Mediation and Moderation Analysis: Zero to Hero Tutorial

This comprehensive tutorial takes you from the foundational concepts of Mediation and Moderation Analysis all the way through advanced conditional process models, estimation, evaluation, and practical usage within the DataStatPro application. Whether you are encountering these methods for the first time or looking to deepen your understanding of process analysis, this guide builds your knowledge systematically from the ground up.

Table of Contents

1. Prerequisites and Background Concepts
2. What is Mediation and Moderation Analysis?
3. The Mathematics Behind Mediation and Moderation
4. Assumptions of Mediation and Moderation Analysis
5. Types of Mediation and Moderation Models
6. Using the Mediation and Moderation Component
7. Mediation Analysis
8. Moderation Analysis
9. Conditional Process Analysis
10. Model Fit and Evaluation
11. Advanced Topics
12. Worked Examples
13. Common Mistakes and How to Avoid Them
14. Troubleshooting
15. Quick Reference Cheat Sheet

1. Prerequisites and Background Concepts

Before diving into mediation and moderation analysis, it is essential to be comfortable with the following foundational statistical concepts. Each is briefly reviewed below.

1.1 Simple and Multiple Linear Regression

Simple linear regression models the relationship between one predictor $X$ and an outcome $Y$:

$Y = b_0 + b_1 X + e$

Where:

• $b_0$ is the intercept — the expected value of $Y$ when $X = 0$.
• $b_1$ is the slope — the expected change in $Y$ for a one-unit increase in $X$.
• $e$ is the residual error — the part of $Y$ not explained by $X$.

Multiple linear regression extends this to include several predictors:

$Y = b_0 + b_1 X_1 + b_2 X_2 + \dots + b_k X_k + e$

Each coefficient $b_j$ represents the partial effect of $X_j$ on $Y$, holding all other predictors constant. Mediation and moderation analyses are built entirely on systems of linear regression equations, so a solid understanding of regression is essential.

1.2 Standardised vs. Unstandardised Coefficients

Unstandardised coefficients ($b$) are expressed in the original units of $X$ and $Y$. They answer: "For a one-unit increase in $X$, how many units does $Y$ change?"

Standardised coefficients ($\beta$) are expressed in standard deviation units and are obtained by standardising all variables (mean = 0, SD = 1) before analysis:

$\beta_j = b_j \cdot \frac{\sigma_{X_j}}{\sigma_Y}$

In mediation analysis, the indirect effect is most meaningfully expressed in unstandardised units (so it represents a real-world quantity). Standardised coefficients facilitate comparison across studies with different measurement scales.

1.3 Causal Diagrams (Path Diagrams)

A path diagram is a graphical representation of a statistical model showing the hypothesised relationships between variables:

• Rectangles (or squares) represent observed variables.
• Ovals (or circles) represent latent variables (not used in basic mediation/moderation).
• Single-headed arrows ($\rightarrow$) represent directional relationships (regression paths).
• Double-headed arrows ($\leftrightarrow$) represent correlations (non-directional associations).

The path coefficient on each arrow is the regression coefficient ($b$ or $\beta$) for that specific relationship.

1.4 The Product of Coefficients

The indirect effect in mediation analysis is computed as the product of two regression coefficients: the effect of $X$ on $M$ (the mediator), and the effect of $M$ on $Y$ (the outcome, controlling for $X$):

$\text{Indirect Effect} = a \times b$

This product of coefficients is the foundation of modern mediation analysis. Understanding that an indirect path through a mediator equals the product of the coefficients on that path is the single most important concept in mediation analysis.

1.5 Interaction Terms

An interaction between two variables $X$ and $W$ is represented by their product:

$X \times W$

When this product term is included in a regression model, it captures how the effect of $X$ on $Y$ depends on the level of $W$. This is the mathematical basis of moderation analysis. The interaction term is NOT the same as the sum or average of $X$ and $W$ — it is specifically their product.

1.6 The Concept of Conditional Effects

A conditional effect is the effect of one variable on another at a specific value of a third variable. For example:

$\theta_{X \rightarrow Y \mid W=w} = b_1 + b_3 \cdot w$

This reads: "The effect of $X$ on $Y$ when $W = w$ equals $b_1 + b_3 \times w$." Understanding conditional effects is the key to interpreting moderation results.

1.7 Bootstrapping

Bootstrapping is a resampling method used to construct confidence intervals for statistics whose sampling distributions are unknown or non-normal (such as the product of two regression coefficients $a \times b$). The algorithm is:

1. Draw $B$ bootstrap samples of size $n$ from the original data with replacement.
2. Compute the statistic of interest (e.g., indirect effect $\hat{a} \times \hat{b}$) in each bootstrap sample.
3. The distribution of the $B$ bootstrap estimates approximates the sampling distribution.
4. The 95% CI is the 2.5th and 97.5th percentiles of this bootstrap distribution.

Bootstrapping is the gold standard for testing indirect effects in mediation analysis. Typically $B = 5{,}000$ to $B = 10{,}000$ bootstrap samples are used.

