Mediation Models
Laura Stapleton
UMBC
Mediation Models
Tasha Beretvas
University of Texas at Austin
Session outline

What is mediation?

Basic single mediator model

Short comment on causality

Tests of the hypothesized mediation effect

Mediation models for cluster randomized trials

Brief mention of advanced issues
What is mediation?

A mediator explains how or why two
variables are related.
 In
the context of interventions, a mediator
explains how or why a Tx effect occurs
A mediator is thought of as the
mechanism or processes through which
a Tx influences an outcome (Barron & Kenny, 1986).
 If X  M and M  Y, then M is a mediator


X causes proximal variable, M, to vary which itself
causes distal, variable,Y, to vary
What is mediation?

Mediational process can be
 Observed
or latent
 Internal or external
 At the individual or cluster level
 Based on multiple or sequential processes

Who cares?!
 Mediation
analyses can identify important
processes/mechanisms underlying effective
(or ineffective!) treatments thereby providing
important focal points for future interventions.
Mediation Examples
Bacterial exposure  Disease
 Bacterial exposure  Infection  Disease
 Stimulus  Response

 Might
work for simple organisms (amoebae!),
however, for more complex creatures:
Stimulus  Organism  Response
 Stimulus  Expectancy  Response

 Monkey
and lettuce example
 Maze-bright, maze-dull rats and maze
performance example
Mediation Examples
Intervention  Outcome
 Intervention  Receptivity  Outcome
 Intervention  Tx Fidelity  Outcome
 Intervention  Tch Confid Outcome
 Intervention  Soc Comp Achievement
 Intervention  Phon Aware  Reading
 Intervention  Peer Affil  Delinq Beh

Mediation  Moderation

A moderator explains when an effect
occurs
 Relationship
between X and Y changes for
different values of M
 More in later session of workshop…
Basic (single-level) mediation model
c
Treatment
Yi   0  1 Ti  ei
Outcome
M i   0  1 Ti  ei
Mediator
a
b
Treatment
Outcome
Yi   0'  1' M i   2' Ti  ei'
c’
total effect = indirect effect + direct effect
c=
ab
+
c’
Causality concerns

Just because you estimate the model
XMY
does not mean that the relationships are
causal
 Unless
you manipulate M, causal inferences
are limited.

Mediation model differs from Mediation
design
Causality concerns – mediation model

Remember, if the mediator is not typically
manipulated, causal interpretations are limited
Z
Mediator
M
a
Treatment
T
Ok!
b

Outcome
Y

Possible misspecification

For now, be sure to substantively justify the causal
direction and “assume or hypothesize that M causes Y
and assuming that, here’s the strength of that effect…”
In future research, manipulate mediator

Tests of the hypothesized mediation
effect
Mediator
a
M
Treatment
T
b
Outcome
c’
Y

The estimate of the indirect effect, ab, is based
on the sample

To infer that a non-zero αβ exists in the
population, a test of the statistical significance of
ab is needed

Several approaches have been suggested and
differ in their ability to “see” a true effect (power)
Tests of the hypothesized mediation
effect

Causal steps approach (Baron & Kenny)

Test of joint significance

z test of ab (with normal theory confidence interval)

Asymmetric confidence interval (Empirical M or
distribution of the product)

Bootstrap resampling
Causal steps approach

Step 1: test the effect of T on Y (path c)
c
Treatment
Outcome
 Step 2: test the effect of T on M (path a)
Mediator
a
Treatment
Causal steps approach
Step 3: test the effect of M on Y, controlling for T
(path b)

Mediator
b
Treatment
Outcome
c’
Step 4: to decide on partial or complete
mediation, test the effect of T on Y, controlling for
M (path c’)

Causal steps approach: performance

Step 1 may be non-significant when true
mediation exists
Mediator
+2
What if…
FdF
+3
Treatment
Outcome
T
Dep
-6
Mediator
+2
or…
FdF
+3
Treatment
T
Outcome
+3
-2
Mediator
SS
Dep
Causal steps approach: performance

