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The indirect effect of an exposure on an outcome through an intermediatevariable can be identified by a product of regression coefficients undercertain causal and regression modeling assumptions. Thus, the null hypothesisof no indirect effect is a composite null hypothesis, as the null holds ifeither regression coefficient is zero. A consequence is that existinghypothesis tests are either severely underpowered near the origin (i.e., whenboth coefficients are small with respect to standard errors) or do not preservetype 1 error uniformly over the null hypothesis space. We propose hypothesistests that (i) preserve level alpha type 1 error, (ii) meaningfully improvepower when both true underlying effects are small relative to sample size, and(iii) preserve power when at least one is not. One approach gives a closed-formtest that is minimax optimal with respect to local power over the alternativeparameter space. Another uses sparse linear programming to produce anapproximately optimal test for a Bayes risk criterion. We provide an R packagethat implements the minimax optimal test.

Optimal tests of the composite null hypothesis arising in mediation analysis