network psychometrics - current state and future directions · introduction markov random fields...

Introduction Markov Random Fields Ising Model Gaussian Graphical Model Time-series Analysis Conclusion

Network PsychometricsCurrent State and Future Directions

Sacha Epskamp

IMPS 2018


The Network Perspective

MDInsomnia

Fatigue

Concentration

Worry

Insomnia

Fatigue

Concentration

Worry

• Cramer, A. O. J., Waldorp, L. J., van der Maas, H., & Borsboom, D.(2010). Comorbidity: A network perspective. Behavioral and BrainSciences, 33, 137-193.

• Borsboom, D. (2017). A network theory of mental disorders. WorldPsychiatry, 16, 513.


Disorders usually first diagnosed in infancy, childhood or adolescenceDelirium, dementia, and amnesia and other cognitive disordersMental disorders due to a general medical conditionSubstance−related disordersSchizophrenia and other psychotic disordersMood disordersAnxiety disordersSomatoform disordersFactitious disordersDissociative disordersSexual and gender identity disordersEating disordersSleep disordersImpulse control disorders not elsewhere classifiedAdjustment disordersPersonality disordersSymptom is featured equally in multiple chapters

Disorders usually first diagnosed in infancy, childhood or adolescenceDelirium, dementia, and amnesia and other cognitive disordersMental disorders due to a general medical conditionSubstance−related disordersSchizophrenia and other psychotic disordersMood disordersAnxiety disordersSomatoform disordersFactitious disordersDissociative disordersSexual and gender identity disordersEating disordersSleep disordersImpulse control disorders not elsewhere classifiedAdjustment disordersPersonality disordersSymptom is featured equally in multiple chapters

Borsboom, D., Cramer, A. O., Schmittmann, V. D., Epskamp, S., & Waldorp, L. J.

(2011). The small world of psychopathology. PloS one, 6(11), e27407.


PhD project (2012 - 2016)

http://sachaepskamp.com/Dissertation

• Funded by NWO Researchtalent grant

• Supervisors:• Promoter: Denny

Borsboom• Co-promoter: Lourens

Waldorp

• Important collaborators:• Gunter Maris, Mijke

Rhemtulla, Eiko Fried,Claudia van Borkulo,Maarten Marsman,Angelique Cramer,Harriette Riese, Date vander Veen, GiulioCostantini, Rene Mottus

http://sachaepskamp.com/Dissertation


A1

A2

A3

A4

A5

C1

C2

C3

C4

C5

E1E2

E3

E4

E5

N1

N2

N3

N4

N5

O1

O2

O3

O4

O5

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.

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.


Markov Random Fields


X1

X2

X3

Graphs

G = V ,EV = 1, 2, 3E = (1, 2), (2, 3)

A graph is a set G consisting of two

sets: V (set of nodes) and E (set of

edges).


Xi ⊥⊥ Xj | X−(i,j) = x−(i,j) ⇐⇒ (i , j) 6∈ E

• Graphical representation: Twovariables are connected if they arenot conditionally independent

• Powerful characterization of jointlikelihood between observedvariables


−0.30.5

X1

X2

X3

Weighted graphs

A weighted graph uses a weights matrixto characterize the strength ofconnection:

ΩΩΩ =

0 0.5 00.5 0 −0.30 −0.3 0

Positive edges are drawn blue (or

green) and negative edges red.


In MRFs, weights should indicateconditional independence:

Xi ⊥⊥ Xj | X−(i,j) = x−(i,j) ⇐⇒ ωij = 0

Furthermore, positive weights should be

comparable to negative weights in

strength of association


Markov Random Field

In a Markov random field (MRF) a joint probability distribution isformed by multiplying potential functions φC (xxxC ) ≥ 0 for everyclique in a graph (e.g., one variable, two connected varaibles, a setof three connected variables):

Pr (XXX = xxx) ∝∏

C∈cl(G)

φC (xxxC )

Pairwise MRFs only include potential function up to pairwiseinteractions:

Pr (XXX = xxx) ∝∏i

φi (xi )∏<ij>

φij (xi , xj)

These are typically used in network psychometrics


X

Suppose random variable X with realization x can take twooutcomes: s1 and s2. We can make use of a potential function tomap a unique weight to each outcome:

φ(x) =

a if x = s1

b if x = s2

We can use this potential function to characterize a likelihoodfunction for X :

