1. Moderation: Interpretation David A. Kenny

2. The Moderated Regression Equation Y = aX + bM + cXM + E a = “main effect” of X b = “main effect” of M c = interaction between X and M Important to include both X and M in the model.

3. Interpretation of the Two Main Effects a: “main effect” of X The effect of X when M is zero. b: “main effect” of M The effect of M when X is zero. For these effects to be meaningful, zero must be a meaningful value.

4. Centering Make sure zero in interpretable for X and M. If not then center: Compute the grand mean of X and M and subtract it (or something close).

5. Interpretation of the Interaction Effect (Moderation) c = the effect of XM For every one unit change of M, the effect of X (i.e., increases (or decreases) by c units. Path c represents then how much a changes.

6. The Moderated Effect of X on Y Y = aX + bM + cXM + E Y = (a + cM)X + bM + E The effect of X on Y is equal to a + cM. If we set a + cM to zero, then X has no effect on Y when M = -a/c.

7. Brief Example stress level or S happiness or H socio-economic status (C) H = -2S + 1C + 0.1SC + E For every one unit change of socioeconomic status, the effect of Stress levels on Happiness weakens by 0.1 units. Note that for C = 20, the effect of stress on happiness is zero.

8. Other Strategies for Interpretation Computing simple effects Graphing the moderation

9. Simple Effects The effect that X has on Y at different levels of M. Sometimes called “pick a point.” Two ways to compute: Plug into a + cM the value for M and solve. Complicated to obtain a standard error for this value.

10. Re-centering M ʹ is desired value of M for which a simple effect is desired. Re-estimate the moderated regression equation using M - M ʹ for M. The new value of a will be the simple effect of X when M = M ʹ. Note this gives a p value and a confidence interval.

11. What To Do If M Is Continuous? Pick two values Typically one standard deviation above and below the mean of M. Make sure the values are possible values.

12. Graphing Better than simple effects because it displays all of the effects. How? Compute the least-squares means for each of the four combinations of two values of X two values of M Place in the graph. Connect the sets of two points with a common value of M.

14. Graphing Programs Paul Jose’s ModGraph http://www.victoria.ac.nz/psyc/paul- jose-files/modgraph/ Andrew Hayes’s ModProbehttp://www.afhayes.com/ spss-sas-and-mplus-macros-and- code.html#modprobe

15. Regions of Significance Johnson-Neyman Potthoff extension

16. Regions of Significance Useful if M is measured at the interval level of measurement. Tells you at what intervals of M is the X  Y effect significant. It might tell you the when M is at least 5, the effect is positive and significant, but when M is less than 3, the effect is negative and significant. Can flip X and M.

17. 17 Additional Webinars Assumptions Effect Size and Power ModText