  Mediation analysis aims to assess if, and how, a certain exposure influences
an outcome of interest through intermediate variables. This problem has
recently gained a surge of attention due to the tremendous need for such
analyses in scientific fields. Testing for the mediation effect is greatly
challenged by the fact that the underlying null hypothesis (i.e. the absence of
mediation effects) is composite. Most existing mediation tests are overly
conservative and thus underpowered. To overcome this significant methodological
hurdle, we develop an adaptive bootstrap testing framework that can accommodate
different types of composite null hypotheses in the mediation pathway analysis.
Applied to the product of coefficients (PoC) test and the joint significance
(JS) test, our adaptive testing procedures provide type I error control under
the composite null, resulting in much improved statistical power compared to
existing tests. Both theoretical properties and numerical examples of the
proposed methodology are discussed.
