APS 2006 1 Mediation Analyses in Longitudinal Studies Patrick E. Shrout New York University
Dec 25, 2015
APS 2006 1
Mediation Analyses in Longitudinal Studies
Patrick E. Shrout
New York University
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Parent training
• Suppose mothers of preschool children are randomly assigned to parent training sessions Active condition emphasizes positive parenting,
structured family life and shared experiences Sessions take six months
• Suppose the preschoolers are followed every eight months for 24 months following treatment Observing externalizing behaviors in two settings
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Parent training
Suppose the treatment group did better than the control
HOW CAN WE UNDERSTAND THE EFFECT?
0
10
20
30
40
50
B P F1 F2
ControlTreatment
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Developmental Processes
• Studies with longitudinal follow-up are embedded in developmental processesChildren are growingSocial context may be changingMothers and children may be changed by
their own interactions
• These processes can make interpretation of effects challenging.
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Identifying Mediating Processes can Help Understand Effects
• Parenting programDo mothers in treatment condition show more
warmth as a result of the intervention?Do mothers refrain from negative parenting as
a result of the program?Do the children who refrain from externalizing
behaviors belong to mothers who show improved parenting?
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Plan for Talk
• Review data analysis options for experiments with longitudinal follow-up
• Review mediation analyses of fixed endpoints
• Briefly consider mediation of trajectory patterns (growth-curve analysis)
• Introduce an easy approximate approach toward longitudinal mediationA kind of mediated moderation analysis
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A note on the plan
• Hope talk gives intuitions about approach
• Details of examples and statistical software syntax are available on webwww.psych.nyu.edu/couples/ShroutOr send e-mail to [email protected]
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Common Data Analysis Approaches
• Ignore baseline (pre-randomization) measuresMultiple endpoint t testsRepeated measures in ANOVA traditionLatent trajectory analyses
• Consider baseline measuresMultiple endpoint ANCOVARepeated measures ANCOVALatent trajectory analyses
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Multiple Endpoint Analyses
• Answers the important questions:Does treatment have an immediate effect?Does the effect last to Followup 1?Does the effect last to Followup 2?
• StrengthsMakes use of available dataAllow simple t tests or ANCOVAMediation analysis easily incorporated
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Multiple Endpoint Analysis
• ShortcomingsPiecemeal approach does not reveal overall
patternPower of tests may be limited at any time
pointType I error becomes a concern with multiple
looks
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Repeated Measures Analysis
• ANOVA or ANCOVATreatment is between persons factorTime is within person factorCan test for Treatment X Time interaction
• Answers interesting questionsDoes treatment have any effect during the
follow-up period?Does the effect vary systematically over time?
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Repeated Measures Analysis
• ShortcomingsHandling of Missing Data
• Listwise deletion eliminates persons with one or two missing time points
Traditionally focus on significance tests rather than process
• Mediation analysis more challenging
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Trajectory Analysis
• Consider sequence of each person’s points to be a unified trajectoryWhen smoothed these are latent trajectoriesUsually represented as intercepts and slopes
• Answers questionsDoes treatment have an effect on trajectory
level (intercept)?Does treatment relate to patterns of
improvement or decline?
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Trajectory Analysis
• Examples of some smoothed trajectories.
