Causal Mediation Tree Model for Feature Identification on High-Dimensional Mediators

Abstract

High-dimensional mediation analysis plays an important role in recent biomedical research as a large number of mediators, such as microbiome, could modulate the effect of exposure to the outcome of interest. Most of the current studies focus on modelling independent mediators, but these methods do not consider the non-linear interactive effect between the mediators. Furthermore, it can be challenging to identify features with mediation effects from the high-dimensional mediator space. We proposed an innovative non-parametric approach to build causal mediation trees (CMT) to select important mediators and assess their non-linear interactive mediation effects on the outcome of the study. The data is recursively partitioned into subpopulations constructed by the mediators with the largest mediation effect. We aim to incorporate these non-linear interactions into the mediation framework using this approach and evaluate the total causal effect. Simulation studies were conducted to assess the performance of the CMT algorithm under different scenarios of interactive mediation effects. We applied the method to analyze vaginal microbiome sequencing data from the reproductive-age women’s study. We investigated the causal relationship between ethnic groups and the vaginal pH levels mediated by the vaginal microbiome. We identified three important microbial taxa with strong mediation effects and estimated the total effect of the mediation tree model.

Appendices

Appendix A Estimating the Total Effect

By the definition, the Eq. (1), the NIE can be expressed as $NIE=Y_{am_a}-Y_{a m_{a^*}}$. The estimation of $Y_{am_a}$ are derived below:

$$\begin{aligned} \begin{aligned} E\left[ Y_{am_a}\mid X=x\right]&= E\left\{ E\left[ Y_{am_a}\mid m_a=o,X=x\right] \right\} \;\;\;\;by \;double\; expectation\\&= \sum _{o=1}^{O}{E\left[ Y_{am_a}\mid m_a=o,X=x\right] }P(m_a=o\mid X=x)\\&= \sum _{o=1}^{O}{E\left[ Y_{ao}\mid m_a=o,X=x\right] }P(m_a=o\mid X=x)\\&= \sum _{o=1}^{O}{E\left[ Y_{ao}\mid X=x\right] }P(m_a=o\mid a,x)\;\;by\;Y_{am} \perp m_{a^*}\mid X\; ; \;m_a \perp A \mid X\\&= \sum _{o=1}^{O}{E\left[ Y_{ao}\mid a,o,x\right] }P(o\mid a,x)\;\;by\;Y_{am} \perp m\mid X\\&= \sum _{o=1}^{O-1}\biggl \{\left( \theta ^*_0+\theta ^*_1a+\theta ^*_{2o}+a\theta ^*_{3o}+\theta ^*_4x\right) \\&\quad \times \left\{ \frac{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a+\beta ^*_{2o}x\right) }{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a+\beta ^*_{2o}x\right) }}\right\} \biggl \}\\&\quad +\left( \theta ^*_0+\theta ^*_1a+\theta ^*_{2O}+a\theta ^*_{3O}+\theta ^*_4x\right) \\&\quad \times \left\{ \frac{1}{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a+\beta ^*_{2o}x\right) }}\right\} \\&= \theta ^*_0+\theta ^*_1a+\theta ^*_4x\\&\quad +\sum _{o=1}^{O-1}\biggl \{\left( \theta ^*_{2o}+a\theta ^*_{3o}\right) \left\{ \frac{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a+\beta ^*_{2o}x\right) }{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a+\beta ^*_{2o}x\right) }}\right\} \biggl \}\\&\quad +\left( \theta ^*_{2O}+a\theta ^*_{3O}\right) \left\{ \frac{1}{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a+\beta ^*_{2o}x\right) }}\right\} \end{aligned} \end{aligned}$$

(A1)

