Lecture Notes – Moderation & Interaction

Prevalence of Moderation and Mediation in Psychology Literature

• Meta-survey of article titles (1970-2010)
  • Counted occurrences of the words “mediation”, “moderator”, “mediates”.
  • Frequencies per decade:
  • 1970s 267
  • 1980s 639
  • 1990s 1182
  • 2000s 2893
• Clear acceleration evidences growing interest in interaction-based explanations.

Conceptual Definition of Moderation

• Moderation occurs when the X ➜ Y relationship varies as a function of a third variable (M).
• The moderator changes the size and/or direction of the association.
• Everyday illustration:
  • Predictor (X): Watching horror films.
  • Outcome (Y): Feeling scared at night.
  • Moderator (M): Imagination vividness.
  • Stronger imagination ⇒ steeper X ➜ Y slope.

Interaction Effects and Factorial ANOVA

• 2×2 factorial ANOVA partitions variance into:
  • Main effect A
  • Main effect B
  • $A \times B$ interaction
  • Error
• Conceptual overlap: “interaction” in ANOVA = “moderation” in regression.

Conceptual vs. Statistical Moderation

• Conceptually: joint influence of two predictors on an outcome.
• Statistically (Moderated Multiple Regression – MMR):
  • Include main effects and product (interaction) term.
  • Moderator alters $b_{XY}$ depending on its value.
  • Example: Effect of provoked anger on aggression differs with trait aggressiveness.

Regression Model with Interaction Terms

• Generic model (two predictors):
  $Y<em>i = (b</em>0 + b<em>1X</em>i + b<em>2M</em>i + b<em>3X</em>iM<em>i) + \varepsilon</em>i$
• Construction of interaction term:
  • Multiply each person’s centred $X<em>i$ and $M</em>i$ scores.
• Key question: “Does $b_3 \neq 0$?”

Simple Effects and Interpretation

• Simple (conditional) effect: effect of X on Y at a specific value of M.
• Classical notion: moderation often described as weakening of causal effect, but can
  • amplify
  • reverse
  • eliminate (complete moderation ⇒ effect = 0 at some M).

Types of Interaction Patterns

• Enhancing (+): higher M strengthens X ➜ Y.
• Buffering (–): higher M weakens X ➜ Y.
• Antagonistic (×): higher M reverses sign of X ➜ Y.

Designing Moderator Studies

• Specify moderator hypotheses a priori (including expected direction/pattern).
• Greatest causal leverage when either X or M is experimentally manipulated.
• Questionnaire-derived variables are seldom fully independent; interpret cautiously.

Coding Categorical Moderators (Dummy Variables)

• Dichotomous moderator: code as 0/1 (e.g., Male = 0, Female = 1).
• For k-level categorical variable ⇒ create $k-1$ dummy variables.

Centering Predictors

• Problem: Interaction term causes multicollinearity; main-effect coefficients become hard to read (b’s are conditional at M = 0).
• Solution: Grand-mean centering
  • Transform each predictor: $X_c = X - \bar{X}$.
• What centering does:
  • Makes 0 a meaningful value (sample mean).
  • Reduces correlation among X, M, and $X!M$.
  • Leaves $b_3$, $R^2$, F-test unchanged.

Simple Slope Analysis

• An interaction tells us slopes differ – that is the primary information.
• Typical practice with continuous M:
  • Probe X ➜ Y at M = (–1 SD), (Mean), (+1 SD).
• Over-reliance on individual simple-slope p-values may obscure broader interaction.

Higher-Order Interactions

• 3-way model example:
  $Y' = \beta<em>0 + \beta</em>1X + \beta<em>2Z + \beta</em>3M + \beta<em>4XZ + \beta</em>5XM + \beta<em>6MZ + \beta</em>7XZM$
• Same centering/dummy-coding rules apply.
• ALWAYS plot to interpret; choice of plot depends on coding and theory.

Selecting Which Variable Is “the” Moderator

• Largely theory-driven.
• Example: If one studies gender, one may frame therapy as the moderator of gender; another may do the reverse.

