Dynamic Structural Equation Modeling of Intensive ......Dynamic Structural Equation Modeling of...

Dynamic Structural Equation Modelingof Intensive Longitudinal Data

Oisín RyanUtrecht University

[email protected]

July, 2017

Slides from Ellen L. Hamaker

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Cattell’s data box

vari

able

s

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Time series data: N=1 and T is large

vari

able

s

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Idiographic (N=1) research in psychology

N=1 research has included:• Cattell’s P-technique: factor analysis of N=1 data• Dynamic factor analysis: considering lagged relationships• Measurement burst design: multiple waves of intensive

measurements• Intervention research: ABAB design etc.

Critique of this kind of research:• within-person fluctuations are just noise• results are not generalizable• no one has these data

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New technology

Secure continuous remote alcohol monitor (SCRAM)

Smart glasses

Smart phones

Smart watches

Smart tattoo

Activity trackers

Implants

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Intensive longitudinal data

Different forms of intensive longitudinal data:• daily diary (DD); self-report end-of-day• experience sampling method (ESM); self-report of subjective

experience• ecological momentary assessment (EMA); healthcare related

self-report• ambulatory assessment (AA); physiological measurements• event-based measurements; self-report after a particular event• observational measurements; expert rater

For more info on methodology, check out:• Seminar of Tamlin Conner and Joshua Smyth on YouTube

(https://www.youtube.com/watch?v=nQBBVp9vBIQ)• Society for Ambulatory Assessment (http://www.saa2009.org/)• Life Data (https://www.lifedatacorp.com/)• Quantified Self (http://quantifiedself.com/)

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Characteristics of these kind of data

Data structure:• one or more measurements per day• typically for multiple days• sometimes multiple waves (i.e., Nesselroade’s measurement-burst

design)

Advantages of ESM, EMA and AA• no recall bias• high ecological validity• physiological measures over a large time span• monitoring of symptoms and behavior, with new possibilities for

feedback and intervention (e-Health and m-Health)• window into the dynamics of processes

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A paradigm shift

Taken from Hamaker and Wichers (2017)

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Outline

• Time series analysis• Multilevel time series analysis• DSEM application 1: Multilevel VAR(1) model• DSEM application 2: Mediation• Discussion

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What is time series analysis?

Time series analysis is a class of techniques that is used ineconometrics, seismology, meteorology, control engineering,and signal processing.

Main characteristics:• N=1 technique• T is large (say >50)• concerned with trends, cycles and autocorrelation structure (i.e., serial

dependency)• goal: forecasting (6= prediction)

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Lags

Y

y1y2y3y4y5y6y7y8. . .yT

Y at lag 1

y1y2y3y4y5y6y7. . .

yT−1yT

Y at lag 2

y1y2y3y4y5y6. . .

yT−2yT−1yT

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Sequence, ACF and PACF

White Noise process

Time

WN

0 200 400 600 800 1000

−3

−1

13

0 5 15 25

0.0

0.4

0.8

Lag

AC

F

Series WN

0 5 15 25

−0.

060.

000.

06

Lag

Par

tial A

CF

Series WN

First−order AR process

Time

AR

1

0 200 400 600 800 1000

−4

02

4

0 5 15 25

0.0

0.4

0.8

Lag

AC

F

Series AR1

0 5 15 25

0.0

0.4

Lag

Par

tial A

CF

Series AR1

Second−order AR process

Time

AR

2

0 200 400 600 800 1000

−15

−5

515

0 5 15 25

−0.

50.

00.

51.

0

Lag

AC

FSeries AR2

0 5 15 25

−0.

50.

5

LagP

artia

l AC

F

Series AR2

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Outline


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Lagged relationships in multilevel data

If we have time series data from multiple individuals, wemay want to study:

• individual differences in lagged relationships between avariable and itself: autoregression• individual differences in lagged relationship between

different variables: cross-lagged relationships

If we use multilevel modeling for this, we could refer to it asmultilevel time series analysis, or dynamic multilevelmodeling.

