Download - Mediation Models
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Mediation Models
Laura Stapleton
UMBC
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Mediation Models
Tasha Beretvas
University of Texas at Austin
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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
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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
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What is mediation? Mediational process can be
Observed or latent Internal or externalAt the individual or cluster levelBased 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.
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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 exampleMaze-bright, maze-dull rats and maze
performance example
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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
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Mediation Moderation
A moderator explains when an effect occurs Relationship between X and Y changes for
different values of MMore in later session of workshop…
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Basic (single-level) mediation model
Outcome
Mediator
Treatment
a b
c’
OutcomeTreatment ciii eTY 10
iii eTM 10
''2
'1
'0 iiii eTMY
total effect = indirect effect + direct effect
c = ab + c’
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Causality concerns
Just because you estimate the model
X M Ydoes not mean that the relationships are causalUnless you manipulate M, causal inferences
are limited. Mediation model differs from Mediation
design
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Causality concerns – mediation model
Remember, if the mediator is not typically manipulated, causal interpretations are limited
Possible misspecification
Outcome
Y
Mediator
MTreatment
T
a b
Ok!
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
Z
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Tests of the hypothesized mediation effect
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)
Outcome
Y
Mediator
MTreatment
T
a b
c’
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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
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Causal steps approach Step 1: test the effect of T on Y (path c)
OutcomeTreatment c
Step 2: test the effect of T on M (path a)
Mediator
Treatment
a
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Causal steps approach
Step 4: to decide on partial or complete mediation, test the effect of T on Y, controlling for M (path c’)
Outcome
Mediator
Treatment
b
c’
Step 3: test the effect of M on Y, controlling for T (path b)
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Causal steps approach: performance
Step 1 may be non-significant when true mediation exists
Outcome
Dep
Mediator
FdFTreatment
T
+2 +3
-6What if…
Outcome
Dep
Mediator
FdFTreatment
T
+2 +3or…
Mediator
SS
+3 -2
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Causal steps approach: performance
Lacks power Power is a function of the product of the
power to test each of the three pathsPower discrepancy worsens for smaller n and
smaller effects Lower Type I error rate than expected
i.e., too conservative
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Test of joint significance Very similar to causal steps approach
2nd: test the effect of M on Y, controlling for T (path b)
If both significant, then infer significant mediation
1st: test the effect of T on M (path a)
Outcome
Mediator
Treatment
a b
c’
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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 single-
level models Power lower than other methods in multilevel
models No confidence interval around the indirect effect is
available
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z test of ab product
Calculate z =
2222ab sebsea
Compare z test value to critical values from the standard normal distribution
Can also calculate confidence interval around ab
CI =
Sobel’s seab =
abse
ab
))(( abcritical sezab
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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?
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0
50
100
150
200
-4 -3 -2 -1 0 1 2 3 4
0
50
100
150
200
-4 -3 -2 -1 0 1 2 3 4
0
50
100
150
200
-4 -3 -2 -1 0 1 2 3 4
Sampled 1,000 a ~ N(0,1) and of b ~ N(0,1)
Distribution of path a Distribution of path b
Distribution of product of axb
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Empirical M-test (asymmetric CI)
Determines empirical (more leptokurtic) distribution of z of the ab product (not assuming normality) αβ=0: 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
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.
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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:
If CI does not include zero, then significant
))(( ablower seCVab))(( abupper seCVab
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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
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Bootstrap resampling methods
Determines empirical distribution of the ab product
Several variationsParametric percentile Non-parametric percentileBias-corrected versions of both
Can bootstrap cases or bootstrap residuals. It is typical in multilevel designs to bootstrap
residuals.
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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
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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
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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
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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
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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
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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
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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
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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 →YUpper-level mediation [2→2→1]Cross-level mediation [2→1→1]Cross-level and upper-level mediation
[2→(1 & 2) →1]
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Measured variable partitioning
First, consider that any variable may be partitioned into individual level components and cluster level components
ijjij ruY 000
Cluster
uoj
Yij
Individual
rijNote: No intercepts depicted
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Mediation model possibilities
TxCluster
Tx
TxIndividual
MCluster
M
MIndividual
YCluster
Y
YIndividual
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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
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Total effect of treatmentBefore we examine mediation, let’s examine the total effect of treatment on the outcome…
jjj uT 001000
TxCluster
Tx Y
YCluster
YCluster
01
ijr
ju0
ijjij eY 0
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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
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Upper-level mediation model (2→2→1)
00 01 0j j jM T u
' '0ij j ijY r
TxCluster
Tx Y
YCluster
YCluster
01
ijr
ju0
M
MCluster
’01
’02
jjjj uTM 00201000
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Upper-level mediation model: Results
00 01 0j j jM T u
To estimate the a path, I ran an OLS regression in SPSS using the Level 2 file
Coefficientsa
9.429 .444 21.228 .000
.714 .628 .312 1.137 .278
(Constant)
T
Model1
B Std. Error
UnstandardizedCoefficients
Beta
StandardizedCoefficients
t Sig.
