Generative Models for Microbiome Mediation Analysis

class: title
background-image: url("figures/interactive-screenshot.png")
background-size: cover

`\(\def\Gsn{\mathcal{N}}\)`
`\(\def\Mult{\text{Mult}}\)`
`\(\def\diag{\text{diag}}\)`
`\(\def\*#1{\mathbf{#1}}\)`
`\(\def\Scal{\mathcal{S}}\)`
`\(\def\exp#1{\text{exp}\left(#1\right)}\)`
`\(\def\logit#1{\text{logit}\left(#1\right)}\)`
`\(\def\absarg#1{\left|#1\right|}\)`
`\(\def\E{\mathbb{E}} % Expectation symbol\)`
`\(\def\Earg#1{\E\left[{#1}\right]}\)`
`\(\def\P{\mathbb{P}} % Expectation symbol\)`
`\(\def\Parg#1{\P\left[{#1}\right]}\)`

.center[
<br/>
# Generative Models for Microbiome <br/> Mediation Analysis
<br/>
<br/>
<br/>
<br/>
]

#### CMStatistics 2022

.large[
Joint work with Xinran Miao and Hanying Jiang <br/>
Kris Sankaran | [krisrs1128.github.io/LSLab](krisrs1128.github.io/LSLab) | 17 December 2022
<br/>
]

---

### Microbiome and Meditation

1. A microbiome is a microbe-scale ecosystem. It can be described by taxonomic
composition, genomic function, and biochemical environment.
 
1. There are known relationships with anxiety and depression [1; 2] which seem to work through the Gut-Brain Axis [3; 4; 5].

1. Meditation is known to induce a variety of physiological changes, so it is
natural to ask whether there is a relationship with the microbiome.

.center[
<img src="figures/microbe-mind.svg" width=420/>
]

---

### Psychometric & Genomic Integration

1. The Center for Healthy Minds designed a longitudinal study to evaluate the
relationship. The treatment group received meditation training.
1. The study generated psychometric and microbiome data, and
ongoing collection is gathering additional immunological and behavioral data
2. Changes in one might be associated with effects across all. To approach this, we use the language of graphical modeling.

---

### Mediation Analysis

Mediation models are a type of graphical model where a treatment `\(T\)` can
influence a response `\(Y\)` either directly or indirectly through a mediator
variable `\(M\)`. This is formalized through a series of chained regression models,

`\begin{align*}
M &= \alpha_{0} + \alpha_{T}T + \alpha_{X}^{T}X + \varepsilon^{M} \text{ (mediation model)}\\
Y &= \beta_{0} + \beta_{T}T + \beta_{X}^{T}X + \beta_{M}^{T}M + \varepsilon^{Y} \text{ (outcome model) }.
\end{align*}`

---

### Counterfactual Perspective

* Typically the direct and indirect effects are read off `\(\alpha_{T}\)` and
`\(\beta_{T}\)`.
* A more general approach considers the counterfactual difference in potential outcomes [6; 7],
`\begin{align}
\tau\left(t'\right) &= \Earg{Y\left(X, 1, M\left(X, t'\right)\right) - Y\left(X,0, M\left(X, t'\right)\right)},\\
\delta\left(t'\right) &= \Earg{Y\left(X, t', M\left(X, 1\right)\right) - Y\left(X, t', M\left(X, 0\right)\right)},
\end{align}`
  where the expectation is over draws `\(X\)`, `\(\varepsilon^{M}\)`, and
  `\(\varepsilon^{Y}\)` from the population.
* `\(\tau\)` and `\(\delta\)` are viewed as intervening on the treatment and mediator,
respectively

---
### Counterfactual Perspective

For example, if there is no mediation effect, `\(M\)` is unaffected by the
treatment. Nonetheless, there can still be a large direct effect.

---
### Counterfactual Perspective

Alternatively, the treatment can influence the response entirely by changing the
typical value of the mediator.

---
### Counterfactual Perspective

Both types of effects can exist simultaneously.

---

### Modeling Components

1. We used the Stan probabilistic programming language to implement a variety of
microbiome-specific mediation model components:

* Logistic normal multinomial
  * Zero-inflation via mixtures
  * Change from baseline
  * Latent factors

1. These can be easily recombined (e.g., LNM-Mixture-Factor) because the code is
dynamically generated.

---

### Logistic Normal Multinomial

All our models are variants of the Logistic Normal Multinomial (LNM),

.pull-left[
`\begin{align*}
Y &\sim \Mult\left(\text{Depth}, \varphi^{-1}\left(X^{T}\beta\right)\right) \\
\beta &\sim \Gsn\left(0, \diag\left(\sigma_{k}^{2}\right)\right)
\end{align*}`
where `\(\varphi^{-1}\left(z\right) \propto\left(\exp{z_{1}}, \dots, \exp{z_{K-1}}, 1\right)\)` and Depth denotes sequencing depth.
]

.pull-right[
<img src="figures/lnm.svg" style="display: block; margin: auto;" />
]

---

### Logistic Normal Multinomial Mediation

We incorporate the mediator path in an LNM model.

