Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Time-series AnalysisNetwork Analysis 2017
Sacha Epskamp
11-12-2017
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Psychological Data
Subject Time Item 1 Item 2 Item 3Subject 1 Time 1 2 2 4Subject 2 Time 1 4 1 2Subject 3 Time 1 1 3 0
• Multiple people measured once: cross-sectional analysis
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Psychological Data
Subject Time Item 1 Item 2 Item 3Subject 1 Time 1 2 2 4Subject 1 Time 2 2 4 4Subject 1 Time 3 1 4 3
• Multiple people measured once: cross-sectional analysis• One person measured multiple times: N = 1 time-series
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Psychological Data
Subject Time Item 1 Item 2 Item 3Subject 1 Time 1 2 2 4Subject 1 Time 2 2 4 4Subject 1 Time 3 1 4 3Subject 2 Time 1 4 1 2Subject 2 Time 2 3 1 2Subject 2 Time 3 3 2 1Subject 3 Time 1 1 3 0Subject 3 Time 2 4 0 1Subject 3 Time 3 0 3 3
• Multiple people measured once: cross-sectional analysis• One person measured multiple times: N = 1 time-series• Multiple people measured multiple times: N > 1 time-series
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Cross-sectional Analysis
A1
A2
A3
A4
A5
C1
C2
C3
C4
C5
E1E2
E3
E4
E5
N1
N2
N3
N4
N5
O1
O2
O3
O4
O5
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AgreeablenessA1: Am indifferent to the feelings of others.A2: Inquire about others' well−being.A3: Know how to comfort others.A4: Love children.A5: Make people feel at ease.
ConscientiousnessC1: Am exacting in my work.C2: Continue until everything is perfect.C3: Do things according to a plan.C4: Do things in a half−way manner.C5: Waste my time.
ExtraversionE1: Don't talk a lot.E2: Find it difficult to approach others.E3: Know how to captivate people.E4: Make friends easily.E5: Take charge.
NeuroticismN1: Get angry easily.N2: Get irritated easily.N3: Have frequent mood swings.N4: Often feel blue.N5: Panic easily.
OpennessO1: Am full of ideas.O2: Avoid difficult reading material.O3: Carry the conversation to a higher level.O4: Spend time reflecting on things.O5: Will not probe deeply into a subject.
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AgreeablenessA1: Am indifferent to the feelings of others.A2: Inquire about others' well−being.A3: Know how to comfort others.A4: Love children.A5: Make people feel at ease.
ConscientiousnessC1: Am exacting in my work.C2: Continue until everything is perfect.C3: Do things according to a plan.C4: Do things in a half−way manner.C5: Waste my time.
ExtraversionE1: Don't talk a lot.E2: Find it difficult to approach others.E3: Know how to captivate people.E4: Make friends easily.E5: Take charge.
NeuroticismN1: Get angry easily.N2: Get irritated easily.N3: Have frequent mood swings.N4: Often feel blue.N5: Panic easily.
OpennessO1: Am full of ideas.O2: Avoid difficult reading material.O3: Carry the conversation to a higher level.O4: Spend time reflecting on things.O5: Will not probe deeply into a subject.
• Concentration network: unique variance between two variables
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
N = 1 Time-series Analysis
relaxed
sad
nervous
concentration
tired
rumination
bodilydiscomfort
(b) Temporal network − Patient 1
relaxed
sad
nervous
concentration
tired
rumination
bodilydiscomfort
(a) Contemporaneous network − Patient 1
• Contemporaneous network: conditional concentration given t − 1• Temporal network: regression coefficients between t − 1 and t
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
N > 1 Time-series Analysis
Outgoing
Energetic
Adventurous
Happy
Exercise
Temporal
Outgoing
Energetic
Adventurous
Happy
Exercise
Contemporaneous
Outgoing
Energetic
Adventurous
Happy
Exercise
Between−subjects
• Between-subjects network: concentration network betweenstationary means
• Two-step multilevel VAR
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
The Psychosystems Ecosystem
Cross−sectional
Longitudinal
Binary
ContinuousOrdinal
Yes
No
Yes No
Yes
No
Yes
No
Yes
No
Mixed
Yes
No
Type ofdata
Scale ofmeasurement
Multiplepersons?