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Creating lagged predictorsID

1111122222. . .NNNNN

yit

y11y12y13. . .y1T

y21y22y23. . .y2T

. . .yN1yN2yN3. . .yNT

yit−1

y11y12. . .

y1T−1

y21y22. . .

y2T−1. . .

yN1yN2. . .

yNT−1

xit−1

x11x12. . .

x1T−1

x21x22. . .

x2T−1. . .

xN1xN2. . .

xNT−1

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Creating lagged predictorsID

1111122222. . .NNNNN

yit

y11y12y13. . .y1T

y21y22y23. . .y2T

. . .yN1yN2yN3. . .yNT

yit−1

y11y12. . .

y1T−1

y21y22. . .

y2T−1. . .

yN1yN2. . .

yNT−1

xit−1

x11x12. . .

x1T−1

x21x22. . .

x2T−1. . .

xN1xN2. . .

xNT−115 / 55

Inertia research based on multilevel AR(1) models

Level 1 model:NAit = ci +φiNAi,t−1 +ζit

Level 2 model:ci = γ00 +u0i

φi = γ01 +u1i

This research line was initiated by Suls, Green and Hillis(1998), and continued by the group of Kuppens.

The focus is on individual differences in the autoregressiveparameter φi (=inertia, carry-over, regulatory weakness), whichis shown to be:• positively related to current depression, neuroticism, and being female• predictive of later depression (Kuppens and Koval)

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Dynamic networks based on multilevel VAR(1) models

Level 1 model:y1it = c1i +φ11iy1it−1 + · · ·+φ1kiykit−1 +ζ1it

y2it = c2i +φ21iy1it−1 + · · ·+φ2kiykit−1 +ζ2it

. . .ykit = cki +φk1iy1it−1 + · · ·+φkkiykit−1 +ζkit

Initiated by Bringmann et al. (2013), and further popularizedby the software from Sacha Epskamp.

The focus is on cross-lagged parameters between variables(=nodes; typically symptoms), and on measures based onthese (e.g., centrality).

Main idea is that stronger connections lead to an increasedrisk of developing and maintaining psychopathology.

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A fundamental problem in a nutshell

Number of words per minute

P e r

c e n t a

g e o

f t y

p o s

Cross-sectional relationship


P e r c

e n t a

g e o

f t y p

o s

Within-person relationship


P e r

c e n t a

g e o

f t y

p o s

Between-person relationship

Taken from Hamaker (2012).

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Three perspectives on data

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.8

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=0

x

y

Cross-sectional

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.8

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=0

x

y

Within Between

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.8

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=0

x

y


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−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.8

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=0

x

y

Cross-sectional

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

xy

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.8

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=0

xy

Within

Between

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.8

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=0

x

y


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−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.8

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=0

x

y

Cross-sectional

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

xy

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.8

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=0

xy

Within Between

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.4

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=−.8

x

y

−6 −2 0 2 4 6

−6−2

02

46

rho=0

x

y


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Between-person differences in within-person slopes

βB

Typing speed

Nu

mb

er o

f ty

po

s

Negative event

Neg

ativ

e af

fect

Cu

rren

t n

egat

ive

affe

ct

Preceding negative affect

βB

βB=1

Taken from Hamaker and Grasman (2014).

In conclusion: To study within-person processes we need• (intensive) longitudinal data• to decompose observed variance into within and between• to consider individual differences in within-person

dynamics

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Disadvantages of using regular multilevel software

If we are interested in dynamic multilevel modeling, we mayrun into the following problems/limitation when using standardmultilevel software:

• negative bias in autoregression when centering the lagged predictor(Nickell’s bias)

• only one outcome variable (thus, separate models for multivariateoutcomes)

• only observed variables (no measurement error, moving averageterms, factor models)

• missing data result in many missing cases• unequally spaced observations

Dynamic structural equation modeling (DSEM) in Mplustackles all these problems.

