Mediator analysis within field trials

Laura Stapleton

UMBC

Session outline

Basic mediation model

Comment on causality

Tests of the hypothesized mediation effect

Examples of mediation models for cluster randomized trials

Brief preview of advanced issues and software

Basic mediation model

Outcome

Y

Mediator

MTreatment

T

a b

c’

Outcome

YTreatment

Tc

iii eTY 10

iii eTM 10

''2

'1

'0 iiii eTMY

total effect = indirect effect + direct effect

c = ab + c’

Causality concerns

Because the mediator is not manipulated, causal interpretations are limited

Possible misspecification

Outcome

Y

Mediator

MTreatment

T

a b

Ok!

In future research, manipulate mediator For now, assume or hypothesize that M causes Y

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 ab exists in the population, a test of the 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

z test of ab (with normal theory confidence interval)

Asymmetric confidence interval (Empirical M or distribution of the product)

Other tests not considered today: Causal steps approach (Baron & Kenny)

Test of joint significance

Bootstrap resampling

z test of ab product Calculate z = 2222

ab sebsea

Compare z test value to critical values on the normal distributionCan also calculate confidence interval around ab

CI = One of the least powerful approachesProblem is that the ab product is not normally distributed, so critical values are inappropriate

seab = abse

ab

))(( abcritical sezab

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

I simulated 1,000 estimates of a and 1,000 estimates of b where mean = 0 and SD=1

Distribution of path a Distribution of path b

Distribution of product of axb

Empirical M-test (asymmetric CI) Determines empirical distribution of z of the ab

product (not assuming normality) Distribution is leptokurtic and symmetric when

αβ=0, but is skewed if αβ > 0 or αβ < 0 Given a, b, and their SEs, PRODCLIN determines

the distribution of ab and critical values Confidence interval limits:

If CI does not include zero, then “significant”

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

Mediation models for cluster randomized trials

Extend basic model to situations when treatment is administered at group level

Model depends on whether mediator is measured at group or individual levelUpper-level mediation (2→2→1 Design)Cross-level mediation (2→1→1 Design)Cross-level and upper-level mediation

(2→1 / 2→1 Design)

Measured variable partitioning

First, consider that any variable may be partitioned into individual level components and cluster level components

CLUSTER process(uoj)

Yij

INDIVIDUAL

process(rij)

ijjij ruY 000

Mediation model possibilities

OutcomeCLUSTER

Outcome

Outcome INDIVIDUAL

MediatorCLUSTER

Mediator

Mediator INDIVIDUAL

Treatment CLUSTER

Treatment

Treatment INDIVIDUAL

Data Example Context

Cluster randomized trial (hierarchical design) 14 pre-schools: ½ treatment, ½ control Socio-emotional curriculum Outcome is child behavior Possible mediators: teacher attitude, child socio-

emotional knowledge Sample data are on posted handout (n=84) Analyses with SPSS (HLM and SAS available)

Total effect of treatment

' '0ij j ijY r

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

'0

'01

'00

'0 jjj uT

OutcomeCLUSTER

Outcome

OutcomeINDIVIDUAL

TreatmentCLUSTER

Treatment

γ’01

'0 ju

'ijr

Total effect of treatment: Results

Given that SD of Y is 4.381, effect size of treatment is large: .97.

MIXED Y WITH T/FIXED= T | SSTYPE(3)/RANDOM = INTERCEPT | SUBJECT(J) COVTYPE(VC)/METHOD = REML/CRITERIA = CIN(95) MXITER(100) MXSTEP(5) SCORING(1) SINGULAR(0.000000000001) HCONVERGE (0,ABSOLUTE) LCONVERGE(0,ABSOLUTE) PCONVERGE(0.000001,ABSOLUTE)/PRINT = CPS G SOLUTION TESTCOV.

Upper-level mediation model (2→2→1)

00 01 0j j jM T u

' '0ij j ijY r ' ' ' ' '0 00 01 02 0j j j jM T u

OutcomeCLUSTER

Outcome

OutcomeINDIVIDUAL

MediatorCLUSTER

Mediator

TreatmentCLUSTER

Treatment

γ01

γ’02

γ’01

'0 ju

ju0

'ijr

Upper-level mediation model: Results

00 01 0j j jM T u

To estimate the a path, I ran an OLS regression is SPSS using a file from the 14 schools

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.

