D1

REX B KLINE CONCORDIA D. MODERATION, MEDIATION

SEM ADVANCED

D2

X

1 DM M

1 DY

Y

D3

moderation

mmr

mpatop

ics

D4

cpm

mod. mediation

med. moderation top

ics

D5

cma

cause × mediator

most general top

ics

D6

MMR

X, W, Y are continuous

XW carries interaction

ˆX W XW

Y B X B W B XW A

D7

D8

Edwards, J. R. (2009). Seven deadly myths of

testing moderation in organizational

research. In C. E. Lance & R. J. Vandenberg

(Eds), Statistical and methodological myths

and urban legends: Doctrine, verity and

fable in the organizational and social

sciences (pp. 143–164). New York: Taylor &

Francis.

D9

Myth

You must center, to reduce extreme

collinearity

D10

Truth

Centering changes nothing

Optional, if 0 is not on scale

Center some, others not

D11

Myth

You must use hierarchical entry

D12

Truth

Not required

Possibly misleading

D13

Myth

You can ignore score reliability

Truth

D14

Truth

Score reliability is critical

rXX > .90

D15

Myth

ˆX W XW

Y B X B W B XW A

X, W are “main effects”

D16

Truth

X, W are linear only

D17

Myth

You can ignore curvilinear effects

D18

Truth

Estimate X2 and W2, too

D19

Myth

Small samples are fine

D20

Truth

Large samples needed

D21

X W Y

2 10 5

6 12 9

8 13 11

11 10 11

4 24 11

7 19 10

8 18 7

11 25 5

M 7.125 16.375

D22

ˆ .112 .064 8.873Y X W

2 .033R

D23

X W x w Y

2 10 −5.125 −6.375 5

6 12 −1.125 −4.375 9

8 13 .875 −3.375 11

11 10 3.875 −6.375 11

4 24 −3.125 7.625 11

7 19 −.125 2.625 10

8 18 .875 1.625 7

11 25 3.875 8.625 5

M 7.125 16.375 0 0

D24

ˆ .112 .064 8.873Y X W

ˆ .112 .064 8.625Y x w

2 .033R

D25

4

5

6

7

8

9

Y

10

11

1 5 4

X

2 3 6 7 8 9 10 11

D26

4

5

6

7

8

9

Y

10

11

1 5 4

X

2 3 6 7 8 9 10 11

W < MW

W > MW

D27

Analyses

Y on X, W, XW

Y on x, w, xw

Y on X, W, XWres

D28

XWres (1)

1. Regress XW on X, W

2. Create XW

3. Create XWres = XW − XW

D29

XWres (2)