2. What is Mediation and Moderation Analysis?

2.1 The Core Questions

Mediation and moderation analysis addresses two complementary questions about how and when relationships between variables exist:

Mediation asks HOW or WHY does $X$ affect $Y$?

"Does the effect of stress ($X$) on depression ($Y$) operate through rumination ($M$)?"

Moderation asks WHEN or FOR WHOM does $X$ affect $Y$?

"Is the effect of stress ($X$) on depression ($Y$) stronger for people with low social support ($W$) compared to people with high social support ($W$)?"

Conditional Process Analysis asks HOW AND WHEN simultaneously:

"Does the indirect effect of stress on depression through rumination depend on the level of social support?"

2.2 The Fundamental Distinction

2.3 Visual Summary of Core Concepts

Mediation (M is on the path from X to Y): X ─────────→ Y ↘ ↗ M (Mediator)

Moderation (W changes the strength/direction of X → Y): W (Moderator) ↓ X ─────────→ Y (effect of X depends on W) Moderated Mediation (Conditional Process): W (Moderator) ↓ X ─────────→ Y ↘ ↗ M (Mediator)

2.4 Real-World Applications

2.5 A Brief History

The Baron and Kenny (1986) causal steps approach was the dominant method for mediation analysis for two decades. It required four conditions to be satisfied for mediation to be claimed. However, it has been largely superseded by the product of coefficients approach combined with bootstrapped confidence intervals, advocated by Preacher & Hayes (2004, 2008) and formalised in Hayes' (2013, 2022) PROCESS macro framework — which is the approach implemented in DataStatPro.

3. The Mathematics Behind Mediation and Moderation

3.1 The Simple Mediation Model — Equations

The simplest mediation model involves three variables: a predictor $X$, a mediator $M$, and an outcome $Y$. It requires two regression equations:

Equation 1 — Predicting the mediator from $X$:

$M = i_M + aX + e_M$

Equation 2 — Predicting the outcome from both $X$ and $M$:

$Y = i_Y + c'X + bM + e_Y$

Where:

• $a$ = effect of $X$ on $M$ (the a-path).
• $b$ = effect of $M$ on $Y$, controlling for $X$ (the b-path).
• $c'$ = effect of $X$ on $Y$, controlling for $M$ (the direct effect of $X$).
• $i_M$, $i_Y$ = intercepts.
• $e_M$, $e_Y$ = residual errors.

3.2 The Three Effects in Mediation

The Total Effect ($c$):

The total effect is the effect of $X$ on $Y$ without including $M$:

$Y = i_Y + cX + e_Y$

$c$ combines the direct and indirect pathways.

The Indirect Effect ($a \times b$):

The indirect effect is the product of the $a$-path and the $b$-path:

$\text{Indirect Effect} = a \times b$

It quantifies how much of the effect of $X$ on $Y$ is transmitted through $M$.

The Direct Effect ($c'$):

The direct effect is the effect of $X$ on $Y$ after accounting for $M$:

$c' = c - ab$

The Fundamental Identity:

The total effect equals the direct effect plus the indirect effect:

$c = c' + ab$

This decomposition is the cornerstone of mediation analysis. Every unit of the total effect can be attributed to either the direct path ($c'$) or the indirect path through the mediator ($ab$).

3.3 The Proportion Mediated

The proportion mediated (also called the indirect effect ratio) estimates what fraction of the total effect passes through the mediator:

$PM = \frac{ab}{c} = \frac{ab}{c' + ab} = 1 - \frac{c'}{c}$

⚠️ The proportion mediated is unreliable when the total effect $c$ is near zero (even if the indirect effect $ab$ is substantial — a phenomenon called inconsistent mediation). Never rely on the proportion mediated as the primary evidence for mediation.

3.4 Multiple Mediators — Parallel Mediation

With $k$ parallel mediators $M_1, M_2, \dots, M_k$, the model uses $k + 1$ equations:

Equations for each mediator $j = 1, \dots, k$:

$M_j = i_{M_j} + a_j X + e_{M_j}$

Outcome equation:

$Y = i_Y + c'X + b_1 M_1 + b_2 M_2 + \dots + b_k M_k + e_Y$

Specific indirect effect through mediator $j$:

$\text{Indirect}_j = a_j b_j$

Total indirect effect:

$\text{Total Indirect} = \sum_{j=1}^{k} a_j b_j$

Total effect:

$c = c' + \sum_{j=1}^{k} a_j b_j$

Contrasts between indirect effects: The difference between two specific indirect effects can be tested:

$\text{Contrast}_{jk} = a_j b_j - a_k b_k$

A bootstrapped CI for this contrast that excludes zero indicates that mediators $j$ and $k$ transmit significantly different portions of the effect of $X$ to $Y$.

3.5 Sequential (Serial) Mediation

With two sequential mediators $M_1$ and $M_2$ (where $M_1$ precedes $M_2$ on the causal chain):

Equation 1:

$M_1 = i_{M_1} + a_1 X + e_{M_1}$

Equation 2:

$M_2 = i_{M_2} + a_2 X + d_{21} M_1 + e_{M_2}$

Equation 3:

$Y = i_Y + c'X + b_1 M_1 + b_2 M_2 + e_Y$

Where $d_{21}$ is the effect of $M_1$ on $M_2$.