Control Group
0
10
20
30
40
P F1 F2
Time Point
Ag
gre
ss
on
Sc
ore C1
C2C3C4C5C6
Treatment Group
0
10
20
30
40
P F1 F2
Time Point
Ag
gre
ss
on
Sc
ore T1
T2
T3
T4
T5
T6
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Trajectory Analysis
• AdvantagesMake use of complete patterns of dataRecognizes heterogeneity of subjects (i.e. can
treat intercepts/slopes as “random effects”)Can be approached from two analytic
perspectives• Multilevel models• Structural equation models
Process oriented, allowing mediation approach (e.g. Cheong et al, 2003)
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Trajectory Analysis
• ShortcomingsNumber of time points must be 3 or moreMediation process is global rather than
dynamicWith missing data, SEM approach challenging
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Mediation Analysis: Fixed Endpoint Analysis
• Parent training exampleLet Y be a measure of aggressionLet X be an indicator of treatment group
• X = -.5 CONTROL• X = +.5 TREATMENT
Let M be the positive parenting mediator
• Question: To what extent does Positive Parenting (M) mediate the effect X→Y
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w
Path Model of Total Effect
Y =Intercept + cX + w
X Yc
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Path Model of Mediated Effect
Mi = d1 + aXi + vi
Yi = d2 + c'Xi + bMi + wi
X M Y
wv
a b
c'
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Mediation Question
• To what extent is the direct effect in the second model (c’) smaller than the effect (c) in the first model? (c – c’)
• MacKinnon has shown that the product of the effects going through M is equal to this difference: (c – c’) = ab
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Detailed Empirical Example
• Laurie Miller Brotman and colleagues recruited 92 high risk families with pre-school children into parenting trial. (Brotman et al, 2005) Random assignment of pre-schoolers led to
• 47 in Intervention condition• 45 in Control condition
Measurements taken at four time points• Pre-randomization (baseline) : B• Post-treatment: P• Followup 1 (8 months): F1• Followup 2 (16 months): F2
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Detailed Example, Continued
• Intervention22 weekly group sessions with parent and
child10 or more home visits
• Outcomes for todayObservations of aggressive acts by children in
standardized settings by blind ratersObservations of mother’s parenting
interactions in standardized settings by blind raters
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Detailed Example, Continued
• Longitudinal follow-up challenging.
• Some children were not observed at certain time points.
• Listwise deletion results in 54 participants 26 Control 28 Intervention
Control (n=47)
Interv (n=45)
Post Interv
37 40
Follow 1
30 32
Follow 2
31 34
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Endpoint Analysis: Followup 2
• Counts of Aggression are skewed Log transformation
of 10 minute rate used as outcome.
• Adjusted for baseline aggression
Endpoint Analysis: Follow 2
00.20.40.60.8
11.21.41.6
Follow 2A
gg
ress
ion
Co
un
t/10
min
Control
Intervention
t(62) = -1.83 p=0.072
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Mediation of F2 Endpoint Effect
X: Intervention (effect coded)M: Observed Positive Parenting Follow 2Y: Log Observed Aggression Rate Follow 2
Total (unadjusted) Effect = -0.65Indirect Effect = (0.29)(-0.38) = -0.11
X Y
Ma=0.29
c’=-0.54
b=-0.38
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Mediation of F2 Endpoint Effect
X: Intervention (effect coded)M: Observed Positive Parenting Follow 2Y: Log Observed Aggression Rate Follow 2
Total (unadjusted) Effect = -0.65Indirect Effect = (0.29)(-0.38) = -0.11
X Y
Ma=0.29
c’=-0.54
b=-0.38
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Moving Beyond Usual Analyses
• Limitations of usual analyses includeHow missing data is handledHow general patterns are fit over timeHow difficult it is to model mediation
processes
• A generalized Repeated Measures Analysis helps with these limitations
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Generalized Linear Models
• Generalized linear models extend the RM ANOVA model toTake into account correlated observation
• GEE: generalized estimating equation approach
Take into account special characteristics of Y• Logistic regression for binary data• Poisson regression for counts and rates
Include all observations in analysisSee Diggle et al. (2002)
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Basic Idea of GEE
• Treat the different time points as if they were independent observations E.g. in software, they become separate lines of data
• Define a linear model, such as an ANOVA or regression model
• Estimate the model and the residuals• Study the degree of correlation among residuals
from single persons• Re-estimate the model taking the correlation into
account.
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Repeated Measures Revisited
• We might be interested in the modelY = B0 + B1X + B2T + B3(X*T) +AY0+ e
• X is an indicator of TreatmentT is an indicator of post intervention TimeY0 is baseline covariate
• B1 is the average treatment effect
• B2 is the average time effect (linear model)
• B3 describes how the treatment effect varies with time.
