Moderator and Mediator Analysis

(1)

Moderator and Mediator Analysis

Marijtje van Duijn

October 18, 2011 Seminar General Statistics

Overview

• What is moderation and mediation?

• What is their relation to statistical concepts?

• Example(s)

(2)

October 18, 2011 Moderator and mediator analysis 3

Mediation and Moderation

X

₁

X

₂

Y

Examples

• Y : test score

• X1 : sex, SES, etc.

• X2 : ability (IQ score)

• Y : test score

• X1 : brain volume, SES parents

• X2 : ability

(3)

Multiple regression

• Goal: to explain variation in Y using Xs

• Assumptions

– Independent observations

– Normality (of residuals) and constant variance

– Linearity (of relationship Y and Xs)

i i

i

X X E

Y = β

₀

+ β

₁ ₁

+ β

₂ ₂

+

Regression Model

X

₁

X

₂

β₁

Y

β₂

E

(4)

Explained variance in regression

Y

X₁ X₂

Circles represent variances

X₁and X₂explain different parts of Y and are independent

Multicollinearity

Y

X₂ X₁

‘competition’ between variables for explaining Y

Degree depends on correlation between X’s

(5)

Association between all (3) variables

• Several correlations to describe association

– Partial correlation: correlation between two variables with the third variable fixed (i.e. corrected for the third variable) (partial in SPSS) ‘unique portion of variance’

– Semi-partial correlation: correlation between two variables with one variable’s association corrected for the third variable (part in SPSS output) ‘extra variance explained’

• Important in regression when variables are entered in a certain order

• Usually the partial correlation is smaller than the uncorrected (‘zero-order’) correlation, and larger than the semi-partial correlation

Multicollinearity/Mediation

X

₁

X

₂

Y

(6)

Mediation as a special case of multicollinearity 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 between X1

and Y disappears when M is added to the regression equation

Mediation

(Baron & Kenny, 1986),

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

X Y

c

b a

c’

M

(7)

Mediation as ‘prescribed’ by Baron and Kenny (1986)

• Estimate regression of Y on only X₁

– Estimated parameter c

• Estimate regression of M on X₁

– Estimated parameter a

• Estimate effect of M on 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 square root of

b²sa2+ a²sb2 (Sobel test)

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

• Some eye-balling also possible

• Or: nonparametric tests (based on bootstrapping)

(8)

Example (Miles and Shevlin)

• Y: read, a measure of the number of books that people have read.

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

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

Idea: how much people enjoy reading books -> the number of books bought -> the number of books read But: is the number of books a complete mediator….?

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

Descriptive Statistics Mean

Std.

Deviation N

buy 15,73 8,165 40

enjoy 9,28 5,354 40

Correlations

read buy enjoy

Pearson Correlation

buy ,747 1,000 ,644

enjoy ,732 ,644 1,000

Coefficients^a Model

Unstandardized Coefficients

Standardized Coefficients

t Sig.

B Std. Error Beta

1 (Constant) 4,331 ,785 5,517 ,000

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

a. Dependent Variable: read

Step 1

(9)

Step 2

Standardiz ed Coefficients

t Sig.

B Std. Error Beta

1 (Constant) 6,616 2,020 3,274 ,002

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

a. Dependent Variable: buy

Step 3

Standardized Coefficients

t Sig.

Correlations

B Std. Error Beta

Zero-

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 s_a=.189 – b=.487 s_b=.074

– ab=.478, s_ab=√(.487²*.189²+.982²*.074²)= .12 – z-test 4.08; p<0.001

• Eye-balling

– c=.487 (.074), then roughly .34 < c < .64

– c’=.286 (.083): partial mediation (because .12 < c’ < .45)

(10)

Model choice is important

• Many other patterns of association are possible

– Arrows between X₁and X₂may be

reversed – not always clear which variable

‘mediates’

– No causal relation, just association

• Explicit assumption of positive

associations and ordering of (semi-) partial correlations. Not guaranteed…

Moderation

X

₁

X

₂

Y

(11)

Moderation is ‘interaction’

• The effect of X2 depends 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 X₁and X₂

• X₁and X₂ may be correlated

• Importance of model formulation

Regression model

• Centering is (usually) advised – Facilitates interpretation

– Reduces the inevitable multicollinearity incurred with interaction terms

i i i i

i

i X X X X E

Y =β₀+β₁ ₁ +β₂ ₂ +β₃ ₁ ⋅ ₂ +

i c i c i c

i c

i

i X X X X E

Y = β₀ + β₁ ₁ + β₂ ₂ +β₃ ₁ ⋅ ₂ +

(12)

X

₁

dichotomous and X

₂

continuous

• X₁= 0or X₁= 1

– (e.g. man/woman; control/experimental group)

• Y = β0+ β1X1+ β2X2+ β3X1X2 – X₁= 0: Y = β₀+ β₂X₂

– X₁= 1: Y = (β0+ β1) + (β2+β3)X₂

• so: intercept and regressioneffect of X2change

– interaction effect represents the change in effect of X₂ or, the difference in the effect of X2between the two groups – interpretation of β1

– General formula: Y = (β0+ β1X1) + (β2+ β3X1)X2

X

₁

and X

₂

dichotomous

• X1 = 0or X1= 1(e.g. man/woman)