The three indirect effects:

Total indirect effect:

$\text{Total Indirect} = a_1 b_1 + a_2 b_2 + a_1 d_{21} b_2$

Total effect identity:

$c = c' + a_1 b_1 + a_2 b_2 + a_1 d_{21} b_2$

3.6 The Simple Moderation (Interaction) Model

The simple moderation model includes the predictor $X$, the moderator $W$, and their product:

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

Where:

• $b_1$ = effect of $X$ on $Y$ when $W = 0$ (the conditional direct effect of $X$ at $W = 0$).
• $b_2$ = effect of $W$ on $Y$ when $X = 0$.
• $b_3$ = the interaction effect — how much the effect of $X$ on $Y$ changes for each one-unit increase in $W$.
• $b_0$ = the intercept when both $X = 0$ and $W = 0$.

The conditional effect of $X$ on $Y$ at a specific value $w$ of $W$:

$\hat{\theta}_{X \rightarrow Y}(w) = b_1 + b_3 w$

This is the simple slope of $Y$ on $X$ at the value $W = w$. The moderation hypothesis is tested by examining whether $b_3 \neq 0$.

3.7 Centering in Moderation Analysis

Mean-centering the predictor and moderator before computing the interaction term is strongly recommended:

$X_c = X - \bar{X}, \quad W_c = W - \bar{W}$

The model becomes:

$Y = b_0 + b_1 X_c + b_2 W_c + b_3 (X_c \times W_c) + e$

Benefits of mean-centering:

• Reduces multicollinearity between $X$, $W$, and $X \times W$ (without changing the interaction coefficient $b_3$).
• Makes the intercept $b_0$ interpretable as the predicted value of $Y$ when $X$ and $W$ are at their means.
• Makes $b_1$ interpretable as the effect of $X$ on $Y$ at the mean of $W$ (not at $W = 0$, which may be outside the observed range).

⚠️ Centering does NOT change $b_3$ (the interaction coefficient) or the model's $R^2$. It only changes the interpretation of $b_1$ and $b_2$ and reduces multicollinearity.

3.8 Probing the Interaction: Simple Slopes

After establishing a significant interaction ($b_3 \neq 0$), the interaction must be probed to understand its nature. The simple slopes method computes the conditional effect of $X$ on $Y$ at representative values of $W$:

Standard values of $W$ (Pick-a-point approach):

• Low: $W = \bar{W} - 1\text{SD}_W$ (one SD below the mean).
• Medium: $W = \bar{W}$ (the mean).
• High: $W = \bar{W} + 1\text{SD}_W$ (one SD above the mean).

Simple slope at each value $w$:

$\hat{\theta}_{X \rightarrow Y}(w) = b_1 + b_3 w$

Standard error of the simple slope:

$SE[\hat{\theta}(w)] = \sqrt{\widehat{\text{Var}}(b_1) + 2w \cdot \widehat{\text{Cov}}(b_1, b_3) + w^2 \cdot \widehat{\text{Var}}(b_3)}$

t-statistic for the simple slope:

$t = \frac{\hat{\theta}(w)}{SE[\hat{\theta}(w)]}$

With $df = n - k - 1$ where $k$ is the number of predictors.

Johnson-Neyman (Floodlight) Method: Rather than evaluating at specific values of $W$, the Johnson-Neyman technique finds the exact value(s) of $W$ at which the simple slope transitions from significant to non-significant (the region of significance):

$w^* = \frac{-b_1 b_3 \pm t_{\alpha/2} \sqrt{b_3^2 \widehat{\text{Var}}(b_1) - (t_{\alpha/2}^2 - b_1^2)\widehat{\text{Var}}(b_3) + 2b_1 b_3 \widehat{\text{Cov}}(b_1, b_3)}}{b_3^2 - t_{\alpha/2}^2 \widehat{\text{Var}}(b_3)}$

The region of significance is the range of $W$ values where the simple slope of $X$ on $Y$ is statistically significant ($p < \alpha$).

3.9 The Conditional Process Model — Moderated Mediation

The moderated mediation model combines mediation and moderation. The indirect effect $a \times b$ becomes a function of the moderator $W$:

General form (moderator on the $a$-path):

$M = i_M + a_1 X + a_2 W + a_3 (X \times W) + e_M$

$Y = i_Y + c'X + bM + e_Y$

The conditional indirect effect at value $w$ of $W$:

$\text{Indirect Effect}(w) = (a_1 + a_3 w) \times b$

This is a linear function of $W$. The index of moderated mediation (Hayes, 2015) is $a_3 b$:

• If the CI for $a_3 b$ excludes zero, the indirect effect is significantly moderated.

