What is it?
• What is the definition of moderation?– “A moderating variable is a third
variable that affects the relationship between two others, so that the nature of the impact of the predictor on the criterion varies according to the level or value of the moderator” (Holmbeck, 1997, p. 599).
• Hmm. Nice definition, but what does it mean in the real world with real variables?
• I think that learning about moderation is helped by learning what mediation is at the same time. So what’s mediation?
Mediation
• “A mediating variable is one which specifies how (or the mechanism by which) a given effect occurs” between an independent variable (IV) and a dependent variable (DV). (Holmbeck, 1997, p. 599).
• Okay, is that clear? Well, maybe not perfectly. Let’s delve into these two phenomena more deeply, and hopefully at the end of my explanation it will be clearer.
My definitions• Moderation: this tells you the
circumstances under which the main effect occurs. If you don’t obtain a significant moderation effect, then you know that the main effect applies about equally at all levels of the IV and moderator. If you do obtain a significant moderation, then you need to look at a figure to see under what joint conditions of the moderator and IV the main effect occurs.
• Mediation: this tells you about the “mechanism” of how the IV affects the DV. Does the effect of the IV “go through” the mediator, or does it merely have a simple direct effect?
• Let’s look at some diagrams.
Statistically, how are they different or similar?
• Let’s begin with moderation.• Let’s choose an IV (e.g., stress), and
a potential moderating variable (e.g., coping), as predictors of a DV (e.g., depression).
• Aiken & West recommend that you centre these two main effects, i.e., subtract the mean from the variables. Reduces multicollinearity. Not a z-score.
• Then you create a product term by multiplying the two main effects. (What do you do when you have a categorical moderator like gender? I’ll show you later.)
• Then you regress your DV on your IVs in a hierarchical fashion.
I’ve always been dubious about this centering thing—
does it matter?
Correlations
1 .481** .929**
. .000 .000
2060 2018 2030
.481** 1 .678**
.000 . .000
2018 2031 2018
.929** .678** 1
.000 .000 .
2030 2018 2030
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
EMUCH
RUMINATE
product w/o centering
EMUCH RUMINATEproduct w/ocentering
Correlation is significant at the 0.01 level (2-tailed).**.
Uncentred variables
Okay, this is with centering
Correlations
1 .481** .317**
. .000 .000
2031 2018 2018
.481** 1 .408**
.000 . .000
2018 2060 2018
.317** .408** 1
.000 .000 .
2018 2018 2018
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
rumination centered
emuch centered
product w/ centering
ruminationcentered
emuchcentered
product w/centering
Correlation is significant at the 0.01 level (2-tailed).**.
Alright, it does seem to make a difference! So don’tforget to center first.
The centering controversy
• I just said that one should centre one’s variables before creating the product term.
• Why? I have read that centering them prevents multicolliearity (and my correlations support this contention).
• I have never actually tested this hypothesis with regard to graphing the results, so I decided to do so.
• I used the centered and uncentered stress and rumination scores in separate regressions, and I found the following:
Results• No differences in beta weights,
significances, or R2s for the main effects.
• No difference in significances or R2s on the third step,
• But the beta weights were different on the third step.
• Why? It has to do with this issue of correlations among the three predictors.
• Is this a reason to be concerned? Only if you report and use the beta weights.
• Most important question: do these two methods yield different figures? Answer: no. See the following pages.
Uncentered variables
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
low med high
Stress
Rumination
high
med
low
Centered variables
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
low med high
Stress
Rumination
high
med
low
So what to do?
• I don’t think that it’s a major issue. You can choose to do it or not as you wish, and it won’t affect the important outcomes: significances, R2s, or the resulting figures.
• Purists will require it, but this is because they hand-computed means for figures. We don’t have to do that anymore.
• So don’t sweat it.
What are you looking for?
• You must have a significant main effect for the IV, in this case, stress. Without this you have nothing to moderate. (must have this)
• Second, you may or may not have a main effect for the moderating variable, coping. If you have a significant main effect, then you know that it has a direct effect on the DV. (usually have this)
• Third, you check to see whether you have significant moderation by determining whether the interaction term is a statistically significant predictor. (rare to find)
Let’s look at the SPSS print-out
Coefficientsa
-5.26E-03 .040 -.133 .895
7.096E-02 .003 .541 26.827 .000
-1.18E-02 .037 -.323 .747
4.788E-02 .003 .365 17.093 .000
9.630E-02 .006 .363 16.992 .000
4.249E-02 .039 1.077 .282
5.123E-02 .003 .391 17.445 .000
9.937E-02 .006 .375 17.407 .000
-1.03E-03 .000 -.076 -3.672 .000
(Constant)
emuch centered
(Constant)
emuch centered
rumination centered
(Constant)
emuch centered
rumination centered
product w/ centering
Model1
2
3
B Std. Error
UnstandardizedCoefficients
Beta
StandardizedCoefficients
t Sig.
Dependent Variable: NEGADJa.
Interpretation please . . .