Regularize?
Gaussian?
qgraphcor_auto() cor()
hugehuge.npn()
Regularize?
qgraphgraph = 'pcor'qgraph
graph = 'glasso'
IsingSamplerEstimateIsing()
IsingFit
Binarize
mlVAR
GraphicalVAR
bootnet
lvnet
mgm
NetworkCom−parnisonTest
Stationary?
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
N = 1: Graphical VAR
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Time-series Data
• One person measured several times in a short period• Cases can not reasonably be assumed to be independent
• Knowing someone’s level of fatigue at a time point helps predicthis or her level of fatigue at the next time point.
• Likelihood not easy to compute without three assumptions:• The time-series factorize according to a graph• The model does not change over time• The first measurement is exogenous
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Y1(p) Y2
(p) Y3(p) Y4
(p) Y5(p)
lag−0
Y1(p) Y2
(p) Y3(p) Y4
(p) Y5(p)
lag−1
Y1(p) Y2
(p) Y3(p) Y4
(p) Y5(p)
lag−2
• We will use the lag-1 factorization
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Graphical Vector Auto-regression (VAR)
YYY t | yyy t−1 ∼ N (BBByyy t−1,ΘΘΘ)
• Variables assumed centered• BBB encodes the temporal network
• Temporal prediction• ΘΘΘ−1 encodes the contemporaneous network
• GGM• Graphical VAR model
• Wild, B., Eichler, M., Friederich, H. C., Hartmann, M., Zipfel, S., &Herzog, W. (2010). A graphical vector autoregressive modellingapproach to the analysis of electronic diary data. BMC medicalresearch methodology, 10(1), 28.
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Temporal effects
• The temporal network shows that one variable predicts anothervariable in the next measurement occasion
• Granger causality• Only temporal network from (graphical) VAR needed in
predicting new responses
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Contemporaneous effects
• The contemporaneous network shows that two variables predictone-another after taking temporal information into account
• Contains effects faster than the time-window of measurement• Somatic arousal → anticipation of panic attack → anxiety
• The temporal network can be seen as a correction fordependent measurements in estimating the GGM
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Exercising
Energetic
Temporal network
Exercising
Energetic
Contemporaneous network
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
• Estimation straightforward using multiple regression• For model selection, we use the graphical VAR model
• Wild et al. (2010). A graphical vector autoregressive modellingapproach to the analysis of electronic diary data. BMC MedicalResearch Methodology 10 (1): 28.
• Estimation via LASSO regularization, using EBIC to selectoptimal tuning parameter
• Abegaz & Wit (2013). Sparse Time Series Chain GraphicalModels for Reconstructing Genetic Networks. Biostatistics:kxt005.
• Rothman, Elizaveta, & Zhu (2010). Sparse MultivariateRegression with Covariance Estimation. Journal ofComputational and Graphical Statistics 19 (4): 947–62.
• We implemented these methods in the R packagegraphicalVAR
• Also implemented in sparseTSCGM
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Assumptions• Stationarity
• Plausible when data is obtained in short time-span, lessplausible if data is obtained in longer time-span
• Stationary means• Can be assessed by regressing each time-series on time itself as
predictor• Detrending is possible: for example one can remove a linear
trend (see practical)• Stationary network model(s)
• Short intro to time-varying networks this afternoon• Equidistant measurements
• With multiple measurments per day, by default violated due tonights
• Remove nights, or model nights as missing observations• Continuous time modeling poses promising ways to deal with
non-equidistant measurements
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Personalized Networks in Clinical Practice
relaxed
sad
nervous
concentrationtired
rumination bodilydiscomfort
(a) Temporal network − Patient 1
relaxed
sad
nervous
concentrationtired
rumination bodilydiscomfort
(b) Contemporaneous network − Patient 1
• Contemporaneous network: conditional concentration given t − 1• Temporal network: regression coefficients between t − 1 and t
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
N > 1: Multi-level VAR
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Multi-level VAR
• Each subject is assumed to have their own temporal andcontemporaneous VAR model
• VAR parameters come from distribution• Fixed effect• Random effect
• Bringmann, L. F., Vissers, N., Wichers, M., Geschwind, N.,Kuppens, P., Peeters, F., ... & Tuerlinckx, F. (2013). Anetwork approach to psychopathology: new insights intoclinical longitudinal data. PloS one, 8(4), e60188.
