Our package offers a suite of functions for performing mediation analysis with high-dimensional mediators. Unlike methods for single-mediator mediation analysis—which have been distributed by packages such as “psych,” “mediation,” “medScan”—our package focuses on settings whether there are many potential mediators that need evaluating simultaneously, a topic which has recently become the focus of prolific and exciting methodological work.

You can install hdmed from GitHub with:

```
# install.packages("devtools")
::install_github("dclarkboucher/hdmed") devtools
```

To see how high-dimensional mediation analysis works mathematically, let \(A\) be an exposure, \(Y\) be an outcome, \(\mathbf{C}\) be a set of \(q\) covariates, and \(\mathbf{M}\) be a set of \(p\) potential mediators in the causal pathway between \(A\) and \(Y\). Then, supposing we have data on \(n\) individuals, we can evaluate the mediating role of \(\mathbf{M}\) with the equations

\[ \begin{equation} E[Y_i|A_i,\mathbf{M}_i,\mathbf{C_i}] = \beta_aA_i+\mathbf{\beta_m}^T\mathbf{M_i} + \mathbf{\beta_c}^T\mathbf{C_i} \end{equation} \]

and

\[ \begin{equation} E[\mathbf{M_i}|A_i,\mathbf{C_i}] =\mathbf{\alpha_a}A_i + \mathbf{\alpha_c}\mathbf{C_i}\text{,} \end{equation} \]

where the first equation is the **outcome model** and
the second equation is the **mediator model**. In the
outcome model, our primary estimands are \(\beta_a\)— the direct effect of the
exposure on the outcome independent of \(\mathbf{M}\)— and \(\mathbf{\beta_m}\), a \(p\)-vector of the association between each
mediator and \(Y\) given \(A\) and \(\mathbf{C}\). (Unlike many methods common
to single-mediator analysis, all the methods included in our package
assume there is no interaction effect between \(\mathbf{M}\) and \(A\) on \(Y\).) Likewise, our primary concern in the
mediator model is \(\mathbf{\alpha_a}\), which is a \(p\)-vector of the conditional associations
between each mediator and the exposure given \(\mathbf{C}\). As for the other
coefficients, \(\mathbf{\beta_c}\) is a
\(q\)-vector of the covariate-outcome
effects, and \(\mathbf{\alpha_c}\) is a
\(p\times q\) matrix of
covariate-mediator associations.

Once the outcome and mediator models have been fitted, mediation analysis can be performed by assessing their estimated coefficients. The chief quantities of interest are:

\(\mathbf{\alpha_a}^T \mathbf{\beta_m}\), the

**global mediation effect**of \(A\) on \(Y\) through \(M\);\(\beta_a\), the

**direct effect**of \(A\) on \(Y\);\(\mathbf{\alpha_a}^T \mathbf{\beta_m} + \beta_a\), the

**total effect**of \(A\) on \(Y\); and\(\frac{\mathbf{\alpha_a}^T \mathbf{\beta_m}}{\mathbf{\alpha_a}^T \mathbf{\beta_m}+\beta_a}\), the proportion of the total effect due to mediation (referred to as the

**proportion mediated**.)

All the methods provided by our package can fit this model except for
HDMM (`mediate_hdmm`

) and LVMA (`mediate_lvma`

),
which instead of the typical model assumptions, assume the mediation
between \(A\) and \(Y\) is transmitted by unmeasured latent
variables. (See the documentation of those functions for more detail.)
The other methods produce, at the very least, estimates of the direct
effect, global mediation effect, and total effect, making them suitable
for performing mediation analysis with the standard assumptions.
Moreover, in the case of BSLMM (`mediate_bslmm`

), HIMA
(`mediate_hima`

), HDMA (`mediate_hdma`

), MedFix
(`mediate_medfix`

), and Pathway LASSO
(`mediate_pathway_lasso`

), we also report estimates of the
**mediation contributions**, which are the contributions
\((\mathbf{\alpha_a})_j(\mathbf{\beta_m})_j\)
of each mediator to \(\mathbf{\alpha_a}^T
\mathbf{\beta_m}\), \(j\) from
\(1\) to \(p\). Though useful for identifying
potentially important mediators, we stress that these contributions
*generally cannot be interpreted as causal effects unless the
mediators are* *independent conditional on* \(A\) *and* \(\mathbf{C}\). Conditions for when \(\mathbf{\alpha_a}^T \mathbf{\beta_m}\) and
\(\beta_a\) can be interpreted causally
are laid out by Song et al. (2019) (see `mediate_bslmm`

for
complete reference). Note also that, as programmed, the methods HIMA
(`mediate_hima`

