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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