1. Moderation: Assumptions
David A. Kenny

2. What Are They? Causality Linearity Homogeneity of Variance
No Measurement Error

3. Causality X and M must both cause Y.
Ideally both X and M are manipulated variables and measured before Y. Of course, some moderators cannot be manipulated (e.g., gender).

4. Causal Direction
Need to know causal direction of the X to Y relationship.
As pointed out by Irving Kirsch, direction makes a difference!

5. Surprising Illustration
Judd & Kenny (2010, Handbook of Social Psychology), pp (see Table 4.1).
A dichotomous moderator with categories A and B
The X  Y effect can be stronger for the A’s than the B’s.
The Y  X effect can be stronger for the B’s than the A’s.

6. Direction of Causality Unclear
In some cases, causality is unclear or the two variables may not even be a direct causal relationship.
Should not conduct a moderated regression analysis.
Tests for differences in variances in X and Y, and if no difference, test for differences in correlation.

7. Crazy Idea? Assume that either X  Y or Y  X.
Given parsimony, moderator effects should be relatively weak.
Pick the causal direction by the one with fewer moderator effects.

8. Proxy Moderator
Say we find that Gender moderates the X  Y relationship.
Is it gender or something correlated with gender: height, social roles, power, or some other variable.
Moderators can suggest possible mediators.

9. Graphing
Helpful to look for violations of linearity and homogeneity of variance assumptions.
M is categorical.
Display the points for M in a scatterplot by different symbols.
See if the gap between M categories change in a nonlinear way.

11. Linearity
Using a product term implies a linear relationship between M and X to Y relationship: linear moderation.
The effect of X on Y changes by a constant amount as M increases or decreases.
It is also assumed that the X  Y effect is linear: linear effect of X.

12. Alternative to Linear Moderation
Threshold model: For X to cause Y, M must be greater (lesser) than a particular value.
The value of M at which the effect of X on Y changes might be empirically determined by adapting an approach described by Hamaker, Grasman, and Kamphuis (2010).

13. Second Alternative to Linear Moderation
Curvilinear model: As M increases (decreases), the effect of X on Y increases but when M gets to a particular value the effect reverses.

14. Testing Linear Moderation
Add M2 and XM2 to the regression equation.
Test the XM2 coefficient.
If positive, the X  Y effect accelerates as M increases.
If negative, then the X  Y effect de-accelerates as M increases.
If significant, consider a transformation of M.

15. The Linear Effect of X Graph the data and look for nonlinearities.
Add X2 and X2M to the regression equation.
Test the X2 and X2M coefficients.
If significant, consider a transformation of X.

16. Nonlinearity or Moderation?
Consider a dichotomous moderator in which not much overlap with X (X and M highly correlated).
Can be difficult to disentangle moderation and nonlinearity effects of X.

17. Nonlinear RelationshipY X
Moderation Y X

18. Homogeneity of Variance
Variance in Moderation Analysis X
Y (actually the errors in Y)

19. Different Variance in X for Levels of M
Not a problem if regression coefficients are computed.
Would be a problem if the correlation between X and Y were computed.
Correlations tend to be stronger when more variance.

20. Equal Error Variance A key assumption of moderated regression.
Visual examination
Plot residuals against the predicted values and against X and Y
Rarely tested
Categorical moderator
Bartlett’s test
Continuous moderator
not so clear how to test

21. Violation of Equal Error Variance Assumption: Categorical Moderator
The category with the smaller variance will have too weak a slope and the category with the larger variance will too strong a slope.
Separately compute slopes for each of the groups, possibly using a multiple groups structural equation model.

22. Violation of Equal Error Variance Assumption: Continuous Moderator
No statistical solution that I am aware of.
Try to transform X or M to create homogeneous variances.

23. Variance Differences as a Form of Moderation
Sometimes what a moderator does is not so much affect the X to Y relationship but rather alters the variances of X and Y.
A moderator may reduce or increase the variance in X.
Stress  Mood varies by work versus home; perhaps effects the same, but much more variance in stress at work than home.

24. Measurement Error
Product Reliability (X and M have a normal distribution)
Reliability of a product: rxrm(1 + rxm2)
Low reliability of the product
Weaker effects and less power
Bias in XM Due to Measurement Error in X and M
Bias Due to Differential X Variance for Different Levels of M

25. Differential Reliability
categorical moderator
differential variances in X
If measurement error in X, then reliability of X varies, biasing the two slopes differentially.
Multiple groups SEM model should be considered

26. Additional Webinars
Effect Size and Power
ModText