Variance component tests of multivariate mediation effects under composite null hypotheses

Mediation effects of multiple mediators are determined by two associations: one between an exposure and mediators ( $$ - $$ ) and the other between the mediators and an outcome conditional on the exposure ( $$ - $$ ). The test for mediation effects is conducted under a composite null hypothesis, that is, either one of the $$ - $$ and $$ - $$ associations is zero or both are zeros. Without accounting for the composite null, the type 1 error rate within a study containing a large number of multimediator tests may be much less than the expected. We propose a novel test to address the issue. For each mediation test $$ , $$ , we examine the $$ - $$ and $$ - $$ associations using two separate variance component tests. Assuming a zero-mean working distribution with a common variance for the element-wise $$ - $$ (and $$ - $$ ) associations, score tests for the variance components are constructed. We transform the test statistics into two normally distributed statistics under the null. Using a recently developed result, we conduct $$ hypothesis tests accounting for the composite null hypothesis by adjusting for the variances of the normally distributed statistics for the $$ - $$ and $$ - $$ associations. Advantages of the proposed test over other methods are illustrated in simulation studies and a data application where we analyze lung cancer data from The Cancer Genome Atlas to investigate the smoking effect on gene expression through DNA methylation in 15 114 genes.