mediator analysis within field trials laura stapleton umbc
TRANSCRIPT
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Mediator analysis within field trials
Laura Stapleton
UMBC
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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
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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’
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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
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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’
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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
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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
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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
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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
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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)
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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
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Mediation model possibilities
OutcomeCLUSTER
Outcome
Outcome INDIVIDUAL
MediatorCLUSTER
Mediator
Mediator INDIVIDUAL
Treatment CLUSTER
Treatment
Treatment INDIVIDUAL
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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)
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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
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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.
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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
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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
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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>>
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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
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Upper-level mediation model: Results Significance test of the indirect effect PRODCLIN http://www.public.asu.edu/~davidpm/ripl/Prodclin/
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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
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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>>
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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>>
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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
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Cross-level mediation model: Results Test of the indirect effect
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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
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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>>
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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
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Test of the indirect effect at the individual level:
Cross-level and upper-level mediation model (2→1 / 2→1)
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Test of the indirect effect at the cluster level:
Cross-level and upper-level mediation model (2→1 / 2→1)
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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:
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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
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Notes on software
SPSS
HLM
SAS (PROC MIXED)
MLwiN
Mplus