On the Optimality of the LR Test for Mediation

Testing for mediation, or indirect effects, is empirically very important in many disciplines. It has two obvious symmetries that the testing procedure should be invariant to. The ordered absolute t-statistics from two ordinary regressions are maximal invariant under the associated groups of transformations. Sobel’s (1982) Wald-type and the LR test statistic are both functions of this maximal invariant and satisfy two logical coherence requirements: (1) size coherence: rejection at level α implies rejection at all higher significance levels; and (2) information coherence: more (less) evidence against the null implies continued (non) rejection of the null. The LR test statistic is simply the smallest of the two absolute t-statistics, and we show that the LR test is the Uniformly Most Powerful (information and size) Coherent Invariant (UMPCI) test. In short: the LR test for mediation is simple and best.