Mediation & Indirect-Effects Analysis

Definition & Core Concept

• Mediation (a.k.a. indirect-effects analysis)

  • Situation in which the relationship between a predictor (independent variable; IV) and an outcome (dependent variable; DV) can be explained by their mutual association with a third variable (the mediator).

  • Conceptual hallmark: when the mediator is included in the model, the strength of the IV–DV link is reduced (sometimes to zero).

  • Evaluated statistically by estimating the indirect effect and inspecting its confidence interval (CI); mediation is inferred when the CI does not include $0$.

• Contrast with Moderation

  • Moderation asks: Does a variable change the strength or direction* of the IV–DV relationship?* (interaction / slope difference)

  • Mediation asks: Does the IV influence the DV through* another mechanism (the mediator)?* ‑ attempts to explain the relationship.

The Basic Statistical Model

• Simple (total-effect) model: IV $\rightarrow$ DV with regression coefficient $c$ .

• Mediated model (Figure 10.11):- Path $a$: IV $\rightarrow$ Mediator

  • Path $b$: Mediator $\rightarrow$ DV (controlling for IV)

  • Path $c'$ (c-prime): IV $\rightarrow$ DV (controlling for Mediator)

  • Indirect effect: $ab$

  • Direct effect: $c'$

  • Total effect: $c = c' + ab$

When to Use Mediation Analysis

• Theorize that a small direct effect ($c'$) exists after accounting for a large indirect effect ($ab$).

• Empirical or theoretical literature supports:- IV $\rightarrow$ DV linkage

  • IV $\rightarrow$ Mediator linkage

  • Mediator $\rightarrow$ DV linkage

• Checklist (Field):- IV variations account for DV variations.

  • IV variations account for mediator variations.

  • Mediator variations account for DV variations.

  • Adding mediator reduces IV $\rightarrow$ DV association.

Statistical Assumptions

• Normality of residuals.

• Linearity among variables.

• Homoscedasticity (equal error variance across combinations of IV & mediator values).

• Non-zero variance in predictors.

• No extreme outliers.

• DV must be continuous; IV continuous; mediator continuous or dichotomous.

• Multicollinearity: expected because IV and mediator are correlated; may inflate SEs but unavoidable conceptually.

Four Paths & Four-Step (Baron & Kenny-style) Approach

1. Step 1 (path $c$): Regress DV on IV — test if total effect is significant.

2. Step 2 (path $a$): Regress Mediator on IV — test significance.

3. Step 3 (path $b$): Regress DV on Mediator and IV — inspect mediator coefficient.

4. Step 4 (path $c'$): Same regression as Step 3 — inspect IV coefficient.- Total mediation: Steps 1–3 significant and $c' = 0$ .

   • Partial mediation: Steps 1–3 significant and \left|c'\right| < \left|c\right| but $c' \neq 0$ .

Obtaining the Coefficients

• Regression 1 (DV ~ IV) $\Rightarrow c$ .

• Regression 2 (Mediator ~ IV) $\Rightarrow a$ .

• Regression 3 (DV ~ IV + Mediator) $\Rightarrow c'$ (for IV) and $b$ (for Mediator).

Quantifying the Indirect Effect

• Point estimate: $ab$ (unstandardized) or $\frac{ab\times SD<em>{IV}}{SD</em>{DV}}$ (fully standardized).

• Significance decision: examine CI for $ab$.- CI excludes $0$ $\Rightarrow$ mediation present.

• Effect-size variants:- Partially standardized indirect effect.

  • Completely standardized indirect effect.

  • $\kappa^2$ (Preacher & Kelley, 2011).

  • $R^2_{\text{med}}$ (variance accounted for uniquely by indirect path).

Testing $ab = 0$ (Modern Approaches)

1. Separate tests of $a$ and $b$- Easy; lacks CI for $ab$; resembles old four-step method.

2. Sobel test- Compute $s<em>_{ab}=\sqrt{a^{2}s</em>b^{2}+b^{2}s<em>a^{2}}$ (with $s</em>a,s_b$ the SEs of $a,b$).

   • $Z = \dfrac{ab}{s_{ab}}$; compare $\left|Z\right|$ to $1.96$ (two-tailed $\alpha=.05$).