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If we are interested in dynamic multilevel modeling, we mayrun into the following problems/limitation when using standardmultilevel software:• negative bias in autoregression when centering the lagged predictor

(Nickell’s bias)

• only one outcome variable (thus, separate models for multivariateoutcomes)




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(Nickell’s bias)• only one outcome variable (thus, separate models for multivariate

outcomes)




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outcomes)• only observed variables (no measurement error, moving average

terms, factor models)



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terms, factor models)• missing data result in many missing cases

• unequally spaced observations


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terms, factor models)• missing data result in many missing cases• unequally spaced observations


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Outline


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Data: Daily measurements affect

Data come from the COGITO study of the MPI in Berlin; goal isto study aging using a younger and older sample.

Analyses here are based on Hamaker et al. (under revision).

Characteristics of the younger and older sample:• aged 20-31; aged 65-80• 101 individuals; 103 individuals• about 100 daily measurements of positive affect (PA) and

negative affect (NA)

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Decomposition

Decomposition into a between part and a within partPAit = µPA,i +PA∗itNAit = µNA,i +NA∗it

where• µPA,i and µNA,i are the individual’s means on PA and NA (i.e., baseline,

trait, or equilibrium scores)⇒ between-person part• PA∗it and NA∗it are the within-person centered (cluster-mean centered)

scores⇒ within-person part

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Total, between-, and within-person variance

0

0.5

1

1.5

2

2.5

PA young PA older NA young NA older

Total Between Within

ICC=.89

ICC=.76

ICC=.68

ICC=.64

Intraclass correlation:σ2

betweenσ2

between+σ2within

=σ2

betweenσ2

total

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Bivariate model: Multilevel vector AR(1) model

ζ𝑃𝑃𝑃𝑃,𝑡𝑡

ζ𝑁𝑁𝑃𝑃,𝑡𝑡

𝜙𝜙𝑃𝑃𝑃𝑃

𝜙𝜙𝑁𝑁𝑃𝑃

𝜙𝜙𝑃𝑃𝑁𝑁

𝜙𝜙𝑁𝑁𝑁𝑁

𝜙𝜙𝑁𝑁𝑁𝑁 𝜙𝜙𝑃𝑃𝑁𝑁 𝜙𝜙𝑁𝑁𝑃𝑃 𝜙𝜙𝑃𝑃𝑃𝑃 𝜇𝜇𝑁𝑁𝑃𝑃 𝜇𝜇𝑃𝑃𝑃𝑃

Within

Between 𝑃𝑃𝑃𝑃𝑡𝑡 𝑁𝑁𝑃𝑃𝑡𝑡

𝑃𝑃𝑃𝑃𝑡𝑡(𝑤𝑤)

𝑁𝑁𝑃𝑃𝑡𝑡(𝑤𝑤)

𝑃𝑃𝑃𝑃𝑡𝑡(𝑤𝑤) 𝑁𝑁𝑃𝑃𝑡𝑡

(𝑤𝑤)

𝜇𝜇𝑁𝑁𝑃𝑃 𝜇𝜇𝑃𝑃𝑃𝑃

Decomposition 𝑃𝑃𝑃𝑃𝑡𝑡−1

(𝑤𝑤)

𝑁𝑁𝑃𝑃𝑡𝑡−1(𝑤𝑤)

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Within-person level model

Lagged within-person model:

PA∗it = φPP,iPA∗i,t−1 +φPN,iNA∗i,t−1 +ζPA,it

NA∗it = φNN,iNA∗i,t−1 +φNP,iPA∗i,t−1 +ζNA,it

where• φPP,i is the autoregressive parameter for PA (i.e., inertia, carry-over)• φNN,i is the autoregressive parameter for NA (i.e., inertia, carry-over)

