Modular software for mediation analysis of microbiome data

# Modular Software for Mediation Analysis of Microbiome Data

<div id="subtitle_left">
Slides: <a href="https://go.wisc.edu/5ms623">go.wisc.edu/5ms623</a><br/>
Paper: <a href="https://go.wisc.edu/ebm917">go.wisc.edu/ebm917</a><br/>
Lab: <a href="https://measurement-and-microbes.org">measurement-and-microbes.org</a> <br/>
</div>
<div id="subtitle_right">
Kris Sankaran <br/>
<a href="https://www.birs.ca/events/2025/5-day-workshops/25w5324/">Novel Approaches for Multi-Omics</a><br/>
17 | July | 2025 <br/>
</div>

`\(\def\Gsn{\mathcal{N}}\)`
`\(\def\Dir{\text{Dir}}\)`
`\(\def\Mult{\text{Mult}}\)`
`\(\def\diag{\text{diag}}\)`
`\(\def\*#1{\mathbf{#1}}\)`
`\(\def\Scal{\mathcal{S}}\)`
`\(\def\exp#1{\text{exp}\left(#1\right)}\)`
`\(\def\logit#1{\text{logit}\left(#1\right)}\)`
`\(\def\absarg#1{\left|#1\right|}\)`
`\(\def\E{\mathbb{E}} % Expectation symbol\)`
`\(\def\Earg#1{\E\left[{#1}\right]}\)`
`\(\def\P{\mathbb{P}} % Expectation symbol\)`
`\(\def\Parg#1{\P\left[{#1}\right]}\)`
`\(\def\m#1{\boldsymbol{#1}}\)`
`\(\def\Unif{\text{Unif}}\)`
`\(\def\win{\tilde{w}_{\text{in}}}\)`
`\(\def\reals{\mathbb{R}}\)`
`\(\newcommand{\wout}{\tilde w_{\text{out}}}\)`

---

### Microbiome as Mediator

We're just beginning to understand how the microbiome mediates the relationship
between environmental exposures and human health.

.center[
<div class="caption">
<img src="figures/chemotherapy.png"/>
Figure adapted from [1].
</div>
]

Chemotherapy-induced microbiome changes can worsen adverse events, like
diarrhea and mucositis, limiting treatment dosage and duration 
[2; 3].

---

### Alternative Mediators

More generally, for microbiome research that seeks to explain mechanisms, it is
often necessary to analyze indirect effects.

.center[
  <div class="caption">
<img src="figures/akkermansia_cholesterol.jpg" width=540/><br/>
Figure adapted from [4].
</div>
]

For example, _Akkermansia muciniphila_ can help regulate cholesterol levels by producing
proteins that activate important pathways for cholesterol absorption and
metabolism [5].

---

### Causal Inference Setup

Causal mediation analysis is often a good match for carrying out these studies 
[6; 7].  Traditional methods
from social science and epidemiology need to be adapted to reflect properties of
microbiome data 
[8; 9; 10].

---

### Multiple Mediators

In most multi-omics applications, the mediators are high-dimensional. Each might
represent a gene, methylation site, or microbial taxon, for example.

---

### Two-Step Linear Regression

During estimation, we might add a feature selection step/penalty to ensure that
most mediators do not play a role for the outcome.

`\begin{align*}
M&=\alpha T+\zeta_M^{\top} X+E\\
Y&=\eta T+\beta^{\top} M+\zeta_Y^{\top} X+\epsilon
\end{align*}`

* `\(\eta \in \mathbb{R}\)`: Direct effect.
* `\(\alpha^\top \beta\)`: Overall indirect effect.
* `\(\alpha_{k}\beta_{k}\)`: The indirect effect through path `\(k\)`.

What if we want to go beyond a linear model?

---

### Counterfactual Notation

Under the counterfactual causal mediation analysis framework 
[11; 12], we imagine
counterfactuals for both mediators and outcomes, depending on treatments that we
may never have seen.

