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Mediation Analysis

Source: vignettes/04_mediation.Rmd
04_mediation.Rmd

Introduction

Causal inference often requires investigating how an effect occurs. Does XX affect YY directly, or does it work through a mediator MM?

The because package provides a fully automated Bayesian mediation analysis tool, because_mediation(), which decomposes the Total Effect of an exposure on an outcome into: 1. Direct Effect: The effect of X→YX \to Y not mediated by other variables in the graph. 2. Indirect Effect(s): The effect propagated through intermediate variables (X→M→YX \to M \to Y).

This is calculated by multiplying the posterior distributions of coefficients along each path, preserving full uncertainty quantification.

Example: Ecological Mediation (Elevation Gradient)

In this example, we investigate how Elevation affects Plant Abundance. We hypothesize that Elevation acts through a causal chain involving Temperature and Soil Moisture:

  1. Elevation determines Temperature (higher elevation →\to lower temperature).
  2. Temperature influences Soil Moisture (lower temperature →\to lower evaporation →\to higher moisture).
  3. Abundance is driven by Moisture, Temperature, and potentially a direct effect of Elevation (e.g., due to UV radiation or partial pressure of gases).

1. Simulate Data

We simulate N=200N=200 plots along an elevation gradient.

library(because)

set.seed(42)
N <- 200

# 1. Elevation (Exogenous variable)
# Ranges roughly from 500m to 1500m
Elevation <- rnorm(N, mean = 1000, sd = 200)

# 2. Temperature (Mediator 1)
# Decreases with Elevation (Lapse rate approx effect)
# Coef: -0.01 implies 100m climb -> -1 degree C
Temp <- 25 - 0.01 * Elevation + rnorm(N, sd = 2)

# 3. Moisture (Mediator 2)
# Driven by Temperature. Cooler -> Moister.
# We model a negative relationship with Temp.
Moisture <- 20 - 2 * Temp + rnorm(N, sd = 5)

# 4. Plant Abundance (Outcome)
# - Positive effect of Moisture (+0.5)
# - Positive effect of Temperature (+1.5)
# - Direct negative effect of Elevation (-0.005) due to harsh conditions
Abundance <- 10 + 0.5 * Moisture + 1.5 * Temp - 0.005 * Elevation + rnorm(N, sd = 10)

eco_data <- data.frame(Elevation, Temp, Moisture, Abundance)
head(eco_data)

2. Standardize Data

Important: For mediation analysis, it is highly recommended to standardize your continuous variables (mean = 0, sd = 1) before fitting.

Standardization ensures that: 1. Coefficients are comparable: All effects are expressed in standard deviation units (standardized effects). 2. Scale Invariance: The calculation of Indirect Effects (product of coefficients) is more interpretable relative to the Total Effect. 3. Convergence: MCMC sampling often behaves better with standardized scales.

# Standardize all variables
eco_data_std <- scale(eco_data)
head(eco_data_std)

3. Fit the Structural Equation Model

We define the structural equations reflecting our causal DAG. Notice the chain: Elevation -> Temp -> Moisture -> Abundance.

# Define the structural equations
eco_eqs <- list(
  Temp ~ Elevation,
  Moisture ~ Temp,
  Abundance ~ Moisture + Temp + Elevation
)

# Fit the model
# We use a short chain for demonstration purposes. Use more iterations for real analysis.
fit <- because(
  equations = eco_eqs,
  data = eco_data_std,
  n.iter = 2000
)
summary(fit)

We can also plot the fitted causal model with its standardised paths:

4. Perform Mediation Analysis

We want to understand the Total Effect of Elevation on Abundance, and decompose it into its direct and indirect components.

# Run Mediation Analysis for Elevation -> Abundance
med_results <- because_mediation(fit, exposure = "Elevation", outcome = "Abundance")

Inspect the Summary 

med_results$summary

Interpretation (Standardized Units): * Total Effect: The net impact of Elevation on Abundance in SD units. * Direct Effect: The path Elevation -> Abundance (independent of mediators). * Total Indirect Effect: The sum of all mediated paths. In our simulation, Elevation lowers Temp, which raises Moisture, which increases Abundance.

Inspect Individual Paths

The because_mediation function automatically traces all valid paths from exposure to outcome in the DAG.

med_results$paths

We expect to see three distinct paths:

  1. Direct: Elevation -> Abundance
  2. Short Indirect: Elevation -> Temp -> Abundance
    • Elevation lowers Temp (negative correlation); Temp increases Abundance (positive correlation).
    • The product of these effects is Negative.
  3. Long Indirect (Chain): Elevation -> Temp -> Moisture -> Abundance
    • Elevation lowers Temp (negative).
    • Lower Temp raises Moisture (negative relationship →\to positive change in moisture).
    • Higher Moisture raises Abundance (positive).
    • The chain involves two negative links and one positive link, resulting in a Positive indirect effect.

The function handles this decomposition automatically, providing credibility intervals for each specific mechanism.

Technical Note

The function calculates the indirect effect as the product of coefficients along the path. For example, for the long chain: Indirectchain=βElev→Temp×βTemp→Moist×βMoist→Abund \text{Indirect}_{\text{chain}} = \beta_{Elev \to Temp} \times \beta_{Temp \to Moist} \times \beta_{Moist \to Abund} This approach is standard for linear models. For non-linear models, this represents an approximation of the average causal effect.