Continuous Moderator
Variables
in Multiple Regression Analysis
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
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.
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.
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.
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.
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;
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;
Regression Output
• R2 = .113 , p < .001
• ZAR =
.303ZMisanth + .067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?
Idealism = Low (-1)
•
•
•
•
Substitute (-1) for ZIdeal
.303 ZMisanth + .067(-1) + (-.153)(-1) ZMisanth
ZAR = .456 ZMisanth - .067
AR increases by .456 SD for each one SD
increase in Misanth
• Now, watch this simple slope decrease as we
increase idealism.
Idealism = Medium (0)
or High (+1)
• Medium Idealism
303 ZMisanth + .067(0) + .153(0) Zmisanth
• ZAR = . 303 ZMisanth
• High Idealism
• .303 ZMisanth + .067(1) + (-.153)(1) ZMisanth
• ZAR = .15 ZMisanth + .067
Find 2 Points for Each Line
• Low Idealism: ZAR = .456 ZMisanth - .067
• Low Idealism, Low Misanthropy:
 ZAR = .456(-1) -.067 = -.523
• Low Idealism, High Misanthropy:
 ZAR = .456(+1) - .067 = .389
• Mean Idealism: ZAR = . 303 ZMisanth
• Mean Idealism, Low Misanthropy:
 ZAR = .303(-1) = -.303
• Mean Idealism, High Misanthropy:
 ZAR = .303(+1) = .303
• High Idealism: ZAR = .15 ZMisanth + .067
• High Idealism, Low Misanthropy:
 ZAR = .15(-1) + .067 = -.083
• High Idealism, High Misanthropy:
 ZAR = .15(+1) + .067 = .217
Plot the Three Lines
Use Italassi
It Comes with Data
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
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.
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
Click on the Equations Tab
• Enter the parameters for the interaction
model.
Click on the 2-D Tab
• Model: Multiple with interaction
• Misanthropy on the abscissa.
• Move the slider to vary the level of
idealism.
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).
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.
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 150.0000 0.0004
Model
coeff
constant -0.0202
IDEAL
0.0672
MISANTH 0.3028
INT_1
-0.1456
INT_1
se
0.0773
0.0777
0.0777
0.0733
t
-0.2615
0.8643
3.8990
-1.9861
p
0.7940
0.3888
0.0001
0.0488
LLCI
-0.1730
-0.0864
0.1494
-0.2905
ULCI
0.1326
0.2207
0.4563
-0.0007
R-square increase due to interaction(s):
R2-chng
F
df1
df2
p
0.0233 3.9446 1.0000 150.0000 0.0488
The Simple Slopes
Conditional effect of X on Y at values of the moderator(s)
IDEAL Effect
se
t
p
LLCI
ULCI
-1.0000 0.4485 0.1074 4.1738 0.0001 0.2361 0.6608
-0.0000
1.0000
0.3028
0.1572
0.0777
0.1062
3.8990
1.4803
0.0001 0.1494
0.1409 -0.0526
Values for “Effect” here are the
standardized simple slopes at three
levels of standardized Idealism.
0.4563
0.3670
Johnson-Neyman Technique
Moderator values(s) defining Johnson-Neyman
significance region(s)
Value
% below
% above
0.7788
77.2727
22.7273
Misanthropy is significantly correlated
with support for animal rights when the
standardized value of idealism is .77 or
lower.
Predicted Values of zar
Data for visualizing conditional effect
of X on Y
MISANTH
IDEAL
yhat
-1.0000
-1.0000
-0.5358
-0.0000
-1.0000
-0.0874
1.0000
-1.0000
0.3611
-1.0000
-0.0000
-0.3230
-0.0000
-0.0000
-0.0202
1.0000
-0.0000
0.2826
-1.0000
1.0000
-0.1102
-0.0000
1.0000
0.0469
1.0000
1.0000
0.2041
Data for Plot of Simple Slopes
data plot; input Misanthropy Idealism
Animal_Rights; cards;
-1.0000
-0.0000
1.0000
-1.0000
-0.0000
1.0000
-1.0000
-0.0000
1.0000
-1.0000
-1.0000
-1.0000
-0.0000
-0.0000
-0.0000
1.0000
1.0000
1.0000
-0.5358
-0.0874
0.3611
-0.3230
-0.0202
0.2826
-0.1102
0.0469
0.2041
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;
Plot of Simple Slopes
Table of Conditional Effects
Conditional effect of X on Y at values of the moderator (M)
IDEAL
Effect
se
t
p
LLCI
ULCI
-2.0275
0.5981
0.1686
3.5483
0.0005
0.2650
0.9312
-1.0140
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 -0.0558
0.3664
2.0267
0.0076
0.1669
0.0458
0.9635 -0.3221
0.3374
I have trimmed this table a lot, so it
would fit on this slide.
2 Leftmost & 2 Rightmost Cols
data plot_JN; input Idealism Effect llci ulci;
cards;
-2.0275 0.5981 0.2650 0.9312
-1.0140 0.4505 0.2368 0.6642
0.7788 0.1894 0.0000 0.3788
1.0131 0.1553 -0.0558 0.3664
I have trimmed out most of the rows
here to make this fit on this slide.
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;
Johnson-Neyman Plot