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Multi-level VAR
Adding superscript p for subject. Level 1 model:
YYY (p)t | yyy(p)t = N
(µµµ(p) +BBB(p)
(yyy (p)t−1 −µµµ
(p)),ΘΘΘ(p)
)Level 2 model: [
µµµ(p)
Vec(BBB(p)
)] ∼ N (fff ,ΩΩΩ) .fff encodes fixed effects and ΩΩΩ the distribution of random effects.
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
µ1
β11
β12
µ2
β21
β22
Y1 Y2
1 1
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Each Parameter has a Distribution
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
x
dnor
m(x
, mea
ns[i]
, SD
s[i])
0
x
dnor
m(x
, mea
ns[i]
, SD
s[i])
x
dnor
m(x
, mea
ns[i]
, SD
s[i])
x
dnor
m(x
, mea
ns[i]
, SD
s[i])
x
dnor
m(x
, mea
ns[i]
, SD
s[i])
dnor
m(x
, mea
ns[i]
, SD
s[i])
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Individual Networks
0.15
0.15
−0.16
0.2
0.28
−0.58
Y1 Y2
1 1
Bob 0.06
0.07
−0.1
0.22
0.29
−0.54
Y1 Y2
1 1
Alice
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Random Effects
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
x
dnor
m(x
, mea
ns[i]
, SD
s[i])
BobAlice
x
dnor
m(x
, mea
ns[i]
, SD
s[i])
x
dnor
m(x
, mea
ns[i]
, SD
s[i])
x
dnor
m(x
, mea
ns[i]
, SD
s[i])
x
dnor
m(x
, mea
ns[i]
, SD
s[i])
dnor
m(x
, mea
ns[i]
, SD
s[i])
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Fixed Effects
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
x
dnor
m(x
, mea
ns[i]
, SD
s[i])
Mean
x
dnor
m(x
, mea
ns[i]
, SD
s[i])
x
dnor
m(x
, mea
ns[i]
, SD
s[i])
x
dnor
m(x
, mea
ns[i]
, SD
s[i])
x
dnor
m(x
, mea
ns[i]
, SD
s[i])
dnor
m(x
, mea
ns[i]
, SD
s[i])
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Fixed Effects
0.07
−0.14
0.17
0.18 0.34
−0.6
Y1 Y2
1 1
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Individual Differences
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
x
dnor
m(x
, mea
ns[i]
, SD
s[i])
95% interval
x
dnor
m(x
, mea
ns[i]
, SD
s[i])
x
dnor
m(x
, mea
ns[i]
, SD
s[i])
x
dnor
m(x
, mea
ns[i]
, SD
s[i])
x
dnor
m(x
, mea
ns[i]
, SD
s[i])
dnor
m(x
, mea
ns[i]
, SD
s[i])
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Parameter correlation Matrix
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Parameter Correlation Matrix
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
1
0.01
−0.54
0.21
−0.18
0.46
0.01
1
−0.24
−0.66
0.77
0.06
−0.54
−0.24
1
−0.12
0.34
−0.75
0.21
−0.66
−0.12
1
−0.76
0.1
−0.18
0.77
0.34
−0.76
1
−0.18
0.46
0.06
−0.75
0.1
−0.18
1
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Stability
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
1
0.01
−0.54
0.21
−0.18
0.46
0.01
1
−0.24
−0.66
0.77
0.06
−0.54
−0.24
1
−0.12
0.34
−0.75
0.21
−0.66
−0.12
1
−0.76
0.1
−0.18
0.77
0.34
−0.76
1
−0.18
0.46
0.06
−0.75
0.1
−0.18
1
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Connectivity
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
1
0.01
−0.54
0.21
−0.18
0.46
0.01
1
−0.24
−0.66
0.77
0.06
−0.54
−0.24
1
−0.12
0.34
−0.75
0.21
−0.66
−0.12
1
−0.76
0.1
−0.18
0.77
0.34
−0.76
1
−0.18
0.46
0.06
−0.75
0.1
−0.18
1
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Between-subject Effects
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
