2011 10 18 moderation mediation

8/9/2019 2011 10 18 Moderation Mediation

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Moderator and MediatorAnalysis

Marijtje van Duijn

October 18, 2011

Seminar General Statistics

October 18, 2011 Moderator and mediator analysis 2

Overview

What is moderation and mediation?

What is their relation to statistical

concepts? Example(s)


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Mediation and Moderation

X1

X2

Y


Examples

Y : test score

X1 : sex, SES, etc.

X2 : ability (IQ score)

Y : test score

X1 : brain volume, SES parents

X2 : ability

More?


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Multiple regression

Goal: to explain variation in Y using Xs

Assumptions Independent observations

Normality (of residuals) and constantvariance

Linearity (of relationship Y and Xs)

iiii EXXY +++=

22110


Regression Model

X1

X2

1

Y

2

E


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Explained variance in regression

Y

X1 X2

Circles represent variancesX1 andX2 explain different parts of Y and are independent


Multicollinearity

Y

X2X1

competition between variables for explaining Y

Degree depends on correlation between XsVariance inflation factor (VIF): worse (= less precise)

parameter estimation


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Mediation as a special case ofmulticollinearity and/or model selection

Causal model defines direction of arrows

X2 is the mediator (M)

Also: intervening or process variable

Or: indirect causal relationship

Relations between all variables are assumed

to be positive Question is whether direct effect betweenX1

and Y disappears when M is added to theregression equation


Mediation(Baron & Kenny, 1986),

http://davidakenny.net/cm/mediate.htm)

X Yc

ba

c

M


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Mediation as prescribed byBaron and Kenny (1986)

Estimate regression of Yon only X1 Estimated parameter c

Estimate regression of Mon X1 Estimated parameter a

Estimate effect of Mon Y, together with X1

Estimated parameters b and c Complete mediation: c=0

Partial mediation: c< c(can be tested)


Testing of change in c

Amount of mediation c-c

Theoretically equal to ab (indirect path)

Standard error of ab is approximately squareroot of

b2sa2 + a2sb2 (Sobel test)

see (do) http://quantpsy.org/sobel/sobel.htmNote: neither c nor c are needed!

Some eye-balling also possible

Or: nonparametric tests (based on bootstrapping)


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Example (Miles and Shevlin)

Y: read, a measure of the number of books thatpeople have read.

X: enjoy, scale score to measure how much peopleenjoy reading books

M: buy, a measure of how many books people havebought in the previous 12 months

Idea: how much people enjoy reading books -> the

number of books bought -> the number of books readBut: is the number of books a complete mediator.?

(People could go to the library or borrow books fromfriends.)


Descriptive Statistics

MeanStd.

Deviation Nread 8,85 3,563 40

buy 15,73 8,165 40

enjoy 9,28 5,354 40

Correlations

read buy enjoyPearsonCorrelation

read 1,000 ,747 ,732

buy ,747 1,000 ,644

enjoy ,732 ,644 1,000

Coefficientsa

Model

Unstandardized

Coefficients

Standardized

Coefficients

t Sig.B Std. Error Beta1 (Constant) 4,331 ,785 5,517 ,000

enjoy ,487 ,074 ,732 6,625 ,000

a. Dependent Variable: read

Step 1


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Step 2Coefficientsa

Model

Unstandardized

Coefficients

Standardiz

ed

Coefficients

t Sig.B Std. Error Beta1 (Constant) 6,616 2,020 3,274 ,002

enjoy ,982 ,189 ,644 5,190 ,000

a. Dependent Variable: buy

Step 3Coefficientsa

Model

Unstandardized

Coefficients

Standardized

Coefficients

t Sig.

Correlations

B Std. Error BetaZero-order Partial Part

1 (Constant) 2,973 ,765 3,887 ,000

buy ,205 ,054 ,471 3,786 ,001 ,747 ,528 ,360

enjoy ,286 ,083 ,429 3,452 ,001 ,732 ,494 ,328

a. Dependent Variable: read


Examples

Baron/Kenny + Sobel a=.982 sa =.189

b=.487 sb =.074

ab=.478, sab =(.4872*.1892+.9822*.0742)= .12

z-test 4.08; p


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Model choice is important

Many other patterns of association arepossible Arrows between X1 and X2 may be

reversed not always clear which variablemediates

No causal relation, just association

Explicit assumption of positiveassociations and ordering of (semi-)partial correlations. Not guaranteed


Moderation

X1

X2

Y


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Moderation is interaction

The effect of X2depends on the value of X1(or vice versa) different slopes for different folks

can be a way to tackle non-linearity in regression

Interpretation depends on type of variable

Interaction usually (but not necessarily)

product of X1 and X2 X1 and X2may be correlated

Importance of model formulation


Regression model

Centering is (usually) advised

Facilitates interpretation Reduces the inevitable multicollinearity

incurred with interaction terms

iiiiii EXXXXY ++++=

21322110

i

c

i

c

i

c

i

c

ii EXXXXY ++++=

21322110


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X1 dichotomous andX2 continuous

