new developments in structural equation modeling -...
TRANSCRIPT
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
A4
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.
A5
X → Y1 → Y2 = ab
c'
a
b c′
1 D1
Y1 X
1 D2
Y2
A6
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
*
A8
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)
A9
Half longitudinal mediation
a
b
M11
O21
X11
1 D12
M12
O22
1 D22
A10
O11
M12
X12
M13
X13
O12 O13
X11
M11
a
b
Full longitudinal mediation
A11
Design
o Concurrent design
o Uncertain directionality
o Equivalent models
A12
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)
A13
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
A14
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
A15
Design
o Minimal:
Time precedence
X is manipulated
M, Y are nonexperimental
A16
X M
Y
X M
Y
U
M Y
M X Y
M U Y
A17
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.
A18
Moderation
o X, W, Z are continuous
o Moderated MR
o = + + +ˆX W XW
Y B X B W B XW A
A19
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.
A20
Myths
You must center, to reduce extreme collinearity
A21
Truths
Centering changes nothing
Optional, if 0 is not on scale
Center some, others not
A22
Myth
You must use hierarchical entry
A23
Truths
Not required
Possibly misleading
A24
Myth
You can ignore score reliability
Truth
A25
Truth
Score reliability is critical
rXX > .90
A26
Myth
= + + +ˆX W XW
Y B X B W B XW A
X, W are “main effects”
A27
Truth
X, W are linear only
A28
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
A29
ˆ .112 .064 8.873Y X W= − +
2 .033R =
A30
4
5
6
7
8
9
Y
10
11
1 5 4
X
2 3 6 7 8 9 10 11
A31
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
A32
ˆ .112 .064 8.873Y X W= − +
2 .033R =
ˆ 1.768 .734 .108 3.118Y X W XW= + − −
2 .829R =
A33
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
A35
ˆ 1.768 .734 .108 3.118Y X W XW= + − −
ˆ .000 .035 .108 8.903Y x w xw= − − +
2 .829R =
A36
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
A37
ˆ (1.768 .108 ) (.734 3.118)Y W X W= − + −
16.38W
M =
4.34 10.36 16.38 22.40 28.42
A38
= − + −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
A39
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
A40
http://graph.seriesmathstudy.com/
MW
+1 SDW
+2 SDW
−1 SDW
−2 SDW
A41
Path models
o Diagram options:
1. Include XW
2. Moderated path
A42
BXW
BXW BXW
W
DY 1
Y
X W
X
DY 1
Y
W
XW
DY 1
Y
X
A43
Process modeling
o Both:
1. Mediated moderation
2. Moderated mediation (conditional indirect)
A44
X
W
XW
DM 1
M
DY 1
Y
Mediated moderation
A45
Process modeling
o Moderated mediation:
1. 1st stage
2. 2nd stage
3. 1st and 2nd stage
A46
1st stage
X → M depends on W
Y
DY 1
W
X
XW
M
DM 1
A47
2nd stage
M → Y depends on W
X
W
M DM 1
Y
DY 1
MW
A48
1st and 2nd stage
X → M, M → Y depend on W
Y
DY 1
M
DM 1
W
X
XW
A49
Mediation in SCM
o Nonparametric model
o Consistent definition
o Linear or nonlinear models
A50
Mediation in SCM
o Assumes X × M:
Conditional direct, indirect
Equations, not diagrams
A51
Mediation in SCM
o Assumes X × M:
> 1 direct, indirect
Counterfactuals (PO)
A52
Mediation in SCM
o Direct effect:
Controlled (CDE)
Natural (NDE)
A53
Mediation in SCM
o Total effect:
TE = NDE + NIE
A54
Mediation in SCM
o CDE:
How much Y changes
Given X = 0 to X = 1
If M = m for all cases
A55
Mediation in SCM
o CDE:
Different value of DE
For every value of M
Mean Y change over M
A56
Mediation in SCM
o NDE:
Allows for variation in M
But at values that would be observed in control
A57
Mediation in SCM
o NDE:
How much Y changes
Given X = 0 to X = 1
If M varies as under X = 0
A58
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
A59
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) ]
A60
Petersen, et al. (2006)
X = 0, control; X = 1, anti-viral therapy
M = viral load
Y = CD4 T-cells
A61
CDE
Mean ∆ T-cells if viral load is same for all cases
NDE Mean ∆ T-cells if viral load were among untreated cases
A62
NIE Mean ∆ T-cells among treated if viral load changed from untreated to treated levels
A63
Mediation in SCM
o Valeri & VanderWeele (2013) SAS/STAT, SPSS
o Muthén (2011) Mplus
A64
Mediation in SCM
o Hicks & Tingley (2012) Stata
o Tingley et al. (2014) R
A65