• φPN,i is the cross-lagged parameter for NA to PA (i.e., spill-over)• φNP,i is the cross-lagged parameter for PA to NA (i.e., spill-over)• ζPA,it is the innovation for PA (residual, disturbance, dynamic error)• ζNA,it is the innovation for NA (residual, disturbance, dynamic error)

Parameters estimated at this level are the residual variancesand covariance:[

ζPA,it

ζNA,it

]∼MN

[[00

],

[θ11θ21 θ22

]]

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where• φPP,i is the autoregressive parameter for PA (i.e., inertia, carry-over)• φNN,i is the autoregressive parameter for NA (i.e., inertia, carry-over)• φPN,i is the cross-lagged parameter for NA to PA (i.e., spill-over)• φNP,i is the cross-lagged parameter for PA to NA (i.e., spill-over)

• ζPA,it is the innovation for PA (residual, disturbance, dynamic error)• ζNA,it is the innovation for NA (residual, disturbance, dynamic error)


ζPA,it

ζNA,it

]∼MN

[[00

],

[θ11θ21 θ22

]]

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where• φPP,i is the autoregressive parameter for PA (i.e., inertia, carry-over)• φNN,i is the autoregressive parameter for NA (i.e., inertia, carry-over)• φPN,i is the cross-lagged parameter for NA to PA (i.e., spill-over)• φNP,i is the cross-lagged parameter for PA to NA (i.e., spill-over)• ζPA,it is the innovation for PA (residual, disturbance, dynamic error)• ζNA,it is the innovation for NA (residual, disturbance, dynamic error)


ζPA,it

ζNA,it

]∼MN

[[00

],

[θ11θ21 θ22

]]

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where• φPP,i is the autoregressive parameter for PA (i.e., inertia, carry-over)• φNN,i is the autoregressive parameter for NA (i.e., inertia, carry-over)• φPN,i is the cross-lagged parameter for NA to PA (i.e., spill-over)• φNP,i is the cross-lagged parameter for PA to NA (i.e., spill-over)• ζPA,it is the innovation for PA (residual, disturbance, dynamic error)• ζNA,it is the innovation for NA (residual, disturbance, dynamic error)


ζPA,it

ζNA,it

]∼MN

[[00

],

[θ11θ21 θ22

]]27 / 55

Between-person level model

Between level: fixed and random effects

µPA,i

µNA,i

φPP,i

φPN,i

φNP,i

φNN,i

=

γP

γN

γPP

γPN

γNP

γNN

+

uP,i

uN,i

uPP,i

uPN,i

uNP,i

uNN,i

ui ∼MN(0,Ψ)

Where:• γP to γNN ⇒ fixed effects• uP,i to uNN,i ⇒ random effects

Parameters estimated at this level are:• 6 fixed effects (i.e., γ ’s)• 6 variances for random effects (i.e., diagonal elements of Ψ)• 15 covariances between the random effects (i.e., off-diagonal elements

in Ψ)

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Between-person level model


µPA,i

µNA,i

φPP,i

φPN,i

φNP,i

φNN,i

=

γP

γN

γPP

γPN

γNP

γNN

+

uP,i

uN,i

uPP,i

uPN,i

uNP,i

uNN,i

ui ∼MN(0,Ψ)

Where:• γP to γNN ⇒ fixed effects• uP,i to uNN,i ⇒ random effects

Parameters estimated at this level are:• 6 fixed effects (i.e., γ ’s)• 6 variances for random effects (i.e., diagonal elements of Ψ)• 15 covariances between the random effects (i.e., off-diagonal elements

in Ψ)28 / 55

Bivariate model: Mplus code

VARIABLE: NAMES ARE id sessdatena1 na2 na3 na4 na5 na6 na7 na8 na9 na10pa1 pa2 pa3 pa4 pa5 pa6 pa7 pa8 pa9 pa10sessionNr age_pre sex CESDpre CESDpost dayNA dayPA older;