X1 = 0 orX1 = 1 (e.g. man/woman; control/experimental group)

Y =0 +1X1 + 2X2 +3X1X2 X1 = 0: Y = 0 +2X2 X1 = 1: Y = (0 +1) + (2 +3)X2

so: intercept and regressioneffect of X2change interaction effect represents the change in effect ofX2 or,

the difference in the effect ofX2 between the two groups interpretation of 1 General formula: Y = (0 + 1X1) + (2+3X1)X2


X1 andX2 dichotomous

X1 = 0 orX1 = 1 (e.g. man/woman)

X2 = 0 orX2 = 1 (e.g. control/experimental grp)

Y =0 +1X1 +2X2 +3X1X2 X1 = 0, X2 = 0 : Y = 0 X1 = 1, X2 = 0 : Y = 0 + 1 X1 = 0, X2 = 1 : Y = 0 + 2 X1 = 1, X2 = 1 : Y = 0 + 1+2+3

so: defines four groups with their own mean

Interaction defines extra effect ofX1 =1 andX2 =1


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X1 nominal andX2 continuous

X1 takes on more than 2 (c) values (e.g. age groups, control/exp1/exp2)

Make dummies, for each contrast, wrt 1reference group (e.g. controls) Leads to c-1 dichotomous variables

And also c-1 interaction terms Also possible: dummies (indicators) for

each group leaving out the intercept


c=3; group 1 reference group

group D1 D2

1 0 0

2 1 0

3 0 1

Y =0 +1dD1 +2dD2 +2X2 +3D1X2 +4D2X2 group1: Y =0 +2X2 group2: Y = (0 +1d) + (2 +3)X2 group3: Y = (0 +2d) + (2 +4)X2


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X1 andX2 continuous

Y =0 +1X1 +2X2 +3X1X2

Y = (0 +1X1) + (2+3X1)X2

Y = (0 +2X2) + (1+3X2)X1

1X

3

2

*

2

2X

3

1

*

1

2X1

X3

2

X2

1

X1

0

*

0

2c1c32c

*

1c

**

222c

111c

XXXXY

XXX

XXX

210

=

=

+=

+++=

=

=

1c

**

22

2c

**

11

XY:XX

XY:XX

10

20

+==

+==


Example (Miles and Shevlin)

Y: grade in statistics course

X1: number of books read (0-4)

X2: number of classes attended (0-20)Descriptive Statistics

Mean Std. Deviation Ngrade 63,5500 16,70552 40

books 2,00 1,432 40

attend 14,10 4,278 40


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Coefficientsa

Model

Unstandardized

Coefficients

Standardiz

ed

Coefficient

s

t Sig.

Correlations

Collinearity

Statistics

B Std. Error Beta

Zero-

order Partial Part

Toleran

ce VIF1 (Constant) 63,422 2,223 28,534 ,000

booksc 4,037 1,753 ,346 2,303 ,027 ,492 ,354 ,310 ,803 1,245

attendc 1,283 ,587 ,329 2,187 ,035 ,482 ,338 ,295 ,803 1,245

2 (Constant) 61,469 2,320 26,494 ,000

booksc 4,081 1,677 ,350 2,433 ,020 ,492 ,376 ,314 ,803 1,245

attendc 1,333 ,562 ,341 2,372 ,023 ,482 ,368 ,306 ,802 1,247

bookscxatte

ndc

,735 ,349 ,271 2,104 ,042 ,241 ,331 ,271 ,997 1,003

a. Dependent Variable: grade


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Other example

(missed) interaction:non-linearity?

3 groups witheach a distinctlinear relation



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Model Summaryb

.405a .164 .134 2.57780

Model1

R R Square

Adjusted

R Square

Std. Error of

the Estimate

Predictors: (Constant), xa.

Dependent Variable: yb.

Coefficientsa

5.724 1.029 5.560 .000

.163 .070 .405 2.341 .027 .405 .405 .405 1.000 1.000

(Constant)

x

Model1

B Std. Error

Unstandardized

Coefficients

Beta

Standardized

Coefficients

t Sig. Zero-order Partial Part

Correlations

Tolerance VIF

Collinearity Statistics

Dependent Variable: ya.

Residuals Statisticsa

5.8864 9.7927 7.8667 1.12042 30

-4.88636 3 .16046 .00000 2.53297 30

-1.767 1.719 .000 1.000 30

-1.896 1.226 .000 .983 30

Predicted Value

Residual

Std. Predicted Value

Std. Residual

M in imum Maximum Mean S td . Deviat ion N


20-2

Regression Standardized Residual

6

4

2

0

Frequency

Mean =-2,78E-17Std. Dev. =0,983

N =30

Histogram

Dependent Variable: y

1,00,80,60,40,20,0

Observed Cum Prob

1,0

0,8

0,6

0,4

0,2

0,0

ExpectedCumProb

Normal P-P Plot of Regression Standardized Residual



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Model Summaryb

.980a .960 .957 .57477

Model1

R R SquareAdjustedR Square

Std. Error ofthe Estimate

Predictors: (Constant), xkwadraat, xa.Dependent Variable: yb.

