Statistics One

Lecture 13 Moderation

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Three segments

•  Moderation Example 1 •  Centering predictors •  Moderation Example 2

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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

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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

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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)

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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

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10 8 6 4 2 0246810

246810

X

Z

Ý

Ý = 2 +.6(X) + .2(Z)

Centered, Additive

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5 3 1 -1 -3 -5-3-1135

-3-1135

Ý = 6 +.6(X) + .2(Z)

Ý

X

Z

Uncentered, Moderation

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10 8 6 4 2 0246810

246810

Ý = 2 +.6(X) + .2(Z) + .4(X*Z)

Ý

X

Z

Centered, Moderation

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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

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Lecture 13 ~ Segment 3

Moderation Example 2

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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)

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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

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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

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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

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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

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Results: Model 1

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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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