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

• Y = β₀+ β₁X₁+ β₂X₂+ β₃ X₁X₂

– X₁= 0, X₂= 0 : Y = β₀ – X₁= 1, X₂= 0 : Y = β₀+ β₁ – X₁= 0, X₂= 1 : Y = β₀+ β₂

– X₁= 1, X₂= 1 : Y = β0 + β1+ β2+ β3

• so: defines four groups with their own mean

• Interaction defines ‘extra’ effect of X =1 and X =1

(13)

X

₁

nominal and X

₂

continuous

• X₁takes on more than 2 (c) values – (e.g. age groups, control/exp1/exp2)

• Make dummies, for each contrast, wrt 1 reference 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 = β₀+ β_1dD₁+ β_2dD₂+ β₂X₂+β₃D₁X₂+β₄ D₂X₂ – group1: Y = β₀+ β₂X₂

– group2: Y = (β₀+ β_1d) + (β₂+β₃)X₂ – group3: Y = (β₀+ β_2d) + (β₂+β₄)X₂

(14)

X

₁

and X

₂

continuous

Y = β0+ β1X1+β2X2+β3X1X2⇔ Y = (β0+ β1X1) + (β2+β3X1)X2⇔

Y = (β0 +β2X2) + (β1+β3X2)X1

X1 β3 β2

* β2

X2 β3 β1

* β1

X2 X1 β3 X2 β2 X1 β1 β0

* β0

2c 1c 3 2c

* 1c

*

* 2 2 2c

1 1 1c

X X β X β X β β Y

X X X

2 1 0

−

=

−

=

+

−

=

+ +

+

=

−

=

−

=

1c

*

* 2

2

2c

*

* 1

1

X β β Y : X X

1 0

2 0

+

=

+

=

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 N

grade 63,5500 16,70552 40

books 2,00 1,432 40

attend 14,10 4,278 40

(15)

Standardiz ed Coefficient

s

t Sig.

Correlations

Collinearity Statistics B Std. Error Beta

Zero-

order Partial Part Toleran

ce VIF

1 (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

(16)

Other example

(missed) interaction:

non-linearity?

3 groups with each a distinct linear relation

(17)

Model Summary^b

.405^a .164 .134 2.57780

Model 1

R R Square

Adjusted R Square

Std. Error of the Estimate

Predictors: (Constant), x a.

Dependent Variable: y b.

Coefficients^a

5.724 1.029 5.560 .000

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

(Constant) x Model 1

B Std. Error Unstandardized

Coefficients

Beta Standardized

Coefficients

t Sig. Zero-order Partial Part

Correlations

Tolerance VIF Collinearity Statistics

Dependent Variable: y a.

Residuals Statistics^a

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

Minimum Maximum Mean Std. Deviation N

2 0

-2

Regression Standardized Residual 6

4

2

0

Frequency

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

N =30 Histogram

Dependent Variable: y

1,0 0,8 0,6 0,4 0,2 0,0

Observed Cum Prob 1,0

0,8

0,6

0,4

0,2

0,0

Expected Cum Prob

Normal P-P Plot of Regression Standardized Residual

(18)

Model Summary^b

.980^a .960 .957 .57477

Model 1

R R Square

Adjusted R Square

Predictors: (Constant), xkwadraat, x a.

Coefficients^a

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 Model 1

Coefficients

Beta Standardized

Coefficients

Correlations

.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

2 0

-2

4

2

0

Frequency

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

N =30 Histogram

15,00 10,00 5,00 0,00 -5,00 -10,00 -15,00

x 2,00

0,00

-2,00

y

Partial Regression Plot

Model Summary^b

.410^a .168 .107 2.61796

Model 1

R R Square

Adjusted R Square

Predictors: (Constant), groep, x a.

Coefficients^a

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 Model 1

Coefficients

Beta Standardized

Coefficients

Correlations

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

6

4

2

Frequency

Histogram

2,50

0,00

-2,50

-5,00

y

(19)

Model Summary^b

.750^a .562 .512 1.93502

Model 1

R R Square

Adjusted R Square

Predictors: (Constant), groep3, groep2, x a.

Coefficients^a

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 Model 1

Coefficients

Beta Standardized

Coefficients

Correlations

4.7273 11.1227 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

2 0

-2

6

4

2

0

Frequency

Mean =3,47E-16 Std. Dev. =0,947 N =30 Histogram

4,00 2,00 0,00 -2,00 -4,00

x 4,00

2,00

0,00

-2,00

-4,00

y

Model Summary^b

.997^a .993 .992 .25050

Model 1

R R Square

Adjusted R Square

Predictors: (Constant), intxgr3, intxgr2, x, groep2, groep3

a.

Coefficients^a

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 Model 1

Coefficients

Beta Standardized

Coefficients

Correlations

1.0000 10.4182 7.8667 2.76031 30

-.41818 .65758 .00000 .22789 30

-2.488 .924 .000 1.000 30

-1.669 2.625 .000 .910 30

2 0 -2

10

8

6

4

2

0

Frequency

Mean =2,81E-15 Std. Dev. =0,91 N =30 Histogram

4,00 2,00 0,00 -2,00 -4,00

x 4,00

2,00

0,00

-2,00

-4,00

y

(20)

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 in detecting mediation and moderation

• Substantive / theoretical considerations should guide the model selection

process!

(21)

Other methods/models

• Mediation sometimes – too – simple – More refined path analysis

• Regression analysis sometimes – too – simple

• More elaborate models for causal modeling

– Structural Equation Models