Lacks power
 Power
is a function of the product of the
power to test each of the three paths
 Power discrepancy worsens for smaller n and
smaller effects

Lower Type I error rate than expected
 i.e.,
too conservative
Test of joint significance

Very similar to causal steps approach
Mediator
a
b
Treatment
Outcome
c’
 1st: test the effect of T on M (path a)
 2nd: test the effect of M on Y, controlling for T (path
b)
 If both significant, then infer significant mediation
Test of joint significance: performance





Better power than causal steps approach
Type I error rate slightly lower than expected
Power nearly as good as newer methods in singlelevel models
Power lower than other methods in multilevel
models
No confidence interval around the indirect effect is
available
z test of ab product

Calculate z =
 Sobel’s
seab =
ab
seab
a 2 seb2  b2 sea2
 Compare z test value to critical values from the
standard normal distribution

Can also calculate confidence interval around ab
CI =
ab  ( zcritical )( seab )
z test of ab product: performance






One of the least powerful approaches
Type I error rate much lower than expected .05.
Single-level models, it approaches the power of
other methods when sample size are 500 or
greater, or effect sizes are large
Multilevel models, it never reaches the levels of
other models although it does get closer
although still lower
Problem is that the ab product is not normally
distributed, so critical values are inappropriate
How is the ab product distributed?
Sampled 1,000 a ~ N(0,1) and of b ~ N(0,1)
200
200
Distribution of path a
Distribution of path b
150
150
100
100
50
50
0
0
-4
-3
-2
-1
0
1
2
3
4
-4
-3
-2
-1
0
1
2
3
200
150
Distribution of
product of axb
100
50
0
-4
-3
-2
-1
0
1
2
3
4
4
Empirical M-test (asymmetric CI)

Determines empirical (more leptokurtic) distribution
of z of the ab product (not assuming normality)
dist’n is leptokurtic and symmetric
 αβ>0: dist’n is less leptokurtic and +ly skewed
 αβ<0: dist’n is less leptokurtic and -ly skewed
 αβ=0:


Due to asymmetry, different upper and lower critical
values needed to calculate asymmetric confidence
intervals (CIs).
Meeker derived tables for various combinations of
Za and Zb values (increments of 0.4) that could be
used to calculate asymmetric CIs.
Empirical M-test (asymmetric CI)

MacKinnon et al created PRODCLIN that,
given a, b, and their SEs, determines the
distribution of ab and relevant critical
values for calculating asymmetric CI.
(MacKinnon & Fritz, 2007, 384-389).

Confidence interval limits:
ab  (CVlower )( seab )
ab  (CVupper )( seab )

If CI does not include zero, then significant
Empirical M-test: performance

Good balance of power while maintaining
nominal Type I error rate

Performed well in both single-level and
multi-level tests of mediation

Only bootstrap resampling methods had
(very slightly) better power than this
method

PRODCLIN software is easy to use
Bootstrap resampling methods



Determines empirical distribution of the ab
product
Several variations
 Parametric percentile
 Non-parametric percentile
 Bias-corrected versions of both
Can bootstrap cases or bootstrap residuals.
 It
is typical in multilevel designs to bootstrap
residuals.
Parametric percentile bootstrap





With original sample, run the analysis and obtain
estimates of variance(s) of residuals
New residuals are then resampled from a
distribution ~N(0,σ2) (thus, the “parametric”).
New values of M are created by using the fixed
effects estimates from the original analysis, T
and the resampled residual(s).
New values of Y are created using the fixed
effects, and T and M values and residual(s).
Then, the analysis is run and the ab product is
estimated
Parametric percentile bootstrap

The process of resampling and estimating ab is
repeated many times (commonly 1,000 times)

The ab estimates are then ordered

With 1,000 estimates, the 25th and the 975th are
taken as the lower and upper limits of the 95%
(empirically derived) CI.