   • Assumes normality of $ab$; low power.

3. Bootstrapping (preferred)- Non-parametric resampling with replacement (e.g. 000-10 000 draws; PROCESS default =5000).

   • Produces bias-corrected & accelerated (BCa) asymmetric CI.

   • No need for normality; higher power; implemented in PROCESS (Model 4 for simple mediation).

4. Monte-Carlo simulation (generate sampling distribution from parameter estimates and SEs).

Example: Pornography Consumption $\rightarrow$ Infidelity Mediated by Relationship Commitment (Lambert et al., 2012)

• Variables:- X (IV): $\ln$-transformed pornography consumption (LnConsum).

  • M (Mediator): Relationship commitment.

  • Y (DV): Infidelity (Infideli).

• Sample: $N=239$ .

Path Estimates

• Path $a$ (X → M)- $b=-0.47$, $SE=.213$, $t=-2.21$, $p=.028$ .

• Path $b$ (M → Y controlling X)- $b=-0.27$, $SE=.059$, $t=-4.61$, p<.001 .

• Path $c'$ (direct X → Y)- $b=0.46$, $SE=.195$, $t=2.35$, $p=.020$ .

• Total effect $c$ (from separate regression)- $b=0.585$, $SE=.201$, $t=2.91$, $p=.004$ .

Indirect-Effect Output (PROCESS Model 4)

• Point estimate: $ab = 0.1273$ .

• Bootstrapped (5000) BCa 95% CI: $[0.0232, 0.3350]$ (does not include $0$).

• Partially standardized effect: $0.1818$, CI $[0.0325, 0.4684]$ .

• Completely standardized effect: $0.0405$, CI $[0.0073, 0.1032]$ .

• Effect-size ratios:- $\frac{ab}{c} = 0.218$ ($\approx$ 22 % of total effect via mediator).

  • $\frac{ab}{c'} = 0.278$ .

• ***$R^2_{\text{med}} = 0.0138$ (CI $\approx$ 0.002–0.048).

• ***$\kappa^2 = 0.041$ (CI $\approx$ 0.008–0.104) — considered small.

• Sobel test: $Z = 1.953$, $p = .051$ (marginal), yet bootstrapped CI is significant $\Rightarrow$ trust bootstrapping.

Interpretation

• Evidence for partial mediation: direct path remains significant but attenuated (\left|0.46\right| < \left|0.585\right|).

• Practical meaning: higher pornography use is related to lower commitment, which in turn predicts greater infidelity; about one-fifth of the total effect is routed through commitment.

Sample Reporting Sentence (Field-style)

• “There was a significant indirect effect of pornography consumption on infidelity through relationship commitment, $b = 0.13$, BCa 95% CI $[0.02, 0.34]$, representing a small effect ($\kappa^2 = .04$, 95% CI $[.01, .10]$).”

Practical & Methodological Notes

• Multicollinearity warning: High correlation between IV and mediator inflates SEs; interpretation focuses on paths rather than unique variance.

• Software: SPSS PROCESS (Model 4), R (e.g., mediation or lavaan packages), Mplus, etc.

• Sample size: Bootstrapping alleviates but does not eliminate the need for adequate N; very small samples can render CIs unstable.

• Ethical/Philosophical: Mediation posits causal ordering; requires theoretical justification and (ideally) longitudinal or experimental design to bolster causal claims.

Quick Reference Equations

• Total effect: $c = c' + ab$

• Indirect effect (unstandardized): $ab$

• SE of Sobel: $s<em>_{ab}=\sqrt{a^{2}s</em>b^{2}+b^{2}s*a^{2}}$

• $Z<em>{Sobel}=\dfrac{ab}{s</em>{ab}}$

• Fully standardized indirect: $\dfrac{ab\, SD<em>{IV}}{SD</em>{DV}}$

These notes cover conceptual distinctions, assumptions, analytic procedures, effect-size calculations, modern testing strategies, and a detailed worked example with full output interpretation—sufficient to replicate or critically appraise a mediation analysis from raw data to publication-ready prose.