Dependent Variable: M1a.
What is the estimate of a and its SE?
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Upper-level mediation model: Results
To estimate the b path, I ran a model in HLM
What is the estimate of b and its SE?
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 c’ and its SE?
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Upper-level mediation model: Results
Direct effect = 3.671 Indirect effect = (.714)(.795) = .568Total effect = DE + IE = 3.671 + .568 = 4.239
TxCluster
Tx Y
YCluster
YCluster
ijr
ju0
M
MCluster
.714
3.671
.795
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Upper-level mediation model: Results
Causal steps approach
Test of joint significance
z test of ab product
Empirical-M test
BC parametric bootstrap
Step 1 significant, but not Steps 2 and 3
Neither path a nor path b are significant
se=.68, z=.83, p=.41 95% CI = -.78 to 1.91
95% CI = -.47 to 2.26
95% CI = -.42 to 3.68
No.
No.
No.
No.
No.
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Upper-level mediation model: Results PRODCLIN http://www.public.asu.edu/~davidpm/ripl/ Prodclin/
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Cross-level mediation model (2→1→1)
0 ,ij j ijM r ' ' '0 1ij j j ij ijY M r
' ' ' '0 00 01 0j j jT u ' '1 10j
OutcomeCLUSTER
Outcome
OutcomeINDIVIDUAL
Mediator
MediatorINDIVIDUAL
TreatmentCLUSTER
Treatment
γ’01
γ’10
MediatorCLUSTER
Mediator
MediatorINDIVIDUAL
TreatmentCLUSTER
Treatment
γ01
Model A Model B
ju0 '0 ju
'ijr
'ijr
jjj uT 001000
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Cross-level mediation model: ResultsTo estimate the a path:
What is a and its SE?
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 ----------------------------------------------------------------------------
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Cross-level mediation model: ResultsTo estimate the b path:
What is b and its SE?
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 ----------------------------------------------------------------------------
And for c’?
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Cross-level mediation model: Results
Direct effect = 2.675 Indirect effect = (2.643)(.592) = 1.564Total effect = 2.675 + 1.564 = 4.239
OutcomeCLUSTER
Outcome
OutcomeINDIVIDUAL
Mediator
MediatorINDIVIDUAL
TreatmentCLUSTER
Treatment
2.675
.592
MediatorCLUSTER
Mediator
MediatorINDIVIDUAL
TreatmentCLUSTER
Treatment
2.643
Model A Model B
ju0 '0 ju
'ijr
'ijr
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Cross-level mediation model: Results
Causal steps approach
Test of joint significance
z test of ab product
Empirical-M test
BC parametric bootstrap
Steps 1, 2 and 3 significant
Paths a and b significant
se=.802, z=1.95, p=.051 95% CI = -.01 to 3.13
95% CI = .19 to 3.32
95% CI = .31 to 3.57
Yes
No
Yes
Yes
Yes
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Cross-level and upper-level mediation model [2→(1 & 2) →1]
0 ,ij j ijM r
0 00 01 0j j jT u
' ' '0 1ij j j ij ijY M r
jjjj uAveMT 00201000
OutcomeCLUSTER
Outcome
OutcomeINDIVIDUAL
Mediator
MediatorINDIVIDUAL
TreatmentCLUSTER
Treatment
γ’01
γ’10
MediatorCLUSTER
Mediator
MediatorINDIVIDUAL
TreatmentCLUSTER
Treatment
γ01
Model A Model B
ju0 '0 ju
'ijrijr
MediatorCLUSTER
Avg M
γ’02
101 j
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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 ----------------------------------------------------------------------------
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Cross-level and upper-level mediation model [2→(1 & 2) →1]
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
OutcomeCLUSTER
Outcome
OutcomeINDIVIDUAL
Mediator
MediatorINDIVIDUAL
TreatmentCLUSTER
Treatment
2.761
.600
MediatorCLUSTER
Mediator
MediatorINDIVIDUAL
TreatmentCLUSTER
Treatment
2.643
Model A Model B
ju0 '0 ju
'ijr'
ijr
MediatorCLUSTER
Avg M
-.041
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Cross-level and upper-level mediation model [2→(1 & 2) →1] Group-mean centered M
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.
OutcomeCLUSTER
Outcome
OutcomeINDIVIDUAL
Mediator
MediatorINDIVIDUAL
TreatmentCLUSTER
Treatment
2.761
.600
MediatorCLUSTER
Mediator
MediatorINDIVIDUAL
TreatmentCLUSTER
Treatment
2.643
Model A Model B
ju0 '0 ju
'ijr'
ijr
MediatorCLUSTER
Avg M
0.559
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Cross- and upper-level mediation model: Results at the individual level
Causal steps approach
Test of joint significance
z test of ab product
Empirical-M test
BC parametric bootstrap
Steps 1, 2 and 3 significant
Paths a and b significant
se=.886, z=1.79, p=.073 95% CI = -.15 to 3.32
95% CI = .19 to 3.44
? Not yet programmed
Yes
Yes
No
Yes
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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
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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.