CLUSTER = id; ! Specify the person id variableUSEVAR = dayPA dayNA; ! Specify which variables are used in the modelMISSING = ALL(-999);

LAGGED = dayPA(1) dayNA(1); ! This creates lagged variablesTINTERVAL = sessdate(1); ! This is to account for unequal intervals

ANALYSIS: TYPE IS TWOLEVEL RANDOM; ! This allows for random slopesESTIMATOR = BAYES; ! DSEM requires Bayesian estimationPROC = 2; ! Using 2 processors makes it fasterBITER = (5000); ! This implies at least 5000 iterations are usedTHIN = 10; ! Thinning helps with getting more stable results

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MODEL: %WITHIN% ! Specify the random lagged relationshipsp_pp | dayPA ON dayPA&1;p_pn | dayPA ON dayNA&1;p_np | dayNA ON dayPA&1;p_nn | dayNA ON dayNA&1;

%BETWEEN% ! Allow all 6 random effects to be correlatedp_pp WITH p_pn-p_nn dayPA dayNA;p_pn WITH p_np-p_nn dayPA dayNA;p_np WITH p_nn dayPA dayNA;p_nn WITH dayPA dayNA;dayPA WITH dayNA;

OUTPUT: TECH1 TECH8 STDYX;

PLOT: TYPE = PLOT3;FACTORS =ALL;

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Mplus results: Within-person (younger sample)

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Mplus results: Between-person (younger sample)

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Comparing cross-lagged parameters

Standardization in multilevel models is a tricky issue.

Schuurman, Ferrer, Boer-Sonnenschein and Hamaker (2016)discuss four forms of standardization in multilevel models,using:• total variance (i.e., grand standardization)• between-person variance (i.e., between standardization)• average within-person variance• within-person variance (i.e., within standardization)

Conclusion: last form is most meaningful, as it parallelsstandardizing when N=1.

Standardized fixed effect should be the average standardizedwithin-person effect.

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Mplus standardized results (younger sample)

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Mplus standardized results (younger sample)

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Between-person level: Correlated random effects

To represent the correlation matrices of the 6 random effectsin each group, we can use the network representation (withqgraph from Sacha Epskamp in R):

log( ) log( )

log (- )

log( ) log( )

log (- )

Young sample Older sample

Mod

el 1

M

odel

2

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Outline


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Including level 2 predictor and outcome

Depression was measured prior to the ILD phase andafterwards, using the CESD; we include these measures at thebetween-person level as a predictor and an outcome.

Between level: Including a level 2 predictorµPA,i = γ00 + γ01CESDprei +u0i

µNA,i = γ10 + γ11CESDprei +u1i

φPP,i = γ20 + γ21CESDprei +u2i

φPN,i = γ30 + γ31CESDprei +u3i

φNN,i = γ40 + γ41CESDprei +u4i

φNP,i = γ50 + γ51CESDprei +u5i

Between level: Including a level 2 outcomeCESDposti = γ60 + γ61CESDprei + γ62µPA,i + γ63µNA,i

+γ64φPP,i + γ65φPN,i + γ66φNN,i + γ67φNP,i +u6i

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Dynamic mediation model

𝐶𝐶𝐸𝐸𝐸𝐸𝐸𝐸𝑝𝑝𝑝𝑝𝑝𝑝

𝜙𝜙𝑃𝑃𝑃𝑃 𝜙𝜙𝑃𝑃𝑃𝑃 𝜙𝜙𝑃𝑃𝑃𝑃 𝜙𝜙𝑃𝑃𝑃𝑃 𝜇𝜇𝑃𝑃𝑃𝑃 𝜇𝜇𝑃𝑃𝑃𝑃

𝐶𝐶𝐸𝐸𝐸𝐸𝐸𝐸𝑝𝑝𝑜𝑜𝑜𝑜𝑜𝑜

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Mplus input mediation model

VARIABLE: NAMES ARE id sessdatena1 na2 na3 na4 na5 na6 na7 na8 na9 na10pa1 pa2 pa3 pa4 pa5 pa6 pa7 pa8 pa9 pa10sessionNr age_pre sex CESDpre CESDpost dayNA dayPA older;CLUSTER = id;USEVAR = dayPA dayNA CESDpre CESDpost; ! Plus level 2 variablesBETWEEN = CESDpre CESDpost; ! Specify these as level 2 variablesLAGGED = dayPA(1) dayNA(1);TINTERVAL = sessdate(1);MISSING = ALL(-999);

