Introduction Network Analysis for Psychologists · Network Analysis for Psychologists Sacha Epskamp Introduction Psychology as a Complex System Building Blocks of a Network Types

Network Analysisfor Psychologists

Sacha Epskamp

IntroductionPsychology as a ComplexSystem

Building Blocks of aNetwork

Types of Networks

Interaction Networks

Causal Networks

Predictive Networks

Estimating networktopologyCross-sectional networks

Association networks

Concentration Networks

Time series

NetworkMethodologyDistance and ShortestPaths

Centrality

Connectivity and Clustering

Network Analysiswith qgraphA very tiny introduction to R

Estimating networks inqgraph

Network inference

References

Extra

Network Analysis for PsychologistsUsing qgraph in R

Sacha Epskamp

University of AmsterdamDepartment of Psychological Methods

22-05-2014APS 2014


Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

Extra

I These sheets are available onhttp://sachaepskamp.com/presentations

I Including an R script containing codes of the lastsection

This workshop is cosponsored by APS and the Society ofMultivariate Experimental Psychology (SMEP).

http://sachaepskamp.com/presentations


Sacha Epskamp



Types of Networks


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References

Extra

http://www.psychosystems.org/

http://www.psychosystems.org/


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Types of Networks


Causal Networks

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Psychosystems

Staff:I Denny BorsboomI Angelique CramerI Lourens WaldorpI Sacha EpskampI Claudia van BorkuloI Mijke Rhemtulla

Students:I Marie DesernoI Jolanda

KossakowskiI Pia Tio

Collaborators:I Giulio CostantiniI Jonathon LoveI Laura BringmannI Francis TuerlinckxI Laura RuzzanoI Hilde GeurtsI . . .


Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


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Network inference

References

Extra

MDInsomnia

Fatigue

Concentration

Worry


Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

Extra

MDInsomnia

Fatigue

Concentration

Worry


Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

Extra

MDInsomnia

Fatigue

Concentration

Worry


Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

Extra

MDInsomnia

Fatigue

Concentration

Worry


Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

Extra

MDInsomnia

Fatigue

Concentration

Worry


Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

Extra

MDInsomnia

Fatigue

Concentration

Worry


Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

Extra

MDInsomnia

Fatigue

Concentration

Worry


Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

Extra

Insomnia

Fatigue

Concentration

Worry


Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

Extra

Insomnia

Fatigue

Concentration

Worry


Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


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Extra

Psychology as a complex system

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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

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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


Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

Extra

What is a network?

I A network is a set of nodes connected by a set ofedges


Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

Extra

Edge

Node 1

Node 2


Sacha Epskamp



Types of Networks


Causal Networks

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What is a network?


I A node represents an entity, this can be anything:I People

I CitiesI Symptoms

I An edge represents some connection between twonodes. Again, this can be anything:

I Friendship / contactI DistanceI Comorbidity


Sacha Epskamp



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What is a network?


I A node represents an entity, this can be anything:I PeopleI Cities

I SymptomsI An edge represents some connection between two

nodes. Again, this can be anything:



Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


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Network inference

References

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What is a network?


I A node represents an entity, this can be anything:I PeopleI CitiesI Symptoms




Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

Extra

What is a network?






Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

Extra

What is a network?




I Friendship / contact

I DistanceI Comorbidity


Sacha Epskamp



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Time series


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What is a network?




I Friendship / contactI Distance

I Comorbidity


Sacha Epskamp



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Causal Networks

Predictive Networks




Time series


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Network inference

References

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What is a network?






Sacha Epskamp



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Causal Networks

Predictive Networks




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Anne is friends with Laura:

Friendship

Anne

Laura


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Anne is friends with Laura and Roger, but Laura is notfriends with Roger:

Anne

LauraRoger


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Networks can be weightedAnne is better friends with met Roger than Laura:

Anne

LauraRoger


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Weights can be signedAnne is friends with Roger and Laura, but Roger andLaura don’t like each other at all!