), HDMA (`mediate_hdma`

), MedFix
(`mediate_medfix`

), and BSLMM allow one to incorporate a
small number of covariates directly, as specified in the above pair of
models, whereas the other as programmed methods do not. If you are
interested in adjusting for covariates with a method that does not allow
them to be inputted to our mediation function directly, consider
regressing those covariates out of the outcome, mediators, and exposures
in advance, when doing so is appropriate. In addition, most functions in
our package assume that the outcome variable is continuous; however,
HIMA and HDMA have options for fitting a binary outcome model with a
standard logistic link.

The `med_dat`

object provided by our package contains a
simple toy dataset for practicing high-dimensional mediation (though in
this case, we are using “high-dimensional” generously, as the dataset
contains only 20 mediators to its 100 observations).

Let us take a look at the data. In `Y`

we have the
outcome, in `A`

we have the exposure, and in `M`

we have a named matrix of mediators.

```
library(hdmed)
# Process data
<- med_dat$Y
Y <- med_dat$M
M <- med_dat$A
A
str(M)
#> num [1:100, 1:20] -0.491 1.339 -0.194 -0.218 -0.108 ...
#> - attr(*, "dimnames")=List of 2
#> ..$ : NULL
#> ..$ : chr [1:20] "m1" "m2" "m3" "m4" ...
```

Now we will perform mediation analysis. For a simple, fast mediation
method we will use the method “high-dimensional mediation analysis” by
Zhang et al. (2016), which we call “HIMA”. HIMA is a straightforward
method that fits the mediator models using ordinary least squares and
the outcome model using penalized regression with the minimax concave
penalty. We don’t have covariates to include, so to use the default
options, we input only `A`

, `M`

, and
`Y`

.

`<- mediate_hima(A, M, Y) hima_out `

Next let’s look at the mediation contributions, which are located in
the `contributions`

table. In this case, the function only
returned one mediator, which happens if the others have an estimated
contribution of zero and do not contribute to the estimated global
mediation effect. Examining the table further, we see `alpha`

as a shorthand for \((\mathbf{\alpha_a})_j\), `beta`

as a shorthand for \((\mathbf{\beta_m})_j\), and
`alpha_beta`

as a shorthand for \((\mathbf{\alpha_a})_j(\mathbf{\beta_m})_j\).
Notice that `ab_pv`

is the \((\mathbf{\alpha_a})_j(\mathbf{\beta_m})_j\)
p-value.

```
$contributions
hima_out#> mediator alpha alpha_pv beta beta_pv alpha_beta ab_pv
#> 1 m3 -0.2737383 0.01438887 0.6040214 1.364285e-07 -0.1653438 0.01438887
```

Finally, the estimated mediation effects are reported in
`effects`

table, which includes the indirect effect (the
global mediation effect), the direct effect, and the total effect. In
theory, one can use these estimates to report the proportion mediated,
as described above, but since the proportion mediated is generally only
useful when \(\mathbf{\alpha_a}^T
\mathbf{\beta_m}\) and \(beta_m\) have the same sign, we will not do
so here.

```
$effects
hima_out#> effect estimate
#> 1 indirect -0.16534380
#> 2 direct 0.01018211
#> 3 total -0.15516169
```

This package serves as companion code for our paper, “Methods for Mediation Analysis with High-Dimensional DNA Methylation Data: Possible Choices and Comparison.” To give our work proper credit, use the citation provided below:

Dylan Clark-Boucher, Xiang Zhou, Jiacong Du, Yongmei Liu, Belinda L Needham, Jennifer A Smith, Bhramar Mukherjee (2023). Methods for Mediation Analysis with High-Dimensional DNA Methylation Data: Possible Choices and Comparison. medRxiv 2023.02.10.23285764