Note that the CI limits may not be symmetric
around the original ab estimate

If CI does not include zero, then significant
mediation
Non-parametric percentile bootstrap

The parametric bootstrap involves the
assumption that the residuals are normally
distributed

Instead, residuals can be resampled with
replacement from the empirical distribution of
actual residuals (saved from the original
sample’s analysis)

The remaining process is the same as for the
parametric version
Bias-corrected bootstrap

With both the parametric and non-parametric
bootstrap, the initial ab product may not be at
the median of the bootstrap ab distribution

Thus, the initial ab estimate is biased

BC-bootstrap procedures “shift” the confidence
interval to adjust for the difference in the initial
estimate and the median
Bootstrap resampling methods:
performance






Resampling methods provide the most power
and accurate Type I error rates of all methods
Parametric has best confidence interval
coverage
BC-parametric had best power, especially with
low effect sizes with normal and non-normally
distributed residuals; Type I error rate was
slightly high for multilevel analyses
Non-parametric had the most accurate Type I
error rates; good overall power
BC Non-parametric had good power
But … complicated to program
Summary: tests of the hypothesized
mediation effect
Causal steps approach
 Test of joint significance
 z test of ab
 Empirical M
 Bootstrap resampling


 OK for single level…

 Yes! Easy!
 Yes! Not quite as easy…
but does have the most power
Example for today
Social-emotional curriculum = Tx
 Child social competence = outcome
 Randomly selected classrooms (one per
school)
 Why would Tx affect outcome?

 Teacher
attitude about importance?
 Child understanding of others’ behaviors?
 Puppet show down-time soothes child?

Researcher should think in advance of
possible mediators to measure
Mediation models for cluster
randomized trials

Extend basic model to situations when treatment
is administered at cluster level

Model depends on whether mediator is
measured at cluster or individual level

Definition (Krull & MacKinnon, 2001) depends on level at
which each variable is measured: T → M →Y
 Upper-level mediation [2→2→1]
 Cross-level mediation [2→1→1]
 Cross-level and upper-level mediation
[2→(1 & 2) →1]
Measured variable partitioning

First, consider that any variable may
be partitioned into individual level
components and cluster level
components
Yij   00  u 0 j  rij
Note: No intercepts depicted
Cluster
uoj
Yij
Individual
rij
Mediation model possibilities
Tx
Cluster
M
Cluster
Y
Cluster
Tx
M
Y
Tx
M
Y
Individual
Individual
Individual
Data Example Context

Cluster randomized trial (hierarchical design)

14 preschools: ½ treatment, ½ control
6
kids per school (/classroom)

Socio-emotional curriculum

Outcome is child social competence behavior

Possible mediators: teacher attitude about
importance of including this kind of training in
classroom, child socio-emotional knowledge

