mediation hypothesis $H0:θ1θ2=0$ poses a theoretical problem, nonregular at the origin, where standard tests have extremely low rejection probabilities. We prove that a similar test only exists if $1/α∈N$⁠. It is unique, but with unfortunate properties. An innovative varying-g method is introduced to derive a nearly similar test, $gopt$⁠, with power close to the envelope without these undesirable properties, and a second, less similar test, $g1.2$⁠, with more empirical appeal. Results are illustrated in an educational setting.