• Stress (emuch: how much stress is caused by everyday life events) is a main effect predictor, = .54, R2 = .29, p < .001. More stress, worse negative adjustment.
• Rumination (amount of thinking about one’s negative affect) is a positive and significant predictor, = .36, R2 change = .10, p < .001. More rumination, worse negative adjustment.
• Stress by rumination (product term) is a negative and significant predictor, = -.08, R2 change = .005, p < .001. What does this mean? Notice the small unique variance. Huge sample (N = 2,500).
The magic of ModGraphTM
• Let me now introduce you to ModGraph, and tell you what it can do. Since I’m inordinately proud of it, I can’t help but show it off.
• The point I’d like to make from the previous page is: “one cannot interpret the interaction from a significant beta weight”. One must graph the interaction to learn what it means.
• How does one graph these? Well, one could laboriously hand-compute 9 algebraic formulae, and then either draw a figure or import the means for PowerPoint, or . . .
• The enlightened researcher could employ ModGraph to do this in a fraction of the time.
Moderation by Rumination
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
low med high
Stress
Rumination
high
med
low
The interpretation of moderation
• There are a multitude of patterns that will yield a statistically significant interaction. (I am in the process of trying to catalogue all of them.)
• Generally one looks for the “spread” or “fan” effect. The means for the three rumination groups are most different under the condition of low stress. Thus, the most easily understandable interpretation is something like,– “rumination worsens the impact of
stress on negative adjustment under conditions of low stress”.
• Notice that the spread is not dramatic; the huge sample size allowed us to find this subtle effect.
Okay, smartypants, what about categorical moderators?
• The previous example (i.e., rumination) featured a continuous moderator, but what about categorical variables like ethnic group, gender, religion, etc.?
• Well, what do you know, ModGraph can handle them too! At this time, only two-level variables can be probed. In the future, we will be able to handle n-level cases.
Moderation of social support by gender
• We know that females report using more social support than males. In our data, it’s:– Males: 18.9– Females: 22.4
• But moderation can tell you how they actually use it. For example, does social support reduce depression equally for the two genders, or is there a difference between them?
• Three IV terms: social support; gender (0 = males; 1 = females); and social support X gender (product term).
By the way . . .
• You can perform an ANOVA that is similar to this regression:– Neg adj as the DV; and– Gender and Social Support are the two
IVs.
• Difference? All IVs in ANOVAs must be categorical (which is fine for gender), so you must dichotomize (high vs. low) or trichotomize (high, medium, and low) social support.
• I trichotomized in this case, and see what the print-out shows.
Two main effects and an interaction:
ANOVA results
Tests of Between-Subjects Effects
Dependent Variable: CDI
4668.055a 5 933.611 16.273 .000
194493.468 1 194493.468 3390.102 .000
2048.343 2 1024.172 17.852 .000
2586.202 1 2586.202 45.079 .000
533.947 2 266.974 4.653 .010
108144.298 1885 57.371
335136.000 1891
112812.353 1890
SourceCorrected Model
Intercept
SOCTRI
GENDER
SOCTRI * GENDER
Error
Total
Corrected Total
Type III Sumof Squares df Mean Square F Sig.
R Squared = .041 (Adjusted R Squared = .039)a.
Same basic findings in the regression format
Coefficientsa
10.878 .176 61.660 .000
-.174 .032 -.126 -5.505 .000
9.315 .287 32.494 .000
-.241 .033 -.174 -7.361 .000
2.573 .375 .162 6.870 .000
9.494 .299 31.780 .000
-.145 .056 -.105 -2.609 .009
2.469 .377 .156 6.544 .000
-.145 .069 -.082 -2.112 .035
(Constant)
social sup centered
(Constant)
social sup centered
Are you male or female?
(Constant)
social sup centered
Are you male or female?
SOCXGEN
Model1
2
3
B Std. Error
UnstandardizedCoefficients
Beta
StandardizedCoefficients
t Sig.
Dependent Variable: CDIa.
The resulting figure
Moderation by Gender
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
low med high
Social support
Gender
females
males
Simple slopes
• Now there is another great feature of ModGraph that we can talk about! It has the ability to compute simple slopes, and tell you whether a particular slope is significantly different from zero.
• You need to obtain four more bits of information:– Variance of social support (SS);– Variance of the interaction (gender by
SS)– Covariance of interaction by SS; and– N of sample.
• You must request SPSS to output the covariance matrix to get the first three items. Not hard to do.
Simple slopes
• The computation of the simple slopes indicates that the females evidenced a steeper slope compared to the males. What does that mean?– Males: slope = -.02, t = -2.61, p
< .01; r = -.10*; and– Females: slope = -.04, t = -7.20, p
< .00001; r = -.20**.
• This result indicates that females benefit from social support more than males.
Okay, so what did we learn here?
(Well, besides the fact that ModGraph is just about the coolest programme this side of Doom . . ?)
• We learned:– That moderation tells you under what
circumstances something has an effect, i.e., it qualifies the main effects.
– That one cannot explain the moderation effect until you have looked at the graph.
– That interpreting interactions is difficult. Practice, practice, practice.