Anne

LauraRoger


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Networks can be directedAnne likes Laura, but Laura doesn’t like Anne:

Anne

Laura


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Directed Undirected

Unw

eigh

ted

Wei

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Interpreting Networks

A network can be interpreted in different ways:I As a model of interacting components

I Information can spread from node to node via edgesI As a causal modelI As a predictive model


Sacha Epskamp



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I Information can spread from node to node via edges

I As a causal modelI As a predictive model


Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

Extra



I Information can spread from node to node via edgesI As a causal model

I As a predictive model


Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

Extra



I Information can spread from node to node via edgesI As a causal modelI As a predictive model


Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

Extra

Interacting components


Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

Extra


Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

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Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

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Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

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Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

Extra


Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


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Network inference

References

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Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


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Network inference

References

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Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


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Network inference

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Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


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Network inference

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Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

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Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


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Network inference

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Psychopathology as a virus...


Sacha Epskamp



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Networks can model causalityA motivated student will get higher grades:

Motivated

Grade


Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

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A difficult class causes students to get lower grades:

DifficultClass

Grade


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Motivated

Grade

DifficultClass


Sacha Epskamp



Types of Networks


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Insomnia

Fatigue

Concentration

Worry


Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

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Causal models

I Methods such as Structural Equation Modeling(SEM) can be used to confirmatory test causalstructures

I Exploratory inference for causal structures is muchharder and mostly not possible without experimentaldesign

I In cross-sectional data only specific structures underthe strong assumption of acyclicness are inferable

I time series could be used to test Granger-causality,but this is again only a confirmatory test


Sacha Epskamp



Types of Networks


Causal Networks

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Time series


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Causal models






Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

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Causal models






Sacha Epskamp



Types of Networks


Causal Networks

Predictive Networks




Time series


Centrality




Network inference

References

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Causal models






Sacha Epskamp



Types of Networks


Causal Networks

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Time series


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Predictive networks

I A network we can infer is a predictive network

I In predictive networks, we draw an edge from node Ato node B if node A predicts node B.

I Underlying causal networks can be directlytranslated into predictive networks

I Equivalent causal networks lead to the samepredictive network

I If a interaction network generated the data thepredictive network can correctly estimate thestructure


Sacha Epskamp



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Causal Networks

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Time series


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Predictive networks

I A network we can infer is a predictive networkI In predictive networks, we draw an edge from node A

to node B if node A predicts node B.

I Underlying causal networks can be directlytranslated into predictive networks




Sacha Epskamp



Types of Networks


Causal Networks

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Predictive networks


to node B if node A predicts node B.I Underlying causal networks can be directly

translated into predictive networks




Sacha Epskamp



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Causal Networks

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Predictive networks



translated into predictive networksI Equivalent causal networks lead to the same

predictive network



Sacha Epskamp



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Predictive networks



translated into predictive networksI Equivalent causal networks lead to the same

predictive networkI If a interaction network generated the data the

predictive network can correctly estimate thestructure


Sacha Epskamp



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True generating model:

A B

I Does A predict B?

I Yes!I Does B predict A?

I Yes!I If node A predicts node B, node B predicts node A.

Hence, predictive networks often are undirectednetworks


Sacha Epskamp



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A B

I Does A predict B?I Yes!

I Does B predict A?I Yes!

I If node A predicts node B, node B predicts node A.Hence, predictive networks often are undirectednetworks


Sacha Epskamp



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A B


I Does B predict A?

I Yes!I If node A predicts node B, node B predicts node A.

Hence, predictive networks often are undirectednetworks


Sacha Epskamp



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Time series


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A B





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A B





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A B

Predictive model:

A B


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A

B

C

I Does A predict B?

I Yes!I Does B predict C?

I Yes!I Does A predict C or vise-versa?

I A and C will be correlated, thus knowing A allowsyou to predict C

I But if we also know B, A will not add informationabout C.

I Thus A only predicts C via B


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A

B

C


I Does B predict C?I Yes!

I Does A predict C or vise-versa?I A and C will be correlated, thus knowing A allows

you to predict CI But if we also know B, A will not add information

about C.I Thus A only predicts C via B


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A

B

C


I Does B predict C?

I Yes!I Does A predict C or vise-versa?





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A

B

C







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A

B

C



I Does A predict C or vise-versa?





