Mediation and Moderation Analysis

Mediation explains how an effect occurs (the mechanism), while moderation explains when or for whom an effect occurs (the boundary condition).

Mediation Model

Types of Effects

| Effect | Definition | Formula |

|--------|-----------|---------|

| Total effect | Overall effect of X on Y | |

| Direct effect | Effect of X on Y not through M | |

| Indirect effect | Effect of X on Y through M | |

Testing the Indirect Effect

Sobel Test

Bootstrapping (Preferred)

Generate bootstrap samples, compute for each, and construct a confidence interval for the indirect effect. If the CI excludes zero, mediation is significant.

Moderation (Interaction)

Mediated Moderation vs Moderated Mediation

| Concept | Definition |

|---------|-----------|

| Mediated moderation | The interaction effect on Y is explained through a mediator |

| Moderated mediation | The indirect effect (mediation) varies across levels of a moderator |

Python Implementation

import numpy as np

import pandas as pd

import statsmodels.api as sm

from statsmodels.formula.api import ols

import matplotlib.pyplot as plt



np.random.seed(42)



# Simulate mediation data

n = 500

X = np.random.randn(n)

M = 0.6 * X + np.random.randn(n)

Y = 0.4 * M + 0.2 * X + np.random.randn(n)



# Path analysis

# Path a: X -> M

model_m = ols('M ~ X', data=pd.DataFrame({'X': X, 'M': M})).fit()

a = model_m.params['X']

se_a = model_m.bse['X']



# Path b: M -> Y (controlling for X)

model_y = ols('Y ~ X + M', data=pd.DataFrame({'X': X, 'M': M, 'Y': Y})).fit()

b = model_y.params['M']

se_b = model_y.bse['M']

c_prime = model_y.params['X']

c_total = ols('Y ~ X', data=pd.DataFrame({'X': X, 'Y': Y})).fit().params['X']



indirect = a * b

direct = c_prime

total = c_total



print(f"Path a: {a:.3f} (SE: {se_a:.3f})")

print(f"Path b: {b:.3f} (SE: {se_b:.3f})")

print(f"Direct effect: {direct:.3f}")

print(f"Indirect effect: {indirect:.3f}")

print(f"Total effect: {total:.3f}")

print(f"Proportion mediated: {indirect/total:.1%}")



# Bootstrap CI for indirect effect

n_boot = 5000

indirect_boots = []

for _ in range(n_boot):

    idx = np.random.choice(n, n, replace=True)

    X_b, M_b, Y_b = X[idx], M[idx], Y[idx]

    a_b = sm.OLS(M_b, sm.add_constant(X_b)).fit().params[1]

    b_b = sm.OLS(Y_b, np.column_stack([X_b, M_b])).fit().params[1]

    indirect_boots.append(a_b * b_b)



ci_lower = np.percentile(indirect_boots, 2.5)

ci_upper = np.percentile(indirect_boots, 97.5)

print(f"\nBootstrap 95% CI for indirect effect: [{ci_lower:.3f}, {ci_upper:.3f}]")

print(f"Significant: {'Yes' if ci_lower > 0 or ci_upper < 0 else 'No'}")



# Moderation

W = np.random.randn(n)

Y_mod = 0.3 * X + 0.4 * W + 0.25 * X * W + np.random.randn(n)

model_mod = ols('Y ~ X * W', data=pd.DataFrame({'X': X, 'W': W, 'Y': Y_mod})).fit()

print(f"\nModeration model:")

print(model_mod.summary().tables[1])

Worked Example

Key Takeaways

Related Topics

• See Causal Inference for establishing causation

• See Regression Discontinuity for quasi-experimental designs

• See Propensity Score Matching for observational studies