_Mediator under treatment `\(t\)`_: 
`\begin{align*}
M\left(t\right)
\end{align*}`

_Outcome under treatment combination_ `\(\left(t, t'\right)\)`:
`\begin{align*}
Y\left(t, M\left(t'\right)\right)
\end{align*}`

Note that this is not observable whenever `\(t \neq t'\)`, but we can reason about
it abstractly.

---

### Direct vs. Indirect Effects

Suppose that the treatments belong to two groups, `\(T \in \{0, 1\}\)`. Then, direct
and indirect effects are defined as:
`\begin{align*}
\frac{1}{2} \sum_{t'=0}^{1} \mathbb{E}\left[Y(1, M(t')) - Y(0, M(t'))\right] \text{   (Direct Effect)}\\
\frac{1}{2} \sum_{t=0}^{1} \mathbb{E}\left[Y(t, M(1)) - Y(t, M(0))\right] \text{   (Indirect Effect)}
\end{align*}`

We may also want to know the pathwise indirect effect through mediator `\(k\)`:
`\begin{align*}
\frac{1}{2} \sum_{t'=0}^{1} \mathbb{E}\left[Y(t', M_k(1), M_{-k}(t')) - Y(t', M_k(0), M_{-k}(t'))\right]
\end{align*}`

---

### Geometric Interpretation

---

### Geometric Interpretation

---

### Geometric Interpretation

---

### Identification

We need to generalize the ignorability to support causal identification of these
three effects. The complete required conditions are given in the backup slides.

---

### Integration Challenge

Our interest in this project came from a re-analysis of a gut-brain axis
microbiome study. Participants were randomized into either a mindfulness
intervention or a control group. In this case, the the taxonomic community
composition is an outcome, and the survey measurements are mediators.

---

### Integration Challenge

We can also use mediation analysis to guide microbiome-metabolome integration.
This this quote comes from a study of inflammatory bowel disease (IBD) 
[13]:

<div style="font-size: 20px">
> The CD-associated compounds eicosatrienoic (ETA) and docosapentaenoic (DPA)
acid were involved in negative associations with control-associated species and
positive associations with IBD-associated species. ETA and DPA are
polyunsaturated long-chain fatty acids (PUFAs) [which] possess bactericidal
activity by virtue of their hydrophobic nature and potential to disrupt
bacterial cell membranes. 
</div>

As in the previous example, we can imagine a mediation analysis with a
high-dimensional microbiome outcome.

---

### Multiple Mediators and Outcomes

These applications require multivariate outcomes. The effect definitions
directly generalize to individual outcomes `\(Y_{j}\left(t\right)\)`.
.center[
<img src="figures/multivariate-mediation-outcomes.png" width=600/>
]

---

---

### Univariate Mediation Analysis

In the case of univariate mediators and outcomes, we have simple expressions for
the two-step regression:

`\begin{align*}
m_{i} = \alpha t_i + \epsilon_{i}^{m_{i}}\\
y_i = \eta t_{i} + \beta m_i + \epsilon_i^y
\end{align*}`

In this case, the direct and indirect effects reduce to:

- Direct effect: `\(\eta\)`
- Indirect effect: `\(\alpha \beta\)`

---

### Code Interface

We could imagine writing a function to implement this analysis.

``` r
model <- mediation(y, m, t)
effects(model)
```
]

But how can we adopt the more general counterfactual perspective, without
restricting ourselves to linear models?

---

### Code Interface

The `mediation` package [11; 12]
solves this problem by instead creating an interface like:

``` r
f1 <- lm(y ~ t + m)
f2 <- lm(m ~ t)
model <- mediation(f1, f2)
effects(model)
```
]

We can now accommodate any mediation or outcome model, not just linear ones!
Indeed, it's easy to use `glm` in the `mediation` package.

---

### Multimedia Interface

`multimedia` generalizes this interface for multivariate responses and outcomes
appropriate to microbiome data. For example, for the mindfulness study, we use a
logistic-normal multinomial model 
[14; 15] for the outcomes
and lasso regression [16] for the
mediators.

``` r
model <- multimedia(
    exper, # dataset
    lnm_model(), # outcome model
    glmnet_model(lambda = 0.5, alpha = 0) # mediation model
)
```
]

We fit each model separately, but this can still accommodate a few commonly used
approaches.