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A

B

C







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I In an association network nodes are connected ifthey predict each other regardless of other nodes

I Edges represent correlations

I In a concentration network nodes are connected ifthey predict each other given that we know othernodes

I Edges represent partial correlations or multipleregression coefficients

I If there is no edge between two nodes they areconditionally independent given all other nodes

I Such networks are also called Markov RandomFields.


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I Edges represent correlationsI In a concentration network nodes are connected if

they predict each other given that we know othernodes





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IQ

Grades

Finishingcollege

I A smart student will get better grades

I A student with good grades is more likely to finishcollege

I IQ is correlated with finishing college and thuspredicts finishing college

I Association

I Given that we know the grades a student got, the IQscore does not improve the prediction of finishingcollege

I Concentration


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IQ

Grades

Finishingcollege

I A smart student will get better gradesI A student with good grades is more likely to finish

college

I IQ is correlated with finishing college and thuspredicts finishing college

I Association


I Concentration


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IQ

Grades

Finishingcollege


collegeI IQ is correlated with finishing college and thus

predicts finishing college

I AssociationI Given that we know the grades a student got, the IQ

score does not improve the prediction of finishingcollege

I Concentration


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IQ

Grades

Finishingcollege



predicts finishing collegeI Association


I Concentration


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IQ

Grades

Finishingcollege





I Concentration


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IQ

Grades

Finishingcollege





I Concentration


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A

B

C

Association network:

A

B

C

Concentration network:

A

B

C


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F

A B C

Where F is a unobservable latent variableI Does A predict B?


I Yes!I Does A predict C?

I Yes!If a latent cause underlies the nodes, all nodes predict allother nodes. Since we can not condition on F theprediction is not mediated. The association andconcentration networks then are the same: fullyconnected clusters.


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F

A B C


I Yes!


I Does A predict C?I Yes!

If a latent cause underlies the nodes, all nodes predict allother nodes. Since we can not condition on F theprediction is not mediated. The association andconcentration networks then are the same: fullyconnected clusters.


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F

A B C






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F

A B C



I Yes!

I Does A predict C?I Yes!



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F

A B C






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F

A B C




I Yes!



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F

A B C






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F

A B C


A

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F

A B C


A

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A

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Predictive networks

I Easily constructable

I Make no assumptions about directionalityI Naturally cyclicI Especially edges concentration networks can identify

causal effectsI Correspond to undirected network that could underlie

the data generating model


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Predictive networks

I Easily constructableI Make no assumptions about directionality

I Naturally cyclicI Especially edges concentration networks can identify




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Predictive networks

I Easily constructableI Make no assumptions about directionalityI Naturally cyclic

I Especially edges concentration networks can identifycausal effects

I Correspond to undirected network that could underliethe data generating model


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Predictive networks

I Easily constructableI Make no assumptions about directionalityI Naturally cyclicI Especially edges concentration networks can identify

causal effects

I Correspond to undirected network that could underliethe data generating model


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Predictive networks

I Easily constructableI Make no assumptions about directionalityI Naturally cyclicI Especially edges concentration networks can identify




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Estimating Network Topology

I In this block we will look at how to estimateassociation and concentration networks forcross-sectional and time-series data.

I All these methods only require a covariance matrixI We will assume that all nodes are normally

distributed

I For ordinal data threshold models can be used(These can easily be computed in qgraph)

I Polychoric correlationsI Polyserial correlations

I nonnormal continuous data could be transformedusing the nonparanormal transformation (Liu,Lafferty, & Wasserman, 2009)

I For binary data we can use the Ising Model

I To start, an association network for cross-sectionaldata is simply a graph of the correlation matrix


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I All these methods only require a covariance matrix

I We will assume that all nodes are normallydistributed



I nonnormal continuous data could be transformedusing the nonparanormal transformation (Liu et al.,2009)




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distributed




I For binary data we can use the Ising ModelI To start, an association network for cross-sectional

data is simply a graph of the correlation matrix


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distributedI For ordinal data threshold models can be used

(These can easily be computed in qgraph)






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(These can easily be computed in qgraph)I Polychoric correlations

I Polyserial correlationsI nonnormal continuous data could be transformed

using the nonparanormal transformation (Liu et al.,2009)




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(These can easily be computed in qgraph)I Polychoric correlationsI Polyserial correlations