`\begin{align*}
Y &\sim \Mult\left(\text{Depth}, \varphi^{-1}\left(\mu\right)\right) \\
\mu &= \beta_0 + \beta_T T + \beta_X^T X + \beta_M^T M + \varepsilon^{\eta} \\ 
M &= \alpha_0 + \alpha_T T + \alpha_X^T X + \varepsilon^m\\
\end{align*}`

---

### Model Comparison

Rather than describing individual models, I would like to focus on model
comparison, because standard approaches are not satisfactory,

.pull-left[
Prediction performance: Good prediction of future composition doesn’t
guarantee accurate inference of mediation effects.
<img src="figures/prediction_forecasting.png"/>
]

.pull-right[
Traditional Simulation: Simulating from one of the assumed model structures
gives it an unfair advantage.
<img src="figures/basic_simulation.png"/>
]

---

### Zero-Inflated Quantiles (ZINQ)

1. We resolve these difficulties by defining a semisynthetic simulator, following [8].
2. This approach estimates a CDF for each species using,
`\begin{align*}
 \logit{\Parg{Y > 0 \vert X}} = \gamma_{0} + \gamma^{T}X \\
 Q_{Y}\left(\tau \vert X, Y > 0\right) =\xi_{0}\left(\tau\right) + \xi\left(\tau\right)^{T}X
\end{align*}`
    where `\(Q_{Y}\left(\tau \vert X, Y > 0\right)\)` is the conditional `\(\tau^{th}\)` quantile of a nonzero count.

.center[
<img src="figures/zinq.png" width=400/>
]

---

### Simplified Setting

* Before getting to the meditation study, let's see how simulation strategies
compare on a simple setup.
* Consider the problem of evaluating an LNM model. We will compare estimation
quality when we simulate from,
 - The LNM itself
 - A simulator based on a pilot dataset

---

### Synthetic Setup

In the first simulation, we simulate from a version of the LNM,

`\begin{align*}
Y &\sim \Mult\left(N_{i}, \varphi^{-1}\left(\xi_{0} + \xi_{T}T\right)\right) \\
\xi_{T} &:= \text{HardThreshold}\left(\tilde{\xi}_{T}, \text{keep 25%}\right) \\
\xi_{0}, \tilde{\xi}_{T} &\sim \Gsn\left(0, I_{K}\right) \\
\end{align*}`

.center[
<img src="figures/lnm-spherical.png" width=500/>
]

---

### Semisynthetic Setup

In the second, we use the exact same `\(\xi_{T}\)`, but now to exponentially tilt
samples from treatment,

`\begin{align*}
Y \sim \Mult\left(N_{i}, \exp{\xi_{T}T}\odot \hat{p}^{\ast}\right)
\end{align*}`
Here, `\(\hat{p}^{\ast}\)` is drawn randomly with replacement from compositions
in an observed pilot dataset (the meditation study data, in this case).

---

### Simulation Comparison

The purely synthetic simulation setup leads to overoptimistic power and FSR
estimates, compared to the semisynthetic setup.

.center[
<img src="figures/semisynthetic_comparison.png" width=950/>
]

---

### Graphical ZINQ

1. We can adapt this to the graphical model setting by estimating nonparametric relationships across edges.
2. We can estimate ground truth direct and indirect effects by simulating from known `\(\gamma\)` and `\(\xi\left(\tau\right)\)`.

.center[
<img src="figures/mediation-assumptions2.svg" width=570/>
]

---

### Graphical ZINQ

.center[
<img src="figures/mediation-assumptions.svg" width=570/>
]

---

### Semisynthetic Simulation Recipe
  
1. **Estimate `\(\hat{\gamma}, \hat{\xi}\left(\tau\right)\)` from real data**.  This defines `\(\hat{F}_{y \vert x, t, m}\)` from which to simulate community profiles.
2. **Define true positives and negatives**. We rank species according to their
estimated effects and set simulation `\(\xi\left(\tau\right), \gamma\)` for all
but the top 25% to 0.
3. **Simulate data from alternative configurations**. We vary the sample size and rescale coefficients `\(\hat{\xi}\left(\tau\right)\)` while constraining relative abundances for negative taxa
4. Estimate models across settings and **compute error rates**.

