# Multivariate Mediation Effects

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This note is based on Huang, Y.-T. (n.d.). Variance component tests of multivariate mediation effects under composite null hypotheses. Biometrics, 0(0).

**mediation**: the state in which an exposure or an intervention affects an outcome through an intermediate variable, that is, the **mediator**.

- Baron and Kenny (1986): first propose a statistical approach to mediation analysis for a single-mediator model using a difference model
- Rubin (1978): further generalize the theory underlying the mediation analysis through a counterfactural outcome framework (?)
- the counterfactural approach extends the mediation analysis to various nonlinear models for dichotomous outcomes and survival outcomes with censoring
- the mediation analysis has advanced toward the setting with multiple mediators

Motivation of the paper: whether the effect of cigarette smoking on gene expression is mediated by affecting the epigenetic DNA methylation of the gene.

Conduct a gene-centric analysis where analyze one gene at a time and for each gene, set up a multimediator model with

- the exposure: cigarette smoking intensity
- the mediators: DNA methylation markers
- the outcome: mRNA expression of the gene

While literature of causal mediation analyses is more about estimation, this paper focuses on hypothesis testing, which is less studied in causal inference literature but important in application such as genome-wide analyses.

- MacKinnon et al. (2002): conduct comprehensive simulation studies to compare the type I error rate and statistical power of various existing tests for the single-mediator model
- Zhao et al. (2014): propose an efficient testing procedure for the marginal mediation effect and illustrate its genomic application
- Huang and Pan (2016): propose to examine both element-wise and marginal mediation effects when the number of mediators $p$ is larger than the sample size $n$ using a series of single-mediator models
- Barfield et al. (2017): in the setting with a large number of tests, hypothesis tests without considering the composite null are likely to be conservative when the true effect in a study is very sparse. Huang (2019) proposes a procedure to account for the null composition to address the issue.
- However, Huang (2019) focuses on tests with only one mediator, and how to conduct a large number of multimediator tests under the composite null hypothesis remains unclear.
**The paper aims to develop a testing procedure that adjusts for the null composition based on score tests of variance components for the multimediation effect.**

## Causal Mediation Model AND Assumptions

- $S$: smoking pack-years
- $\M$: DNA methylation values of CpG loci within a gene
- $Y$: mRNA expression of the gene
- $\X$: covariates

For a subject $i=1,\ldots,n$, assume the following model for a gene $j$ that contains the DNA methylation marker $k, k=1,\ldots,p_j$, where $p_j$ is the total number of markers within the gene $j$:

where

- $Y_{ij}$ follows a distribution in the exponential family
- $\varepsilon_{Mij} = (\varepsilon_{Mij1},\ldots,\varepsilon_{Mijp_j})^T$ follows a $p_j$ dimension multivariate normal with mean zeros and a covariance $\Sigma_j$.
- $g(\cdot)$: monotone link function

Define

- $Y(s, \m)$: the
**counterfactual outcome (??)**$Y$ has the exposure $S$ and the mediators $\M$ been set to $s$ and $\m$, respectively - $\M(s)$: the counterfactual mediator value had $S$ been set to $s$

Assumptions for identifiability of mediation:

- $Y(s)\ind S\mid \X$: no unmeasured confounding for the association of $Y$ and $S$
- $Y(s, \m)\ind \M\mid (\X, S)$: no unmeasured confounding for the association of $Y$ and $\M$ given $S$
- $M(s)\ind S\mid \X$: no unmeasured confounding for the association of $\M$ and $S$
- $Y(s, \m)\ind \M(s^*)\mid \X$: no $S$-induced confounder for the $\M$-$Y$ association.

Under the above assumptions, the marginal mediation effect of gene $j$ for $S=s_1$ vs. $s_0$ defined as

The null hypothesis, $H_0:\Delta=0$ is equivalent to the composite null hypothesis:

Two special case where the marginal mediation effect $\Delta = 0$ under $\alpha_S\neq 0$ and $\beta_M\neq 0$.

- perfect cancellation of the element-wise mediation effects
- the disjoint effect