Multiple Mediator Models. Most behaviors are affected by multiple mediators. Straightforward extension of the single mediator case but interpretation can be more difficult.
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MEDIATOR
M1
MEDIATOR
M2
DEPENDENT VARIABLE
INDEPENDENT VARIABLE
c
Y
X
MEDIATOR
M3
MEDIATOR
M4
MEDIATOR
M1
a1
a2
MEDIATOR
M2
DEPENDENT VARIABLE
INDEPENDENT VARIABLE
Y
X
a3
MEDIATOR
M3
a4
MEDIATOR
M4
2. The independent variable causes the potential mediators: M1 = a1X + e2,M2 = a2X + e3, M3 = a3X + e4, M4 = a4X + e5
MEDIATOR
M1
b1
a1
b2
a2
MEDIATOR
M2
DEPENDENT VARIABLE
INDEPENDENT VARIABLE
c’
Y
X
a3
b3
MEDIATOR
M3
a4
b4
MEDIATOR
M4
a1b1
Mediated effects =a1b1,a2b2,a3b3,a4b4
Standard error =
Total mediated effect=a1b1+a2b2+a3b3+a4b4=c  c’
Direct effect=c’ Total effect=a1b1+a2b2+a3b3+a4b4+c’=c
Test for significant mediation:
z’= Compare to empirical distribution
of the mediated effect
proc reg;
model y=x;
model y=x m1 m2/covb;
model m1=x;
model m2=x;
Regression
/variables= x y m1 m2
/dependent=y
/enter=x.
regression
/variables= x y m1 m2
/dependent=y
/enter=x m1 m2.
regression
/variables= x y m1
/dependent=m1
/enter= x.
regression
/variables x y m2
/dependent=m2
/enter= x.
MEDIATOR
.8401
(.1580)
.5690
(.1568)
M1
.1122
(.2073)
DEPENDENT VARIABLE
INDEPENDENT VARIABLE
Y
X
MEDIATOR
.5297
(.1696)
M2
.2219
(.1460)
a1b1= (.8401) (.5690) = .4781 for mediation through social climate and
a2b2= (.2219) (.5297) = .1175 for mediation through feedback.
The total mediated effect of a1b1 ( .4781) plus a2b2 (.1175) equals .5956 which is equal to cc’ =.7078.1122 =.5956.
The a1b1 mediated effect (sa1b1 = .1499) was statistically significant (ta1b1 = 3.183) and the a2b2 mediated effect (sa2b2 = .0838) was not (ta2b2 = 1.403).
The standard error of the total mediated effect is equal to .1717 yielding a z statistic of 3.468.
Mediation through social climate,
Asymmetric LCL= .2079 and UCL = .8284. Using the delta standard error, LCL= .1654 and UCL = .7906.
Mediation through feedback,
Asymmetric LCL = .0261 and UCL = .3106. Using the delta standard error, LCL= .0510 and UCL = . 2861.
Sa1b1a2b2 =
Add 2b1b2sa1a2 to the equation if there is a covariance between a1and a2, sa1a2 if covariance structure modeling is used, for example. There may also be other covariances that are needed but these are typically very small.
The difference between the two mediated effects is equal to .3605 with a standard error of .1717 yielding a z statistic of 2.099.
Contrasts can be used to test pairs of mediated effects in any model.
See MacKinnon (2000) Contrasts in Multiple Mediator Models
Knowledge of the effects of AAS use
.083
.02 (.006)
.236
2.42 (.258)
Team as information source
.079
.08 (.006)
.217
.52 (.061)
.000
.001 (.056)
Group
Intentions
Perceived risks of AAS use
.168
.44 (.066)
.265
.25 (.024)
.149
.62 (.108)
.155
.09 (.014)
Reasons to use AAS
Effect Estimate Estimate/ LCL UCL
(Std Error) SE
Knowledge .046 3.00 .075 .017
(.015)
Team as .041 2.97 .068 .014
Information (.014)
Perceived Severity .108 5.56 .145 .071
(.013)
Reasons to use .056 4.29 .031 .081
Anabolic Steroid (.031)
Direct Effect of .001 0.017 .109 .111
Program on Intentions (.056)
Knowledge of the effects of AAS use
b1
a1
Team as information source
b2
a2
Intentions to use AAS
c’
Group
Perceived risks of AAS use
a3
b3
a4
b4
Reasons to use AAS
Contrast Examplesa1b1a2b2
2(a2b2) (a3b3+a4b4)
a2b2+c’ – 2(a4b4)
Sa1b1a2b2 =
Add 2b1b2sa1a2 to the equation if there is a covariance between a1and a2, sa1a2 if covariance structure modeling is used, for example. There may also be other covariances that are needed but these are typically very small.
Effect Estimate Estimate/ LCL UCL
(Std Error) SE
Pairwise Contrast : .005 0.22 .046 .036
Of Knowledge vs. (.021)
Team as Information
Pairwise Contrast : .066 2.67 .115 .017
Of Team as (.025)
Information vs.
Perceived Severity
From MacKinnon (2000) Contrasts in Multiple Mediator Models.
Inconsistent mediation models are models where at least one of the mediated effects and direct effects have different signs (see MacKinnon, Krull, & Lockwood 2000).
If the overall effect of X on Y is zero but there is a significant mediated effect, then it is an inconsistent mediation model. These effects are sometimes called suppressor effects. In these models the effect of X on Y actually increases when the mediator is included in the model.
“one may be equally misled in assuming that an absence of relation between two variables is real, whereas it may be due .. to the intrusion of a third variable” (Rosenberg, 1968, p. 84).