Pr (X = x) ∝ φ(x)


X

Suppose random variable X with realization x can take twooutcomes: s1 and s2. We can make use of a potential function tomap a unique weight to each outcome:

φ(x) =

a if x = s1

1/a if x = s2

We can use this potential function to characterize a likelihoodfunction for X :

Pr (X = x) ∝ φ(x)


X1 X2

We can add a different potential function for X2:

Pr (X1 = x1,X2 = x2) ∝ φ1(x1)φ2(x2)

This model implies X1 and X2 are independent, since:

Pr (X1 = x1,X2 = x2) = Pr (X1 = x1) Pr (X2 = x2)

Two parameters (one per potential function) used to model threeprobabilities (4 possible outcomes): not a saturated model.


X1 X2

We can add a potential function to model the pairwise interactionbetween X1 and X2:

φ12(x1, x2) =

c if x1 = s1 ∧ x2 = s1

d if x1 = s2 ∧ x2 = s1

e if x1 = s1 ∧ x2 = s2

f if x1 = s2 ∧ x2 = s2

To obtain:

Pr (X1 = x1,X2 = x2) ∝ φ1(x1)φ2(x2)φ12(x1, x2)


X1 X2

We can add a potential function to model the pairwise interactionbetween X1 and X2:

φ12(x1, x2) =

c if x1 = s1 ∧ x2 = s1

1/c if x1 = s2 ∧ x2 = s1

1/c if x1 = s1 ∧ x2 = s2

c if x1 = s2 ∧ x2 = s2

To obtain:

Pr (X1 = x1,X2 = x2) ∝ φ1(x1)φ2(x2)φ12(x1, x2)


X1

X2

X3

Let XXX> =[X1 X2 X3

]with realization xxx . No interaction

between X1 and X3, φ13(x1, x3) = 1∀x1,x3, gives:

Pr (XXX = xxx) ∝ φ1(x1)φ2(x2)φ3(x3)φ12(x1, x2)φ23(x2, x3)

X1 and X3 are conditionally independent:

Pr(XXX−(2) = xxx−(2) | X2 = x2

)∝ φ1(x1)φ12(x1, x2)φ3(x3)φ23(x2, x3)

∝ φ∗1(x1)φ∗3(x3)

But not marginally independent:

Pr(XXX−(2) = xxx−(2)

)∝∑x2

φ1(x1)φ2(x2)φ3(x3)φ12(x1, x2)φ23(x2, x3)


This Ising Model


Pairwise Markov Random Field:

Pr (XXX = xxx) ∝∏i

φi (xi )∏<ij>

φij (xi , xj)

Now encode S1 = 1 and S2 = −1, we can define without loss ofinformation (in binary setting):

lnφi (xi ) = τixi

lnφij(xi , xj) = ωijxixj

This gives the Ising Model:

Pr (XXX = xxx) ∝ exp

∑i

τixi +∑<ij>

ωijxixj

in which τi encodes the potential of Xi to be in state 1 or −1 andωij encodes the strength of connection between Xi and Xj .


The Ising Model


Estimating an Ising model

Due to the intractable normalizing constant, standard methods forestimating an Ising model (e.g., maximum likelihood) are notpractical. We can, however, easily estimate conditionaldistributions for node i given that we observe all other nodes:

Pr(Xi = Xi | XXX−(i) = xxx−(i)

)∝ exp

τi +∑j ,j 6=i

ωijxj

xi

• This is a multiple logistic regression model!

• The Ising model can thus be estimating by performing severalmultiple logistic regressions


ω12

ω34

ω14

ω23

X1

X2

X3

X4


ω12

ω34

ω14

ω23

X1

X2

X3

X4

Pr (X1 = 1) ∝ exp (τ1 + ω12x2 + ω14x4)


ω12

ω34

ω14

ω23

X1

X2

X3

X4

Pr (X2 = 1) ∝ exp (τ2 + ω12x1 + ω23x3)


ω12

ω34

ω14

ω23

X1

X2

X3

X4

Pr (X3 = 1) ∝ exp (τ3 + ω23x2 + ω34x4)


ω12

ω34

ω14

ω23

X1

X2

X3

X4

Pr (X4 = 1) ∝ exp (τ4 + ω14x1 + ω34x3)


Estimating an Ising model

Multivariate estimation options:

• Optimize the pseudolikelihood∏i Pr

(Xi = Xi | XXX−(i) = xxx−(i)

)• Phrase the model as a loglinear model and use ML estimation

Methods for choosing which edges to include:

• Model selection

• Significance thresholding

• LASSO regularization• van Borkulo, C. D., Borsboom, D., Epskamp, S., Blanken, T. F.,

Boschloo, L., Schoevers, R. A., & Waldorp, L. J. (2014). A new

method for constructing networks from binary data. Scientific

reports, 4 (5918).