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Represent each variable with a node:

1

2


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Connect them with a green edge if they are positivelycorrelated:

0.5

1

2


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Or a red edge if they are negatively correlated:

−0.5

1

2


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The weaker the correlation the vaguer/thinner the edge:

0.2

1

2


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Real data example

Included in qgraph is a dataset in which the Dutchtranslation of a commonly used personality test, theNEO-PI-R (Costa & McCrae, 1992; Hoekstra, de Fruyt, &Ormel, 2003), was administered to 500 first yearpsychology students (Dolan, Oort, Stoel, & Wicherts,2009). The NEO-PI-R consists of 240 items designed tomeasure the five central personality factors:

I NeuroticismI ExtroversionI AgreeablenessI Openness to ExperienceI Conscientiousness


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Toy dataset

Worry (W) Concentration (C) Fatigue (F) Insomnia (I)4 2 3 11 6 2 15 4 4 64 2 8 82 9 1 32 6 2 58 2 4 81 5 4 14 4 5 34 2 5 4


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Insomnia

Fatigue

Concentration

Worry


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Data

## W C F I## 1 4 2 3 1## 2 1 6 2 1## 3 5 4 4 6## 4 4 2 8 8## 5 2 9 1 3## 6 2 6 2 5## 7 8 2 4 8## 8 1 5 4 1## 9 4 4 5 3## 10 4 2 5 4


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Association

round(cor(Data),2)

## W C F I## W 1.00 -0.67 0.40 0.70## C -0.67 1.00 -0.73 -0.38## F 0.40 -0.73 1.00 0.52## I 0.70 -0.38 0.52 1.00


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Association

W C F IWorry 1.00Concentration -0.67 1.00Fatigue 0.40 -0.73 1.00Insomnia 0.70 -0.38 0.52 1.00


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AssociationW C F I

Worry 1.00Concentration -0.67 1.00Fatigue 0.40 -0.73 1.00Insomnia 0.70 -0.38 0.52 1.00

W

C

F

I


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Association

library("qgraph")qgraph(cor(Data))

W

C

F

I


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Association networks...I Allow for a powerful visualization of the correlational

structure

I Indicates the presence of latent variablesI But also include many spurious connectionsI And thus not well suited for network analysis


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structureI Indicates the presence of latent variables

I But also include many spurious connectionsI And thus not well suited for network analysis


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structureI Indicates the presence of latent variablesI But also include many spurious connections

I And thus not well suited for network analysis


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structureI Indicates the presence of latent variablesI But also include many spurious connectionsI And thus not well suited for network analysis


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Concentration

We can construct a concentration network by predictingall variables from all other variables:

W = β21C + β31F + β41I + εW

Where εW ∼ N(0, σW ) and is assumed to be independentof all variables that are not W

C = β12W + β32F + β42I + εC

F = β13W + β23C + β43I + εF

I = β14W + β24C + β34F + εI


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Concentration


W = β21C + β31F + β41I + εW


C = β12W + β32F + β42I + εC

F = β13W + β23C + β43I + εF

I = β14W + β24C + β34F + εI


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Concentration


W = β21C + β31F + β41I + εW


C = β12W + β32F + β42I + εC

F = β13W + β23C + β43I + εF

I = β14W + β24C + β34F + εI


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Concentration


W = β21C + β31F + β41I + εW


C = β12W + β32F + β42I + εC

F = β13W + β23C + β43I + εF

I = β14W + β24C + β34F + εI


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Concentration


W = β21C + β31F + β41I + εW


C = β12W + β32F + β42I + εC

F = β13W + β23C + β43I + εF

I = β14W + β24C + β34F + εI


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fitW <- lm(W ~ C + F + I, data = Data)fitW

#### Call:## lm(formula = W ~ C + F + I, data = Data)#### Coefficients:## (Intercept) C F## 6.606 -0.723 -0.564## I## 0.518

fitC <- lm(C ~ W + F + I, data = Data)fitF <- lm(F ~ W + C + I, data = Data)fitI <- lm(I ~ W + C + F, data = Data)