---

### Available Models

The LNM model is designed to have multivariate responses. The remaining options
apply the same type of model in parallel across all outcomes.

1. `lnm_model`: A logistic normal multinomial model [17].

1. `glmnet_model`: A call to `glmnet` for lasso or elastic net regression 
[18].

1. `rf_model`: A call to `ranger` for random forest regression 
[19].

1. `brms_model`: A call to `brms`, which can be customized likelihoods for
`lognormal()`, `hurdle_negbinomial()`, `cox()`, etc. [20].

---

### Data Formats

Besides modeling, `multimedia` has data structures that make it easier to
manipulate real and counterfactual data. For example, we can use `tidy`
selection syntax [21] to categorize variables.

``` r
head(ibd_data)
```

```
## # A tibble: 6 × 341
##   m0031_phenyllactate m0045_azelate m0171_palmitic_acid m0219_2_hydroxyhexadecanoate m0244_alpha_linolenic_a…¹ m0246_linoleic_acid m0250_oleic_acid m0256_stearic_acid
##                 <dbl>         <dbl>               <dbl>                        <dbl>                     <dbl>               <dbl>            <dbl>              <dbl>
## 1                9.44          7.18                8.26                        10.0                       7.02                8.86             9.28               8.13
## 2                8.92          6.47                8.19                         9.57                      6.29               10.4             11.1                9.10
## 3                6.36          8.16                7.39                        10.4                       7.49                8.88             8.65               7.60
## 4                8.24          5.46                9.44                         9.83                      5.78                8.53             8.68               8.83
## 5                9.73          6.50                7.69                         9.03                      5.73               10.4             10.8                6.90
## 6                0             4.44                7.47                         5.65                      5.50                9.03             9.73               7.44
## # ℹ abbreviated name: ¹m0244_alpha_linolenic_acid
## # ℹ 333 more variables: m0354_arachidonic_acid <dbl>, m0393_1213_diHOME <dbl>, m0472_docosahexaenoic_acid <dbl>, m0731_lithocholic_acid <dbl>,
## #   m0830_hyodeoxycholateursodeoxycholate <dbl>, m1129_chenodeoxycholate <dbl>, m1360_chenodeoxycholatedeoxycholate <dbl>, m0833_deoxycholic_acid <dbl>,
## #   m0900_ketodeoxycholate <dbl>, m0930_cholate <dbl>, m1258_cholate <dbl>, m1303_lithocholate <dbl>, m1403_cholate <dbl>, m0253_sphingosine <dbl>,
## #   m0906_threosphingosine <dbl>, m0998_threosphingosine <dbl>, m0927_piperine <dbl>, m0264_linoleoyl_ethanolamide <dbl>, m0339_linoleoyl_ethanolamide <dbl>,
## #   m0410_C180e_MAG <dbl>, m0416_cholesterol <dbl>, m0467_C180_MAG <dbl>, m0478_cholestenone <dbl>, m0554_cholestenone <dbl>, m0878_C160_LPC <dbl>,
## #   m0915_C200_LPE <dbl>, m0949_C180_LPC <dbl>, m1271_C342_DAG <dbl>, m1291_C342_DAG <dbl>, m1277_C341_DAG <dbl>, m1351_C364_DAG <dbl>, m1367_C364_DAG <dbl>, …
```

These are the data from the motivating IBD study as a raw `data.frame`.