The estimate is .714 with a standard error of .628

Upper-level mediation model: ResultsTo estimate the b path, I ran a mixed model

The estimate is .795 with a SE of .656

MIXED Y WITH T M1/FIXED= T M1 | SSTYPE(3)/RANDOM = INTERCEPT | SUBJECT(J) COVTYPE(VC)<<remainder of syntax same as before>>

Upper-level mediation model: Results

OutcomeCLUSTER

Outcome

OutcomeINDIVIDUAL

MediatorCLUSTER

Mediator

TreatmentCLUSTER

Treatment

.714

3.671

.795

'0 ju

ju0

'ijrDirect effect = 3.671

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

Upper-level mediation model: Results Significance test of the indirect effect PRODCLIN http://www.public.asu.edu/~davidpm/ripl/Prodclin/

Cross-level mediation model (2→1→1)

OutcomeCLUSTER

Outcome

OutcomeINDIVIDUAL

Mediator

MediatorINDIVIDUAL

TreatmentCLUSTER

Treatment

γ’01

γ’10

MediatorCLUSTER

Mediator

MediatorINDIVIDUAL

TreatmentCLUSTER

Treatment

γ01

Model A Model Bju0 '

0 ju

'ijr

0 ,ij j ijM r

0 00 01 0j j jT u

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

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

'ijr

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

The estimate is 2.643 with SE of 1.195

MIXED M2_GrandC WITH T/FIXED= T | SSTYPE(3)/RANDOM = INTERCEPT | SUBJECT(J) COVTYPE(VC)<<remainder of syntax same as before>>

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

The estimate is .592 with a SE of .143

MIXED Y WITH M2_GrandC T/FIXED= M2_GrandC T | SSTYPE(3)/RANDOM = INTERCEPT | SUBJECT(J) COVTYPE(VC)<<remainder of syntax same as before>>

Cross-level mediation model: Results

OutcomeCLUSTER

Outcome

OutcomeINDIVIDUAL

Mediator

MediatorINDIVIDUAL

TreatmentCLUSTER

Treatment

2.675

.592

MediatorCLUSTER

Mediator

MediatorINDIVIDUAL

TreatmentCLUSTER

Treatment

2.643

Model A Model Bju0 '

0 ju

'ijr

'ijr

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

Cross-level mediation model: Results Test of the indirect effect

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

' '1 10j

'0

'02

'01

'00

'0 jjjj uAveMT

MediatorCLUSTER

Mediator

MediatorINDIVIDUAL

TreatmentCLUSTER

Treatment

γ01

Model A Model Bju0

'ijr

Outcome

OutcomeINDIVIDUAL

M

MediatorINDIVIDUAL

Treatment

γ’10

OutcomeCLUSTER

MediatorCLUSTER

TreatmentCLUSTER γ’01

γ’02

Ave. M

'0 ju

'ijr

Cross-level and upper-level mediation model: Results

Path a is the same as in the prior model. For the b paths:MIXED Y WITH M2_AVE M2_GrandC T/FIXED= M2_AVE M2_GrandC T | SSTYPE(3)/RANDOM = INTERCEPT | SUBJECT(J) COVTYPE(VC)<<remainder of syntax same as before>>

Cross-level and upper-level mediation model (2→1 / 2→1)

Outcome

OutcomeINDIVIDUAL

M

MediatorINDIVIDUALGRAND_C

Treatment

.600

OutcomeCLUSTER

MediatorCLUSTER

TreatmentCLUSTER 2.761

-.041

Ave. M

'0 ju

'ijr

MediatorCLUSTER

Mediator

MediatorINDIVIDUAL

TreatmentCLUSTER

Treatment

2.643

ju0

'ijr

Note that there are now TWO mediation paths: abindividual = (2.643)(.600) = 1.586abcluster = (2.643)(-.041) = -.109

Test of the indirect effect at the individual level:

Cross-level and upper-level mediation model (2→1 / 2→1)

Test of the indirect effect at the cluster level:

Cross-level and upper-level mediation model (2→1 / 2→1)

Cross-level and upper-level mediation model (2→1 / 2→1)

Outcome

OutcomeINDIVIDUAL

M

MediatorINDIVIDUALGROUP_C

Treatment

.600

OutcomeCLUSTER

MediatorCLUSTER

TreatmentCLUSTER 2.761

.559

Ave. M

'0 ju

'ijr

MediatorCLUSTER

Mediator

MediatorINDIVIDUAL

TreatmentCLUSTER

Treatment

2.643

ju0

'ijr

abcluster = (2.643)(.559) = 1.477 with GROUP_Cabcluster = (2.643)(-.041) = -.109 with GRAND_C

The level-2 effect of the mediator differs with group- versus grand-mean centering:

Brief preview of advanced issues

Multisite / randomized blocks (1→1 →1) Testing mediation in 3-level models Including multiple mediators Examining moderated mediation Dichotomous or polytomous outcomes Measurement error in mediation models Bayesian estimation of indirect effects

Notes on software

SPSS

HLM

SAS (PROC MIXED)

MLwiN

Mplus

stapleton@umbc.edu