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W = β21C+β31F + β41I + εW

C = β12W+β32F + β42I + εC

F = β13W + β23C + β43I + εF

I = β14W + β24C + β34F + εI

The regression coefficients have a symmetricalrelationship:

βijσ2j = βjiσ

2i


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W = β21C+β31F + β41I + εW

C = β12W+β32F + β42I + εC

F = β13W + β23C + β43I + εF

I = β14W + β24C + β34F + εI

The regression coefficients have a symmetricalrelationship:

βijσ2j = βjiσ

2i


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beta_WC <- coef(fitW)[2]beta_CW <- coef(fitC)[2]sigma_W <- summary(fitW)$sigmasigma_C <- summary(fitC)$sigma

beta_WC * sigma_C^2

## C## -1.163

beta_CW * sigma_W^2

## W## -1.163


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W = β21C + β31F + β41I + εW

C = β12W + β32F + β42I + εC

F = β13W + β23C + β43I + εF

I = β14W + β24C + β34F + εI


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W = β21C + β31F + β41I + εW

C = β12W + β32F + β42I + εC

F = β13W + β23C + β43I + εF

I = β14W + β24C + β34F + εI

These regression parameters can be standardized toobtain the partial correlation coefficient:

ρij =βijσj

σi=βjiσi

σj

Through mathematical magic, these correspond directlyto the negative standardized off-diagonal elements of theinverse of the correlation matrix!


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W = β21C + β31F + β41I + εW

C = β12W + β32F + β42I + εC

F = β13W + β23C + β43I + εF

I = β14W + β24C + β34F + εI


ρij =βijσj

σi=βjiσi

σj



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W = β21C + β31F + β41I + εW

C = β12W + β32F + β42I + εC

F = β13W + β23C + β43I + εF

I = β14W + β24C + β34F + εI


ρij =βijσj

σi=βjiσi

σj



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beta_WC * sigma_C / sigma_W

## C## -0.7651

beta_CW * sigma_W / sigma_C

## W## -0.7651

-cov2cor(solve(cov(Data)))

## W C F I## W -1.0000 -0.7651 -0.5889 0.7754## C -0.7651 -1.0000 -0.7993 0.5871## F -0.5889 -0.7993 -1.0000 0.6468## I 0.7754 0.5871 0.6468 -1.0000


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These partial correlations give us the concentrationgraph:

library("qgraph")qgraph(cor(Data), graph = "concentration")

W

C

F

I

Why does this still not look good?


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These partial correlations give us the concentrationgraph:

library("qgraph")qgraph(cor(Data), graph = "concentration")

W

C

F

I

Why does this still not look good?


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n = 10!

W C F I4 2 3 11 6 2 15 4 4 64 2 8 82 9 1 32 6 2 58 2 4 81 5 4 14 4 5 34 2 5 4


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Coefficients for Worry:

Table:

Dependent variable:W

C −0.723∗∗ (0.248)F −0.564 (0.316)I 0.518∗∗ (0.172)Constant 6.606∗∗ (2.057)

Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01


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Coefficients for Concentration:

Table:

Dependent variable:C

W −0.810∗∗ (0.278)F −0.810∗∗ (0.249)I 0.415 (0.234)Constant 8.451∗∗∗ (0.992)

Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01


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Coefficients for Fatigue:

Table:

Dependent variable:F

W −0.615 (0.345)C −0.789∗∗ (0.242)I 0.451∗ (0.217)Constant 7.460∗∗∗ (1.810)

Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01


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Coefficients for Insomnia:

Table:

Dependent variable:I

W 1.161∗∗ (0.386)C 0.830 (0.467)F 0.927∗ (0.446)Constant −7.071 (4.176)

Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01


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I We are specifically interested in identifying whichpartial correlations are zero

I These elements correspond to missing edges in thegraph

I We could identify zeroes by significance testing orstepwise model-selection, but...

I Problem of multiple comparisonI Intractable model spaceI p-values and NHST are evil

I A well accepted approach for jointly model selectionand parameter estimation is the least absoluteshrinkage and selection operator (LASSO)

I Limits the sum of absolute regression weights, whichcauses insignificant edges to shrink to zero

I Useful in any regression case. For graphs thegraphical lasso (Friedman, Hastie, & Tibshirani,2008) is a very fast algorithm.

