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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

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mediation models

Mediation Models

Laura Stapleton

UMBC

mediation models2

Mediation Models

Tasha Beretvas

University of Texas at Austin

session outline
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
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
what is mediation5
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
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 examples7
Mediation Examples

Intervention  Outcome

Intervention  Receptivity  Outcome

Intervention  Tx Fidelity  Outcome

Intervention  TchConfid Outcome

Intervention  Soc Comp Achievement

Intervention  Phon Aware  Reading

Intervention Peer Affil  DelinqBeh

mediation moderation
Mediation  Moderation
  • A moderator explains when an effect occurs
    • Relationship between X and Ychanges for different values of M
    • More in later session of workshop…
basic single level mediation model
Basic (single-level) mediation model

c

Treatment

Outcome

Mediator

a

b

Treatment

Outcome

c’

total effect = indirect effect + direct effect

c= ab+ c’

causality concerns
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
causality concerns mediation model
Causality concerns – mediation model
  • Remember, if the mediator is not typically manipulated, causal interpretations are limited

Z

Mediator

M

a

b

Treatment

T

Ok!

Outcome

Y

  • Possible misspecification
  • For now, be sure to substantively justify the causal direction and “assumeor 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
Tests of the hypothesized mediation effect

Mediator

M

a

b

  • The estimate of the indirect effect, ab, is based on the sample
  • To infer that a non-zero αβexists 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)

Treatment

T

Outcome

Y

c’

tests of the hypothesized mediation effect13
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
Causal steps approach
  • Step 1: test the effect of T on Y (path c)

c

Treatment

Outcome

  • Step 2: test the effect of T on M (path a)

Mediator

a

Treatment

causal steps approach15
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
Causal steps approach: performance
  • Step 1 may be non-significant when true mediation exists

Mediator

FdF

+2

+3

What if…

Treatment

T

Outcome

Dep

-6

Mediator

FdF

+2

+3

or…

Treatment

T

Outcome

Dep

+3

-2

Mediator

SS

causal steps approach performance17
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
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
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
z test of ab product
  • Calculate z =
  • Sobel’sseab=
  • Compare z test value to critical values from the standard normal distribution
  • Can also calculate confidence interval around ab

CI =

z test of ab product performance
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 abproduct distributed?
sampled 1 000 a n 0 1 and of b n 0 1
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

empirical m test asymmetric ci
Empirical M-test (asymmetric CI)
  • Determines empirical (more leptokurtic) distribution of zof the abproduct (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 Zbvalues (increments of 0.4) that could be used to calculate asymmetric CIs.
empirical m test asymmetric ci24
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

empirical m test performance
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
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.
parametric percentile bootstrap
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 bootstrap28
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
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
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
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

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

example for today
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
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
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

mediation model possibilities
Mediation model possibilities

Tx

Cluster

M

Cluster

Y

Cluster

Tx

M

Y

Tx

Individual

M

Individual

Y

Individual

data example context
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
Total effect of treatment

Before we examine mediation, let’s examine the total effect of treatment on the outcome…

Tx

Cluster

Y

Cluster

01

Tx

Y

Y

Cluster

total effect of treatment fe results
Total effect of treatment: FEResults

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
Upper-level mediation model (2→2→1)

M

Cluster

01

’01

Tx

Cluster

Y

Cluster

’02

Tx

M

Y

Y

Cluster

upper level mediation model results
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?

upper level mediation model results42
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 results43
Upper-level mediation model: Results

M

Cluster

.714

.795

Tx

Cluster

Y

Cluster

3.671

Tx

M

Y

Y

Cluster

  • Direct effect = 3.671
  • Indirect effect = (.714)(.795) = .568
  • Total effect = DE + IE = 3.671 + .568 = 4.239
upper level mediation model results44
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 bare 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

upper level mediation model results45
Upper-level mediation model: Results
  • PRODCLINhttp://www.public.asu.edu/~davidpm/ripl/ Prodclin/
cross level mediation model 2 1 1
Cross-level mediation model (2→1→1)

Model A

Model B

Mediator

CLUSTER

γ01

Treatment

CLUSTER

Treatment

CLUSTER

Outcome

CLUSTER

γ’01

Mediator

Mediator

Treatment

Treatment

Outcome

Mediator

INDIVIDUAL

Mediator

INDIVIDUAL

γ’10

Outcome

INDIVIDUAL

cross level mediation model results
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 results48
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 results49
Cross-level mediation model: Results

Model A

Model B

Mediator

CLUSTER

2.643

Treatment

CLUSTER

Treatment

CLUSTER

Outcome

CLUSTER

2.675

Mediator

Mediator

Treatment

Treatment

Outcome

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
cross level mediation model results50
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 bsignificant

se=.802, z=1.95, p=.051 95% CI = -.01 to 3.13

No

Yes

95% CI = .19 to 3.32

95% CI = .31 to 3.57

Yes

cross level and upper level mediation model 2 1 2 1
Cross-level and upper-level mediation model [2→(1 & 2) →1]

Model A

Model B

Mediator

CLUSTER

γ’02

γ01

Mediator

CLUSTER

γ’01

Treatment

CLUSTER

Treatment

CLUSTER

Outcome

CLUSTER

Avg M

Mediator

Mediator

Treatment

Treatment

Outcome

Mediator

INDIVIDUAL

Mediator

INDIVIDUAL

γ’10

Outcome

INDIVIDUAL

cross level and upper level mediation model results
Cross-level and upper-level mediation model: Results

Path a is the same as in the prior model. For the band 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 2 153
Cross-level and upper-level mediation model [2→(1 & 2) →1]

Model A

Model B

Mediator

CLUSTER

-.041

Mediator

CLUSTER

2.643

Treatment

CLUSTER

Treatment

CLUSTER

Outcome

CLUSTER

2.761

Avg M

Mediator

Mediator

Treatment

Treatment

Outcome

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 = 1.477+2.761 = 4.238
cross level and upper level mediation model 2 1 2 1 group mean centered m
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

Treatment

CLUSTER

Treatment

CLUSTER

Outcome

CLUSTER

2.761

Avg M

Mediator

Mediator

Treatment

Treatment

Outcome

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.
cross and upper level mediation model results at the individual level
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 bsignificant

se=.886, z=1.79, p=.073 95% CI = -.15 to 3.32

No

95% CI = .19 to 3.44

Yes

? Not yet programmed

brief review of advanced issues
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
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.
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