Moderation Analysis

Tests whether the strength or direction of an X→Y relationship depends on a third variable W — e.g., does supervisor support buffer the stress→burnout link?

Data requirements

Assumptions

• All multiple-regression assumptions for the full model including the product term.
• Both X and W (and their product) measured reliably — the product's reliability is roughly the product of the components' reliabilities.
• Adequate variance in W across its range; sparse regions make simple slopes there meaningless.

Hypotheses

The concept

Estimate Y = b₀ + b₁X + b₂W + b₃(X×W). The interaction coefficient b₃ is the change in X's slope per one-unit increase in W. Mean-centering X and W before forming the product makes b₁ interpretable as the X effect at the average W (it does not change b₃ or its test).

A significant b₃ is only the beginning: probe it. Simple-slopes analysis re-estimates the X→Y slope at chosen W values (mean ± 1 SD, or meaningful values); the Johnson–Neyman technique finds the exact W range where the X effect is significant. Plot the simple slopes — reviewers and readers understand pictures of interactions far better than coefficients.

Worked example

Does supervisor support (W) buffer the effect of workload (X) on burnout (Y)? N = 385 nurses; X and W mean-centered.

Result: b₃ = −0.14, SE = 0.05, p = .004, ΔR² = .018. Simple slopes: at low support (−1 SD) workload → burnout b = 0.48 (p < .001); at high support (+1 SD) b = 0.20 (p = .03) — support significantly weakens the workload–burnout link.

How to run it

df$Xc <- scale(df$workload, scale = FALSE)   # mean-center
df$Wc <- scale(df$support,  scale = FALSE)

model <- lm(burnout ~ Xc * Wc, data = df)     # * adds product term
summary(model)

library(interactions)
sim_slopes(model, pred = Xc, modx = Wc)        # simple slopes + J-N
interact_plot(model, pred = Xc, modx = Wc)     # the picture

import statsmodels.formula.api as smf

df["Xc"] = df.workload - df.workload.mean()
df["Wc"] = df.support - df.support.mean()

model = smf.ols("burnout ~ Xc * Wc", data=df).fit()
print(model.summary())

# simple slopes at W = -1SD, 0, +1SD: refit with shifted W or compute
# b1 + b3*w and its SE via the covariance matrix (model.cov_params()).

Interpreting the output

• b₃ significant → the effect of X is conditional; report ΔR² of the product term.
• Simple slopes at meaningful W values tell the substantive story ('significant only for low-support employees').
• Johnson–Neyman gives the W threshold where the effect switches on/off.
• With centered predictors, b₁ = X effect at average W — say so explicitly.
• Non-significant interactions in modest samples are inconclusive, not evidence of no moderation.

APA-style reporting

Common mistakes

Related methods