Coefficientsa

8.512 .259 32.839 .000

.140 .016 .349 9.042 .000 .405 .867 .348 .996 1.004

-.054 .002 -.894 -23.156 .000 -.916 -.976 -.892 .996 1.004

(Constant)

x

xkwadraat

Model1

B Std. Error

Unstandardized

Coefficients

Beta

Standardized

Coefficients


Correlations

Tolerance VIF




.5888 10.4412 7.8667 2.71361 30

-.82773 1.16910 .00000 .55460 30

-2.682 .949 .000 1.000 30-1.440 2.034 .000 .965 30

Predicted Value

Residual

Std. Predicted ValueStd. Residual



20-2


6

4

2

0

Frequency

Mean =-3,61E-16Std. Dev. =0,965

N =30

Histogram


15,0010,005,000,00-5,00-10,00-15,00

x

2,00

0,00

-2,00

y

Partial Regression Plot



Model Summaryb

.410a .168 .107 2.61796

Model1



Predictors: (Constant), groep, xa.


Coefficientsa

6.021 1.301 4.629 .000

.220 .166 .548 1.329 .195 .405 .248 .233 .181 5.515

-.528 1.375 -.158 -.384 .704 .337 -.074 -.067 .181 5.515

(Constant)

x

groep

Model1

B Std. Error

Unstandardized

Coefficients

Beta

Standardized

Coefficients


Correlations

Tolerance VIF




5.7131 9.9467 7.8667 1.13588 30

-4.71313 3 .17007 .00000 2.52607 30

-1.896 1.831 .000 1.000 30

-1.800 1.211 .000 .965 30

Predicted Value

Residual


Std. Residual



20-2


6

4

2

0

Frequency

Mean =5,69E-16Std. Dev. =0,965

N =30

Histogram


4,002,000,00-2,00-4,00

x

2,50

0,00

-2,50

-5,00

y




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Model Summaryb

.750a .562 .512 1.93502

Model1



Predictors: (Constant), groep3, groep2, xa.Dependent Variable: yb.

Coefficientsa

4.556 .912 4.994 .000

.172 .123 .427 1.396 .174 .405 .264 .181 .180 5.552

3.476 1.310 .602 2.653 .013 .645 .462 .344 .327 3.057

-.326 2.038 -.056 -.160 .874 -.030 -.031 -.021 .135 7.394

(Constant)

x

groep2

groep3

Model1

B Std. Error

UnstandardizedCoefficients

Beta

StandardizedCoefficients


Correlations

Tolerance VIF




4.7273 1 1.1227 7.8667 2.07709 30

-3.72727 3 .72727 .00000 1.83220 30

-1.511 1.568 .000 1.000 30-1.926 1.926 .000 .947 30

Predicted Value

Residual

Std. Predicted ValueStd. Residual



20-2


8

6

4

2

0

Frequency

Mean=3,47E-16Std.Dev. =0,947

N =30

Histogram


4,002,000,00-2,00-4,00

x

4,00

2,00

0,00

-2,00

-4,00

y




Model Summaryb

.997a .993 .992 .25050

Model1

R R Square

Adjusted

R Square

Std. Error of

the Estimate

Predictors: (Constant), intxgr3, intxgr2, x, groep2,

groep3

a.


Coefficientsa

4.97E-014 .171 .000 1.000

1.000 .028 2.485 36.259 .000 .405 .991 .609 .060 16.657

10.145 .417 1.756 24.309 .000 .645 .980 .408 .054 18.505

18.000 .596 3.116 30.201 .000 -.030 .987 .507 .026 37.737

-.985 .039 -2.378 -25.250 .000 .626 -.982 -.424 .032 31.455

-1.500 .039 -5.401 - 38.458 .000 -.081 -.992 -.646 .014 69.919

(Constant)

x

groep2

groep3

intxgr2

intxgr3

Model1

B Std. Error

Unstandardized

Coefficients

Beta

Standardized

Coefficients


Correlations

Tolerance VIF




1.0000 1 0.4182 7.8667 2.76031 30

-.41818 .65758 .00000 .22789 30

-2.488 .924 .000 1.000 30

-1.669 2.625 .000 .910 30

Predicted Value

Residual


Std. Residual



20-2


12

10

8

6

4

2

0

Frequency

Mean =2,81E-15Std.Dev.=0,91

N =30

Histogram


4,002,000,00-2,00-4,00

x

4,00

2,00

0,00

-2,00

-4,00

y




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Model choice important

Missing interaction may result in

Violations of linearity assumption

non-constant variance (heterogeneity)

Incorrect model choice and interpretation


Conclusion

Model choice and selection crucial indetecting mediation and moderation

Substantive / theoretical considerationsshould guide the model selectionprocess!


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Other methods/models

Mediation sometimes too simple

More refined path analysis

Regression analysis sometimes too simple

More elaborate models for causalmodeling

Structural Equation Models

2011 10 18 moderation mediation

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