General form (moderator on the $b$-path):

$M = i_M + a X + e_M$

$Y = i_Y + c'X + b_1 M + b_2 W + b_3 (M \times W) + e_Y$

The conditional indirect effect at value $w$ of $W$:

$\text{Indirect Effect}(w) = a \times (b_1 + b_3 w)$

Index of moderated mediation: $a \times b_3$

4. Assumptions of Mediation and Moderation Analysis

4.1 Causal Ordering (Temporal Precedence)

The most fundamental assumption of mediation analysis is that the hypothesised causal ordering is correct: $X$ must precede $M$ in time, and $M$ must precede $Y$ in time.

Why it matters: Mediation analysis estimates causal indirect effects. If the temporal order is wrong (e.g., $M$ actually precedes $X$), the computed $a \times b$ product does not represent a causal indirect effect — it is merely a partial correlation.

How to check:

• Use longitudinal data where $X$, $M$, and $Y$ are measured at different time points.
• In experimental designs, randomise $X$ to ensure it causally precedes $M$.
• In cross-sectional data, rely on theory and prior literature to justify the causal order.

⚠️ Cross-sectional data alone cannot establish causal mediation. Mediation from cross-sectional data should always be described as "consistent with a mediation hypothesis" rather than "demonstrating causal mediation."

4.2 No Unmeasured Confounding

Both the $X \rightarrow M$ relationship and the $M \rightarrow Y$ relationship must be free of unmeasured confounders — variables that affect both the predictor/mediator and the outcome simultaneously.

Why it matters: If an unmeasured variable $C$ affects both $M$ and $Y$, the $b$-path estimate (effect of $M$ on $Y$) is biased, and the indirect effect $ab$ does not represent a causal effect.

How to address:

• Measure and control for known confounders.
• Use experimental manipulation of $M$ when possible.
• Conduct sensitivity analysis (e.g., Imai et al.'s sensitivity parameter $\rho$) to assess how large an unmeasured confounder would need to be to nullify the indirect effect.

4.3 Linearity

The model assumes that all relationships (paths) are linear. Non-linear relationships (e.g., curvilinear effects, threshold effects) are not captured by standard mediation and moderation models.

How to check:

• Plot residuals against fitted values (should show no pattern).
• Add polynomial terms (e.g., $X^2$) to the model and test their significance.
• Use partial regression plots (added variable plots) for each predictor.

4.4 Normally Distributed Residuals

The residuals $e_M$ and $e_Y$ should be approximately normally distributed. This is required for the validity of t-tests and F-tests of regression coefficients.

How to check:

• Q-Q plots of residuals.
• Shapiro-Wilk test of residuals.
• Histograms of residuals.

Remedy when violated: Use bootstrapped confidence intervals for indirect effects — bootstrapping does not require normality of the indirect effect and is robust to non-normal residuals.

4.5 Homoscedasticity

The variance of the residuals should be constant across all levels of the predictors (homoscedasticity). Heteroscedasticity (non-constant variance) inflates or deflates standard errors, affecting the validity of significance tests.

How to check:

• Breusch-Pagan test.
• White test.
• Plot residuals vs. fitted values: look for a fan shape.

Remedy when violated: Use heteroscedasticity-consistent (HC) standard errors (White's robust SEs). DataStatPro implements HC3 robust standard errors as an option.

4.6 Independence of Observations

Each observation must be independent of all others. Violations occur with clustered data (e.g., students within schools), repeated measures (same subject at multiple times), or dyadic data (e.g., couples).

How to address:

• Clustered data: Use multilevel mediation/moderation models.
• Longitudinal/repeated measures: Use cross-lagged panel models or cross-lagged mediation.
• Dyadic data: Use Actor-Partner Interdependence Model (APIM) with mediation extensions.

4.7 No Perfect Multicollinearity

Predictors should not be perfectly correlated with each other. In moderation analysis, the interaction term $X \times W$ is often highly correlated with $X$ and $W$ individually (multicollinearity), which can inflate standard errors and destabilise coefficient estimates.

How to check:

• Variance Inflation Factor (VIF): $\text{VIF} > 10$ is typically a concern.
• Condition index from eigenvalue decomposition.

Remedy: Mean-center $X$ and $W$ before forming the interaction term $X \times W$ (as described in Section 3.7). This substantially reduces multicollinearity without changing $b_3$.

4.8 Adequate Sample Size and Statistical Power

Mediation analysis (especially bootstrapped indirect effects) requires sufficient sample size for stable and reproducible results:

💡 Use Monte Carlo power analysis (e.g., via the pwrss or pwr2ppl packages in R, or the DataStatPro power module) to determine the required sample size for your specific model and effect size prior to data collection.

5. Types of Mediation and Moderation Models

5.1 Mediation Models

5.2 Moderation Models

5.3 Conditional Process Models (Moderated Mediation / Mediated Moderation)

5.4 Key Terminology Clarification

6. Using the Mediation and Moderation Component

The Mediation and Moderation component in DataStatPro provides a complete workflow for specifying, estimating, evaluating, and visualising all mediation, moderation, and conditional process models.