Assumptions for Moderated Multiple Regression

1. Dependent variable continuous (interval/ratio).
2. Independent X continuous; M can be continuous or dichotomous.
3. Independence of residuals
   • Durbin-Watson or residual sequence plot.
4. Linearity within each subgroup (for dichotomous M) – check scatterplots.
5. Homoscedasticity – equal error variances across combinations of X and M.
6. No problematic multicollinearity – inspect Tolerance / VIF.
7. No influential outliers – examine Mahalanobis distance, studentized residuals.
8. Residuals approximately normal – Q-Q plot, Shapiro–Wilk.

Advantages & Disadvantages of Moderation Analysis

Advantages

• When X or M is manipulated, can support causal moderation claims.
• Can explain conditional effects ignored by simple main-effect models.
  Disadvantages
• Frequent confusion between statistical interaction and theoretical moderation.
• Without experimental manipulation, still correlational → causality uncertain.

Practical Tips for Moderated Regression

• Always include main effects.
• Never dichotomize continuous predictors.
• Use full regression equation for interpretation.
• Consider centering when zero-points are arbitrary.
• Be cautious with ordinal predictors.
• Significant interaction ⇒ moderation present.

Timing of Measurement

• Ideally measure moderator before manipulating/measuring X.
  • Ensures M is unaffected by X when X is manipulated.
• Time-invariant moderators (e.g., race) less sensitive to timing.

Relationship Between X and M

• If X is randomized, $\text{Cov}(X,M) \approx 0$.
• If not randomized, X and M may correlate; that correlation has no special meaning (unlike mediation).
• Excessive X–M correlation inflates collinearity in $XM$ term.

Worked Example: Video Games, Callous Traits, Aggression

• Research Question: Do violent video games raise aggression more for youths with callous-unemotional traits?
• Variables
  • $X$ = Weekly hours of violent video-game play (centred)
  • $M$ = Callous-unemotional traits (centred)
  • $Y$ = Aggression score

Statistical Model Specification

$Aggression<em>i = (b</em>0 + b<em>1\,Gaming</em>i + b<em>2\,Callous</em>i + b<em>3\,Gaming</em>i\,Callous<em>i) + \varepsilon</em>i$

PROCESS Macro Output (Model 1)

• Sample size = 442.
• Model summary
  • $R = .614$, $R^2 = .377$ (≈38 % variance explained)
  • Overall F(3,438) = 90.53, p < .001.
• Coefficients
  • Intercept $b_0 = 39.97$ (SE =.48)
  • Callous traits $b_2 = .76$ (SE =.047), $t=16.30$, p<.001
  • Gaming $b_1 = .17$ (SE =.076), $t=2.23$, $p=.026$
  • Interaction $b_3 = .027$ (SE =.007), $t=3.71$, p<.001
  • → Significant moderation.

Conditional (Simple) Effects Table

• For Callous = Mean–1 SD (–9.62): $b = -0.09$, n.s.
• For Callous = Mean (0): $b = 0.17$, p =.026.
• For Callous = Mean+1 SD (+9.62): $b = 0.43$, p
• Interpretation: Video-game aggression link strengthens as callous traits rise.

Johnson–Neyman Technique

• Identifies region where X ➜ Y is significant.
• Critical moderator values:
  • $M < -17.10$ and $M > -0.72$ ⇒ effect significant at $\alpha=.05$.
  • Shows continuous range, not just ±1 SD.

Graphing & Simple Slopes

• Create predicted values for combinations of Gaming (Low = –1 SD, Mean, High = +1 SD) and Callous (Low/Mean/High).
• Plot illustrates diverging lines:
  • Low-callous youths: almost flat relation.
  • High-callous youths: steep positive slope (aggression rises sharply with gaming).

Reporting Results

"A hierarchical regression tested whether callous-unemotional traits moderated the association between violent video-game play and aggression. After centring predictors, the interaction term was significant, $b = .027$, $SE =.007$, $t(438) = 3.71$, p < .001, $\Delta R^2 = .02$. Simple-slope analyses revealed that video-game exposure predicted aggression only at average or high levels of callous traits (Mean: $b=.17$, p =.026; +1 SD: $b=.43$, p

Summary Cheat-Sheet

• Moderation = interaction; ask whether effect of X depends on M.
• Include X, M, and $X\times M$ in regression.
• Centre continuous predictors; dummy-code categorical.
• Check assumptions (normality, homoscedasticity, multicollinearity, etc.).
• Probe significant interactions with simple slopes or Johnson–Neyman.
• Remember: causal claims require at least one experimental manipulation and correct temporal ordering.