DEFINE: CENTER CESDpre CESDpost (GRANDMEAN);! Grand mean centering

ANALYSIS: TYPE IS TWOLEVEL RANDOM;ESTIMATOR = BAYES;PROCESSORS = 2;BITER = (5000);THIN = 10;

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MODEL: %WITHIN% ! Same as beforep_pp | dayPA ON dayPA&1;p_pn | dayPA ON dayNA&1;p_np | dayNA ON dayPA&1;p_nn | dayNA ON dayNA&1;

%BETWEEN% ! Mediation model with parameter namesp_pp-p_nn dayPA dayNA ON CESDpre (a1-a6);CESDpost ON p_pp-p_nn dayPA dayNA CESDpre (b1-b7);

MODEL CONSTRAINT: ! Compute the indirect effectsnew (ab_p_pp); ab_p_pp=a1*b1;new (ab_p_pn); ab_p_pn=a2*b2;new (ab_p_np); ab_p_np=a3*b3;new (ab_p_nn); ab_p_nn=a4*b4;new (ab_dayPA); ab_dayPA=a5*b5;new (ab_dayNA); ab_dayNA=a6*b6;

OUTPUT: TECH1 TECH8 STDYX;

PLOT: TYPE = PLOT3;FACTOR =ALL;

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Mplus output mediation model (younger sample)

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Mplus output mediation model (older sample)

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Outline


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Advantages of using DSEM in Mplus (thus far)

Compared to standard multilevel software:• Multiple outcome variables: this allows for correlated residuals and

correlated random effects• Unequal time interval: can be handled by choosing a grid for inserting

missings• Outcomes at between-person level• Person-mean centering integral part of model estimation (solves

Nickell’s bias)

Compared to other Bayesian software (e.g., WinBUGS, jags,Stan):• Easy to use due to tailor-made code• Default uninformative priors for parameters (even for small variances)• Fast (which makes a difference in case of Bayes)

Other recent developments: mlVAR, ctsem and open Mx (inR); Bayesian Ornstein-Uhlenbeck Model (BOUM); GIMME.

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More advantages of using DSEM in Mplus

Other options offered by DSEM in Mplus version 8:• Diverse plotting options: allows for inspection of data and results• Latent variables: allows for measurement error to be split off and for

moving average terms• Cross-classified models: allows for random effects of time• Random variance: allows for individual difference in variability

Future options Mplus will offer:• Regime-switching models: allows for a process to switch between

distinct states• Residual dynamic modeling: allows for easy combination of time

trends and residual lagged relationships

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Random innovation variance (univariately)

Within level: AR(1) with random φi

NA∗it = φiNA∗i,t−1 +ζit ζit ∼ N(0,σ2i )


µi = γµ +u0i

φi = γφ +u1i

log(σ2i ) = γlog(σ2)+u2i

u0i

u1i

u2i

∼MN

000

,ψ11

ψ21 ψ22ψ31 ψ32 ψ33

Reasons to assume individual differences for σ2:• individuals may differ with respect to the variability in exposure to

external factors• individuals may differ with respect to their reactivity to external

influences (see reward experience and stress sensitivity research)

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Random innovation variances and covariance

In the bivariate case, we want random innovation variancesAND random innovation covariance.

