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 an interaction between a dichotomous variable and a continuous variable.
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. Data AR1; infile <point to data file, \Moderate.dat';
*MODERATE.SAS;
Data AR1; infile <point to data file, \Moderate.dat';
input AR Misanth Ideal;
Proc Means stddev min max nmiss;
var AR Misanth Ideal; run;
Proc Standard mean=0 std=1 out=Zs;
Data Interaction; set Zs;
Interact = Misanth*Ideal;
Proc Corr; var AR Misanth Ideal;
Proc Reg;
Model AR = Misanth Ideal interact / tol;
run; QUIT;

6. 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.

7. 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.

8. 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;

9. Create the Interaction Term(s) & Run the Regression
Data Interaction; set Zs; Interact = Misanth*Ideal; Proc Corr; var AR Misanth Ideal; Proc Reg; Model AR = Misanth Ideal interact / tol; run; QUIT;

10. Regression Output R2 = .113 , p < .001
ZAR = .303ZMisanth ZIdeal  .146Zinteract -.02
The interaction is significant, p = .049.
What does the regression look like for low (-1), medium (0), and high (+1) values of the moderator?

11. Do It Yourself
Obtaining the simple slopes yourself is a bit of a pain in the arse.
See
It is much easier to get them by using Hayes’ Process software.

12. 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).

13. 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.

14. Process Hayes
Here I shall use Version 2.16, which is installed on the computers in our labs.
Version 3.0 is available. It covers some models not covered by 2.16 and has syntax that is a bit different from in 2.16.
You must run the Process program, within SAS or SPSS, prior to invoking the macro.

15. 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

16. 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

17. The Simple Slopes
Conditional effect of X (Misanthropy) 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.

18. 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 or lower.

19. 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

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

21. 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;

22. Plot of Simple Slopes

23. 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.

24. 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.

25. 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;

26. Johnson-Neyman Plot

27. Version 3 of Process
%process (data=Zs,y=ar,x=Misanth,w=Ideal,model=1,jn=1,plot=1);
With Version 3 of Process the simple slopes are obtained at the 16th, 50th, and 84th percentiles of the moderator.
If the moderator were perfectly normally distributed in the sample, one standard deviation below the mean would be the 15.87th percentile, the 50th percentile would be the mean, and one standard deviation above the mean would be the 84.13th percentile.

29. 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.

30. 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.

31. Mitigation With No Moderation

32. Mitigation With Moderation