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Repeated Measures Revisited
• We might be interested in the modelY = B0 + B1X + B2T + B3(X*T) +AY0+ e
• X is an indicator of TreatmentT is an indicator of post intervention TimeY0 is baseline covariate
• B1 is the average treatment effect
• B2 is the average time effect (linear model)
• B3 describes how the treatment effect varies with time.
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Repeated Measures Revisited
• We might be interested in the modelY = B0 + B1X + B2T + B3(X*T) +AY0+ e
• X is an indicator of TreatmentT is an indicator of post intervention TimeY0 is baseline covariate
• B1 is the average treatment effect
• B2 is the average time effect (linear model)
• B3 describes how the treatment effect varies with time.
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Repeated Measures Revisited
• We might be interested in the modelY = B0 + B1X + B2T + B3(X*T) +AY0+ e
• X is an indicator of TreatmentT is an indicator of post intervention TimeY0 is baseline covariate
• B1 is the average treatment effect
• B2 is the average time effect (linear model)
• B3 describes how the treatment effect varies with time.
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Repeated Measures Revisited
• We might be interested in the modelY = B0 + B1X + B2T + B3(XT) +AY0+ e
• X is an indicator of TreatmentT is an indicator of post intervention TimeY0 is baseline covariate
• B1 is the average treatment effect
• B2 is the average time effect (linear model)
• B3 describes how the treatment effect varies with time.
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Interaction of Treatment and Time
• Time moderates the Treatment effect. With longer followup,
the treatment effect is larger
• Question: Does the Mother’s positive behavior explain (mediate) the moderation?
20
25
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B P F1 F2
ControlTreatment
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20
25
30
B P F1 F2
ControlTreatment
20
25
30
B P F1 F2
ControlTreatment
What about Baseline?
• Including the baseline in the trajectory introduces extraneous moderation
• When the effect of the intervention is immediate, the plot with the baseline has a quadratic component.
• I recommend ADJUSTING for baseline as in ANCOVA.
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Mediated Moderation
• Muller, Judd & Yzerbyt (2005) considered mediated moderation In the case when all observations were strictly
independent
• Their approach can be extended to the case of mediation in longitudinal studies
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MJY(2005) approach
• They wrote the moderation model asY = B0 + B1X + B2T + B3(XT) + e
• Question: Can a mediation model be written to determine if B3 can be reduced (i.e. explained) by a mediation process?
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MJY(2005) approach
• MJY describe two ways that the mediator can explain the Intervention x Time effect.
The effect of the intervention on M can change with T while M has a direct effect on Y.
The effect of the intervention on M can be constant over time while the effect of M on Y changes with T.
X Y
T
X Y
T
M
X Y
T
M
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Example of Mediated Moderation
• Software systems such as SAS and STATA have generalized estimation software.
• Syntax in SAS is pretty simple
PROC GENMOD; CLASS SUBJID; MODEL Y = X T X*T Y0/ DIST=Poisson type3; REPEATED SUBJECT=SUBJID /TYPE=EXCH; RUN;
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Example of Parent Training
• DV: Count of aggressive actsHighly skewed, and modeled with Poisson link
functionEliminated one control child whose
aggression acts at all time points were 5 times higher than the next person.
• IVs: Intervention, Time and InteractionAdjusted for pre-randomization aggression
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Example Continued
• N=91 participants included
• Total of 201 person-times used
• GEE analysis indicated that there was essentially no residual correlation“Working correlation” = -0.01.
