Abstract: Mediation analysis using structural equation models has become a widely used tool to study whether the effect of an exposure on an outcome is mediated by some intermediate factors. When multiple mediators are present, a statistical inference on the joint mediation effect is challenging because of the composite null hypotheses with a large number of parameter configurations. We propose a simultaneous likelihood ratio test that uses a block coordinate descent algorithm to solve the constrained likelihood under the irregular null parameter space using the Lagrange multiplier approach. We establish the asymptotic null distribution and examine the performance of the proposed joint test statistic using extensive simulations and a comparison with existing tests. The simulation results show that our method controls the type I error properly and, in general, provides better power than that of existing test methods. We apply our method to examine whether a group of glucose metabolites and acetylamino acids mediate the effect of nutrient intakes on insulin resistance.

Key words and phrases: Constrained maximum likelihood, directed acyclic graph, Lagrange multiplier, multi-dimensional mediators, structural equation model.