Similarly,

$$\begin{aligned} \begin{aligned} E\left[ Y_{am_{a^*}}\mid X=x\right]&= E\left\{ E\left[ Y_{am_{a^*}}\mid m_{a^*}=o,X=x\right] \right\} \\&=\sum _{o=1}^{O}{E\left[ Y_{am_{a^*}}\mid m_{a^*}=o,X=x\right] }P(m_{a^*}=o\mid X=x)\\&= \theta ^*_0+\theta ^*_1a+\theta ^*_4x\\&\quad +\sum _{o=1}^{O-1}\biggl \{\left( \theta ^*_{2o}+a\theta ^*_{3o}\right) \left\{ \frac{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a^*+\beta ^*_{2o}x\right) }{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a^*+\beta ^*_{2o}x\right) }}\right\} \biggl \} \\&\quad +\left( \theta ^*_{2O}+a\theta ^*_{3O}\right) \left\{ \frac{1}{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a^*+\beta ^*_{2o}x\right) }}\right\} \end{aligned} \end{aligned}$$

(A2)

.

Therefore, the average NIE can be expressed as:

$$\begin{aligned} \begin{aligned} Avg\;NIE_*&=E\left[ Y_{am_a}\mid X=x\right] - E\left[ Y_{am_{a^*}}\mid X=x\right] \\&=\theta ^*_0+\theta ^*_1a+\theta ^*_4x+\sum _{o=1}^{O-1} \biggl \{ \left( \theta ^*_{2o}+a\theta ^*_{3o}\right) \left\{ \frac{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a+\beta ^*_{2o}x\right) }{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a+\beta ^*_{2o}x\right) }}\right\} \biggl \}\\&\quad +\left( \theta ^*_{2O}+a\theta ^*_{3O}\right) \left\{ \frac{1}{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a+\beta ^*_{2o}x\right) }}\right\} \\&\quad - \Bigg ( \theta ^*_0+\theta ^*_1a+\theta ^*_4x\\&\quad +\sum _{o=1}^{O-1}\biggl \{\left( \theta ^*_{2o}+a\theta ^*_{3o}\right) \left\{ \frac{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a^*+\beta ^*_{2o}x\right) }{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a^* \beta ^*_{2o}x\right) }}\right\} \biggl \} \\&\quad +\left( \theta ^*_{2O}+a\theta ^*_{3O}\right) \left\{ \frac{1}{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a^*+\beta ^*_{2o}x\right) }}\right\} \Bigg )\\&= \sum _{o=1}^{O-1}\biggl \{ \left( \theta ^*_{2o}+a\theta ^*_{3o}\right) \left\{ \frac{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a+\beta ^*_{2o}x\right) }{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a+\beta ^*_{2o}x\right) }}\right\} \biggl \}\\&\quad +\left( \theta ^*_{2O}+a\theta ^*_{3O}\right) \left\{ \frac{1}{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a+\beta ^*_{2o}x\right) }}\right\} \\&\quad - \sum _{o=1}^{O-1}\biggl \{\left( \theta ^*_{2o}+a\theta ^*_{3o}\right) \left\{ \frac{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a^*+\beta ^*_{2o}x\right) }{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a^*+\beta ^*_{2o}x\right) }}\right\} \biggl \}\\&\quad -\left( \theta ^*_{2O}+a\theta ^*_{3O}\right) \left\{ \frac{1}{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a^*+\beta ^*_{2o}x\right) }}\right\} \\&= \sum _{o=1}^{O-1}\biggl \{\left( \theta ^*_{2o}+a\theta ^*_{3o}\right) \Biggl \{\frac{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a+\beta ^*_{2o}x\right) }{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a+\beta ^*_{2o}x\right) }}\\&\quad -\frac{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a^*+\beta ^*_{2o}x\right) }{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a^*+\beta ^*_{2o}x\right) }}\Biggl \}\biggl \}\\&\quad + \left( \theta ^*_{2O}+a\theta ^*_{3O}\right) \Biggl \{ \frac{1}{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a+\beta ^*_{2o}x\right) }}\\&\quad - \frac{1}{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a^*+\beta ^*_{2o}x\right) }}\Biggl \}\\&=\sum _{o=1}^{O-1}\biggl \{\left( \theta ^*_{2o}+a\theta ^*_{3o}\right) \Biggl \{\frac{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a+\beta ^*_{2o}x\right) }{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a+\beta ^*_{2o}x\right) }}\\&\quad -\frac{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a^*+\beta ^*_{2o}x\right) }{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a^*+\beta ^*_{2o}x\right) }}\Biggl \}\biggl \} \;\; where\;\theta ^*_{2O}=\theta ^*_{3O}=0 \end{aligned} \end{aligned}$$