The Ising Model & Psychometrics

• The Ising model can be shown to be mathematicallyequivalent to a multidimensional IRT model.• e.g., Marsman, M., Borsboom, D., Kruis, J., Epskamp, S., van

Bork, R., Waldorp, L. J., van der Maas, H. L. J. & Maris, G.K. J. (2018). An introduction to Network Psychometrics:Relating Ising network models to item response theory models.Multivariate Behavioral Research, 53(1), 15-35.

• Rank-1 clusters in the network correspond to latent variablesin the MIRT model

• This equivalence allows for re-interpretation of MIRT modelsand network models alike


Adding (arbitrary) variance potential function lnφii (xi ) = ωiix2i , we

can write the Ising model as:


(τττ>xxx +

1

2xxx>ΩΩΩxxx

)

We can arbitrarily set the diagonal such that ΩΩΩ 0, allowing us totake the eigenvalue decomposition:

ΩΩΩ = QQQΛΛΛQQQ>,

Now, reparameterize τi = −δi and −2√

λj2 qij = αij to obtain:

Pr(XXX = xxx) =

∫ ∞−∞

f (θθθ) Pr(XXX = xxx |ΘΘΘ = θθθ)dθθθ

With:

Pr(Xi = xi |ΘΘΘ = θθθ) =exp

(xi(ααα>i θθθ − δi

))∑xi

exp(xi(ααα>i θθθ − δi

))ΘΘΘ | XXX = xxx ∼ N

(±1

2A>x ,

√1

2III

)


Adding (arbitrary) variance potential function lnφii (xi ) = ωiix2i , we

can write the Ising model as:


(τττ>xxx +

1

2xxx>ΩΩΩxxx

)We can arbitrarily set the diagonal such that ΩΩΩ 0, allowing us totake the eigenvalue decomposition:

ΩΩΩ = QQQΛΛΛQQQ>,

Now, reparameterize τi = −δi and −2√

λj2 qij = αij to obtain:

Pr(XXX = xxx) =

∫ ∞−∞

f (θθθ) Pr(XXX = xxx |ΘΘΘ = θθθ)dθθθ

With:

Pr(Xi = xi |ΘΘΘ = θθθ) =exp

(xi(ααα>i θθθ − δi

))∑xi

exp(xi(ααα>i θθθ − δi

))ΘΘΘ | XXX = xxx ∼ N

(±1

2A>x ,

√1

2III

)


I: irritability, C: chronic worrying, W: weight problems, D: depressed mood, S:

sleep problems. Example from Marsman, M., Borsboom, D., Kruis, J.,

Epskamp, S., van Bork, R., Waldorp, L. J., van der Maas, H. L. J. & Maris, G.

K. J. (2018). An introduction to Network Psychometrics: Relating Ising

network models to item response theory models. Multivariate Behavioral

Research, 53(1), 15-35.


The Gaussian Graphical Model


Now assume Xi ∈ R ∀i and the log of all potential functions arelinear, and include variance potential function φii (xi ):

lnφi (xi ) = τixi

lnφij(xi , xj) = −κijxixjlnφii (xi ) = −κiix2

i

We obtain:


(τττ>xxx − 1

2xxx>KKKxxx

)Now let ΣΣΣ = KKK−1 and µµµ = ΣΣΣτττ :


(−1

2(xxx −µµµ)>ΣΣΣ−1(xxx −µµµ)

)Which implies XXX ∼ N(µµµ,ΣΣΣ)!



lnφi (xi ) = τixi


i

We obtain:


(τττ>xxx − 1

2xxx>KKKxxx

)

Now let ΣΣΣ = KKK−1 and µµµ = ΣΣΣτττ :


(−1





lnφi (xi ) = τixi


i

We obtain:


(τττ>xxx − 1

2xxx>KKKxxx

)Now let ΣΣΣ = KKK−1 and µµµ = ΣΣΣτττ :


(−1




The Gaussian Graphical Model (GGM)

The precision matrix KKK = ΣΣΣ−1 encodes a network structure!Furthermore, this matrix can be standardized to partial correlationcoefficients:

Cor(Xi ,Xj | XXX−(i ,j)

)= −

κij√κii√κjj

= ωij

Combining this information, we can form a psychometric model:

ΣΣΣ = ∆∆∆ (III −ΩΩΩ)−1 ∆∆∆

• ∆∆∆ is a diagonal scaling matrix

• ΩΩΩ is a symmetrical matrix with 0 on the diagonal and partialcorrelation coefficients on offdiagonal elements


GGM estimation

• The GGM can be fitted similarly to SEM models, allowing formodel search strategies

• The conditional distribution of a GGM is a multiple regressionmodel, which allows for nodewise estimation similar to Isingmodel estimation

• Edge selection via significance thresholding or regularization

• The graphical LASSO is a fast way to obtain a range of GGMmodels to choose the best model from

• Bayesian estimation methods are also promising

• See the symposium: Network Psychometrics 1: Advances inNetwork Structure Estimation, Thursday 9:45-11:15


GGM estimation


• The conditional distribution of a GGM is a multiple regressionmodel, which allows for nodewise estimation similar to Isingmodel estimation• Edge selection via significance thresholding or regularization


• Bayesian estimation methods are also promising

• See the symposium: Network Psychometrics 1: Advances inNetwork Structure Estimation, Thursday 9:45-11:15


GGM estimation


• The conditional distribution of a GGM is a multiple regressionmodel, which allows for nodewise estimation similar to Isingmodel estimation• Edge selection via significance thresholding or regularization


• Bayesian estimation methods are also promising• See the symposium: Network Psychometrics 1: Advances in

Network Structure Estimation, Thursday 9:45-11:15


SEM and GGM

Structural Equation Modeling Gaussian Graphical Model

• Both models imply a variance-covariance matrix ΣΣΣ, aimed toclosely resemble the sample variance-covariance matrix SSS withpositive degrees of freedom.


Easiness of Class Intelligence

Grade IQ

Diploma

Causal model

Easiness of Class Intelligence

Grade IQ

Diploma

Markov Random Field

• MRF is uniquely identified and well parameterized

• MRF allows for exploratory hypothesis-generating insight inpossible causal structures


Implementing networks in SEM

The variance-covariance matrices in a SEM model can be modeledas a network.

ΣΣΣ = ΛΛΛΨΨΨΛΛΛ> + ΘΘΘ

Residual networks:

ΘΘΘ = ∆∆∆ΘΘΘ (III −ΩΩΩΘΘΘ)−1 ∆∆∆ΘΘΘ

Latent networks:

ΨΨΨ = ∆∆∆ΨΨΨ (III −ΩΩΩΨΨΨ)−1 ∆∆∆ΨΨΨ


Residual Network Modeling (RNM)

ΣΣΣ = ΛΛΛΨΨΨΛΛΛ> + ∆∆∆ΘΘΘ (III −ΩΩΩΘΘΘ)−1 ∆∆∆ΘΘΘ

• Network is formed at the residuals of SEM

• Model a network while not assuming no unobserved commoncauses

• Model a latent variable structure without the assumption oflocal independence


Latent Network Modeling (LNM)

ΣΣΣ = ΛΛΛ∆∆∆ΨΨΨ (III −ΩΩΩΨΨΨ)−1 ∆∆∆ΨΨΨΛΛΛ> + ΘΘΘ

• Models conditional independence relations between latentvariables as a network

• Model networks between latent variables

• Exploratory search for conditional independence relationshipsbetween latents


Time-series Analysis


Graphical Vector Auto-regression (VAR)

XXX t | xxx t−1 ∼ N (BBBxxx 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.


