Mindfulness and Microbiome

-- A Causal Mediation Perspective

By Xinran Miao in Causal inference Microbiome

October 26, 2022

Does mindfulness training affect microbiome? If so, how? In this blog, we motivate and formulate these problems in a causal mediation framework This analysis is a joint work with Hanying Jiang, mentored by Prof. Kris Sankaran. 1 following Imai, K., Keele, L. and Yamamoto, T.(2010).

Motivation

The gut microbiome refers to the community of bacteria and their genetic material living in the gut. They have documented relationships with human health and diseases. One with emerging attention is the gut-brain axis: researches have demonstrated the association between microbiota composition and stress / anxiety related behavior.

On the other hand, recent studies exhibit an effect of mindfulness-based intervention on anxiety and depression.

Putting two pieces together, a natural question comes up:

(Q1) Does mindfulness training affect microbiota composition?

Thinking deeper, we may ask:

(Q2) If an effect exists, how does it work? For instance, does mindfulness training affect microbiome by changing people’s diet habits?

The figure below provides an illustration 2.

Problem Rephrasing

We regard (Q1) as a question for causal effect: How can we quantify the the effect of an treatment (mindfulness training) on the microbita composition?

(Q2) can be viewed within a causal mediation framework: To what extent can we attribute the treatment effect to mediators (e.g., diet)?

In the rest of this blog, we will formulate (Q2) with tools in causal inference, where (Q1) will be automatically included.

Problem Formulation: Causal Mediation Framework

We introduce notations:

  • \(T\in\{0, 1\}\): treatment indicator variable,

  • \(Y(t)\in\mathbb{N}^K\): potential taxa count profile with treatment \(t\in\{0,1\}\), and

  • \(K\): number of taxa.

Then the average treatment effect can be defined as the difference in expected potential outcomes, \[\begin{align*} \mathbb{E}[Y(1)] - \mathbb{E}[Y(0)]. \end{align*}\]

For better understanding, mediation analysis adds an intermediate variable (mediator) to the causal path between treatment and outcome.

It splits the treatment effect into two parts:

  • indirect effect: the effect that works through mediators, and

  • direct effect: the effect unrelated to mediators.

Let \(M(t')\) be mediators as a function of treatment \(t'\in\{0,1\}\), then the potential outcome becomes a function of both treatment \(T\) and mediator \(M(t')\), denoted by \(Y(t, M(t'))\) .

The average (total) treatment effect becomes \[\begin{align*} \mathbb{E}[Y(1, M(1))] - \mathbb{E}[Y(0, M(0))]. \end{align*}\]

In a randomized experiment, we can only observe the outcome where \(t=t'\) being either \(0\) or \(1\).

But conceptually, we can ask What If questions answered by \(Y(t, M(t'))\), even when \(t\not=t'\).

  • Q: hat if we intervene on mediator only? A: \(Y(0, M(1))\).

  • Q: What if we intervene with controlled mediator? A: \(Y(1, M(0))\).

We will define indirect and direct effects \(\delta(t)\) and \(\tau(t')\) following Imai et al (2010). Before we get into this, let’s first introduce the observed variables.

Identification

For generality, we consider the identifiability of natural direct / indirect effects without specification of data types. The proof follows from Appendix in Imai, K., Keele, L. and Yamamoto, T.(2010).

Set-up

Consider a random sample of size \(n\), where we observe \((X_i, T_i,M_i, Y_i)\) for each sample \(i=1,\cdots,n\), where

  • \(X_i\) : pretreatment covariate with support \(\mathcal{X}\)
  • \(T_i\) : binary treatment variable,
  • \(M_i\) : mediator variable with support \(\mathcal{M}\),
  • \(Y_i\) : outcome variable with support \(\mathcal{Y}\).

In order to examine how treatment affect the outcome, we split the treatment effect into two parts depending on whether it works through the mediator. We control for measured confounders \(X\).

For each sample \(i\), we define its natural indirect effect

\[\delta_i(t) = Y_i(t,M_i(1)) - Y_i(t,M_i(0))\] and natural direct effect.

\[\zeta_i(t) = Y_i(1,M_i(t) - Y_i(0,M_i(t)).\]

Our interest are the average natural indirect effect

\[\begin{equation} \bar\delta(t) = \mathbb{E}[Y_i(t,M_i(1)) - Y_i(t,M_i(0))] \tag{1} \end{equation}\]

and the average natural direct effect \[\begin{equation} \bar\zeta(t) = \mathbb{E}[Y(1, M(t)) - Y(0, M(t))] \tag{2} \end{equation}\]

with \(t=0,1\).

