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Mediation Models

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Title: Mediation Models


1
Mediation Models
  • Laura Stapleton
  • UMBC

2
Mediation Models
  • Tasha Beretvas
  • University of Texas at Austin

3
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

4
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

5
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.

6
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

7
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

8
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

9
Basic (single-level) mediation model
c
Outcome
Treatment
Mediator
a
b
Outcome
Treatment
c
total effect indirect effect direct effect
c ab c
10
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

11
Causality 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

12
Tests 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)

13
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

14
Causal 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
15
Causal 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)

16
Causal 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
17
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

18
Test 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

19
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

20
z test of ab product
  • Calculate z
  • Sobels seab
  • Compare z test value to critical values from the
    standard normal distribution
  • Can also calculate confidence interval around ab
  • CI

21
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?

22
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
23
Empirical 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.

24
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

25
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

26
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.

27
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,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

28
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

29
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
    samples analysis)
  • The remaining process is the same as for the
    parametric version

30
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

31
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

32
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
33
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

34
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

35
Measured 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
36
Mediation model possibilities
Tx Cluster
M Cluster
Y Cluster
Tx
M
Y
Tx Individual
M Individual
Y Individual
37
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

38
Total 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
39
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
40
Upper-level mediation model (2?2?1)
M Cluster
?01
?01
Tx Cluster
Y Cluster
?02
Tx
Y
M
Y Cluster
41
Upper-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?
42
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?
43
Upper-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

44
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
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
45
Upper-level mediation model Results
  • PRODCLIN http//www.public.asu.edu/davidpm/ripl/
    Prodclin/

46
Cross-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
47
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?
48
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?
49
Cross-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

50
Cross-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
51
Cross-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
52
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
--------------------------------------------------
--------------------------
53
Cross-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

54
Cross-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.

55
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

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
56
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

57
Notes 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
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