Step-by-Step Guide

Step 1 — Select Dataset

Choose the dataset from the "Dataset" dropdown. Ensure:

• All variables (predictor, mediator(s), moderator(s), outcome, covariates) are in separate columns.
• All variables are numeric (continuous or binary coded 0/1).
• The dataset has been screened for missing data and outliers.

💡 Tip: Run descriptive statistics (means, SDs, skewness) before mediation/moderation analysis. Extreme skewness ($|z| > 2$) may affect the normality of residuals and make bootstrapped CIs preferable.

Step 2 — Select Analysis Type

Choose from the "Analysis Type" dropdown:

• Mediation Analysis — for pure mediation models (simple, parallel, sequential).
• Moderation Analysis — for pure moderation models (simple, multiple, three-way).
• Conditional Process Analysis — for combined moderated mediation / mediated moderation models.

Step 3 — Specify the Model Structure

• Predictor (X): Select the independent variable.
• Outcome (Y): Select the dependent variable.
• Mediator(s) (M): Select one or more mediator variables (for mediation models). Specify the type of mediation:
  • Parallel (mediators are not causally connected).
  • Sequential (specify the causal order of mediators).
• Moderator(s) (W, V): Select one or more moderator variables (for moderation and conditional process models).
• Covariates (C): Select any control variables to include in all equations.

⚠️ Important: Specify the model structure based on your theory, NOT based on what produces the best statistics. Choosing the model after seeing the data is a form of HARKing (Hypothesising After Results are Known) and invalidates statistical inference.

Step 4 — Select the Conditional Process Model Type

If running a Conditional Process Analysis, select the specific model from the dropdown:

• First-Stage Moderated Mediation — $W$ moderates the $X \rightarrow M$ path.
• Second-Stage Moderated Mediation — $W$ moderates the $M \rightarrow Y$ path.
• Both-Stages Moderated Mediation — $W$ moderates both $a$ and $b$ paths.
• Mediated Moderation — $M$ mediates the $X \times W \rightarrow Y$ effect.
• Sequential Moderated Mediation — sequential mediators with moderation.

Step 5 — Select Probing Options (for Moderation)

Specify how to probe significant interactions:

• Spotlight Analysis (Pick-a-Point): Compute simple slopes at Low ($-1\text{SD}$), Mean, and High ($+1\text{SD}$) values of $W$.
• Johnson-Neyman (Floodlight) Analysis: Compute the exact region(s) of significance across all values of $W$.
• Custom values of $W$: Enter specific percentiles or theoretically meaningful values.

💡 Tip: Always request both spotlight and Johnson-Neyman analyses. The pick-a-point approach is useful for visualisation; Johnson-Neyman gives a complete picture of where the interaction is and is not significant.

Step 6 — Configure Bootstrap Settings

• Number of Bootstrap Samples ($B$): Default = 5,000. Increase to 10,000 for publication.
• Confidence Level: Default = 95% (BCa bootstrapped CIs). Change to 90% or 99% as needed.
• Bootstrap Method:
  • Percentile CI: Simple and widely used.
  • Bias-Corrected and Accelerated (BCa) CI: Corrects for bias and skewness in the bootstrap distribution. Recommended for indirect effects.

💡 Recommendation: Use BCa bootstrapped CIs with $B = 10{,}000$ for indirect effects in published work. This is the gold standard for mediation analysis.

Step 7 — Mean-Centering Options

For models involving interaction terms (moderation or conditional process):

• Mean-center all continuous predictors and moderators: Recommended — reduces multicollinearity.
• Standardise all variables: Alternative — produces fully standardised coefficients.
• No centering: Use only when $X = 0$ and $W = 0$ are meaningful reference points.

Step 8 — Display Options

Select which outputs and visualisations to display:

• ✅ Path diagram with estimated coefficients.
• ✅ Regression equation summaries for each equation in the model.
• ✅ Direct, indirect, and total effects table.
• ✅ Bootstrapped confidence intervals for indirect effects.
• ✅ Simple slopes table and plot (for moderation).
• ✅ Johnson-Neyman plot with region of significance.
• ✅ Interaction plot (Y vs. X at Low/Mean/High values of W).
• ✅ Conditional indirect effects table (for conditional process models).
• ✅ Index of moderated mediation with bootstrapped CI.

Step 9 — Run the Analysis

Click "Run Analysis". The application will:

1. Estimate all regression equations using OLS.
2. Compute direct, indirect, and total effects.
3. Generate $B$ bootstrap samples and compute the bootstrap distribution of indirect effects.
4. Construct BCa bootstrapped confidence intervals for all indirect effects.
5. Compute simple slopes at all specified values of $W$.
6. Compute the Johnson-Neyman region of significance.
7. Generate all selected visualisations and output tables.

7. Mediation Analysis

7.1 Simple Mediation

Simple mediation tests whether the effect of $X$ on $Y$ is transmitted through a single mediator $M$. This is the foundational mediation model.