I Implemented in glasso package (Friedman, Hastie, &Tibshirani, 2011)

I Even usable when you have more variables thenmeasures!


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I Useful in any regression case. For graphs thegraphical lasso (Friedman et al., 2008) is a very fastalgorithm.

I Implemented in glasso package (Friedman et al.,2011)



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I Problem of multiple comparison

I Intractable model spaceI p-values and NHST are evil







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I Problem of multiple comparisonI Intractable model space

I p-values and NHST are evilI A well accepted approach for jointly model selection

and parameter estimation is the least absoluteshrinkage and selection operator (LASSO)






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glasso penalty: 0

## With rho=0, there may be convergence problems if the input matrix s## is not of full rank

0.59−0.59

0.65

−0.770.78

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glasso penalty: 0.1

−0.320.33

0.44

−0.60.64

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glasso penalty: 0.25

−0.140.16

0.31

−0.480.55

−0.57

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glasso penalty: 0.3

−0.010.03

0.22

−0.390.49

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glasso penalty: 0.4

0.21

−0.370.47

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glasso penalty: 0.5

0.21

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glasso penalty: 1

0.19

−0.31

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glasso penalty: 2

0.09

−0.18

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glasso penalty: 3

−0.04

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glasso penalty: 4

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We can compute an information criterion for each penaltyparameter, and choose the model with the best (lowest)value. In concentration networks the extended BayesianInformation Criterium (EBIC) works well (Foygel & Drton,2010):

EBIC = −2L + |E | log n + 4|E |γ log p

Where L is the log-likelihood, |E | the number of edges inthe graph, n the sample size and p the number ofvariables in the graph. γ is typically set to 0.5 or 0 toobtain regular BIC.In essence this is the likelihood function with a penalty forthe amount of parameters.


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Find optimal penalty given extended BIC:

130

135

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0 2 4 6Penalty

EB

IC


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Find optimal penalty given extended BIC:

130

135

140

145

0 2 4 6Penalty

EB

IC


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Find a sparse concentration graph using glasso:

qgraph(cor(Data), graph = "glasso", sampleSize = 10)

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C. Adaptive lasso network

Paper by Giulio Costantini, Sacha Epskamp, Denny Borsboom, MarcoPerugini, René Mõttus, Lourens J. Waldorp & Angelique O. J. Cramersubmitted


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Partial correlations (using adaptive lasso)

Paper by Jolanda J. Kossakowski, Jacobien M. Kieffer, SachaEpskamp & Denny Borsboom in preparation


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Zooming in: Physical Functioning en Item 2


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If the data is binary we can use the Ising model to estimate aconcentration network. We have implemented a similar method asdescribed in these slides for continuous data in the IsingFit package(van Borkulo, Epskamp, & with contributions fromAlexander Robitzsch, 2014)

fal slewak

hyp

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rmo

mod

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app

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Paper by Claudia D. van Borkulo, Denny Borsboom, Sacha EpskampTessa F. Blanken, Lynn Boschloo, Robert A. Schoevers & Lourens J.Waldorp submitted


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Time series

I In time-series data, we do not want to predict everynode given every other node

I It is senseless to predict the past from the future, orpredict the future given other variables in the future

I We don’t care how well a patient being suicidal todaypredicts how depressed he was yesterday

I We don’t care how well a patient being depressedtomorrow will predict how suicidal he will be tomorrow

I Therefore we only set up a predictive model betweentime points, which can be condensed in a directednetwork

I Looking only between consecutive time points iscalled a Lag-1 process. We will not look at morecomplicated models today


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True causal model between days

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Simulated for 100 days:

−4

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2

0 25 50 75 100Time

Val

ue

Variable

Worry

Concentration

Fatigue

Insomnia


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head(Data)

## W C F I## 1 0.0 0.0 0.0 0.0## 2 -0.6 0.2 -0.8 1.6## 3 0.0 -0.8 0.5 1.4## 4 0.8 -0.6 2.1 1.1## 5 -0.1 -2.2 2.4 0.7## 6 0.5 0.5 2.2 0.9


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We can compute the correlations between consecutivetime-points:

corL1 <- cor(Data[-nrow(Data),],Data[-1,])round(corL1,2)

## W C F I## W 0.50 -0.21 0.22 0.42## C -0.33 0.40 -0.07 -0.21## F 0.14 0.18 0.49 0.18## I 0.10 0.02 0.44 0.58

Which gives us the Lag-1 correlations, and correspond tothe association network


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Lag-1 correlations

qgraph(corL1, layout = "circle")

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A subject over time...