Sample data are on handout
Total effect of treatment
Before we examine mediation, let’s examine
the total effect of treatment on the outcome…
u 0 j
Tx
Cluster
Tx
Yij   0 j  eij
   01
 T j  u0 j
 0 j   00
01
Y
Cluster
Y
Y
Cluster
rij
Total effect of treatment: FE Results
Final estimation of fixed effects:
---------------------------------------------------------------------------Standard
Approx.
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
34.357143
1.029102
33.386
12
0.000
T, G01
4.238095
1.455370
2.912
12
0.014
----------------------------------------------------------------------------
c
Upper-level mediation model (2→2→1)
01
M
Cluster
Tx
Cluster
Tx
’01
’02
M
M j   00   01T j  u0 j
Yij   '0 j  r 'ij
   01
 M j   02
 T j  u0 j
 0 j   00
u 0 j
Y
Cluster
Y
Y
Cluster
rij
Upper-level mediation model: Results
To estimate the a path, I ran an OLS
regression in SPSS using the Level 2 file
M j   00   01T j  u0 j
Coeffi cientsa
Model
1
(Const ant)
T
Unstandardized
Coeffic ients
B
St d. Error
9.429
.444
.714
.628
St andardiz ed
Coeffic ients
Beta
.312
t
21.228
1.137
Sig.
.000
.278
a. Dependent Variable: M1
What is the estimate of a and its SE?
Upper-level mediation model: Results
To estimate the b path, I ran a model in HLM
Final estimation of fixed effects:
---------------------------------------------------------------------------Standard
Approx.
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
34.640907
1.036530
33.420
11
0.000
M1, G01
0.794540
0.656229
1.211
11
0.252
T, G02
3.670567
1.502879
2.442
11
0.033
----------------------------------------------------------------------------
What is the estimate of b and its SE?
What is the estimate of c’ and its SE?
Upper-level mediation model: Results
M
Cluster
Tx
Cluster
Tx
u 0 j
Y
Cluster
3.671
M
Y
Y
Cluster
 Direct
effect = 3.671
rij
 Indirect effect = (.714)(.795) = .568
 Total effect = DE + IE = 3.671 + .568 = 4.239
Upper-level mediation model: Results

Causal steps approach
No
.
Step 1 significant, but
not Steps 2 and 3

Test of joint significance
No
.
Neither path a nor path
b are significant

z test of ab product
No
.
se=.68, z=.83, p=.41
95% CI = -.78 to 1.91

Empirical-M test
No
.
95% CI = -.47 to 2.26

BC parametric bootstrap
No
.
95% CI = -.42 to 3.68
Upper-level mediation model: Results

PRODCLIN http://www.public.asu.edu/~davidpm/ripl/ Prodclin/
Cross-level mediation model (2→1→1)
Model A
Model B
u0 j
γ01
u0' j
Mediator
CLUSTER
Treatment
CLUSTER
Treatment
CLUSTER
γ’01
Mediator
Outcome
CLUSTER
Mediator
Treatment
Treatment
Outcome
Mediator
Mediator
INDIVIDUAL
INDIVIDUAL
γ’10
Outcome
INDIVIDUAL
rij'
M ij   0 j  rij ,
 0 j   00   01T j  u0 j
Yij   '0 j   '1 j M ij  r 'ij
 '0 j   '00   '01Tj  u '0 j
 '1 j   '10
rij'
Cross-level mediation model: Results
To estimate the a path:
Final estimation of fixed effects:
---------------------------------------------------------------------------Standard
Approx.
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
39.309524
0.845210
46.509
12
0.000
T, G01
2.642857
1.195308
2.211
12
0.047
----------------------------------------------------------------------------
What is a and its SE?
Cross-level mediation model: Results
To estimate the b path:
Final estimation of fixed effects:
---------------------------------------------------------------------------Standard
Approx.
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
35.138955
0.941637
37.317
12
0.000
T, G01
2.674528
1.358185
1.969
12
0.072
For M2_GRAND slope, B1
INTRCPT2, G10
0.591620
0.142895
4.140
81
0.000
----------------------------------------------------------------------------
What is b and its SE?
And for c’?
Cross-level mediation model: Results
Model A
Model B
u0 j
u0' j
Mediator
CLUSTER
Treatment
CLUSTER
Treatment
CLUSTER
Mediator
2.675
Outcome
CLUSTER
Mediator
Treatment
Treatment
Outcome
Mediator
Mediator
INDIVIDUAL
INDIVIDUAL
Outcome
INDIVIDUAL
rij'
 Direct
effect = 2.675
 Indirect effect = (2.643)(.592) = 1.564
 Total effect = 2.675 + 1.564 = 4.239
rij'
Cross-level mediation model: Results