1. Regress XW on X, W

2. Save residuals

3. Rename as XWres

D30

D31

D32

X W x w

XW .747 .706 xw −.138 .050

XWres 0 0

D33

Products

BXW = Bxw = BXWres

Same interaction

Same R2

D34

D35

ˆ .112 .064 8.873Y X W

Unconditional linear

2 .033R

D36

ˆ 1.768 .734 .108 3.118Y X W XW

2 .829R

D37

ˆ 1.768 .734 .108 3.118Y X W XW

If W ↑ 1pt,

slope Y on X ↓ .108

D38

ˆ 1.768 .734 .108 3.118Y X W XW

If X ↑ 1pt,

slope Y on W ↓ .108

D39

100

15

200

25

30

W

0

5

10

15

20

Y

2 4 6 8 10 12 14

X

ˆ 1.768 .734 .108 3.118Y X W XW

D40

ˆ 1.768 .734 .108 3.118Y X W XW

Conditional linear

Slope, Y on X is 1.768, if W = 0

Slope, Y on W is .734, if X = 0

D41

Centering

x = X − MX, w = W – MW

x = 0 says X = MX

w = 0 says X = MW

D42

ˆ .112 .064 8.625Y x w

2 .033R

ˆ .000 .035 .108 8.903Y x w xw

2 .829R

D43

ˆ .112 .064 8.873Y X W

2 .033R

resˆ .112 .064 .108 8.873Y X W XW

2 .829R

D44

Simple regressions

Simple slopes

Simple intercepts

Generate equations

D45

Y on X as a function of W

ˆ 1.768 .734 .108 3.118Y X W XW

ˆ 1.768 .108 .734 3.118Y X XW W

ˆ (1.768 .108 ) .(734 3.118)Y W X W

D46

ˆ (1.768 .108 ) .(734 3.118)Y W X W

16.38W

M

4.34 10.36 16.38 22.40 28.42

D47

ˆ (1.768 .108 ) .(734 3.118)Y W X W

22.40ˆ .651 13.324

WY X

D48

W

Level Score Regression equation

+2 SD 28.42 ˆ 1.301 17.712Y X

+1 SD 22.40 ˆ .651 13.324Y X

Mean 16.38 ˆ .001 8.905Y X

−1 SD 10.36 ˆ .649 4.486Y X

−2 SD 4.34 ˆ 1.299 .068Y X

D49 1 5 4

X

2 3 6 7 8 9 10 11 4

5

6

7

8

9

Y

10

11

MW

−2 SDW

+SDW

−SDW

+2 SDW

D50

MW

+1 SDW

+2 SDW

−1 SDW

−2 SDW

http://graph.seriesmathstudy.com/

D51

Other horizons

X, W, XW

X, X2, W, W2, XW

X, X2, W, W2, XW, X2W

D52

Other horizons

X, W, Z, XW, XZ, WZ, XWZ

E.g., XW over Z

Really?

D53

Dawson, J. F., & Richter, A. W. (2006). Probing

three-way interactions in moderated

multiple regression: Development and

application of a slope difference test.

Journal of Applied Psychology, 91, 917–926.

D54

(a) Regression perspective

BX

BW

BXW

Y

1 D

X

XW

W

(b) Compact symbolism

BX

BW

BXW

Y

1 D

X

W

D55

(d) W as focal variable,

X as moderator

BW

BX BXW

W

X

Y

1 D

(c) X as focal variable,

W as moderator

BX

BW BXW

X

W

Y

1 D

D56

D57

X

W

Y

1 D

X

W

Y

1 D

D58

Kline, R. B. (2015). The mediation myth. Basic

and Applied Social Psychology, 37, 202–

213.

Little, T. D. (2013). Longitudinal structural

equation modeling. New York: Guilford.

D59

Design

Time precedence: X → M → Y

Experimental X

What about M → Y?

D60

MacKinnon, D. P., & Pirlott, A. G. (2015). Statistical

approaches for enhancing causal interpretation

of the M to Y relation in mediation analysis.

Personality and Social Psychology Review, 19,

30–43.

Stone–Romero, E. F., & Rosopa, P. J. (2011).

Experimental tests of mediation models:

Prospects, problems, and some solutions.

Organizational Research Methods, 14, 631–646.

D61

Design

Time precedence: X → M → Y

Longitudinal

D62

M1

O1

X1

1 D12

M2

O2

1 D22

a

b

D63

Selig, J. P., & Preacher, K. J. (2009).

Mediation models for longitudinal data in

developmental research. Research in

Human Development, 6, 144–164.

D64

No design

Indirect effect

Mediation

D65

Hayes, A. F. (2013a). Conditional process modeling:

Using structural equation modeling to examine

contingent causal processes. In G. R. Hancock & R.

O. Mueller (Eds.), Structural equation modeling: A

second course (2nd ed.) (pp. 219–266). Greenwich,

CT: IAP.

Hayes, A. F. (2013b). Introduction to mediation,

moderation, and process control analysis: A

regression-based approach. New York: Guilford.

D66

CPM

Mediated moderation

Moderated mediation

Cause × mediator

D67

Mediated moderation

W

Y

1 DY

X

1 DM

M

D68

Lance, C. E. (1988). Residual centering,

exploratory and confirmatory moderator

analysis, and decomposition of effects in

path models containing interaction

effects. Applied Psychological

Measurement, 12, 163–175.

D69

D70

Moderated mediation (1)

1st-stage moderation, X → M → Y

X → M depends on W

W

Y

1 DY

X

1 DM

M

D71

Mediated moderation (2)

1st-stage moderation, W → M → Y

W → M depends on X

W

Y

1 DY

X

1 DM

M

D72

Moderated mediation

2nd-stage moderation, X → M → Y

M → Y depends on W

X

W

M 1

DM

Y

1 DY

D73

Edwards, J. R., & Lambert, L, S. (2007).

Methods for integrating moderation and

mediation: A general analytical

framework using moderated path

analysis. Psychological Methods, 12, 1–22.

D74

Curran, T., Hill, A. P., & Niemiec, C. P. (2013).

