Mediation and Moderation Analysis

Statistics

Understanding How and When Effects Occur

Mediation reveals the mechanism through which X affects Y, while moderation identifies boundary conditions. Together they transform simple causal questions into nuanced explanations of process and context.

Psychology — Identify psychological mechanisms that explain therapeutic interventions
Management — Determine whether leadership style effects depend on team culture
Public Health — Understand how education affects health through behavioral pathways

Mediation answers how; moderation answers when — both are essential for complete understanding.

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

DfMediation

A variable $M$ mediates the effect of $X$ on $Y$ if $X$ influences $M$ , which in turn influences $Y$ . The mediator is the mechanism through which $X$ affects $Y$ .

DfModeration

A variable $W$ moderates the effect of $X$ on $Y$ if the strength or direction of the $X \to Y$ relationship depends on the level of $W$ .

Mediation Model

Mediation Path Equations

M = aX + e_1

Here,

$a$ =Effect of X on mediator M
$b$ =Effect of M on Y (controlling for X)
$c'$ =Direct effect of X on Y
$ab$ =Indirect (mediated) effect
$c$ =Total effect: $c = c' + ab$

Types of Effects

| Effect | Definition | Formula |

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

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

| Direct effect | Effect of X on Y not through M | $c'$ |

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

Complete vs Partial Mediation

Complete mediation: $c' = 0$ (all effect is through M)
Partial mediation: $c' \neq 0$ (some effect direct, some through M)

Testing the Indirect Effect

Sobel Test

Sobel Test Statistic

z = \frac{ab}{\sqrt{b^2 s_a^2 + a^2 s_b^2}}

Here,

$a, b$ =Path coefficients
$s_a, s_b$ =Standard errors of a and b

Sobel Test Limitation

The Sobel test assumes the product $ab$ is normally distributed, which is often not true in practice. Bootstrapping is preferred.

Bootstrapping (Preferred)

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

Moderation (Interaction)

Moderation Model

Y = b_0 + b_1 X + b_2 W + b_3 (X \times W) + e

Here,

$b_1$ =Main effect of X
$b_2$ =Main effect of W (moderator)
$b_3$ =Interaction effect (moderation)
$X \times W$ =Product term (interaction)

Interpreting Interactions

If $b_3 \neq 0$ , the effect of $X$ on $Y$ depends on $W$ . Plot the relationship at different levels of $W$ (e.g., ±1 SD) to visualize the 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

Example: Therapy -> Coping -> Depression

Does coping mediate the effect of therapy on depression?

X: Therapy (1=treatment, 0=control)
M: Coping skills score
Y: Depression score

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

| a (Therapy -> Coping) | 1.85 | 0.32 | < 0.001 |

| b (Coping -> Depression) | -0.42 | 0.08 | < 0.001 |

| c' (Direct) | -1.23 | 0.41 | 0.003 |

| ab (Indirect) | -0.777 | — | — |

Bootstrap 95% CI: [-1.21, -0.41] (excludes zero -> significant mediation)

Conclusion: Coping significantly mediates the therapy effect. 39% of the total effect is mediated through coping skills.

Key Takeaways

Summary: Mediation and Moderation

Mediation explains how X affects Y (through M)
Moderation explains when the X -> Y effect changes (depending on W)
The indirect effect ( $ab$ ) quantifies mediation
Use bootstrapping for the indirect effect (preferred over Sobel test)
Moderation is tested via interaction terms: $X \times W$
Always mean-center variables before creating interaction terms
Mediated moderation and moderated mediation combine both concepts

Mediation and Moderation Analysis

Mediation and Moderation Analysis

Understanding How and When Effects Occur

DfMediation

DfModeration

Mediation Model

Mediation Path Equations

Types of Effects

Testing the Indirect Effect

Sobel Test

Sobel Test Statistic

Bootstrapping (Preferred)

Moderation (Interaction)

Moderation Model

Mediated Moderation vs Moderated Mediation

Python Implementation

Worked Example

Example: Therapy -> Coping -> Depression

Key Takeaways

Summary: Mediation and Moderation

Related Topics

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