Moderation Analysis — Hayes/PROCESS Model 1

Dr. R. Düsing · University of Osnabrück
Example
XStress level (PSS score, standardized) — predictor WSocial support (BSSS score) — moderator YDepressive symptoms (PHQ-9 score)
Does social support (W) buffer the effect of stress (X) on depression (Y)? A negative interaction coefficient b₃ would mean: the higher the social support, the weaker the impact of stress on depressive symptoms.
① Scatterplot
② Regression Equation
③ Coefficients
④ Simple Slopes / Spotlight Analysis
▶ Extension — Johnson-Neyman Technique (Floodlight Plot)
Johnson-Neyman Technique
The spotlight analysis shows simple slopes only at selected W values. The Johnson-Neyman technique analytically determines at which W values the effect of X on Y crosses the significance boundary:

θ(X→Y|W) = b̂₁ + b̂₃·W SE(θ) = √[ s²·(v₁₁ + W²·v₃₃ + 2W·v₁₃) ] W* from: (b̂₃² − t²·s²·v₃₃)·W² + 2(b̂₁b̂₃ − t²·s²·v₁₃)·W + (b̂₁² − t²·s²·v₁₁) = 0 The floodlight plot shows θ(W) with a 95% CI across the entire W range. The red shading marks the non-significant region (CI contains 0).
Concepts
What is moderation?
Moderation occurs when a third variable W changes the strength or direction of the effect of X on Y. This is modeled via an interaction term: Y = b₀ + b₁X + b₂W + b₃(X·W). The decisive coefficient is b₃: if it is ≠ 0, the X effect depends on the value of W. (Hayes/PROCESS Model 1.) Special case: if X and W are both binary (group × time), this exact model is the econometrics' difference-in-differences analysis — the same interaction term, just framed differently. → Difference-in-Differences
Simple slopes / spotlight
The effect of X on Y at a fixed W value is called the simple slope: θ(X→Y|W) = b₁ + b₃·W. The spotlight analysis examines it at selected W values — classically M−1SD, M, M+1SD, or more robustly the percentile ranks 16, 50, 84. For a normally distributed moderator (not a requirement of moderation!) both give the same result; for skewed distributions, the percentile-rank approach guarantees that the chosen values always lie within the moderator's actual range. Each simple slope is a point estimate with a confidence interval and t-test — showing at which support level stress significantly affects depression.
Johnson-Neyman / floodlight
Instead of individual W values, the Johnson-Neyman technique answers the question across the entire W range: at what W* does the X effect switch from significant to non-significant? The floodlight plot shows θ(W) with a 95% CI; wherever the band includes zero, the effect is not meaningful. This avoids the arbitrariness of fixed spotlight points.
Centering
X and W should be centered (or z-standardized) before the analysis. Then b₁ and b₂ are the effects at the mean of the respective other variable — instead of at an often meaningless value of 0. The interaction coefficient b₃ is unaffected by this. Centering also reduces the (numerically harmless but confusing) collinearity of the product term X·W. This can increase power for the conditional effects b₁ and b₂ (the effect of X or W at the value 0 of the respective other, not main effects!) because their standard errors usually get smaller — it has no effect, however, on the power of the interaction term b₃.
Ordinal vs. disordinal interaction
Ordinal: the rank order of the simple-slope lines stays the same across the X range — W only dampens or amplifies the effect. Disordinal: the lines cross within the data range, the X effect reverses its sign. In the example, a strongly negative b₃ produces a disordinal interaction — at high social support, stress can act in the opposite direction.
Moderation vs. mediation
Easy to confuse, but fundamentally different: moderation asks when / for whom an effect occurs (W changes the X→Y effect, interaction b₃). Mediation asks why / through what path (a mediator M transmits the effect, indirect path a·b). Moderation = Hayes Model 1, mediation = Model 4. → Mediation Analysis
Moderation Analysis — Help
Example

X = stress level (PSS, standardized), W = social support (BSSS score), Y = depressive symptoms (PHQ-9).
Question: does social support dampen the effect of stress on depression? A negative b₃ would suggest a buffering effect.

Model structure — Hayes/PROCESS Model 1

The moderation model (Model 1 by Hayes) extends simple regression with an interaction term X·W:

Y = b₀ + b₁·X + b₂·W + b₃·(X·W) + ε

b₁ is the effect of X when W = 0 (conditional main effect).
b₂ is the effect of W when X = 0.
b₃ is the interaction coefficient: it indicates how much the effect of X on Y changes when W increases by one unit. If b₃ ≠ 0, moderation is present.

