1. Continuous Moderator Variables
in Multiple Regression Analysis

2. What is a Moderator?
A variable that alters the relationship between two or more other variables.
If the relationship between X and Y varies across levels of M, then M is a moderator.
“Moderation” is nothing more than what we called “interaction” in factorial ANOVA

3. Misanthropy, Idealism, and Attitudes About Animals
Same data I used to illustrate a Potthoff analysis.
But idealism will not be dichotomized.
The criterion variable is score on the first subscale of the Animal Attitudes scale.
The Animal Rights subscale
12 Likert-type items
Cronbach alpha = .87.

4. Download Moderate.dat from my data files page.
Moderate.sas from my SAS programs page.
Point the program to the data file.
Run the program.

5. Center the Variables ? Subtract mean from each score
For all predictor variables that are involved in the interaction(s)
This is commonly done and believed to prevent problems with
Multicollinearity
And other things (see Howell)
May center the outcome variable too, but that is not necessary.

6. Don’t Center the Variables
As recently demonstrated by Andrew Hayes, it is NOT necessary to center the predictors.
It may, however, be easier to interpret the results if they are centered.

7. Standardize the Variables
Unstandardized regression coefficients are rarely useful for the psychologist.
Just standardize all of the variables to z scores.
Which, of course, are centered.
proc standard mean=0 std=1 out=Zs;

8. Create the Interaction Term(s) & Run the Regression
data Interaction; set Zs;
Interact = Misanth*Ideal;
proc corr; var AR Misanth Ideal Interact;
proc reg; model AR = Misanth Ideal interact / stb tol; run;

9. Regression Output R2 = .113 , p < .001
ZAR = .303ZMisanth ZIdeal  .153Zinteract
The interaction is significant, p = .049.
What does the regression look like for low (-1), medium (0), and high (+1) values of the moderator?

10. Idealism = Low (-1) Substitute (-1) for ZIdeal
.303 ZMisanth (-1) + (-.153)(-1) ZMisanth
ZAR = .456 ZMisanth
AR increases by .456 SD for each one SD increase in Misanth
Now, watch this simple slope decrease as we increase idealism.

11. Idealism = Medium (0) or High (+1)
Medium Idealism 303 ZMisanth (0) (0) Zmisanth
ZAR = . 303 ZMisanth
High Idealism
.303 ZMisanth (1) + (-.153)(1) ZMisanth
ZAR = .15 ZMisanth

12. Find 2 Points for Each Line
Low Idealism: ZAR = .456 ZMisanth
Low Idealism, Low Misanthropy:
ZAR = .456(-1) = -.523
Low Idealism, High Misanthropy:
ZAR = .456(+1) = .389

13. Mean Idealism: ZAR = . 303 ZMisanth Mean Idealism, Low Misanthropy:
Mean Idealism, High Misanthropy:
ZAR = .303(+1) = .303

14. High Idealism: ZAR = .15 ZMisanth + .067
High Idealism, Low Misanthropy:
ZAR = .15(-1) = -.083
High Idealism, High Misanthropy:
ZAR = .15(+1) = .217

15. Plot the Three Lines

16. Use Italassi

17. It Comes with Data

18. Click the Equations Tab
You get Y predicted from X1
Y predicted from X2
Y predicted from X1 and X2
Y predicted from X1 and X2 and the interaction term

19. Click on the 2-D View Tab
Select the predictor variable to display on the abscissa.
Select “Multiple with interaction.”
Move the slider to change the value of the moderator variable.

20. Click on the Variables Tab
Enter these values
Dependent = AR
Independent X1 = Misanthropy
Minimum = -1.97
Maximum = 2.5
Independent X2 = Idealism
Minimum = -2.54
Maximum = 2.54

21. Click on the Equations Tab
Enter the parameters for the interaction model.

22. Click on the 2-D Tab Model: Multiple with interaction
Misanthropy on the abscissa.
Move the slider to vary the level of idealism.

26. Process Hayes
Hayes, A. F. (2013). Introduction to mediation, moderation, and conditional process analysis. New York, NY: Guilford.
Highly recommended for those into mediation, moderation, and moderated mediation (aka conditional process analysis).

27. Process Hayes
Hayes provides SAS and SPSS macros that make it much easier to conduct these analyses.
Download the macros, data files, etc.
The first step is to identify the model that matches the analysis you wish to do.
Run Process.sas.
If the data are not already in SAS, bring them in.