REASONS TO USE AAS
XM
.573 (.105)
.073 (.014)
PROGRAM
INTENTION TO USE AAS
.181 (.056)
X
Y
Mediated effect = .042
Standard error = .011
Power
Perceptions
+
+
M1
0
Harassment
Organizational Status
Y
X
Social
Dominance

+
M2
Reaction
Time
+

M1
0
Typing
Proficiency
Age
Y
X
Skill
+
+
M2
Occupation
X1
Resp.
Education
h1
Father
Education
X2
Resp.
Income
h3
No. of
Siblings
X3
Resp.
Occupation
h2
g11
.0385
(.0025)
B21
4.3767
(.1202)
B31
.1998
(.0364)
g21
.1352
(.0175)
g12
.1707
(.0156)
g31
.0114
(.0045)
g32
.0712
(.0275)
g22
.0490
(.1082)
g13
.2281
(.0176)
g33
.0373
(.0314)
B32
.0704
(.0045)
g23
.4631
(.1231)
.0385
(.0025)
Father
Occupation
X1
Resp.
Education
h1
Resp.
Occupation
h2
B21
4.3767
(.1202)
X1–>η1–> η2
γ11β21
(.0385) (4.3747) = .1685
sγ11β21= Square Root[
(.0385)2 (.1202)2 + (4.3747)2 (.0025)2 ]=.0118
Effect Parameters Estimate SE
FEDUC > REDUC > ROCC
X1–>η1–> η2γ11β21 .1685 .0118
FEDUC > ROCC > RINC
X1 –> η2–> η3γ21β32 .0095 .0014
FEDUC > REDUC > RINC
X1–> η1–> η3γ11β31 .0077 .0015
FEDUC > REDUC >ROCC > RINC
X1–> η1–> η2 –> η3 γ11β21β32 .0119 .0011
FOCC > REDUC > ROCC
X2–> η1> η2γ12β21 .7473 .0713
Occupation
X1
Resp.
Education
h1
Resp.
Income
h3
Resp.
Occupation
h2
g11
.0385
(.0025)
B21
4.3767
(.1202)
B32
.0704
(.0045)
b4
b1
b2
b3
X
M1
M2
Y
Mediated effect = b1b2b3
Var(b1b2b3) = b12b22sb32+ b12b32sb22+ b22b32sb12+ 2 b1b2b32sb2b12+
2 b1b22b3sb1b32 + 2 b12b2b3sb2b32
Standard Error(b1b2b3)=
Occupation
X1
Resp.
Education
h1
Resp.
Income
h3
Resp.
Occupation
h2
g11
.0385
(.0025)
B21
4.3767
(.1202)
B31
.1998
(.0364)
g21
.1352
(.0175)
B32
.0704
(.0045)
The keyword EF command on the OUTPUT line in LISREL requests output of total mediated effects and their standard errors. The keyword EFFECTS=YES on the /PRINT line has EQS print out total mediated effects and standard errors.
These programs print the total mediated effect of X on Y. For example,with this model the total mediated effect of X1 on 2 is the same as the specific mediated effect, X1 > 1, > 2, = .1683. The total mediated effect of X1 on 3 equals X1 > 2 > 3 plus X1 > 1 > 3, plus X1 > 1 > 2 > 3 or the sum of three specific indirect effects.
You will need to apply the formulas above to find specific mediated effects and their standard errors.
DECOMPOSITION OF EFFECTS WITH NONSTANDARDIZED VALUES
PARAMETER INDIRECT EFFECTS
__________________________
INC1961 =V1 = .308*V3 + .148*V4 + .029*V5 + .090*V6
.021 .014 .002 .012
14.403 10.286 13.186 7.413
.070 E2 + .508 E3
.004 .031
15.682 16.601
OCC1962 =V2 = .998*V4 + .168*V5 + .747*V6 + 4.377 E3
.082 .012 .071 .120
12.197 14.281 10.492 36.402
Indirect Effects of X on Y
FATHOCC FATHEDUC NUMSIB
________ ________ ________
EDUC _ _ _ _ _ _
OCC1962 0.1683 0.7473 0.9982
(0.0118) (0.0713) (0.0819)
14.2746 10.4868 12.1916
INC1961 0.0291 0.0902 0.1485
(0.0022) (0.0121) (0.0143)
13.3260 7.4621 10.3749
Mplus 3.0 will compute biascorrected bootstrap confidence intervals. Specify the number of bootstrap samples, BOOTSTRAP =500 and include CINTERVAL on the OUTPUT line.
Mplus 3.0 now computes standard errors and confidence intervals for tests of specific indirect effects with the “MODEL INDIRECT” statement!
MODEL INDIRECT
INC1961 IND FATHOCC;
Requests the three indirect effects from father’s occupation to income in 1961.
INC1961 IND EDUC FATHEDUC;
Requests specific indirect effect from father’s education to 1961 income.
Equations for standard errors of mediated effects are more complicated because they include the measurement models for the variables in the model.
Covariance between a and b may be nonzero so use formula that includes covariance between a and b. SEM programs compute the values of total mediated effect and Mplus 3.0 will compute specific mediated effects that include appropriate covariances in the standard error calculations. Resampling methods can also be used to obtain confidence intervals such as in Mplus 3.0 by specifying the number of bootstrap samples, BOOTSTRAP =500 and CINTERVAL on the OUTPUT line.
There are methods to incorporate multiple mediators and latent variables in mediator models. These models require a covariance structure analysis program to estimate the models. Standard errors of mediated effects can contrasts among mediated effects can be evaluated.
However, remember the assumptions of the single mediator model apply to the multiple mediator model. The additional variables address the omitted variable assumption. But other assumptions still apply. Specificity of significant mediation paths improve interpretation.
The results from a multiple mediator model may shed light on the true underlying mechanisms but there are alternative explanations of results. Remember that the path relating the mediators to Y is correlation.