Moderation analysis (Y ~ X × W)

What is Moderation analysis (Y ~ X × W)?

Moderation asks: does the effect of X on Y depend on the level of W? The model Y = β₀ + β₁X + β₂W + β₃(X·W) + ε has β₃ as the moderator effect — significant ⇒ X's slope changes across W.

Two complementary displays: simple slopes at low (W = mean − 1SD), mean (W = mean), and high (W = mean + 1SD) values, and the Johnson-Neyman region — the values of W at which X's effect crosses significance. JN is more honest than the ±1SD convention because it doesn't depend on arbitrary cut-points.

Mean-centring X and W before forming the interaction removes nuisance collinearity and makes β₁ and β₂ interpretable as main effects at the mean of the other; we do this automatically.

When should I use Moderation analysis (Y ~ X × W)?

• Theory predicts the X → Y relationship varies across W (e.g. treatment effect depends on baseline severity).
• Exploring interactions discovered in a preliminary analysis (with appropriate confirmation in a separate sample).

What data does it need?

X (predictor) + W (moderator) + Y (outcome), all numeric.

What does it report?

Coefficient table, simple slopes at three W values, J-N boundaries, plot of conditional X effect across W.

How do I interpret the result?

If β₃ is significant but the simple slope at each ±1SD point is non-significant, the moderation is weak — the effect of X just barely changes across W.

JN intervals can be empty (X is significant at every W in the range) or cover everything (X is never significant). Both are informative.

See also

• Mediation analysis (X → M → Y)
• Linear regression

References

• Aiken & West (1991). Multiple Regression: Testing and Interpreting Interactions.