7.1.1 Model Specification

The model requires two regression equations:

Path a equation:

$M = i_M + a X + e_M$

Path b and c' equation:

$Y = i_Y + c'X + bM + e_Y$

Total effect equation (for reference):

$Y = i_Y + cX + e_Y$

The path diagram is: a b X ─────────────→ M ─────────────→ Y ↘ ↗ ──────────── c' ───────────

7.1.2 The Four Conditions (Baron & Kenny, Historical)

For historical context, the Baron-Kenny causal steps approach required:

1. $X$ significantly predicts $Y$ (total effect $c$ significant).
2. $X$ significantly predicts $M$ (a-path significant).
3. $M$ significantly predicts $Y$ controlling for $X$ (b-path significant).
4. The effect of $X$ on $Y$ is reduced when $M$ is included ($c' < c$).

⚠️ The Baron-Kenny approach is now considered outdated and is NOT recommended. Condition 1 is not required (mediation can exist without a significant total effect — called "inconsistent mediation"). Use bootstrapped indirect effects instead.

7.1.3 Modern Approach: Bootstrapped Indirect Effects

The modern approach tests mediation by constructing a bootstrapped confidence interval for the indirect effect $ab$:

1. Estimate $\hat{a}$ and $\hat{b}$ from the data.
2. Compute $\hat{a} \times \hat{b}$ (point estimate of indirect effect).
3. Generate $B = 5{,}000$ bootstrap samples.
4. Compute $\hat{a}^*_b \times \hat{b}^*_b$ in each bootstrap sample $b$.
5. Construct the 95% BCa CI from the bootstrap distribution.
6. Decision: If the 95% CI for $ab$ excludes zero → significant mediation.

7.1.4 Types of Mediation Based on Effect Sizes

7.1.5 Worked Calculation — Simple Mediation

Suppose $n = 250$, and we test whether self-efficacy ($M$) mediates the effect of exercise ($X$, hours/week) on wellbeing ($Y$, 0–100 scale).

Path a equation (predicting self-efficacy from exercise):

$\hat{M} = 4.20 + 0.58 X$ → $\hat{a} = 0.58$, $SE = 0.11$, $p < .001$

Path b and c' equation (predicting wellbeing from exercise and self-efficacy):

$\hat{Y} = 21.30 + 0.31 X + 0.44 M$ → $\hat{b} = 0.44$, $\hat{c'} = 0.31$

Total effect (predicting wellbeing from exercise only):

$\hat{Y} = 23.15 + 0.57 X$ → $\hat{c} = 0.57$

Indirect effect:

$\hat{a} \times \hat{b} = 0.58 \times 0.44 = 0.255$

Verification: $c' + ab = 0.31 + 0.255 = 0.565 \approx c = 0.57$ ✅

Bootstrap 95% BCa CI for indirect effect: $[0.132, 0.403]$

Conclusion: The indirect effect of exercise on wellbeing through self-efficacy is $ab = 0.255$ (95% BCa CI [0.132, 0.403]). Since the CI excludes zero, self-efficacy significantly mediates the exercise-wellbeing relationship. This is partial mediation — the direct effect ($c' = 0.31$) remains significant. Each additional hour of exercise per week is associated with a 0.255-point increase in wellbeing through increased self-efficacy.

7.2 Parallel Multiple Mediation

Parallel multiple mediation tests whether $X$ affects $Y$ through two or more simultaneous mediators that are not causally connected to each other.

7.2.1 Model Specification (Two Mediators)

Path equations:

$M_1 = i_{M_1} + a_1 X + e_{M_1}$

$M_2 = i_{M_2} + a_2 X + e_{M_2}$

$Y = i_Y + c'X + b_1 M_1 + b_2 M_2 + e_Y$

Path diagram:

      a₁         b₁
 ────────→ M₁ ────────↘

X ──────↗ Y ────────→ M₂ ────────↗ a₂ b₂ ↘────────────↗ c'

7.2.2 Effects Decomposition

7.2.3 Testing Contrasts Between Indirect Effects

A key advantage of parallel mediation over separate simple mediations is the ability to directly compare specific indirect effects. The contrast tests whether mediator $M_1$ transmits significantly MORE or LESS of the effect than $M_2$:

$C_{12} = a_1 b_1 - a_2 b_2$

A bootstrapped 95% CI for $C_{12}$ that excludes zero indicates that the two indirect effects differ significantly. This allows statements such as: "The indirect effect through self-efficacy was significantly larger than the indirect effect through motivation ($C_{12} = 0.18$, 95% BCa CI [0.04, 0.35])."

7.2.4 Why Parallel Mediation is Preferred Over Separate Analyses

⚠️ Never run separate simple mediation analyses for each mediator and compare results informally. Always include all mediators simultaneously in a single model to correctly estimate specific indirect effects.

7.3 Sequential (Serial) Mediation

Sequential mediation (also called serial mediation) tests a causal chain hypothesis where $X$ affects $M_1$, which then affects $M_2$, which then affects $Y$. This model makes stronger theoretical claims than parallel mediation.