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Concentration

We can again construct a concentration network byrepeated multiple regressions. This time however weregress on the previous time point

Wt = β11Wt−1 + β21Ct−1 + β31Ft−1 + β41It−1 + εW

Ct = β12Wt−1 + β22Ct−1 + β32Ft−1 + β42It−1 + εC

Ft = β13Wt−1 + β23Ct−1 + β33Ft−1 + β43It−1 + εF

It = β14Wt−1 + β24Ct−1 + β34Ft−1 + β44It−1 + εI

Where εi ∼ N(0, σi)


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fitW <- lm(W[-1] ~ W[-100] + C[-100] + F[-100] + I[-100], data = Data)fitW

#### Call:## lm(formula = W[-1] ~ W[-100] + C[-100] + F[-100] + I[-100], data = Data)#### Coefficients:## (Intercept) W[-100] C[-100]## -0.1490 0.4450 -0.2455## F[-100] I[-100]## 0.0629 -0.0573

fitC <- lm(C[-1] ~ W[-100] + C[-100] + F[-100] + I[-100], data = Data)fitF <- lm(F[-1] ~ W[-100] + C[-100] + F[-100] + I[-100], data = Data)fitI <- lm(I[-1] ~ W[-100] + C[-100] + F[-100] + I[-100], data = Data)


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This model is called a Vector-autoregression (VAR) model

y t = By t−1 + ε

In qgraph the VARglm() function can be used toestimate the regression parameters. The transpose of theestimates correspond to the graph structure (for now).Model selection and LASSO can be used, but have notyet been implemented automatically in qgraph


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VAR network

Beta <- VARglm(Data)$graphqgraph(t(Beta), layout = "circle", labels = names(Data))

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C

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!"#$%&%! + "' (')$*'%+ "+ (+)$*' + ... + #!

Bringmann, L., Vissers, N., Wichers, M., Geschwind, N., Kuppens, P., Peeters, F., Borsboom, D., & Tuerlinckx, F. (2013). A network approach to psychopathology: New insights into clinical longitudinal data. PLoS ONE.

Wednesday, October 2, 13


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Network estimation

Cross-sectional Time-seriesAssociation Correlations Partial correlations∗

Concentration Lag-1 correlations VAR∗

* model selection / LASSO can be used to estimate sparse structure


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Network Methodology

Besides providing a interpretable structure of a complexsystem, we can also use networks to compute uniquemeasures such as:

I Distance

I How long does it take for a node to influence anothernode?

I Centrality

I Which node is the most important?

I Connectivity

I How well are nodes connected?


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Network Methodology


I DistanceI How long does it take for a node to influence another

node?

I Centrality


I Connectivity



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node?I Centrality

I Which node is the most important?I Connectivity



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node?I Centrality


I Connectivity



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node?I Centrality




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node?I Centrality




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I Each edge can be interpreted as having a length

I Length is inversely related to weight

I Two strongly connected nodes are closer: it is easierfor information to go from one node to the other

I The path between two unconnected nodes isinfinitely long: you cannot walk directly from onenode to the other

I In weighted networks the absolute value of edgeweights is used in computing the length


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I Each edge can be interpreted as having a lengthI Length is inversely related to weight





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The distance between Storrs and San Francisco is2638.29 miles (4245.80 km)

2638.29 miles StorrsSan Francisco


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Amsterdam is much farther from San Francisco:

5449.02 mile3523.18 mile

2638.29 mile StorrsSan Francisco

Amsterdam


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We can place nodes however we want:

5449.02 mile

3523.18 mile 2638.29 mile

Storrs

San FranciscoAmsterdam


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Shortest path length

I The distance between two nodes is defined by thelength of the shortest possible path between twonodes

I In unweighted graphs this is simply the amount ofedges on a path

I In weighted graphs the length of a path is computedas the sum of all edge lengths on a path


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Which node is the most influential?


