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A1
New developments in structural equation modeling
Rex B Kline
Concordia University
Montréal
Set B: Mediation UNL Methodology Workshop
A2
A3
Topics
o Mediation:
Design requirements
Conditional process modeling
Cause × mediator
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Design
o Baron, R. M., & Kenny, D. A. (1986). The moderator-
mediator variable distinction in social psychological research: Conceptual, strategic and statistical considerations. Journal of
Personality and Social Psychology, 51, 1173–1182.
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X → Y1 → Y2 = ab
c'
a
b c′
1 D1
Y1 X
1 D2
Y2
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c, a
c′ not (c′ < c)
ab
c
X
1 D2
Y2
a
b c′
1 D1
Y1 X
1 D2
Y2
* *
*
A7
Design
o Newer view:
insufficient
Time precedence
Changes
*
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Mediation refers to the causal
hypothesis that one variable
causes changes in another
variable, which in turn leads to
changes in the outcome
variable.
Little (2013)
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Half longitudinal mediation
a
b
M11
O21
X11
1 D12
M12
O22
1 D22
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O11
M12
X12
M13
X13
O12 O13
X11
M11
a
b
Full longitudinal mediation
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Design
o Concurrent design
o Uncertain directionality
o Equivalent models
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1 D1
Y1 X
1 D2
Y2
1 DX
X
1 D2
Y2
Y1
1 D2
Y2 X
1 D1
Y1
1 DX
X
1 D1
Y1
Y2
1 D1
Y1
1 DX
X
Y2
1 D2
Y2
1 DX
X
Y1
Concurrent measurement (1)
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Concurrent measurement (2)
1 D1
Y1 X
1 D2
Y2
1 D2
Y2 X
1 D1
Y1
1 DX
X
1 D2
Y2
Y1
1 D1
Y1
1 DX
X
Y2
1 D2
Y2
1 DX
X
Y1
1 DX
X
1 D1
Y1
Y2
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Concurrent measurement* (3)
*Equality-constrained reciprocal effects
1 D1
Y1 X
1 D2
Y2
1 D2
Y2 X
1 D1
Y1
1 DX
X
1 D1
Y1
Y2
1 D1
Y1
1 DX
X
Y2
1 D2
Y2
1 DX
X
Y1
1 DX
X
1 D2
Y2
Y1
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Design
o Minimal:
Time precedence
X is manipulated
M, Y are nonexperimental
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X M
Y
X M
Y
U
M Y
M X Y
M U Y
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Design
o Bullock, J. G., Green, D. P., & Ha, S. E. (2010). Yes, but what’s the mechanism? (Don’t expect an easy answer). Journal of
Personality and Social Psychology, 98, 550–558.
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Moderation
o X, W, Z are continuous
o Moderated MR
o = + + +ˆX W XW
Y B X B W B XW A
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Moderation
o 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.
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Myths
You must center, to reduce extreme collinearity
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Truths
Centering changes nothing
Optional, if 0 is not on scale
Center some, others not
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Myth
You must use hierarchical entry
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Truths
Not required
Possibly misleading
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Myth
You can ignore score reliability
Truth
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Truth
Score reliability is critical
rXX > .90
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Myth
= + + +ˆX W XW
Y B X B W B XW A
X, W are “main effects”
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Truth
X, W are linear only
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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
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ˆ .112 .064 8.873Y X W= − +
2 .033R =
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4
5
6
7
8
9
Y
10
11
1 5 4
X
2 3 6 7 8 9 10 11
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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
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ˆ .112 .064 8.873Y X W= − +
2 .033R =
ˆ 1.768 .734 .108 3.118Y X W XW= + − −
2 .829R =
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ˆ 1.768 .734 .108 3.118Y X W XW= + − −
A34
Centering
o XW with X, W
o x = X – MX, w = w – Mw
o xw with x, w
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ˆ 1.768 .734 .108 3.118Y X W XW= + − −
ˆ .000 .035 .108 8.903Y x w xw= − − +
2 .829R =
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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
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ˆ (1.768 .108 ) (.734 3.118)Y W X W= − + −
16.38W
M =
4.34 10.36 16.38 22.40 28.42
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= − + −ˆ (1.768 .108 ) .(734 3.118)Y W X W
4.34, 10.36, 16.38, 22.40, and 28.42
== − + −
22.40ˆ (1.768 .108 * 22.40) (.734 * 22.40 3.118)
WY X
== − +
22.40ˆ .651 13.324
WY X
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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
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http://graph.seriesmathstudy.com/
MW
+1 SDW
+2 SDW
−1 SDW
−2 SDW
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Path models
o Diagram options:
1. Include XW
2. Moderated path
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BXW
BXW BXW
W
DY 1
Y
X W
X
DY 1
Y
W
XW
DY 1
Y
X
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Process modeling
o Both:
1. Mediated moderation
2. Moderated mediation (conditional indirect)
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X
W
XW
DM 1
M
DY 1
Y
Mediated moderation
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Process modeling
o Moderated mediation:
1. 1st stage
2. 2nd stage
3. 1st and 2nd stage
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1st stage
X → M depends on W
Y
DY 1
W
X
XW
M
DM 1
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2nd stage
M → Y depends on W
X
W
M DM 1
Y
DY 1
MW
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1st and 2nd stage
X → M, M → Y depend on W
Y
DY 1
M
DM 1
W
X
XW
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Mediation in SCM
o Nonparametric model
o Consistent definition
o Linear or nonlinear models
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Mediation in SCM
o Assumes X × M:
Conditional direct, indirect
Equations, not diagrams
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Mediation in SCM
o Assumes X × M:
> 1 direct, indirect
Counterfactuals (PO)
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Mediation in SCM
o Direct effect:
Controlled (CDE)
Natural (NDE)
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Mediation in SCM
o Total effect:
TE = NDE + NIE
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Mediation in SCM
o CDE:
How much Y changes
Given X = 0 to X = 1
If M = m for all cases
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Mediation in SCM
o CDE:
Different value of DE
For every value of M
Mean Y change over M
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Mediation in SCM
o NDE:
Allows for variation in M
But at values that would be observed in control
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Mediation in SCM
o NDE:
How much Y changes
Given X = 0 to X = 1
If M varies as under X = 0
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Mediation in SCM
o NIE:
How much Y changes in X = 1
If M changes from values observed in X = 0 to values it would be obtained in X = 1
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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) ]
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Petersen, et al. (2006)
X = 0, control; X = 1, anti-viral therapy
M = viral load
Y = CD4 T-cells
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CDE
Mean ∆ T-cells if viral load is same for all cases
NDE Mean ∆ T-cells if viral load were among untreated cases
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NIE Mean ∆ T-cells among treated if viral load changed from untreated to treated levels
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Mediation in SCM
o Valeri & VanderWeele (2013) SAS/STAT, SPSS
o Muthén (2011) Mplus
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Mediation in SCM
o Hicks & Tingley (2012) Stata
o Tingley et al. (2014) R
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