1
0.01
−0.54
0.21
−0.18
0.46
0.01
1
−0.24
−0.66
0.77
0.06
−0.54
−0.24
1
−0.12
0.34
−0.75
0.21
−0.66
−0.12
1
−0.76
0.1
−0.18
0.77
0.34
−0.76
1
−0.18
0.46
0.06
−0.75
0.1
−0.18
1
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Between-subjects Network
The random effects variance-covariance matrix can be divided infour blocks:
[RRRµµµRRRBBB
]∼ N
(000,[
ΩΩΩµµµ ΩΩΩµµµBBBΩΩΩBBBµµµ ΩΩΩBBB
]).
• Block ΩΩΩµµµ encodes the between-subject relationships betweenmeans
• These can be used to estimate a GGM• Between-subjects network of partial correlations
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Typingspeed
Spellingerrors
Within−subjects
Typingspeed
Spellingerrors
Between−subjects
Example based on Hamaker, E. L. (2012). Why Researchers ShouldThink ‘Within-Person’: A Paradigmatic Rationale. Handbook ofResearch Methods for Studying Daily Life. The Guilford Press NewYork, NY, 43–61.
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Physicalactivity
Heartrate
Within−subjects
Physicalactivity
Heartrate
Between−subjects
Example kindly provided by Ellen Hamaker and based on Hoffman,L. (2015). Longitudinal analysis: Modeling within-person fluctuationand change. New York, NY, USA: Routledge.
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Temporal Estimation• Multi-variate multi-level MLE regression estimation is
complicated and not yet well implemented in open sourcesoftware
• lme4 packages implements univariate multi-level regression• Douglas Bates, Martin Maechler, Ben Bolker, Steve Walker
(2015). Fitting Linear Mixed-Effects Models Using lme4.Journal of Statistical Software, 67(1), 1-48.doi:10.18637/jss.v067.i01
• lmer function
• A multi-level VAR model can be estimated by sequentiallyestimating univariate models
• Estimate all incoming edges per node• Bringmann et al. (2013). A network approach to
psychopathology: new insights into clinical time-series data.PloS one, 8(4), e60188.
doi:10.18637/jss.v067.i01
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
µ1
β11
β12
µ2
β21
β22
Y1 Y2
1 1
µ1
β11
β12
µ2
β21
β22
Y1 Y2
1 1
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Temporal Estimation
• Correlated estimation:• Needs to integrate out a high-dimensional distribution over
parameters• Only feasible for up to ~6 nodes• Does not estimate all parameter covariances
• Not all parameters together in the same model
• Orthogonal estimation• Alternatively, parameter covariances can be fixed to zero• Fast, and works for high dimensions (e.g., 20 nodes)• But, does not return any parameter correlation
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Correlated Estimation
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Orthogonal Estimation
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Y1 Y2
1 1
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Between-subject Estimation
• Between subject effects can be obtained by centering predictorsand adding the person-means as level 2 predictors
• Hamaker, E. L., & Grasman, R. P. (2015). To center or not tocenter? Investigating inertia with a multilevel autoregressivemodel. Frontiers in psychology, 5, 1492.
