Get documents about "Mediator analysis within field trials
Shared by: malj
-
Stats
- views:
- 1
- posted:
- 2/3/2013
- language:
- Unknown
- pages:
- 58
Document Sample
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.
Inthe 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
Treatment c Outcome
Yi 0 1 Ti ei
Mediator M i 0 1 Ti ei
a b
Treatment Outcome
c’ Yi 0' 1' M i 2' Ti ei'
total effect = indirect effect + direct effect
c= ab + c’
Causality concerns
Just because you estimate the model
XMY
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
Mediator
Z
a M b
Treatment
Ok! Outcome
T 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 b
Treatment Outcome
T 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)
Treatment c 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 FdF +3
What if… Treatment Outcome
T -6 Dep
Mediator
+2 FdF +3
or… Treatment Outcome
T +3 -2 Dep
Mediator
SS
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 single-
level models
Power lower than other methods in multilevel
models
No confidence interval around the indirect effect is
available
z test of ab product
ab
Calculate z =
seab
Sobel’s seab = a 2 seb b 2 sea
2 2
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 4
200
150
Distribution of
100
product of axb
50
0
-4 -3 -2 -1 0 1 2 3 4
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.
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.
Itis 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 OK for single level…
z test of ab
Empirical M Yes! Easy!
Bootstrap resampling 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 Cluster
be partitioned into individual level uoj
components and cluster level
components
Yij 00 u 0 j rij Yij
Note: No intercepts depicted Individual
rij
Mediation model possibilities
Tx M Y
Cluster Cluster 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…
u0 j
Tx 01 Y
Cluster Cluster
Tx Y
Yij 0 j eij Y
Cluster
0 j 00 01 T j u0 j
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 ’01
u0 j
Cluster
Tx Y
Cluster ’02 Cluster
Tx M Y
M j 00 01T j u0 j Y
Cluster
Yij '0 j r 'ij rij
0 j 00 01 M j 02 T j u0 j
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
a
Coe fficients
Unstandardiz ed Standardized
Coef f icients Coef f icients
Model B Std. Error Beta t Sig.
1 (Cons tant) 9.429 .444 21.228 .000
T .714 .628 .312 1.137 .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
u0 j
Cluster
Tx Y
Cluster 3.671 Cluster
Tx 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
'
u0 j
γ01 Mediator
Treatment
CLUSTER
Treatment
γ’01 Outcome
CLUSTER CLUSTER CLUSTER
Mediator Mediator
Treatment Treatment Outcome
Mediator Mediator γ’10
INDIVIDUAL INDIVIDUAL
Outcome
INDIVIDUAL
rij'
Yij '0 j '1 j Mij r 'ij rij'
M ij 0 j rij ,
'0 j '00 '01Tj u'0 j
0 j 00 01T j u0 j
'1 j '10
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
Treatment
CLUSTER
Treatment
2.675 Outcome
CLUSTER CLUSTER CLUSTER
Mediator Mediator
Treatment Treatment Outcome
Mediator Mediator
INDIVIDUAL INDIVIDUAL
Outcome
INDIVIDUAL
rij'
rij'
Direct effect = 2.675
Indirect effect = (2.643)(.592) = 1.564
Total effect = 2.675 + 1.564 = 4.239
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
u0 j Mediator
CLUSTER '
u0 j
γ01
γ’01
Mediator
CLUSTER
Treatment Treatment Outcome
CLUSTER CLUSTER CLUSTER
Mediator Avg M Mediator
Treatment Treatment Outcome
Mediator Mediator
INDIVIDUAL INDIVIDUAL
Outcome
INDIVIDUAL
M ij 0 j rij , rij
Yij 0 j Mij r ij
' ' ' rij'
1j
0 j 00 01T j u0 j
0 j 00 01 T j 02 AveM j u0 j
1 j 10
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 '
u0 j
Mediator
Treatment
CLUSTER
Treatment
2.761 Outcome
CLUSTER CLUSTER CLUSTER
Mediator Avg M Mediator
Treatment Treatment Outcome
Mediator Mediator
INDIVIDUAL INDIVIDUAL
Outcome
rij' INDIVIDUAL 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
Cross-level and upper-level mediation model
[2→(1 & 2) →1] Group-mean centered M
Model A Model B
u0 j Mediator
CLUSTER '
u0 j
Mediator
Treatment
CLUSTER
Treatment
2.761 Outcome
CLUSTER CLUSTER CLUSTER
Mediator Avg M Mediator
Treatment Treatment Outcome
Mediator Mediator
INDIVIDUAL INDIVIDUAL
Outcome
rij' INDIVIDUAL rij'
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.
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
