Title: Mediation Models
1Mediation Models
2Mediation Models
- Tasha Beretvas
- University of Texas at Austin
3Session 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
4What 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
5What 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.
6Mediation 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
7Mediation 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
8Mediation ? 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
9Basic (single-level) mediation model
c
Outcome
Treatment
Mediator
a
b
Outcome
Treatment
c
total effect indirect effect direct effect
c ab c
10Causality 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
11Causality concerns mediation model
- Remember, if the mediator is not typically
manipulated, causal interpretations are limited
Z
Mediator M
a
b
?
Outcome Y
Treatment T
Ok!
- Possible misspecification
- For now, be sure to substantively justify the
causal direction and assume or hypothesize that
M causes Y and assuming that, heres the strength
of that effect - In future research, manipulate mediator
12Tests of the hypothesized mediation effect
Mediator M
a
b
Outcome Y
Treatment T
c
- The estimate of the indirect effect, ab, is based
on the sample - To infer that a non-zero aß 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)
13Tests 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
14Causal steps approach
- Step 1 test the effect of T on Y (path c)
c
Outcome
Treatment
- Step 2 test the effect of T on M (path a)
Mediator
a
Treatment
15Causal steps approach
- Step 3 test the effect of M on Y, controlling
for T (path b)
Mediator
b
Outcome
Treatment
c
- Step 4 to decide on partial or complete
mediation, test the effect of T on Y, controlling
for M (path c)
16Causal steps approach performance
- Step 1 may be non-significant when true mediation
exists
Mediator FdF
2
3
What if
Outcome Dep
Treatment T
-6
Mediator FdF
2
3
or
Outcome Dep
Treatment T
3
-2
Mediator SS
17Causal 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
18Test of joint significance
- Very similar to causal steps approach
Mediator
a
b
Outcome
Treatment
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
19Test 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
20z test of ab product
- Compare z test value to critical values from the
standard normal distribution - Can also calculate confidence interval around ab
- CI
21z 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?
22Sampled 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
23Empirical M-test (asymmetric CI)
- Determines empirical (more leptokurtic)
distribution of z of the ab product (not assuming
normality) - aß0 distn is leptokurtic and symmetric
- aßgt0 distn is less leptokurtic and ly skewed
- aßlt0 distn 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.
24Empirical 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
25Empirical 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
26Bootstrap 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.
27Parametric 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,s2) (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
28Parametric 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
29Non-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
samples analysis) - The remaining process is the same as for the
parametric version
30Bias-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
31Bootstrap 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
32Summary 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
33Example 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
34Mediation 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
35Measured variable partitioning
Cluster uoj
- First, consider that any variable may be
partitioned into individual level components and
cluster level components
Yij
Individual rij
Note No intercepts depicted
36Mediation model possibilities
Tx Cluster
M Cluster
Y Cluster
Tx
M
Y
Tx Individual
M Individual
Y Individual
37Data 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
38Total effect of treatment
Before we examine mediation, lets examine the
total effect of treatment on the outcome
Tx Cluster
Y Cluster
?01
Tx
Y
Y Cluster
39Total 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
40Upper-level mediation model (2?2?1)
M Cluster
?01
?01
Tx Cluster
Y Cluster
?02
Tx
Y
M
Y Cluster
41Upper-level mediation model Results
To estimate the a path, I ran an OLS regression
in SPSS using the Level 2 file
What is the estimate of a and its SE?
42Upper-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?
43Upper-level mediation model Results
M Cluster
.714
.795
Tx Cluster
Y Cluster
3.671
Tx
Y
M
Y Cluster
- Direct effect 3.671
- Indirect effect (.714)(.795) .568
- Total effect DE IE 3.671 .568 4.239
44Upper-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
No.
Neither path a nor path b are significant
No.
se.68, z.83, p.41 95 CI -.78 to 1.91
No.
No.
95 CI -.47 to 2.26
No.
95 CI -.42 to 3.68
45Upper-level mediation model Results
- PRODCLIN http//www.public.asu.edu/davidpm/ripl/
Prodclin/
46Cross-level mediation model (2?1?1)
Model A
Model B
Mediator CLUSTER
?01
Outcome CLUSTER
Treatment CLUSTER
Treatment CLUSTER
?01
Mediator
Mediator
Outcome
Treatment
Treatment
Mediator INDIVIDUAL
Mediator INDIVIDUAL
?10
Outcome INDIVIDUAL
47Cross-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?
48Cross-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?
49Cross-level mediation model Results
Model A
Model B
Mediator CLUSTER
2.643
Outcome CLUSTER
Treatment CLUSTER
Treatment CLUSTER
2.675
Mediator
Mediator
Outcome
Treatment
Treatment
Mediator INDIVIDUAL
Mediator INDIVIDUAL
.592
Outcome INDIVIDUAL
- Direct effect 2.675
- Indirect effect (2.643)(.592) 1.564
- Total effect 2.675 1.564 4.239
50Cross-level mediation model Results
- Causal steps approach
- Test of joint significance
- z test of ab product
- Empirical-M test
- BC parametric bootstrap
Yes
Steps 1, 2 and 3 significant
Yes
Paths a and b significant
se.802, z1.95, p.051 95 CI -.01 to 3.13
No
Yes
95 CI .19 to 3.32
95 CI .31 to 3.57
Yes
51Cross-level and upper-level mediation model 2?(1
2) ?1
Model A
Model B
Mediator CLUSTER
?02
?01
Mediator CLUSTER
?01
Outcome CLUSTER
Treatment CLUSTER
Treatment CLUSTER
Avg M
Mediator
Mediator
Outcome
Treatment
Treatment
Mediator INDIVIDUAL
Mediator INDIVIDUAL
?10
Outcome INDIVIDUAL
52Cross-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
--------------------------------------------------
--------------------------
53Cross-level and upper-level mediation model 2?(1
2) ?1
Model A
Model B
Mediator CLUSTER
-.041
Mediator CLUSTER
2.643
Outcome CLUSTER
Treatment CLUSTER
Treatment CLUSTER
2.761
Avg M
Mediator
Mediator
Outcome
Treatment
Treatment
Mediator INDIVIDUAL
Mediator INDIVIDUAL
.600
Outcome INDIVIDUAL
- abind (2.643)(.600) 1.586
- abcluster (2.643)(-.041) -.109
- Total indirect effect 1.586 0.109 1.477
- Total effect 1.4772.761 4.238
54Cross-level and upper-level mediation model 2?(1
2) ?1 Group-mean centered M
Model A
Model B
Mediator CLUSTER
0.559
Mediator CLUSTER
2.643
Outcome CLUSTER
Treatment CLUSTER
Treatment CLUSTER
2.761
Avg M
Mediator
Mediator
Outcome
Treatment
Treatment
Mediator INDIVIDUAL
Mediator INDIVIDUAL
.600
Outcome 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.
55Cross- 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
Yes
Steps 1, 2 and 3 significant
Yes
Paths a and b significant
se.886, z1.79, p.073 95 CI -.15 to 3.32
No
95 CI .19 to 3.44
Yes
? Not yet programmed
56Brief 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
57Notes on software
- HLM,SPSS
- Plug results into PRODCLIN
- SAS (PROC MIXED)
- See handout
- Can use Stapletons macros for bootstrapping
- MLwiN, MPlus
- Have limited bootstrapping capacity but still
have to summarize results - SEM software
- Provide test of ?? but using Sobel.
58- tasha.beretvas_at_mail.utexas.edu