• This can be seen as node-wise estimation of a GGM• Thus, an estimate for the between-subjects GGM can be
obtained by averaging the level-2 predictive effects standardizedwith the residual variances
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Contemporaneous Estimation
• Contemporaneous networks need to be estimated post-hoc byinvestigating the residuals
• Either inverting the sample variance-covariance matrix ofresiduals:
• Fixed• Unique
• Or as a second multi-level model using nodewise estimation ofa GGM:
• Correlated• Orthogonal
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
• Mplus 8 contains dynamic structural equation models• statmodel.com/download/DSEM.pdf
• Multi-level VAR is a special case• statmodel.com/download/usersguide/Chapter9.pdf,
example 9.32• Contemporaneous and between-subject networks are not
obtained by default, can be computed from the Bayesiansamples (BPARAMETERS option)
• Automated wrapper to come in mlVAR (implemented in develversion)
statmodel.com/download/DSEM.pdfstatmodel.com/download/usersguide/Chapter9.pdf
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Empirical Example 1
• Two datasets• Original: 26 subjects, 51 measurements on average, 1323 total
observations• Replication: 65 subjects, 35.5 measurements on average, 2309
total observations
• 16 indicators of neuroticism, extroversion, conscientiousness• Orthogonal estimation of temporal and contemporaneous
effects• Only significant effects shown
• Alpha = 0.05 and using the “or” rule
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Fixed effects
Outgoing
Energetic
Adventurous
Happy
Exercise
Maximum: 0.2
Temporal
Outgoing
Energetic
Adventurous
Happy
Exercise
Maximum: 0.5
Contemporaneous
Outgoing
Energetic
Adventurous
Happy
Exercise
Maximum: 0.5
Between−subjects
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Individual Differences
Outgoing
Energetic
Adventurous
Happy
Exercise
Maximum: 0.18
Temporal
Outgoing
Energetic
Adventurous
Happy
Exercise
Maximum: 0.01
Contemporaneous
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
1
2
34
5
6
7
89
1011
12
13
1415
16
17
Maximum: 0.17
Temporal
1
2
34
5
6
7
89
1011
12
13
1415
16
17
Maximum: 0.55
Contemporaneous
1
2
34
5
6
7
89
1011
12
13
1415
16
17
●
●
●
●
NeuroticismConscientiousnessExtraversionExercise
●
●
●
●
NeuroticismConscientiousnessExtraversionExercise
Maximum: 0.55
Between−subjects
1 = “Worried”; 2 = “Organized”; 3 = “Ambitious”; 4 = “Depressed”; 5 =“Outgoing”; 6 = “Self-Conscious”; 7 = “Self-Disciplined”; 8 = “Energetic”; 9 =“Frustrated”; 10 = “Focused”; 11 = “Guilty”; 12 = “Adventurous”; 13 =“Happy”; 14 = “Control”; 15 = “Achieved”; 16 = “Angry”; 17 = “Exercise.”
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Empirical Example 2
• Re-analysis of original Bringmann example• Bringmann, L. F., Vissers, N., Wichers, M., Geschwind, N., Kuppens,
P., Peeters, F., Borsboom, D., & Tuerlinckx, F. (2013). A networkapproach to psychopathology: new insights into clinical longitudinaldata. PloS one, 8(4), e60188.
• Geschwind, N., Peeters, F., Drukker, M., van Os, J., & Wichers, M.(2011). Mindfulness training increases momentary positive emotionsand reward experience in adults vulnerable to depression: arandomized controlled trial. Journal of consulting and clinicalpsychology, 79(5), 618-628.