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


Estimation

• Multivariate multiple regression on lagged variables

• Edge selection via significance thresholding, model selection(LNM) or LASSO regularization

Important assumptions

• Stationarity• Means• Network structure

• Equidistant measurements


Multi-level VAR

Individual network model per person:

XXX(p)t | xxx (p)

t−1 = N(µµµ(p) +BBB(p)

(xxx

(p)t−1 −µµµ

(p)),ΘΘΘ(p)

)• Temporal and Contemporaneous network per person

• Fixed effects temporal and contemporaneous effects

• Correlations of the means can be used to construct abetween-subjects network• Partial correlation network between means


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


Estimation


Typingspeed

Spellingerrors

Within−subjects

Typingspeed

Spellingerrors

Between−subjects

Example based on Hamaker, E. L. (2012). Why ResearchersShould Think ‘Within-Person’: A Paradigmatic Rationale.Handbook of Research Methods for Studying Daily Life. TheGuilford Press New York, NY, 43–61.


Physicalactivity

Heartrate

Within−subjects

Physicalactivity

Heartrate

Between−subjects

Example provided by Ellen Hamaker and based on Hoffman, L.(2015). Longitudinal analysis: Modeling within-person fluctuationand change. New York, NY, USA: Routledge.


Future Directions & Conclusion


Current state of Network Psychometrics

• Applied in over 100 empirical papers

• Many software packages available• e.g., qgraph, bootnet, lvnet, graphicalVAR, mlVAR,

IsingSampler, IsingFit, mgm, NetworkToolbox,JASP

• Many methodological labs working on network psychometricsor related topics

• Well attended workshops and winter/summer schools

• Popular Facebook group!

• Psychological Dynamics

• Many talks at IMPS!


Current state of Network Psychometrics

• Applied in over 100 empirical papers

• Many software packages available• e.g., qgraph, bootnet, lvnet, graphicalVAR, mlVAR,

IsingSampler, IsingFit, mgm, NetworkToolbox,JASP

• Many methodological labs working on network psychometricsor related topics

• Well attended workshops and winter/summer schools

• Popular Facebook group!• Psychological Dynamics

• Many talks at IMPS!


Title Speaker / Chair Day TimeSymposium 1: Cognitive Development:Network Psychometric Approaches

Maarten Marsman Tuesday 11:00 - 12:00

Symposium 3: Psychometric Developmentsin JASP

Don van den Bergh Tuesday 13:30 - 15:00

Symposium 5: Modeling Intensive Longitu-dinal Data: Perks and Pitfalls

Laura Bringmann Tuesday 15:20 - 16:50

Diagnosing Diagnostic Models: From vonNeumanns Elephant to Model Equivalen-cies and Network Psychometrics

Matthias von Davier Tuesday 17:00 - 17:50

Network Psychometrics: Current State andFuture Directions

Sacha Epskamp Tuesday 17:50 - 18:30

Symposium 8: Non-Cognitive Psychomet-ric Theory and Assessment

Gunter Maris Wednesday 08:30 - 10:00

Equivalent Dynamic Models Peter Molenaar Wednesday 10:15 - 11:00Symposium 9: Timely Perspectives on Dy-namic Models for Time Series and Panels

Eva Ceulemans; Janne Adolf Wednesday 15:00 - 16:30

Network Analysis Han van der Maas Wednesday 16:30 - 18:00Symposium 13: Network Psychometrics 1:Advances in Network Structure Estimation

Denny Borsboom Thursday 09:45 - 11:15

Symposium 17: Network Psychometrics II:Psychometric Extensions to Network Mod-eling

Sacha Epskamp Friday 08:30 - 10:00


Future Directions

• Methodological aspects in MRF estimation• MRFs for ordinal data• Missing data handling• Choosing the right estimation method

• Philosophical issues

• Within- and Between-subjects analysis

• Dynamical network modeling

• Time-varying network models• Incorporating prior knowledge• Multi-level generalized network models (RNM & LNM)

• Building network models from theory

• Network-based adaptive assessment (See symposium onFriday)

• Network meta analysis


Future Directions


• Philosophical issues• Within- and Between-subjects analysis

• Dynamical network modeling

• Time-varying network models• Incorporating prior knowledge• Multi-level generalized network models (RNM & LNM)





Future Directions


• Philosophical issues• Within- and Between-subjects analysis

• Dynamical network modeling• Time-varying network models• Incorporating prior knowledge• Multi-level generalized network models (RNM & LNM)





Thank you for your attention!

• Website: sachaepskamp.com

• Slides at: sachaepskamp.com/presentations• Dissertation at: sachaepskamp.com/dissertation

• Twitter: twitter.com/SachaEpskamp

• Join our Facebook group!• facebook.com/groups/PsychologicalDynamics

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