---

### Data Formats

Besides modeling, `multimedia` has data structures that make it easier to
manipulate real and counterfactual data. For example, we can use `tidy`
selection syntax [21] to categorize variables.

``` r
library(multimedia)
mediation_data(
  ibd_data,
  matches("^m[0-9]{4}"), # outcomes
  "Study.Group", # treatments
  starts_with("g") # mediators
)
```

```
## [Mediation Data] 
## 220 samples with measurements for, 
## 1 treatment: Study.Group 
## 173 mediators: gSutterella, gMarvinbryantia, ... 
## 155 outcomes: m0031_phenyllactate, m0045_azelate, ...
```

---

### Bootstrap

We can gauge uncertainty in the estimated effects by re-estimating both mediator
and outcome models on bootstrap resampled versions of the original data. Here
are the metabolites with the largest indirect and direct effects in the IBD
example. 
.center[
<img src="figures/lasso_bootstrap_vis-1.svg" width=900/>
]

---

### Model Alteration

The package provides syntax for altering models after they've been estimated.
We can set specific coefficients to zero or re-estimate with new model
specifications.

---

``` r
fit <- multimedia(exper, outcome = rf_model(num.trees = 1e3)) |>
    estimate(exper)

altered_m <- nullify(fit, "T->M") |>
    estimate(exper)
altered_ty <- nullify(fit, "T->Y") |>
    estimate(exper)
```
.center[
<img src="figures/visualize_long-1.png"/>
]

---

### Synthetic Null Mediators

Here is the same principle applied to the mindfulness study. The synthetic null
survey responses have been generated from mediation models with `\(\alpha = 0\)`.

.center[
<img src="figures/mindfulness-altered.png" width=780/><br/>
<span style="font-size: 24px;">
The middle panel comes from a synthetic null: `\(T \nrightarrow M \to Y\)`.
</span>
]

---

.pull-three-quarters-right[
These are analogous comparisons for the simulated microbiomes. Sampling from the
fitted mediation models helps with model checking.
]

---

### Sensitivity Analysis

Sensitivity analysis can show which conclusions might be changed if any
identification assumptions are violated (listed in the appendix).  For example,
once we have fit mediation and outcome models, we can simulate according to:

`\begin{align*}
Y^*(t, m)=\hat{Y}(t, m)+\epsilon^y \\
M^*(t)=\hat{M}(t)+\epsilon^m .
\end{align*}`

A caveat is that this only makes sense for continuous outcomes and mediators.

---

### Sensitivity Analysis

We draw `\(\left(\epsilon^y, \epsilon^m\right)\)` from a Gaussian with mean zero and covariance,
`\begin{align*}
\Sigma(\rho, G):=\left(\begin{array}{cc}
\operatorname{diag}\left(\hat{\sigma}_M^2\right) & \rho \hat{\sigma}_M \hat{\sigma}_Y^{\top} \odot \mathbf{1}_G \\
\rho \hat{\sigma}_Y \hat{\sigma}_M^{\top} \odot \mathbf{1}_G^{\top} & \operatorname{diag}\left(\hat{\sigma}_Y^2\right)
\end{array}\right)
\end{align*}`
where `\(\mathbf{1}_{G} \in \{0, 1\}^{K \times J}\)` is an indicator over
mediator-outcome pairs `\(G\)` over which to test sensitivity.

---

---

### Study Background

1. The study [13] carried out an integrative analysis
to discover metabolite and taxonomic markers with diagnostic or therapeutic
potential for IBD.

1. They gathered untargeted metabolomics + whole genome sequencing microbiome
data on a cohort of 220 patients with the disease.

1. This resulted in 8.8K and 11.7K metabolite and genus features, respectively,
all available on the 
[Curated Metabolome-Microbiome repository](https://github.com/borenstein-lab/microbiome-metabolome-curated-data/tree/main)
[22].

---

### Model Setup

1. We applied centered log-ratio and `\(\log\left(1 + x\right)\)` transformations to
the microbiome and metabolome data, respectively. Both were then filtered to
between 150 - 200 of the most abundant features.

1. We treat the microbiome as the mediators and the metabolome as the outcome,
following the discussion from our quote above.

1. Taxonomic composition depends only on treatment status. Each metabolite's
abundance is treated as a sparse linear function of a few microbes.

``` r
model <- multimedia(exper, glmnet_model(lambda = 0.1))
```
]

---

### Direct Effects

.pull-left[
These are the metabolites with the largest direct effects from our bootstrap
analysis. Their variation in abundance can't be attributed to microbiome
community changes.
]

---

### Indirect Effects

This multidimensional scaling (MDS) is based on only microbiome data. Point size
reflects metabolite abundances. Associations between metabolites and the MDS
appear only for outcomes with large indirect effects.