7.3.1 Model Specification (Two Sequential Mediators)

Equation 1:

$M_1 = i_{M_1} + a_1 X + e_{M_1}$

Equation 2:

$M_2 = i_{M_2} + a_2 X + d_{21} M_1 + e_{M_2}$

Equation 3:

$Y = i_Y + c'X + b_1 M_1 + b_2 M_2 + e_Y$

Path diagram: a₁ d₂₁ b₂ X ────────→ M₁ ───────→ M₂ ───────→ Y ↘─────────↗ ↘──── a₂ ───────→ M₂ ↘──── a₁b₁ ───────→ Y c'

7.3.2 The Three Indirect Effects

7.3.3 When to Use Sequential vs. Parallel Mediation

💡 Use sequential mediation ONLY when there is strong theoretical (and ideally temporal) justification for a causal chain between mediators. If the causal connection between $M_1$ and $M_2$ is ambiguous, use parallel mediation and add the $M_1 \rightarrow M_2$ path as a sensitivity check.

7.3.4 Extending to Three or More Sequential Mediators

With three sequential mediators $M_1 \rightarrow M_2 \rightarrow M_3$, the model contains five equations and seven indirect effects (one for each possible sub-path combination), including the full chain $a_1 d_{21} d_{32} b_3$.

For $k$ sequential mediators, the number of indirect effects is $2^k - 1$.

⚠️ Models with more than two sequential mediators require very large samples ($n \geq 500$) and should have very strong theoretical justification. The risk of overfitting and non-replication increases substantially with model complexity.

8. Moderation Analysis

8.1 Simple Moderation

Simple moderation tests whether the relationship between $X$ and $Y$ depends on the level of a third variable $W$ (the moderator).

8.1.1 Model Specification

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

The hypothesis of moderation is tested by $H_0: b_3 = 0$.

A significant $b_3 \neq 0$ indicates that the effect of $X$ on $Y$ varies as a function of $W$.

8.1.2 Interpreting the Interaction Coefficient

The interaction coefficient $b_3$ tells us:

"For each one-unit increase in $W$, the effect of $X$ on $Y$ changes by $b_3$ units."

Or equivalently:

"The conditional effect of $X$ on $Y$ is $b_1 + b_3 W$."

The sign and magnitude of $b_3$ determine the pattern of moderation:

8.1.3 Simple Slopes Analysis (Spotlight Analysis)

After finding a significant interaction, compute simple slopes at three values of $W$:

Low $W$: $\hat{\theta}_{XY}(W_L) = b_1 + b_3(M_W - SD_W)$

Mean $W$: $\hat{\theta}_{XY}(M_W) = b_1 + b_3(M_W)$

High $W$: $\hat{\theta}_{XY}(W_H) = b_1 + b_3(M_W + SD_W)$

For each simple slope, compute:

• The unstandardised coefficient (the simple slope value).
• The standard error.
• The t-statistic and p-value.
• The 95% confidence interval.

8.1.4 Visualising Moderation

The interaction should always be visualised with an interaction plot (also called a moderation plot or floodlight plot):

• X-axis: The predictor $X$ (typically at two values: $X \pm 1\text{SD}$).
• Y-axis: The predicted outcome $\hat{Y}$.
• Lines: One line for each level of $W$ (Low, Mean, High).

Non-parallel lines indicate moderation. Crossing lines indicate that the direction of the $X \rightarrow Y$ relationship reverses across values of $W$.

8.1.5 Moderator Variable Types

For binary moderators: The interaction test compares the simple slope of $X$ on $Y$ in Group 0 vs. Group 1. The simple slope in Group 0 is $b_1$ (since $W = 0$); in Group 1 it is $b_1 + b_3$ (since $W = 1$).

8.2 Multiple Moderation

Multiple moderation tests whether two (or more) moderators $W_1$ and $W_2$ independently moderate the $X \rightarrow Y$ relationship in the same model.

8.2.1 Model Specification

$Y = b_0 + b_1 X + b_2 W_1 + b_3 W_2 + b_4 (X \times W_1) + b_5 (X \times W_2) + e$

Where:

• $b_4$ = interaction between $X$ and $W_1$ (adjusting for $W_2$ and $X \times W_2$).
• $b_5$ = interaction between $X$ and $W_2$ (adjusting for $W_1$ and $X \times W_1$).

The conditional effect of $X$ on $Y$ given specific values of $W_1 = w_1$ and $W_2 = w_2$:

$\hat{\theta}(w_1, w_2) = b_1 + b_4 w_1 + b_5 w_2$

8.2.2 Interpreting Multiple Moderation Results

Each interaction coefficient ($b_4$ and $b_5$) is interpreted conditional on the other moderator being held at zero (or at its mean, if centred):

• $b_4$: How the effect of $X$ on $Y$ changes per unit of $W_1$, at $W_2 = 0$ (or $\bar{W}_2$).
• $b_5$: How the effect of $X$ on $Y$ changes per unit of $W_2$, at $W_1 = 0$ (or $\bar{W}_1$).

Simple slopes are probed at all combinations of $W_1$ and $W_2$ levels (e.g., $3 \times 3 = 9$ combinations for Low/Mean/High values of both moderators).

