Abstract
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.
Highlights
Testing for mediation is empirically extremely important in many scientific disciplines
P[CRLR(α) | Ha] is larger than for any other critical region (CR)(α) ∈ Cα that is not equal to CRLR(α) a.s. This holds uniformly for all values of μ under H0. This optimality property of the Likelihood Ratio (LR) test is derived under coherence requirements that are very weak: it seems more than reasonable to require that any test continues to reject if more extreme outcomes are observed or if the level of the test is increased
We have developed a coherence framework to formulate and analyze the requirement that increasing or decreasing information against the null leads to coherent decisions
Summary
Testing for mediation is empirically extremely important in many scientific disciplines. The test can be expressed in terms of the absolute values of ordinary t-statistics (|T1|, |T2|) in (1) and (2) This renders a test that is invariant to the parameter value β and to the variances of u and v. That a uniformly most powerful test exists within a class of procedures that are information coherent and invariant. Information coherence is the logical requirement introduced, that, when information against the null increases, a test should continue to reject It differs from the common size coherence that requires the same when the size (maximum probability of a Type I error) is increased. As a result the standard t-statistics, T1 and T2 for θ1 and θ2 are asymptotically independent and normally distributed
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