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Analysis without Mediator
• Only Intervention by Time interaction is significant (z=2.01, p=0.044)
Intervention and Time Effects on Aggression
-6.0
-5.5
-5.0
-4.5
-4.0
-3.5
P F1 F2
Time Point
Ln
Rat
e A
gg
ress
ion
Control
Intervention
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Analysis of Mediator
• Intervention effect on mediator is trend (z=1.77, p=.077)• No evidence of Time (z=-0.71, p=.48) or Intervention by
Time interaction (z=-0.22, p=0.77)
Intervention & Time Effects on Positive Parenting
3.20
3.30
3.40
3.50
3.60
3.70
P F1 F2
Time Point
Po
siti
ve A
cts
Control
Intervention
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Analysis After Adjusting for Mediator
• Some indication of moderation remains at the trend level (z=-1.86, p=.063)
Adjusted Intervention & Time Effects on Aggression
-6.0
-5.5
-5.0
-4.5
-4.0
-3.5
P F1 F2
Time Points
Ln
Rat
e A
gg
ress
ion
Control
Intervention
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Analysis After Adjusting for Mediator
• Amount of mediation is modest
Before AfterIntervention and Time Effects on Aggression
-6.0
-5.5
-5.0
-4.5
-4.0
-3.5
P F1 F2
Time Point
Ln
Rat
e A
gg
ress
ion
Control
Intervention
Adjusted Effects on Aggression
-6.0
-5.5
-5.0
-4.5
-4.0
-3.5
P F1 F2
Time Points
Ln
Rat
e A
gg
ress
ion
Control
Intervention
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Decomposition of Mediated Effect
• Muller, Judd and Yzerbyt (2005) suggest fitting the following model to examine mediated moderation:
Y = B0 + B1X + B2T + B3(X*T) + B4M + B5(M*T) + e
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Decomposition of Mediated Effect
• Muller, Judd and Yzerbyt (2005) suggest fitting the following model to examine mediated moderation:
Y = B0 + B1X + B2T + B3(X*T) + B4M + B5(M*T) + e
• In our example, the Parenting by Time interaction was strong and significant (z=-2.76, p=0.006). The beneficial effect of Positive Parenting on
Aggression seemed to increase over time
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Decomposition of Mediated Effect
• MJY(2005) show how to use the estimates of the linear models to decompose the mediation effect
• In our case Only one of the two mediation mechanisms operated Intervention had a stable effect on Positive Parenting Positive Parenting had an effect on Aggression that
increased with time
X Y
T
M
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Some Technical Comments
• MJY(2005) show that under certain circumstances the decomposition of effects adds up (simply) to the total moderation effect.
• The Poisson model does not match those circumstances.
• Nonetheless, the decomposition is a useful tool for interpretation.
• Bootstrap inference methods can be used to study the decomposition (Shrout & Bolger, 2002)
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Summary
• When outcomes of experiments have several follow-up pointsGeneralized linear models that implement
GEE for repeated measures can be useful
• When the effect of a treatment varies over follow-up timeA formal analysis of mediated moderation can
help clarify the causal processes.
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Acknowledgments
Keng-Yen Huang, Ph.D.
Laurie Miller Brotman, Ph.D.
Rachel G. Klein, Ph.D.
Niall Bolger, Ph.D.
Masumi Iida, M.A.
Christopher Burke
Grant: R01 MH 55188 (Dr. Brotman, PI)
www.psych.nyu.edu/couples/Shrout
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References• Brotman, L. M., et. al. (2005). Prevention for preschoolers at high risk for
conduct problems: Immediate outcomes on parenting practices and child social competence. Journal of Clinical Child and Adolescent Psychology, 34(4), 724-734.
• Cheong, J.W., MacKinnon, D.P. & Khoo, S.T. (2003). Investigation of mediational processes using parallel process latent growth curve modeling. Structural Equation Modeling, 10(2), 238-262.
• Diggle, P.J., Heagerty, P., Liang, KY., Zeger, S.L. (2002) Analysis of Longitudinal Data (2nd Edition). New York: Oxford U. Press.
• Muller, D., Judd, C. M. & Yzerbyt, V. Y. (2005). When moderation is mediated and mediation is moderated. Journal of Personality and Social Psychology, 89(6), 852-863.
• Shrout, P. E., & Bolger, N. (2002). Mediation in experimental and nonexperimental studies: New procedures and recommendations. Psychological Methods, 7(4), 422-445. www.psych.nyu.edu/couples/Shrout