---

### FSR and Power against Direct Effects

* For FSR `\(\leq\)` 25%, we find that the LNM-mediation model has the highest power
* Across all models, we are better powered to detect direct rather than indirect effects

.center[
<img src="figures/direct_direct.png" width=900/>
]

---

### FSR and Power against Indirect Effects

* For FSR `\(\leq\)` 25%, we find that the LNM variants have the highest power
* Across all models, we are better powered to detect direct rather than indirect
effects

.center[
<img src="figures/indirect_indirect.png" width=900/>
]

---

### Estimated Direct Effects

We use the results from this simulation to provide FSR guarantees for estimated effects on the real data.

.center[
<img src="figures/lnm_effect_decision_fsr.png" width = 750/>
]

---

### R Package

We have written an R package that to support these modeling and evaluation
techniques.

```r
library(LNMmediation)
library(phyloseq)
data(mindfulness)

var_names <- colnames(sample_data(mindfulness))
mediator_ix <- grepl("mediator", var_names)
id_vars <- c("subject", "timepoint")
data_list <- phyloseq_mediators(mindfulness, var_names[mediator_ix], id_vars)
```

---

```r
fit <- lnm_mediation(model_conf(), data_list)
```

```
## Finished in  87.2 seconds.
```

```r
plot_interval(fit, "direct")
```

---

### Interactive Visualization

---

### References

[1] G. Winter, R. A. Hart, R. P. Charlesworth, et al. "Gut microbiome
and depression: what we know and what we need to know". In: _Reviews in
the Neurosciences_ 29.6 (2018), pp. 629-643.

[2] S. Dash, G. Clarke, M. Berk, et al. "The gut microbiome and diet in
psychiatry: focus on depression". In: _Current opinion in psychiatry_
28.1 (2015), pp. 1-6.

[3] J. A. Foster and K. M. Neufeld. "Gut-brain axis: how the microbiome
influences anxiety and depression". In: _Trends in neurosciences_ 36.5
(2013), pp. 305-312.

---

### References

[4] M. Carabotti, A. Scirocco, M. A. Maselli, et al. "The gut-brain
axis: interactions between enteric microbiota, central and enteric
nervous systems". In: _Annals of gastroenterology: quarterly
publication of the Hellenic Society of Gastroenterology_ 28.2 (2015),
p. 203.

[5] E. A. Mayer, K. Tillisch, A. Gupta, et al. "Gut/brain axis and the
microbiota". In: _The Journal of clinical investigation_ 125.3 (2015),
pp. 926-938.

[6] K. Imai, L. Keele, and D. Tingley. "A general approach to causal
mediation analysis." In: _Psychological methods_ 15.4 (2010), p. 309.

[7] M. B. Sohn and H. Li. "Compositional mediation analysis for
microbiome studies". In: _The Annals of Applied Statistics_ 13.1
(2019), pp. 661-681.

---

### References

[8] W. Ling, N. Zhao, A. M. Plantinga, et al. "Powerful and robust
non-parametric association testing for microbiome data via a
zero-inflated quantile approach (ZINQ)". In: _Microbiome_ 9.1 (2021),
pp. 1-19.

---

### Simulation Comparison

To compare strategies, we compute the false sign rate (FSR) and power across
species with increasingly large estimated effect sizes, analogous to an ROC
curve.

.center[
<img src="figures/threshold_ci.png" width=500/>
]

---

### Simulation Comparison

To compare strategies, we compute the false sign rate (FSR) and power across
species with increasingly large estimated effect sizes, analogous to an ROC
curve.

.center[
<img src="figures/threshold_ci-2.png" width=500/>
]

---

### ZINQ Simulation Fidelity

This model generates fairly realistic data.

.center[
<img src="figures/comparison.png" width=700/>
]

---

### ZINQ Simulation Fidelity

This is the same plot, but restricting to nonnegative counts.

.center[
<img src="figures/comparison_no_zeros.png" width=700/>
]

---

### LNM Goodness-of-Fit

.center[
<img src="figures/lnm_bar.png" width=700/>
]

---

### Comparison across Models

.center[
<img src="figures/realdata_conclusion_direct.png" width=850/>
]

---

### Comparison across Models

.center[
<img src="figures/realdata_conclusion_direct-2.png" width=850/>
]

---

### Interactive Visualization (Mediators)