• Re-analysis using three software packages• Two-step multi-level VAR (mlVAR)• LASSO regularization (graphicalVAR)• Bayesian multivariate estimation (Mlus 8, generated by mlVAR)
• arxiv.org/abs/1609.04156
arxiv.org/abs/1609.04156
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
cheerful
pleasant
worry
fearful
sad
relaxed
Temporal
cheerful
pleasant
worry
fearful
sad
relaxed
Contemporaneous
cheerful
pleasant
worry
fearful
sad
relaxed
Between−subjects
mlVAR estimation
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
cheerful
pleasant
worry
fearful
sad
relaxed
Temporal
cheerful
pleasant
worry
fearful
sad
relaxed
Contemporaneous
cheerful
pleasant
worry
fearful
sad
relaxed
Between−subjects
graphicalVAR estimation
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
cheerful
pleasant
worry
fearful
sad
relaxed
Temporal
cheerful
pleasant
worry
fearful
sad
relaxed
Contemporaneous
cheerful
pleasant
worry
fearful
sad
relaxed
Between−subjects
Mplus estimation
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Pre-print online at http://arxiv.org/abs/1609.04156
http://arxiv.org/abs/1609.04156
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Structural VAR & GIMME
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Structural VAR• Another option for modeling N = 1 data including
contemporaneous effects is structural VAR• Models contemporaneous effects as a directed network• Contemporaneous directed model does not need to be assumed
acyiclic: cycles potentially identified• If the temporal network is not empty (autoregressions), lagged
variables act as exogenous predictors identifying cycles• Rigdon, E. E. (1995). A necessary and sufficient identification rule
for structural models estimated in practice. Multivariate BehavioralResearch, 30(3), 359-383.
• Equivalent to graphical VAR• Temporal edges in GVAR may be induced by a chain connecting
a temporal and contemporaneous edge in SVAR• Contemporaneous edges in GVAR may be induced by a
contemporaneous collider• Can be estimated by performing stepwise model search (e.g.,
using SEM)
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Structural VAR Graphical VAR
Example
At−1
Bt−1
At
Bt
At−1
Bt−1
At
Bt
Pros Causally interpretable; allowscausal modeling of contempora-neous effects; temporal networkestimation takes contemporane-ous effects into account; cycliceffects are identified with auto-regressions.
Well identified; well defined satu-rated model; allows sparse mod-eling of contemporaneous ef-fects; latent variables result inclusters of undirected edges.
Cons Potentially poorly identified;multiple solutions possible;strong reliance on the assump-tion of no latent variables;strong interpretation of thedirection of effect.
Potentially spurious edges inboth the temporal and contem-poraneous networks; No direc-tion of effect in contemporane-ous network. Limited softwarefor confirmatory estimation.
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Group iterative multiple model estimation (GIMME)
• The GIMME algorithm is designed to estimate structural VARmodels for N > 1 data
• Gates, K. M., & Molenaar, P. C. (2012). Group search algorithmrecovers effective connectivity maps for individuals in homogeneousand heterogeneous samples. Neuroimage, 63(1), 310-319.
• Estimates a structural VAR model per subject, checks whichedges are common at group levels and include those edges inall subjects
• Shows which edges are common in all participants and whichedges differ between participants
• Information borrowed from other subjects in terms of structure,not in terms of parameters
• No multi-level modeling!• gimme package in R (http://gimme.web.unc.edu/)
http://gimme.web.unc.edu/
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Lane, S. T., & Gates, K. M. (2017). Automated Selection of RobustIndividual-Level Structural Equation Models for Time Series Data. StructuralEquation Modeling: A Multidisciplinary Journal, 1-15.
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Cross-sectional data revisited
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
• Only under extremely unrealistic assumptions are parametersestimated from cross-sectional data the same as parametersyou would estimate from time-series data
• Molenaar, P. C. (2004). A manifesto on psychology as idiographicscience: Bringing the person back into scientific psychology, this timeforever. Measurement, 2(4), 201-218.
• More and more critisism towards cross-sectional analysisbecause of this
• “[T]he utility of cross-sectional psychopathology networksfundamentally relies on the assumption of ergodicity: that thebetween-person structure at one time is the same as thewithin-person structure over time (Molenaar, 2004)” —- Forbes et al.(in press)
• “We think it is time to investigate networks at the proper level ofinvestigation (ie, at the intraindividual level). As long as we keepinvestigating cross-sectional group-level networks, the results willremain compatible with a traditional disease model, will not beinformative of symptom interactions within individuals, and willobscure scientific reasoning” — Bos & Wanders (2016)
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
“If both [cross-sectional GGM and temporal VAR] result in similar conclusions,this would greatly facilitate future research and clinical applications (...). If not,then cross-sectional symptom networks are unlikely to reflect causal symptomdynamics, as postulated by this network theory.”
a: first observation network (n = 104); person-mean network (n = 104); c: thedynamic network (n = 104 × 50)
• Bos, F. M., Snippe, E., de Vos, S., Hartmann, J. A., Simons, C. J., van derKrieke, L., De Jonge, P. & Wichers, M. (2017). Can We Jump fromCross-Sectional to Dynamic Interpretations of Networks? Implications for theNetwork Perspective in Psychiatry. Psychotherapy and Psychosomatics, 86(3),175-177.