28. Moderate_Process.sas
%process (data=Zs,vars=AR Misanth Ideal, y=ar,x=Misanth,m=Ideal,model=1,jn=1,plot=1);
Model Summary R
R-sq F
df1
df2 p
0.3362
0.1130
6.3721
3.0000
0.0004

29. R-square increase due to interaction(s):
Model
coeff se t p
LLCI
ULCI
constant
0.0773
0.7940
0.1326
IDEAL
0.0672
0.0777
0.8643
0.3888
0.2207
MISANTH
0.3028
3.8990
0.0001
0.1494
0.4563
INT_1
0.0733
0.0488
R-square increase due to interaction(s):
R2-chng F
df1
df2 p
INT_1
0.0233
3.9446
1.0000
0.0488

30. Conditional effect of X on Y at values of the moderator(s)
The Simple Slopes
Conditional effect of X on Y at values of the moderator(s)
IDEAL
Effect se t p
LLCI
ULCI
0.4485
0.1074
4.1738
0.0001
0.2361
0.6608
0.3028
0.0777
3.8990
0.1494
0.4563
1.0000
0.1572
0.1062
1.4803
0.1409
0.3670
Values for “Effect” here are the standardized simple slopes at three levels of standardized Idealism.

31. Johnson-Neyman Technique
Moderator values(s) defining Johnson-Neyman significance region(s)
Value
% below
% above
0.7788
Misanthropy is significantly correlated with support for animal rights when the standardized value of idealism is .77 or lower.

32. Predicted Values of zar
Data for visualizing conditional effect of X on Y
MISANTH
IDEAL
yhat
1.0000
0.3611
0.2826
0.0469
0.2041

33. Data for Plot of Simple Slopes
data plot; input Misanthropy Idealism Animal_Rights; cards;

34. Code for Plot of Simple Slopes
proc sgplot; reg x = misanthropy y = Animal_Rights / group = Idealism nomarkers;
yaxis label='Standardized Support of Animal Rights';
xaxis label='Standardized Misanthropy'; run;

35. Plot of Simple Slopes

36. Table of Conditional Effects
Conditional effect of X on Y at values of the moderator (M)
IDEAL
Effect se t p
LLCI
ULCI
0.5981
0.1686
3.5483
0.0005
0.2650
0.9312
0.4505
0.1082
4.1651
0.0001
0.2368
0.6642
0.7597
0.1922
0.0950
2.0220
0.0450
0.0044
0.3800
0.7788
0.1894
0.0959
1.9759
0.0500
0.0000
0.3788
1.0131
0.1553
0.1068
1.4533
0.1482
0.3664
2.0267
0.0076
0.1669
0.0458
0.9635
0.3374
I have trimmed this table a lot, so it would fit on this slide.

37. 2 Leftmost & 2 Rightmost Cols
data plot_JN; input Idealism Effect llci ulci; cards;
I have trimmed out most of the rows here to make this fit on this slide.

38. Code for Johnson-Neyman Plot
proc sgplot; series x=Idealism y=ulci/curvelabel = '95% Upper Limit' linesattr=(color=red pattern=ShortDash); series x=Idealism y=effect/curvelabel = 'Point Estimate' linesattr=(color=blue pattern=Solid); series x=Idealism y=llci/curvelabel = '95% Lower Limit' linesattr=(color=red pattern=ShortDash); xaxis label = 'Idealism'; yaxis label = 'Conditional effect of misanthropy'; refline 0/axis=y transparency=0.5; refline .7788/axis=x transparency=0.5; run;

39. Johnson-Neyman Plot

40. Main Effects vs Moderation
Imagine a study where the variables are
Level of stress (state)
Dose, in mg, of a new drug (nopressor) designed to reduce blood pressure.
SBP, patients resting systolic blood pressure minus the systolic blood pressure considered to be normal/healthy for a person of the subject’s age and sex.

41. I frequently find my students writing statements (hypotheses) like this: “Dose of nopressor will moderate the effect of stress on systolic blood pressure such that those with high stress will exhibit lower blood pressure when the dose of nopressor is high. That is, nopressor will mitigate the hypertension caused by high stress.” Then I have to explain that the presence of a mitigating effect does not establish moderation.

42. Mitigation With No Moderation

43. Mitigation With Moderation