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Lecture slides stats1.13.l13.air
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Statistics One
Lecture 13 Moderation
1
Three segments
• Moderation Example 1 • Centering predictors • Moderation Example 2
2
Lecture 13 ~ Segment 1
Moderation Example 1
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Moderation & Mediation
• Moderation and Mediation may sound alike but they are quite different – Moderation (Lecture 13) – Mediation (Lecture 14) – Both demonstrated in R (Lab 7)
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Moderation
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Mediation
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Mediator
An example
• X: Experimental manipulation – Stereotype threat
• Y: Behavioral outcome – IQ test score
• Z: Moderator – Working memory capacity (WMC)
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Moderation
• A moderator variable (Z) will enhance a regression model if the relationship between X and Y varies as a function of Z
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Moderation
• Experimental research – The manipulation of an IV (X) causes change in
a DV (Y) – A moderator variable (Z) implies that the effect
of the IV on the DV (X on Y) is NOT consistent across the distribution of Z
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Moderation
• Correlational research – Assume a correlaton between X and Y – A moderator variable (Z) implies that the
correlation between X and Y is NOT consistent across the distribution of Z
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Moderation
• If X and Y are correlated then we can use regression to predict Y from X
• Y = B0 + B1X + e • CAUTION! • If there is a moderator, Z, then B1 will NOT be
representative across all Z – The relationship between X and Y is different at different
levels of Z
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Moderation model
• If both X and Z are continuous
– Y = B0 + B1X + B2Z + B3(X*Z) + e
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Moderation model
• If X is categorical* and Z is continuous
– Y = B0 + B1(D1) + B2(D2) + B3Z + B4(D1*Z) + B5(D2*Z) e
*3 levels of X
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How to test for moderation
• If both X and Z are continuous – Model 1: No moderation
• Y = B0 + B1X + B2Z + e
– Model 2: Moderation • Y = B0 + B1X + B2Z + B3(X*Z) + e
14
How to test for moderation
• If X is categorical* and Z is continuous – Model 1: No moderation
• Y = B0 + B1(D1) + B2(D2) + B3Z + e
– Model 2: Moderation • Y = B0 + B1(D1) + B2(D2) + B3Z +
B4(D1*Z) + B5(D2*Z) + e
15
How to test for moderation
• Compare Model 1 and Model 2 in terms of overall variance explained, that is, R2
– NHST available for this comparison • Evaluate B values for predictors associated
with the moderation effect – (X*Z) – (D1*Z) and (D2*Z)
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Back to the example
• X: Experimental manipulation – Stereotype threat
• Y: Behavioral outcome – IQ test score
• Z: Moderator – Working memory capacity (WMC)
17
Simulated experiment & data
• Students completed a working memory task • Students then randomly assigned to one of
three experimental conditions – Explicit threat (n = 50) – Implicit threat (n = 50) – Control (n = 50)
• Students then completed an IQ test
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Simulated experiment & data
• Experimental condition is categorical so dummy coding is required – Let the Control group be the referent – Let D1 = Explicit threat – Let D2 = Implicit threat
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Results: Summary statistics
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Results: Summary statistics
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Results: Model 1
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Results: Model 2
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Results: Model comparison
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Results: Scatterplot
• Next slide depicts moderation visually
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END SEGMENT
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Lecture 13 ~ Segment 2
Centering predictors
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Centering predictors
• To center means to put in deviation form • XC = X - M
• Why center? – Two reasons
• Conceptual • Statistical
Centering predictors
• Conceptual reason – Suppose
• Y = child’s verbal ability • X = mother’s vocabulary • Z = child’s age
Centering predictors
• Conceptual reason – The intercept, B0, is the predicted score on Y when all
predictors (X, Z) are zero – If X = zero or Z = zero is meaningless, or impossible,
then B0 will be difficult to interpret – In contrast, if X = zero and Z = zero, are the average
then B0 is easy to interpret
Centering predictors
• Conceptual reason – The regression coefficient B1 is the slope for X
assuming an average score on Z – No moderation effect implies that B1 is consistent
across the entire distribution of Z
Centering predictors
• Conceptual reason – In contrast, a moderation effect implies that B1 is NOT
consistent across the entire distribution of Z – Where in the distribution of Z is B1 most
representative of the relationship between X & Y? – Let’s look at this graphically…
Uncentered, Additive
0
10
20
30
40
50
10 8 6 4 2 0246810
246810
X
Z
Ý
Ý = 2 +.6(X) + .2(Z)
Centered, Additive
0
10
20
30
40
50
5 3 1 -1 -3 -5-3-1135
-3-1135
Ý = 6 +.6(X) + .2(Z)
Ý
X
Z
Uncentered, Moderation
0
10
20
30
40
50
10 8 6 4 2 0246810
246810
Ý = 2 +.6(X) + .2(Z) + .4(X*Z)
Ý
X
Z
Centered, Moderation
0
10
20
30
40
50
5 3 1 -1 -3 -5-3-1135
-3-1135
Ý = 16 + 2.6(X) + 2.2(Z) + .4(X*Z)
Ý
X
Z
Centering predictors
• Statistical reason – The predictors, X and Z, can become highly
correlated with the product, (X*Z) • Multicolinearity: when two predictor variables in a
GLM are so highly correlated that they are essentially redundant and it becomes difficult to estimate B values associated with each predictor
Segment Summary
• Centering predictors – Convert raw scores to deviation scores
• XC = X – M
• Reasons for centering – Conceptual
• Regression constant will be more meaningful – Statistical
• Avoid multicolinearity
END SEGMENT
40
Lecture 13 ~ Segment 3
Moderation Example 2
41
Back to the example
• X: Experimental manipulation – Stereotype threat
• Y: Behavioral outcome – IQ test score
• Z: Moderator – Working memory capacity (WMC)
42
How to test for moderation
• If X is categorical* and Z is continuous – Model 1: No moderation
• Y = B0 + B1(D1) + B2(D2) + B3Z + e
– Model 2: Moderation • Y = B0 + B1(D1) + B2(D2) + B3Z +
B4(D1*Z) + B5(D2*Z) + e
43
WAIT! Center continuous predictor
• If X is categorical* and Z is continuous – Model 1: No moderation
• Y = B0 + B1(D1) + B2(D2) + B3Z.center + e
– Model 2: Moderation • Y = B0 + B1(D1) + B2(D2) + B3Z.center +
B4(D1*Z.center) + B5(D2*Z.center) + e
44
Simulated experiment & data
• Students completed a working memory task • Students then randomly assigned to one of
three experimental conditions – Explicit threat (n = 50) – Implicit threat (n = 50) – Control (n = 50)
• Students performed an IQ test
45
Simulated data
• Experimental condition is categorical so dummy coding is required – Let the Control group be the referent – Let D1 = Explicit threat – Let D2 = Implicit threat
46
Results: Model 1
47
Results: Model 1, Centered
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Results: Model 2
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Results: Model 2, Centered
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Results: Model comparison
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Results: Model comparison, Centered
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END SEGMENT
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END LECTURE 13
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