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
But what is a cross-sectional network really?
• See arxiv.org/abs/1609.04156 (current revision draft inDropbox) for a methodological investigation
• Mathematically, if VAR models generated the data,cross-sectional networks are a complicated function ofwithin-subject effects (temporal and contemporaneous) andbetween-subject effects
• Simulations show that cross-sectional networks mostly retrievewithin-subject or between-subject edges, depending on whichhas more variance
• Hypothesis: the variance can be assumed dominant dependingon the type of questions asked
• Clinical interviews, questions relating to a long period of time,questions relating to within-person constants: between-subjectsvariance likely to be dominant
• Questions such as “Did you experience more or less than youraverage ...” potentially feature dominant within-subjectsvariance
arxiv.org/abs/1609.04156
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Isvoranu, A. M., Borsboom, D., van Os, J. & Guloksuz, S. (2016). A NetworkApproach to Environmental Impact in Psychotic Disorder: Brief TheoreticalFramework. Schizophrenia Bulletin, 42(4), 870-873.
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Repeated measures
• With repeated measures, within-person centering all data,pooling and estimating a GGM on the pooled data leads to anetwork that can be interpreted as within-subjects network
• Assuming no auto-regressions; arxiv.org/abs/1609.04156• Described among other developments in recent tutorial paper
• Costantini, G., Richetin, J., Preti, E., Casini, E., Epskamp, S., &Perugini, M. (2017). Stability and variability of personality networks.A tutorial on recent developments in network psychometrics.Personality and Individual Differences.
arxiv.org/abs/1609.04156
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Conclusion
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Conclusion
• Temporal ordering of data can be taken into account• graphical VAR
• Contemporaneous network (GGM)• Temporal network (VAR)• Between-subjects network (GGM)
• structural VAR estimates contemporaneous effects as a directednetwork
• Multi-level modeling or GIMME can be used when there aremultiple subjects
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Limitations and Future Directions
• A lot of problems with VAR models• Multivariate normality• Stationarity• Lag interval• Model complexity (lag-2, day effects, etcetera)
• A lot of potential problems with multi-level estimation• Multivariate estimation• Modeling random contemporaneous effects• Parameter variance-covariances• Model selection
• Possibly move away from multi-level• LASSO variants?
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
The Limit of Observational Data
• Network structures are only hypothesis generating• Highlighting potential causal pathways
• Observational data can never confirm causality• Mixture of experimental and observational data needed
• We need to completely rethink the modeling framework to doso
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
The Psychosystems Ecosystem
Cross−sectional
Longitudinal
Binary
ContinuousOrdinal
Yes
No
Yes No
Yes
No
Yes
No
Yes
No
Mixed
Yes
No
Type ofdata
Scale ofmeasurement
Multiplepersons?
Regularize?
Gaussian?
qgraphcor_auto() cor()
hugehuge.npn()
Regularize?
qgraphgraph = 'pcor'qgraph
graph = 'glasso'
IsingSamplerEstimateIsing()
IsingFit
Binarize
mlVAR
GraphicalVAR
bootnet
lvnet
mgm
NetworkCom−parnisonTest
Stationary?
Introduction N = 1: Graphical VAR N > 1: Multi-level VAR SVAR / GIMME Cross-sectional Conclusion
Thank you for your attention!
IntroductionIntroduction
N=1: Graphical VARn=1
N > 1: Multi-level VARMulti-level VAREstimation
SVAR / GIMMESVAR / GIMME
Cross-sectionalCross-sectional
ConclusionConclusion
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