(A3)

The average NDE is

$$\begin{aligned} \begin{aligned} Avg\;NDE_*&=E\left[ Y_{am_{a^*}}\mid X=x\right] - E\left[ Y_{a^*m_{a^*}}\mid X=x\right] \\&= \theta ^*_0+\theta ^*_1a+\theta ^*_4x\\&\quad +\sum _{o=1}^{O-1}\biggl \{\left( \theta ^*_{2o}+a\theta ^*_{3o}\right) \left\{ \frac{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a^*+\beta ^*_{2o}x\right) }{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a^*+\beta ^*_{2o}x\right) }}\right\} \biggl \}\\&\quad +\left( \theta ^*_{2O}+a\theta ^*_{3O}\right) \left\{ \frac{1}{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a^*+\beta ^*_{2o}x\right) }}\right\} \\&\quad - \Bigg ( \theta ^*_0+\theta ^*_1a^*+\theta ^*_4x\\&\quad +\sum _{o=1}^{O-1}\biggl \{\left( \theta ^*_{2o}+a^*\theta ^*_{3o}\right) \left\{ \frac{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a^*+\beta ^*_{2o}x\right) }{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a^*+\beta ^*_{2o}x\right) }}\right\} \biggl \}\\&\quad +\left( \theta ^*_{2O}+a^*\theta ^*_{3O}\right) \left\{ \frac{1}{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a^*+\beta ^*_{2o}x\right) }}\right\} \Bigg )\\&= \theta ^*_1(a-a^*)+\sum _{o=1}^{O-1}\Biggl \{\left( a-a^*\right) \theta ^*_{3o} \left\{ \frac{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a^*+\beta ^*_{2o}x\right) }{1+\sum _{o=1}^{O-1}{exp\left( \beta ^*_{0o}+\beta ^*_{1o}a^*+\beta ^*_{2o}x\right) }}\right\} \Biggl \} \end{aligned} \end{aligned}$$

(A4)

.

Appendix B Additional Simulations

See Tables 6, 7 and 8.

Table 6 Performance of CMT at the different proportions of exposure by the percentage of building the correct tree structure across four scenarios with an effect size of 3 and sample size of 500

Table 7 Impact of correlated mediators on CMT performance in four scenarios with varying correlation values ($\rho$) for $M_1$ and $M_2$ ($\rho =0,0.3 \; and \;0.5$), consistent effect size of 3, and sample size of 500. Data generation: (1) Generate $Z_1$ and $Z_2$ from a bivariate standard normal distribution with a correlation $\rho$; (2) Compute probabilities $p_1$ and $p_2$ using equation (16); (3) Generate quantiles $g_1$ and $g_2$ from standard normal distributions using $p_1$ and $p_2$ as probabilities; (4) Assign 1 if $Z_1 < g_1$ for $M_1$ and 0 otherwise; Assign 1 if $Z_2 < g_2$ for $M_2$ and 0 otherwise

Table 8 Impact of cut-off values on CMT performance in four scenarios with varying cut-off values ($c=80\%,90\% \; and \;95\%$), consistent effect size of 3, and sample size of 500

Appendix C OTU Distribution for the Microbiome Data

See Table 9.

Table 9 Microbiome information for non-zero proportion by ethnicity