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

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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
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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 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

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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

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
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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

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
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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

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?
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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

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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
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
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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

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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

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
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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
--------------------------------------------------
--------------------------
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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

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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
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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

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.

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• tasha.beretvas_at_mail.utexas.edu