Assumptions

For \(t,t'=0,1\) and any \(x\in\mathcal{X}\), we assume

\[\begin{equation} \{Y_i(t',m), M_i(t)\}\perp T_i\mid X_i=x \tag{3} \end{equation}\]

\[\begin{equation} Y_i(t'm) \perp M_i(t)\mid T_i=t, X_i=x \tag{4} \end{equation}\]

\[\begin{equation} \text{Pr}(T_i=t\mid X_i=x)>0, \mathbb{P}(M_i(t)=m\mid T_i=t,X_i=x)>0 \tag{5} \end{equation}\]

Identifiability

Theorem 1 \({}\) Under the above sequential ignorability assumption, the average natural direct and indirect effects can be identifed as follows for \(t=0,1\).

\[\bar\delta(t) = \iint\mathbb{E}(Y_i\mid M_i=m, T_i=t,X_i=x)\{dF_{M_i\mid T_i=1,X_i=x}(m) - dF_{M|i\mid T_1=0,X_i=x}(m)\}dF_{X_i}(x)\]

\[\bar\zeta(t) = \iint\{\mathbb{E}(Y_i\mid M_i=m,T_i=1,X_i=x)-\mathbb{E}(Y_i\mid M_i=m,T_i=0,X_i=x)\}dF_{M_i\mid T_i=t,X_i=x}(m)dF_{X_i}(x),\] where \(F_Z(\cdot)\) and \(F_{X\mid W}(\cdot)\) represent the distribution function of a random variable \(Z\) and the conditional ditribution function of \(Z\) given \(W\).

Proof. \({}\) Notice that the sequential ignorability assumption implies

\[\begin{equation} Y_i(t',m)\perp T_i\mid M_i(t)=m', X_i=x \tag{6} \end{equation}\]

For any \(t,t'=0,1\), we have \[\begin{align*} \mathbb{E}(Y_i(t, M_i(t'))\mid X_i=x) &= \int\mathbb{E}(Y_i(t,m)\mid M_i(t')=m, X_i=x)dF_{M_i(t')\mid X_i=x}(m) \\ &=\int \mathbb{E}(Y_i(t,m)\mid M_i(t')=m,T_i=t',X_i=x)dF_{M_i(t')\mid X_i=x}(m)\\ &= \int\mathbb{E}(Y_i(t,m)\mid T_i=t', X_i=x)dF_{M_i(t')\mid X_i=x}(m)\\ &=\int\mathbb{E}(Y_i(t,m)\mid T_i=t, X_i=x)dF_{M_i(t')\mid T_i=t',X_i=x}(m)\\ &=\int\mathbb{E}(Y_i(t,m)\mid M_i(t)=m,T_i=t,X_i=x)dF_{M_i(t')\mid T_i=t',X_i=x}(m)\\ &=\int \mathbb{E}(Y_i\mid M_i=m, T_i=t, X_i=x)dF_{M_i(t')\mid T_i=t',X_i=x}(m)\\ &=\int\mathbb{E}(Y_i\mid M_i=m, T_i=t,X_i=x)dF_{M_i\mid T_i=t',X_i=x}(m), \end{align*}\] where the second equality follows from (6), the third and fifth equalities follow from (3), the fourth and last equalities follow from (4), and the sixth equality follows from the fact that \(M_i=M_i(T_i)\) and \(Y_i=Y_i(T_i, M_i(T_i))\).

Therefore, we have \[\begin{align*} \mathbb{E}(Y_i(t,M_i(t')))=\iint \mathbb{E}(Y_i\mid M_i=m,T_i=t,X_i=x)dF_{M_i\mid T_i=t',X_i=x}(m)dF_{X_i}(x) \end{align*}\] Identifiability follows by plugging in the definition of average natural indirect and direct effects defined in (1) and (2). \(\hspace{12.5cm} \square\)

Takeaways

  1. We are interested in the causal effect of mindfulness intervention on microbiome, with intermediate variables in between.

  2. The statistical distillation is a causal mediation framework, which is not limited to this data analysis problem.

  3. The parameters can be identified from observed data with sequential ignorability assumptions

  1. Some descriptions and illustrative plots are inspired by recent talks of Prof. Kris Sankaran and Dr. Margaret Thairu, and course lectures delivered by Prof. Amy Cochran.↩︎

  2. Links to the plots are provided in the reference.↩︎

Posted on:
October 26, 2022
Length:
5 minute read, 946 words
Categories:
Causal inference Microbiome
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