The latter is modeled with an additional factor ηt :

𝑒𝑒𝑃𝑃𝑃𝑃,𝑡𝑡

𝑒𝑒𝑁𝑁𝑃𝑃,𝑡𝑡

η𝑡𝑡

1

1

1

-1

𝜙𝜙𝑃𝑃𝑃𝑃

𝜙𝜙𝑁𝑁𝑃𝑃

𝜙𝜙𝑃𝑃𝑁𝑁

𝜙𝜙𝑁𝑁𝑁𝑁

𝜙𝜙𝑁𝑁𝑁𝑁 𝜙𝜙𝑃𝑃𝑁𝑁 𝜙𝜙𝑁𝑁𝑃𝑃 𝜙𝜙𝑃𝑃𝑃𝑃 𝜇𝜇𝑁𝑁𝑃𝑃 𝜇𝜇𝑃𝑃𝑃𝑃 log(𝜎𝜎𝑒𝑒𝑁𝑁2 ) log(𝜎𝜎𝑒𝑒𝑃𝑃2 ) log (−𝜎𝜎)

Within

Between

log(𝜎𝜎𝑒𝑒𝑁𝑁2 )

log(𝜎𝜎𝑒𝑒𝑃𝑃2 )

log (−𝜎𝜎)

𝑃𝑃𝑃𝑃𝑡𝑡 𝑁𝑁𝑃𝑃𝑡𝑡

𝑃𝑃𝑃𝑃𝑡𝑡(𝑤𝑤)

𝑁𝑁𝑃𝑃𝑡𝑡(𝑤𝑤)

𝑃𝑃𝑃𝑃𝑡𝑡(𝑤𝑤) 𝑁𝑁𝑃𝑃𝑡𝑡

(𝑤𝑤)

𝜇𝜇𝑁𝑁𝑃𝑃 𝜇𝜇𝑃𝑃𝑃𝑃

Decomposition 𝑃𝑃𝑃𝑃𝑡𝑡−1(𝑤𝑤)

𝑁𝑁𝑃𝑃𝑡𝑡−1(𝑤𝑤)

Where:• -ηt is the shared part (we assume a negative covariance)• ePA,t and eNA,t are the unique parts

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Mplus code: Within model

MODEL: %WITHIN%p_pp | dayPA ON dayPA&1;p_pn | dayPA ON dayNA&1;p_np | dayNA ON dayPA&1;p_nn | dayNA ON dayNA&1;

! Create latent variable that represents negative covarianceCov BY dayPA1 dayNA-1;

! Create random (log) varianceslogvarPA | dayPA;logvarNA | dayNA;logCov | Cov;

%BETWEEN%p_pp-p_nn WITH p_pn-p_nn logvarPA logvarNA logCov dayPA dayNA;logvarPA WITH logvarNA logCov dayPA dayNA;logvarNA WITH logCov dayPA dayNA;logCov WITH dayPA dayNA;dayPA WITH dayNA;

OUTPUT: TECH1 TECH8 STDYX FSCOMPARISON;

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What about many variables?

Emilio Ferrer obtained data from 193 dyads for 52-108 days on8 variables (i.e., general and relationship specific PA and NA).

Within level: Vector autoregressive model

GPAM∗itGNAM∗itRSPAM∗itRSNAM∗itGPAF∗itGNAF∗itRSPAF∗itRSNAF∗it

=

φ11 φ12 φ13 φ14 φ15 φ16 φ17 φ18φ21 φ22 φ23 φ24 φ25 φ26 φ27 φ28φ31 φ32 φ33 φ34 φ35 φ36 φ37 φ38φ41 φ42 φ43 φ44 φ45 φ46 φ47 φ48φ51 φ52 φ53 φ54 φ55 φ56 φ57 φ58φ61 φ62 φ63 φ64 φ65 φ66 φ67 φ68φ71 φ72 φ73 φ74 φ75 φ76 φ77 φ78φ81 φ82 φ73 φ84 φ85 φ86 φ87 φ88