Causal steps approach
Yes
Steps 1, 2 and 3 significant

Test of joint significance
Yes
Paths a and b significant

z test of ab product
No
se=.802, z=1.95, p=.051
95% CI = -.01 to 3.13

Empirical-M test
Yes
95% CI = .19 to 3.32

BC parametric bootstrap
Yes
95% CI = .31 to 3.57
Cross-level and upper-level
mediation model [2→(1 & 2) →1]
Model A
Model B
γ01
u0 j
Mediator
CLUSTER
γ’01
Mediator
CLUSTER
Treatment
CLUSTER
u0' j
Treatment
CLUSTER
Outcome
CLUSTER
Avg M
Mediator
Treatment
Mediator
Treatment
Outcome
Mediator
Mediator
INDIVIDUAL
INDIVIDUAL
Outcome
INDIVIDUAL
M ij   0 j  rij ,
 0 j   00   01T j  u0 j
rij
Yij   0 j   M ij  r ij
'
'
1j
'
   01
 T j   02
 AveM j  u0 j
 0 j   00
1j   10
rij'
Cross-level and upper-level
mediation model: Results
Path a is the same as in the prior model. For the b and c’
paths:
Final estimation of fixed effects:
---------------------------------------------------------------------------Standard
Approx.
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
35.095622
1.047773
33.495
11
0.000
T, G01
2.761188
1.602238
1.723
11
0.112
M2_AVE, G02
-0.041278
0.363535
-0.114
11
0.912
For
M2 slope, B1
INTRCPT2, G10
0.600111
0.160566
3.737
80
0.001
----------------------------------------------------------------------------
Cross-level and upper-level mediation model
[2→(1 & 2) →1]
Model A
Model B
u0 j
Mediator
CLUSTER
Mediator
CLUSTER
Treatment
CLUSTER
Treatment
CLUSTER
2.761
Outcome
CLUSTER
Avg M
Mediator
Treatment
u0' j
Mediator
Treatment
Outcome
Mediator
Mediator
INDIVIDUAL
INDIVIDUAL
Outcome
rij'
 abind
= (2.643)(.600) = 1.586
 abcluster = (2.643)(-.041) = -.109
 Total indirect effect = 1.586 – 0.109 = 1.477
 Total effect = 1.477+2.761 = 4.238
INDIVIDUAL
rij'
Cross-level and upper-level mediation model
[2→(1 & 2) →1] Group-mean centered M
Model A
Model B
u0 j
Mediator
CLUSTER
Mediator
CLUSTER
Treatment
CLUSTER
Treatment
CLUSTER
2.761
Outcome
CLUSTER
Avg M
Mediator
Treatment
u0' j
Mediator
Treatment
Outcome
Mediator
Mediator
INDIVIDUAL
INDIVIDUAL
Outcome
rij'

INDIVIDUAL
If the level one predictor had been group-mean centered, then
the L2 path would have been 0.559 not -0.041.
 This path would be interpreted as the sum of the average
individual and contextual effects of M.
 Under grand-mean centering, the path represents the unique
contextual effect.
rij'
Cross- and upper-level mediation
model: Results at the individual level

Causal steps approach
Yes
Steps 1, 2 and 3 significant

Test of joint significance
Yes
Paths a and b significant

z test of ab product
No
se=.886, z=1.79, p=.073
95% CI = -.15 to 3.32

Empirical-M test
Yes
95% CI = .19 to 3.44

BC parametric bootstrap
?
Not yet programmed
Brief review of advanced issues

Multisite / randomized blocks (1→1 →1)
 More





complicated!
Testing mediation in 3-level models
Including multiple mediators
Examining moderated mediation
Dichotomous or polytomous outcomes
Measurement error in mediation models
Notes on software


HLM,SPSS
 Plug results into PRODCLIN
SAS (PROC MIXED)




See handout
Can use Stapleton’s macros for bootstrapping
MLwiN, MPlus
 Have limited bootstrapping capacity but still
have to summarize results
SEM software
 Provide test of  but using Sobel.
tasha.beretvas@mail.utexas.edu