---

### Pathwise Indirect Effects

The top pathwise indirect effects are similar to the geometric interpretation we
saw in the introduction!

---

### Sensitivity Analysis

In this sensitivity analysis, we simulate unmeasured confounding between the
abundances of the _Enterocloster_ genus (mediator) and the metabolites
(outcomes) hydrocinnamici acid, lithocholate, and arginine.

---

### Takeaways

1. Mediation analysis is a useful framework for relating environment, host, and
microbiome features. More generally, counterfactual language can guide
multi-omics data integration.

1. Modular software design can ensure that a few core statistical components can
be applied to a wide range of study applications.

---

### Thank you!

Paper: [go.wisc.edu/ebm917](https://go.wisc.edu/ebm917)

Package: [go.wisc.edu/830110](https://go.wisc.edu/830110) (also on CRAN)

* Contact: ksankaran@wisc.edu
* Lab Members: Margaret Thairu, Shuchen Yan, Yuliang Peng, Langtian Ma, Helena Huang
* Funding: NIGMS R01GM152744, NIAID R01AI184095, Gates 072185

---

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---

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---

### Example Model: Logistic-Normal Multinomial

The logistic-normal multinomial (LNM) model has the form:

.pull-left[
`\begin{align*}
y_{i} \sim \Mult\left(N_{i}, \varphi^{-1}\left(z_{i}^{T}\beta\right)\right) \\
\beta \sim \Gsn\left(0, \diag\left(\sigma_{k}^{2}\right)\right)
\end{align*}`
where `\(\varphi^{-1}\left(z\right) \propto\left(\exp{z_{1}}, \dots, \exp{z_{K-1}}, 1\right)\)`
]

---

### Identification of Overall Direct/Indirect Effects

These sequential ignorability assumptions are sufficient for identification of
overall direct and indirect effects [11]. For any 
`\(t, t', x\)`, we require,

1. Treatment ignorability: `\(\left\{Y\left(t^{\prime}, m\right), M(t)\right\} \perp T \mid X=x\)`
1. Mediator ignorability: `\(Y\left(t^{\prime}, m\right) \perp M(t) \mid T=t, X=x\)`
1. Positivity: `\(\mathbb{P}(T=t \mid X=x)>0\)`
1. Positivity for mediator: `\(p_{M(t)}(m \mid T=t, X=x)>0\)`

---

### Identification of Pathwise Indirect Effects

Pathwise indirect effects require a generalization of sequential ignorability
assumptions [23; 24]. For any 
`\(t, t', t'', m, x, w\)` we require,

1. Treatment ignorability: `\(\left\{Y(t, m, w), M_k\left(t^{\prime}\right), M_{-k}\left(t^{\prime \prime}\right)\right\} \perp T \mid X=x\)`
1. Mediator ignorability: `\(Y\left(t^{\prime}, m, M_{-k}\left(t^{\prime}\right)\right) \perp M_k \mid T=t, X=x\)`
1. Mediator `\(k\)` ignorability: `\(Y\left(t^{\prime}, M_k\left(t^{\prime}\right), w\right) \perp M_{-k} \mid T=t, X=x\)`
1. Positivity: `\(\mathbb{P}(T=t \mid X=x)>0\)`
1. Mediator positivity: `\(p_{\left(M_k t, M_{-k}(t)\right)}(m, w \mid T=t, X=x)>0\)`

---

### Alternative: Hurdle Model

The interface makes it easy to swap in new models. For example, here we replaced
the lasso regression on log-normalized metabolite abundances with a hurdle model
that directly models zero-inflated nonnegative data.

``` r
model <- multimedia(
    exper2, # version without log transformation
    brms_model(family = hurdle_lognormal()) # new outcome model
) |>
    estimate(exper2)
```
]

---

### Alternative: Hurdle Model

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### Identification