APS 20061 Mediation Analyses in Longitudinal Studies Patrick E. Shrout New York University.

APS 2006 1

Mediation Analyses in Longitudinal Studies

Patrick E. Shrout

New York University

APS 2006 2

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

APS 2006 3

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

APS 2006 4

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.

APS 2006 5

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?

APS 2006 6

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

APS 2006 7

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]

APS 2006 8

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

APS 2006 9

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

APS 2006 10

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

APS 2006 11

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?

APS 2006 12

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

APS 2006 13

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?

APS 2006 14

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

APS 2006 15

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)

APS 2006 16

Trajectory Analysis

• ShortcomingsNumber of time points must be 3 or moreMediation process is global rather than

dynamicWith missing data, SEM approach challenging

APS 2006 17

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

APS 2006 18

w

Path Model of Total Effect

Y =Intercept + cX + w

X Yc

APS 2006 19

Path Model of Mediated Effect

Mi = d1 + aXi + vi

Yi = d2 + c'Xi + bMi + wi

X M Y

wv

a b

c'

APS 2006 20

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

APS 2006 21

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

APS 2006 22

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

APS 2006 23

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

APS 2006 24

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

APS 2006 25

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

APS 2006 26

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

APS 2006 27

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

APS 2006 28

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)

APS 2006 29

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.

APS 2006 30

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.

APS 2006 31







APS 2006 32







APS 2006 33







APS 2006 34


• We might be interested in the modelY = B0 + B1X + B2T + B3(XT) +AY0+ e





APS 2006 35

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

30

B P F1 F2

ControlTreatment

APS 2006 36

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.

APS 2006 37

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

APS 2006 38

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?

APS 2006 39

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

APS 2006 40

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;

APS 2006 41

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

APS 2006 42

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.

APS 2006 43

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

APS 2006 44

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

APS 2006 45

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

APS 2006 46

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

APS 2006 47

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

APS 2006 48


• 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

APS 2006 49


• 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

APS 2006 50

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)

APS 2006 51

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.

APS 2006 52

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

APS 2006 53

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

APS 20061 Mediation Analyses in Longitudinal Studies Patrick E. Shrout New York University.

Documents

Transcript of APS 20061 Mediation Analyses in Longitudinal Studies Patrick E. Shrout New York University.