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Moderation X-Y relationship is different (stronger, weaker, different sign) for different values of Z. Arrow from Z to Y is necessary → model doesn’t work without it. Symmetric: Z moderates X-Y = X moderates Z-Y. Interaction in ANOVA
= difference between differences, indicated by nonparallel lines, always Z + Z*X + X predicts Y. 4 cases of moderation based on measurement level, Y always interval. Case 1. Regression analysis → X and Z both interval. Start with main
effects model (𝒀 ̂ = 𝒃𝟎 + 𝒃𝟏 𝑿 + 𝒃𝟐 𝒁 ; X → Y, controlling for Z), X & Z don’t depend on eachother, different constants + identical regression weights. Add product term (𝒀 ̂ = 𝒃𝟎 + 𝒃𝟏 𝑿 + 𝒃𝟐 𝒁 + 𝒃𝟑 𝑿𝒁 ; X → Y, controlling for Z), testing b3
= 0, sig = interaction → different constants (b0 + b2Z) + different regression weights (b1 + b3Z) for each Z. Interval = infinite amount of Z values, with different effect of X on Y → non-parallel regression lines (means interaction!). Choose
values suiting research purposes, or if no a priori, conventional Z = M Z – SDZ, MZ and MZ + SDZ. All lines go through P. Linear-by- linear interaction describes a linear relationship between Z and the regression weight for the X-Y relationship.
Product term is not interaction, becomes the interaction only when the lower-order effects X and Z are also in the regression equation. Always include all relevant lower-order effects. Centering → preventing multicollinearity (lowers correlation
XZ with X & Z) + interpreting main effects (after centering, b1 gives effect of X on Y for mean of Z, approximately average effect of X on Y). Centre main effects, make interaction of centred main effects. Now, X is 0 at it’s mean → interpret
effect of Z at the mean of X and effect of X at mean of Z. Interaction is tested hierarchically. First test effect of X and Z → Y, then test X, Z, and XZ → Y. If b3 not sig, stick to simpler model (lecturer doesn’t agree). Interpretation: R square
* 100% = % variance explained. Use unstandardized coefficients (b, weights) for interpretation (e.g., .101 = x + 1, y + .101, also check sig). Negative interaction weight = positive X-Y relation weaker for higher Z. Results for main effects =
average person. Case 2 and 3. Interaction in ANCOVA, one nominal predictor and one continuous predictor. Sig moderation effect → at least 2 groups differ in their relationship between X and Y. Case 2 = nominal X, interval Z →
differences between groups (defined by X) on Y are different for different values of Z. Case 3 = interval X, nominal Z → regression slopes of X and Y are different in the different groups defined by Z. Testing is same bc interaction is symmetric
→ ANCOVA with covariate*factor. Start ANCOVA main effects of factor & covariate, add covariate*factor (include but ignore main effects! → misleading). If interaction is sig, compute regression lines for groups. Case 4. Interaction in
ANOVA → nominal predictor and moderator. Interaction is predicted → group differences on Y are different for different levels of Z. Problems and Limitations in Moderation. • Statistical power (low reliability of product variables) power
for testing interactions is low, need large N for testing moderation. • Post-hoc probing: inspecting and testing what sig interactions mean. • Nonlinear, three-way and higher-order • Other cases: binary/other categorical dependent variables.
Moderation X-Y relationship is different (stronger, weaker, different sign) for different values of Z. Arrow from Z to Y is necessary → model doesn’t work without it. Symmetric: Z moderates X-Y = X moderates Z-Y. Interaction in ANOVA
= difference between differences, indicated by nonparallel lines, always Z + Z*X + X predicts Y. 4 cases of moderation based on measurement level, Y always interval. Case 1. Regression analysis → X and Z both interval. Start with main
effects model (𝒀 ̂ = 𝒃𝟎 + 𝒃𝟏 𝑿 + 𝒃𝟐 𝒁 ; X → Y, controlling for Z), X & Z don’t depend on eachother, different constants + identical regression weights. Add product term (𝒀 ̂ = 𝒃𝟎 + 𝒃𝟏 𝑿 + 𝒃𝟐 𝒁 + 𝒃𝟑 𝑿𝒁 ; X → Y, controlling for Z), testing b3
= 0, sig = interaction → different constants (b0 + b2Z) + different regression weights (b1 + b3Z) for each Z. Interval = infinite amount of Z values, with different effect of X on Y → non-parallel regression lines (means interaction!). Choose
values suiting research purposes, or if no a priori, conventional Z = M Z – SDZ, MZ and MZ + SDZ. All lines go through P. Linear-by- linear interaction describes a linear relationship between Z and the regression weight for the X-Y relationship.
Product term is not interaction, becomes the interaction only when the lower-order effects X and Z are also in the regression equation. Always include all relevant lower-order effects. Centering → preventing multicollinearity (lowers correlation
XZ with X & Z) + interpreting main effects (after centering, b1 gives effect of X on Y for mean of Z, approximately average effect of X on Y). Centre main effects, make interaction of centred main effects. Now, X is 0 at it’s mean → interpret
effect of Z at the mean of X and effect of X at mean of Z. Interaction is tested hierarchically. First test effect of X and Z → Y, then test X, Z, and XZ → Y. If b3 not sig, stick to simpler model (lecturer doesn’t agree). Interpretation: R square
* 100% = % variance explained. Use unstandardized coefficients (b, weights) for interpretation (e.g., .101 = x + 1, y + .101, also check sig). Negative interaction weight = positive X-Y relation weaker for higher Z. Results for main effects =
average person. Case 2 and 3. Interaction in ANCOVA, one nominal predictor and one continuous predictor. Sig moderation effect → at least 2 groups differ in their relationship between X and Y. Case 2 = nominal X, interval Z →
differences between groups (defined by X) on Y are different for different values of Z. Case 3 = interval X, nominal Z → regression slopes of X and Y are different in the different groups defined by Z. Testing is same bc interaction is symmetric
→ ANCOVA with covariate*factor. Start ANCOVA main effects of factor & covariate, add covariate*factor (include but ignore main effects! → misleading). If interaction is sig, compute regression lines for groups. Case 4. Interaction in
ANOVA → nominal predictor and moderator. Interaction is predicted → group differences on Y are different for different levels of Z. Problems and Limitations in Moderation. • Statistical power (low reliability of product variables) power
for testing interactions is low, need large N for testing moderation. • Post-hoc probing: inspecting and testing what sig interactions mean. • Nonlinear, three-way and higher-order • Other cases: binary/other categorical dependent variables.