A conditional process model of children's

behavioral engagement and behavioral

disaffection in sport based on self-

determination theory. Journal of Sport &

Exercise Psychology, 35, 30–43.

D75

D76

Desrosiers, A., Vine, V., Curtiss, J., &

Klemanski, D. H. (2014). Observing

nonreactively: A conditional process

model linking mindfulness facets,

cognitive emotion regulation strategies,

and depression and anxiety symptoms.

Journal of Affective Disorders, 165, 31–37.

D77

D78

Hayes, A. F., & Preacher, K. J. (2013).

Conditional process modeling: Using

structural equation modeling to examine

contingent causal processes. In G. R.

Hancock & R. O. Mueller (Eds.), Structural

equation modeling: A second course (2nd

ed.) (pp. 219–266). Greenwich, CT: IAP.

D79

Baron-Kenny

Continuous variables

Linear model

No interaction

D80

a

b c

X M

1 DM

Y

1 DY

D81

Product estimator

X → M, X → Y, M → Y

No omitted confounders

rXX = 1.0

D82

X

Y

1 DY

M

1 DM

D83

1 1M B X A

2 3 4 2Y B X B M B XM A

D84

X × M

No single direct

No single indirect, total

D85

X × M

Effect decomposition?

Nonlinear models?

D86

Pearl, J. (2014). Interpretation and

identification of causal mediation.

Psychological Methods, 19, 459–481.

D87

Causal mediation

Assumes X × M

Linear or nonlinear

Total = direct + indirect

D88

Causal mediation

Counterfactuals

What if Tx were not treated?

What if Cn were treated?

D89

Counterfactuals

Rubin Causal Model

Missing data inference

Latent variables

D90

Example

Experimental X = 0, 1

M, Y are continuous

D91

Direct effects

Controlled (CDE)

Natural (NDE)

No X × M? CDE = NDE

D92

CDE

How much Y changes

As X = 0 to X = 1

If M = m for all cases

D93

CDE

Estimate for m = M

Policy: Lift all to m

D94

NDE

How much Y changes

As X = 0 to X = 1

If M varies as under X = 0

D95

NIE

How much Y changes in X = 1

As M changes from in

X = 0 to X = 1

D96

Total Effect

TE = NDE + NIE

D97

Counterfactuals

CDE = E [ Y (X = 1, M = m) ] – E [ Y (X = 0, M = m) ]

NDE = E [ Y (X = 1, M = m0) ] – E [ Y (X = 0, M = m0) ]

NIE = E [ Y (X = 1, M = m1) ] – E [ Y (X = 1, M = m0) ]

TE = E [ Y (X = 1) ] – E [ Y (X = 0) ]

D98

Petersen, M. L., Sinisi, S. E., & van der Laan,

M. J. (2006). Estimation of direct causal

effects. Epidemiology, 17, 276–284.

D99

X = 0, control; X = 1, AVT

M = viral load

Y = CD4 T-cells

D100

CDE

Mean Δ T-cells if viral load were

the same for all cases

D101

NDE

Mean Δ T-cells if viral load were

as among untreated cases

D102

NIE

Mean Δ T-cells among treated if

viral load changed from

untreated to treated levels

D103

0 1ˆ β βM X

0 1 2 3ˆ θ θ θ θY X M XM

D104

1 3CDE θ θ m

1 3 0NDE θ θ β

2 3 1NIE (θ θ )β

D105

0 1 2 3ˆ θ θ θ θY X M XM

If θ3 = 0:

1CDE θ

1NDE θ

D106

ˆ 1.70 .20M X

ˆ 450.00 50.00 20.00 10.00Y X M XM

D107

β0 = 1.70 and β1 = −.20

θ0 = 450.00,

θ1 = 50.00, θ2 = −20.00,

and θ3 = −10.00

D108

CDE = 50.00 − 10.00 m

NDE = 50.00 − 10.00 (1.70) = 33.00

NIE = (−20.00 − 10.00) −.20 = 6.00

TE = 33.00 + 6.00 = 39.00

D109

Valeri, L., & VanderWeele, T. J. (2013).

Mediation analysis allowing for exposure–

mediator interactions and causal

interpretation: Theoretical assumptions

and implementation with SAS and SPSS

macros. Psychological Methods, 2, 137–

150.

D110

Imai, K., Keele, L., & Tingley, D. (2010). A

general approach to causal mediation

analysis. Psychological Methods, 15, 309–

334.

D111