mediation models

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. 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 mediation? Mediational process can be

Observed or latent Internal or externalAt the individual or cluster levelBased 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 exampleMaze-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 MMore in later session of workshop…

Basic (single-level) mediation model

Outcome

Mediator

Treatment

a b

c’

OutcomeTreatment ciii eTY 10

iii eTM 10

''2

'1

'0 iiii eTMY

total effect = indirect effect + direct effect

c = ab + c’

Causality concerns

Just because you estimate the model

X M Ydoes not mean that the relationships are causalUnless 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

Possible misspecification

Outcome

Y

Mediator

MTreatment

T

a b

Ok!

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

Z

Tests of the hypothesized mediation effect

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)

Outcome

Y

Mediator

MTreatment

T

a b

c’

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)

OutcomeTreatment c

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

Mediator

Treatment

a

Causal steps approach

Step 4: to decide on partial or complete mediation, test the effect of T on Y, controlling for M (path c’)

Outcome

Mediator

Treatment

b

c’

Step 3: test the effect of M on Y, controlling for T (path b)

Causal steps approach: performance

Step 1 may be non-significant when true mediation exists

Outcome

Dep

Mediator

FdFTreatment

T

+2 +3

-6What if…

Outcome

Dep

Mediator

FdFTreatment

T

+2 +3or…

Mediator

SS

+3 -2

Causal steps approach: performance

Lacks power Power is a function of the product of the

power to test each of the three pathsPower 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

2nd: test the effect of M on Y, controlling for T (path b)

If both significant, then infer significant mediation

1st: test the effect of T on M (path a)

Outcome

Mediator

Treatment

a b

c’

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

Calculate z =

2222ab sebsea

Compare z test value to critical values from the standard normal distribution

Can also calculate confidence interval around ab

CI =

Sobel’s seab =

abse

ab

))(( abcritical sezab

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?

0

50

100

150

200

-4 -3 -2 -1 0 1 2 3 4

0

50

100

150

200

-4 -3 -2 -1 0 1 2 3 4

0

50

100

150

200

-4 -3 -2 -1 0 1 2 3 4

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)

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:

If CI does not include zero, then significant

))(( ablower seCVab))(( abupper seCVab

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 variationsParametric percentile Non-parametric percentileBias-corrected versions of both

Can bootstrap cases or bootstrap residuals. It is 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 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

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 →YUpper-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 be partitioned into individual level components and cluster level components

ijjij ruY 000

Cluster

uoj

Yij

Individual

rijNote: No intercepts depicted

Mediation model possibilities

TxCluster

Tx

TxIndividual

MCluster

M

MIndividual

YCluster

Y

YIndividual

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

jjj uT 001000

TxCluster

Tx Y

YCluster

YCluster

01

ijr

ju0

ijjij eY 0

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)

00 01 0j j jM T u

' '0ij j ijY r

TxCluster

Tx Y

YCluster

YCluster

01

ijr

ju0

M

MCluster

’01

’02

jjjj uTM 00201000

Upper-level mediation model: Results

00 01 0j j jM T u

To estimate the a path, I ran an OLS regression in SPSS using the Level 2 file

Coefficientsa

9.429 .444 21.228 .000

.714 .628 .312 1.137 .278

(Constant)

T

Model1

B Std. Error

UnstandardizedCoefficients

Beta

StandardizedCoefficients

t Sig.

Dependent Variable: M1a.

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

What is the estimate of b and its SE?

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 c’ and its SE?

Upper-level mediation model: Results

Direct effect = 3.671 Indirect effect = (.714)(.795) = .568Total effect = DE + IE = 3.671 + .568 = 4.239

TxCluster

Tx Y

YCluster

YCluster

ijr

ju0

M

MCluster

.714

3.671

.795

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

Neither path a nor path b are significant

se=.68, z=.83, p=.41 95% CI = -.78 to 1.91

95% CI = -.47 to 2.26

95% CI = -.42 to 3.68

No.

No.

No.

No.

No.

Upper-level mediation model: Results PRODCLIN http://www.public.asu.edu/~davidpm/ripl/ Prodclin/

Cross-level mediation model (2→1→1)

0 ,ij j ijM r ' ' '0 1ij j j ij ijY M r

' ' ' '0 00 01 0j j jT u ' '1 10j

OutcomeCLUSTER

Outcome

OutcomeINDIVIDUAL

Mediator

MediatorINDIVIDUAL

TreatmentCLUSTER

Treatment

γ’01

γ’10

MediatorCLUSTER

Mediator

MediatorINDIVIDUAL

TreatmentCLUSTER

Treatment

γ01

Model A Model B

ju0 '0 ju

'ijr

'ijr

jjj uT 001000

Cross-level mediation model: ResultsTo estimate the a path:

What is a and its SE?

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

Cross-level mediation model: ResultsTo estimate the b path:

What is b and its SE?

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

And for c’?

Cross-level mediation model: Results

Direct effect = 2.675 Indirect effect = (2.643)(.592) = 1.564Total effect = 2.675 + 1.564 = 4.239

OutcomeCLUSTER

Outcome

OutcomeINDIVIDUAL

Mediator

MediatorINDIVIDUAL

TreatmentCLUSTER

Treatment

2.675

.592

MediatorCLUSTER

Mediator

MediatorINDIVIDUAL

TreatmentCLUSTER

Treatment

2.643

Model A Model B

ju0 '0 ju

'ijr

'ijr

Cross-level mediation model: Results

Causal steps approach

Test of joint significance

z test of ab product

Empirical-M test

BC parametric bootstrap

Steps 1, 2 and 3 significant

Paths a and b significant

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

95% CI = .19 to 3.32

95% CI = .31 to 3.57

Yes

No

Yes

Yes

Yes

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

0 ,ij j ijM r

0 00 01 0j j jT u

' ' '0 1ij j j ij ijY M r

jjjj uAveMT 00201000

OutcomeCLUSTER

Outcome

OutcomeINDIVIDUAL

Mediator

MediatorINDIVIDUAL

TreatmentCLUSTER

Treatment

γ’01

γ’10

MediatorCLUSTER

Mediator

MediatorINDIVIDUAL

TreatmentCLUSTER

Treatment

γ01

Model A Model B

ju0 '0 ju

'ijrijr

MediatorCLUSTER

Avg M

γ’02

101 j

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]

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

OutcomeCLUSTER

Outcome

OutcomeINDIVIDUAL

Mediator

MediatorINDIVIDUAL

TreatmentCLUSTER

Treatment

2.761

.600

MediatorCLUSTER

Mediator

MediatorINDIVIDUAL

TreatmentCLUSTER

Treatment

2.643

Model A Model B

ju0 '0 ju

'ijr'

ijr

MediatorCLUSTER

Avg M

-.041

Cross-level and upper-level mediation model [2→(1 & 2) →1] Group-mean centered M

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.

OutcomeCLUSTER

Outcome

OutcomeINDIVIDUAL

Mediator

MediatorINDIVIDUAL

TreatmentCLUSTER

Treatment

2.761

.600

MediatorCLUSTER

Mediator

MediatorINDIVIDUAL

TreatmentCLUSTER

Treatment

2.643

Model A Model B

ju0 '0 ju

'ijr'

ijr

MediatorCLUSTER

Avg M

0.559

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

Steps 1, 2 and 3 significant

Paths a and b significant

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

95% CI = .19 to 3.44

? Not yet programmed

Yes

Yes

No

Yes

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