💡 Multiple moderation models require substantially larger samples than simple moderation ($n \geq 300$ recommended) because two interaction terms consume more degrees of freedom and interactions are typically harder to detect statistically.

8.3 Moderated Moderation (Three-Way Interaction)

Moderated moderation (also called a three-way interaction) tests whether the moderation effect of $W$ on the $X \rightarrow Y$ relationship is itself moderated by a third variable $V$. In other words: does the strength of the interaction between $X$ and $W$ depend on $V$?

8.3.1 Model Specification

$Y = b_0 + b_1 X + b_2 W + b_3 V + b_4 (X \times W) + b_5 (X \times V) + b_6 (W \times V) + b_7 (X \times W \times V) + e$

Where:

• $b_7$ is the three-way interaction coefficient — the test of moderated moderation.

The conditional interaction effect (how the $X \times W$ interaction varies with $V$):

$\hat{\theta}_{XW}(v) = b_4 + b_7 v$

The conditional simple slope of $X$ on $Y$ at specific values of $W = w$ and $V = v$:

$\hat{\theta}_{X}(w, v) = b_1 + b_4 w + b_5 v + b_7 wv$

8.3.2 Probing Three-Way Interactions

To probe a significant three-way interaction ($b_7 \neq 0$):

Step 1: Pick values of $V$ (e.g., $V_{Low}$, $V_{Mean}$, $V_{High}$).

Step 2: At each value of $V$, compute the two-way interaction between $X$ and $W$:

$\hat{\theta}_{XW}(v) = b_4 + b_7 v$

Step 3: At each combination of $W$ and $V$ values, compute the simple slope of $X$:

$\hat{\theta}_X(w, v) = b_1 + b_4 w + b_5 v + b_7 wv$

This produces $3 \times 3 = 9$ simple slopes (at Low/Mean/High of both $W$ and $V$), organised into three two-way interaction plots, one for each level of $V$.

8.3.3 Visualising Three-Way Interactions

A three-way interaction requires three two-way interaction plots arranged side by side:

• Left panel: $X \rightarrow Y$ at $V_{Low}$, with lines for $W_{Low}$, $W_{Mean}$, $W_{High}$.
• Centre panel: $X \rightarrow Y$ at $V_{Mean}$, with lines for $W_{Low}$, $W_{Mean}$, $W_{High}$.
• Right panel: $X \rightarrow Y$ at $V_{High}$, with lines for $W_{Low}$, $W_{Mean}$, $W_{High}$.

The three-way interaction is significant when the pattern of the two-way $X \times W$ interaction visibly changes across the three panels (e.g., the interaction lines are more or less divergent, or the direction of moderation reverses).

⚠️ Three-way interactions require very large samples ($n \geq 400$ minimum, ideally $n \geq 600$) to achieve adequate statistical power. They are also notoriously difficult to replicate. Always interpret with caution and seek replication.

9. Conditional Process Analysis

Conditional process analysis (Hayes, 2013) refers to models that simultaneously incorporate mediation and moderation — the goal is to understand both how (mediation) and when/for whom (moderation) $X$ affects $Y$.

9.1 Moderated Mediation — First-Stage (W Moderates the a-Path)

In first-stage moderated mediation, the moderator $W$ changes the strength of the $X \rightarrow M$ relationship (the a-path). Different levels of $W$ produce different magnitudes of the effect of $X$ on $M$, which in turn produces different indirect effects.

9.1.1 Model Equations

Equation 1 (a-path moderated by W):

$M = i_M + a_1 X + a_2 W + a_3 (X \times W) + e_M$

Equation 2 (b-path unmoderated):

$Y = i_Y + c'X + bM + e_Y$

9.1.2 The Conditional Indirect Effect

The indirect effect varies with $W$ because the a-path is moderated:

$\text{Indirect Effect}(w) = (a_1 + a_3 w) \times b$

At specific values of $W$:

$\text{IE}(W_{Low}) = (a_1 + a_3 w_L) \times b$

$\text{IE}(W_{Mean}) = (a_1 + a_3 w_M) \times b$

$\text{IE}(W_{High}) = (a_1 + a_3 w_H) \times b$

9.1.3 Index of Moderated Mediation

The index of moderated mediation (IMM) is the rate at which the indirect effect changes as $W$ increases by one unit:

$\text{IMM} = a_3 \times b$

• If the bootstrapped 95% CI for IMM excludes zero → the indirect effect is significantly moderated by $W$.
• The sign of IMM indicates the direction: positive IMM means the indirect effect increases with $W$; negative means it decreases.

9.1.4 Path Diagram

    W
    ↓ (moderates a-path)

X ─────────────→ M ─────────→ Y a₁ + a₃W b ↘────────────────────────↗ c'

9.2 Moderated Mediation — Second-Stage (W Moderates the b-Path)

In second-stage moderated mediation, the moderator $W$ changes the strength of the $M \rightarrow Y$ relationship (the b-path). The mechanism by which $M$ transmits the effect to $Y$ depends on the level of $W$.

9.2.1 Model Equations

Equation 1 (a-path unmoderated):

$M = i_M + aX + e_M$