In the example (Scenario C): PHQ-9 = b₀ + 0.5·stress − 0.5·support − 0.5·(stress·support) + ε.
b₁ = 0.5: at average support (W = 0), one SD more stress raises the PHQ-9 by 0.5 points.
b₂ = −0.5: social support directly lowers depression (at average stress).
b₃ = −0.5: per SD more support, the stress effect drops by 0.5 — the buffering effect.

Interaction term b₃ — interpretation

b₃ > 0: the X effect gets stronger the larger W is — amplifying moderation.
b₃ < 0: the X effect gets weaker the larger W is — dampening moderation (buffering effect).
Ordinal interaction: the simple-slope lines run parallel, offset from each other (no crossing point within the data range).
Disordinal interaction: the lines cross within the data range — the direction of the X effect reverses for different W values.

In the example: b₃ = −0.5 produces a disordinal interaction. The simple slope of stress is computed as θ = 0.5 + (−0.5)·W:
W = −1 (little support): θ = 1.0 — strong positive stress effect.
W = 0 (average support): θ = 0.5 — moderate effect.
W = +1 (high support): θ = 0 — stress has no effect anymore.
W > +1: θ < 0 — stress lowers depression (crossing point, disordinal).

Important: b₁ and b₂ are conditional effects (when the respective other variable = 0), not marginal main effects as in an ANOVA. Standardizing X and W before the analysis makes interpretation considerably easier.

Simple slopes / spotlight analysis

The spotlight analysis computes the effect of X on Y for selected W values (typically: M−SD, M, M+SD, or P16, P50, P84):

θ(X→Y | W = w) = b̂₁ + b̂₃·w

Each simple slope is a point estimate with a CI and t-test. The t-test checks whether the X effect is significantly different from zero for this specific W value. The three regression lines in the scatterplot visualize these conditional effects.

In the example: the three spotlight lines show the stress→depression effect for people with low (P16 ≈ W = −1), average (P50 ≈ W = 0), and high social support (P84 ≈ W = +1). The crossing of the lines makes the disordinal nature visible: at high support, the line flattens or even tips over.

Johnson-Neyman technique (floodlight plot)

The spotlight analysis fixes W values arbitrarily. The Johnson-Neyman technique analytically determines at which W value W* the simple slope crosses the significance boundary (α = .05):

W* from: (b̂₃² − t²·Var(b̂₃))·W² + 2(b̂₁·b̂₃ − t²·Cov(b̂₁,b̂₃))·W + (b̂₁² − t²·Var(b̂₁)) = 0

The floodlight plot shows the conditional effect θ(W) with a 95% CI across the entire W range. The red-shaded area marks where the effect is statistically not significant (CI contains 0). There can be 0, 1, or 2 transition points W*.

In the example: the floodlight plot shows two transition points W*. In the red (non- significant) region between them, the stress effect cannot be statistically distinguished from zero — a classic buffer. Above the second W*, θ becomes significantly negative: at very high social support, higher stress is associated with lower depression scores. This is the disordinal consequence — social support doesn't just buffer, but past a threshold acts as an active protective factor that reverses the stress response into a positive one.

Scenarios A–D

A — No moderation: b₃ = 0; parallel regression lines, no interaction term needed. Social support may have a main effect, but it doesn't buffer.
B — Ordinal interaction: b₃ ≠ 0, no crossing point within the data range; stress always affects depression positively, just to varying degrees depending on support.
C — Disordinal interaction (example): b₁ = 0.5, b₂ = −0.5, b₃ = −0.5; the slopes cross — at very high support the stress effect reverses.
D — Dichotomous moderator: W ∈ {0, 1}; e.g. support present vs. absent; equivalent to two separate regressions with a test for the difference between slopes.

Causality & limitations

A significant interaction term b₃ statistically demonstrates that the relationship between X and Y depends on W — not that W causally moderates the X effect. That would require an experimental manipulation of W. With observational data, confounding remains possible: third variables could influence both W and Y and produce an apparent moderation. Furthermore, statistical power for interaction terms is markedly lower than for main effects — even in larger samples, b₃ is hard to estimate.

References

Hayes, A. F. (2022). Introduction to Mediation, Moderation, and Conditional Process Analysis (3rd ed.). Guilford.
Johnson, P. O. & Neyman, J. (1936). Tests of certain linear hypotheses and their application to some educational problems. Statistical Research Memoirs, 1, 57–93.