GPAM∗it−1GNAM∗it−1RSPAM∗it−1RSNAM∗it−1GPAF∗it−1GNAF∗it−1RSPAF∗it−1RSNAF∗it−1

+

ζ1itζ2itζ3itζ4itζ5itζ6itζ7itζ8it

which gives:GPAM∗it = φ11GPAM∗it−1 +φ12GNAM∗it−1 + · · ·+φ18RSNAF∗it−1 +ζ1it. . .RSNAF∗it = φ81GPAM∗it−1 +φ82GNAM∗it−1 + · · ·+φ88RSNAF∗it−1 +ζ8it

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Multilevel VAR(1)

Within level: Residual covariance matrixζ1it

ζ2it

. . .ζ8it

∼MN(0,Θ∗)

Hence, we estimate 8×8 = 64 lagged parameters, and 8×9/2 = 36 variancesand covariances at the within-person level.

Between level: Fixed and random effectsµ1i

µ2i

. . .µ8i

∼MN(γ,Ψ)

Hence, we estimate 8 grand means, and 8×9/2 = 36 variances andcovariances at the between-person level. In total: 144 parameters.

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Three networks

Lagged, within-person (residual), and between-person:

G−PA.M

G−NA.M

RS−NA.M

RS−PA.MRS−PA.F

RS−NA.F

G−NA.F

G−PA.F G−PA.M

G−NA.M

RS−NA.M

RS−PA.MRS−PA.F

RS−NA.F

G−NA.F

G−PA.F G−PA.M

G−NA.M

RS−NA.M

RS−PA.MRS−PA.F

RS−NA.F

G−NA.F

G−PA.F

Note:• lagged network = within-person standardized lagged relationships• within-person residual network = correlations of within-person residuals• between-person network = correlations of within-person means

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References

• Bringmann, Vissers, Wichers, Geschwind, Kuppens, Peeters, Borsboom &Tuerlinckx (2013). A network approach to psychopathology: New insights intoclinical longitudinal data. PLoS ONE, 8, e60188, 1-13.

• Hamaker (2012). Why researchers should think “within-person”: A paradigmaticrationale. In M. R. Mehl & T. S. Conner (Eds.). Handbook of Research Methodsfor Studying Daily Life, 43-61. New York, NY: Guilford Publications.

• Hamaker, Asparouhov, Brose, Schmiedek & Muthén (submitted). At the frontiersof modeling intensive longitudinal data: Dynamic structural equation models forthe affective measurements from the COGITO study. Multivariate BehavioralResearch.

• Hamaker & Grasman (2015). To center or not to center? Investigating inertiawith a multilevel autoregressive model. Frontiers in Psychology, 5, 1492.

• Hamaker & Wichers (2017). No time like the present: Discovering the hiddendynamics in intensive longitudinal data. Current Directions in PsychologicalScience, 26, 10-15.

• Jongerling, Laurenceau & Hamaker (2015). A multilevel AR(1) model: Allowingfor inter-individual differences in trait-scores, inertia, and innovation variance.Multivariate Behavioral Research, 50, 334-349.

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References

• Koval, Kuppens, Allen & Sheeber (2012). Getting stuck in depression: The rolesof rumination and emotional inertia. Cognition & Emotion, 26, 1412-1427.

• Kuppens, Allen & Sheeber (2010). Emotional inertia and psychologicalmaladjustment. Psychological Science, 21, 984-991.

• Kuppens, Sheeber, Yap, Whittle, Simmons & Allen (2012). Emotional inertiaprospectively predicts the onset of depressive Multilevel AR(1) model 33 disorderin adolescence. Emotion, 12, 283-289.

• Schuurman, Ferrer, de Boer-Sonnenschein & Hamaker (2016). How to comparecross-lagged associations in a multilevel autoregressive model. PsychologicalMethods, 21, 206-221.

• Suls, Green & Hillis (1998). Emotional reactivity to everyday problems, affectiveinertia, and neuroticism. Personality and Social Psychology Bulletin, 24,127-136.

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