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Centrality

A node is central/important/influential if...I ...it has many connections

I Degree / strengthI In concentration network the node with the highest

strength directly predicts the most other nodesI ...it is close to all other nodes

I ClosenessI In concentration networks the node that (indirectly)

has the best predictive value on all other nodes

I ...it connects other nodes

I Betweenness

I ...it is connected to other important nodes

I Eigenvector centrality


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Centrality


I Degree / strength

I In concentration network the node with the higheststrength directly predicts the most other nodes

I ...it is close to all other nodes




I Betweenness




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strength directly predicts the most other nodes

I ...it is close to all other nodes




I Betweenness




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has the best predictive value on all other nodesI ...it connects other nodes

I Betweenness




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I Closeness

I In concentration networks the node that (indirectly)has the best predictive value on all other nodes


I Betweenness




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I Betweenness




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I BetweennessI ...it is connected to other important nodes



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I Betweenness




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insomnia / difficu...psychomotor agitat...

psychomotor retard...

depressed mood

tachycardia / acce...

often easily distr...

is often touchy or...

anxietynausua

sweating / perspir...Weight loss

difficulty concent...

transient visual, ...

fatigue / fatigue ...

increased appetite

Clinically signifi...

delusions

coma

extreme negativism

wei

ght g

ain

Hypersomnia

feelings of worthl...

Loss of appetite

excessive motor ac...peculiarities of v...

mar

kedl

y di

min

ishe

...

echolalia

echopraxia

recurrent thoughts...

disorganized speech

flight of ideas or...

disorganized behav...

inflated self−este...decreased need for...

more talkative tha...

increase in goal−d...

excessive involvem...

A d

istin

ct p

erio

d ...

motoric immobilitypupillary dilation

mutism

derealization

depersonalization

palpitations

vomiting

fear of one or mor...nystagmus / vertic...

pounding heart

sensations of shor...

abdominal distressThe situations are...

chills or hot flus...

There is evidence ...

fear of losing con...

stupor

elevated blood pre... trembling or shaking

ches

t pai

n or

dis

c...

fear of dying

paresthesias

persistent concern...

worry about the im...

a significant chan...

incoordination

memory impairment ...

The person recogni...

Marked and persist...

rambling flow of t...

slur

red

spee

ch

blurring of vision

tremors

Anxiety about bein...

A distinct period ...

often lies to obta...

often bullies, thr...

often initiates ph...

has used a weapon ... has been physicall...

has been physicall...

has stolen while c...

has forced someone...

has deliberately e...

has deliberately d...

has broken into so...

has stolen items o...

often stays out at...

has run away from ...is often truant fr...

The development of...

often fails to giv...

often has difficul...

often does not see...

ofte

n do

es n

ot fo

l...

often has diff icu...

often avoids, disl...

often loses t hing...is often forgetful...

often fidgets with...

often leaves seat ...

often runs about o...


is often "on the g...

often talks excess...

often blurts out a...


often interrupts o...

Excessive anxiety ...

Disturbance of con...

unsteady gait

bradycardia

chills

confusion, seizure...

flat or inappropri...

muscular weakness,...

lowered blood pres...

generalized muscle...

the person experie...

dizziness

lethargy

depressed reflexes

euphoria

Presence of two or...

lacks close friend...

cardiac arrhythmia

piloerection

nervousness

excitement

flushed face

diuresis

gastroint estinal ...

mus

cle

twitc

hing

periods of inexhau...

muscle aches

lacr

imat

ion

or r

hi...

diarrhea

yawning

fever

seiz

ures

Focal neurological...

aphasia

apraxia

agnosia disturbance in exe...

numbness or dimini...

ataxia

dysa

rthr

ia

mus

cle

rigid

ity

hyperacusis

Confusion about pe...

The

per

son

finds

i...

mind going blank

mus

cle

tens

ion

odd beliefs or mag...unusual perceptual...

decreased heart rate

vivid, unpleasant ...

suspiciousness or ...

The traumatic even...

transient, stress ...

Marked avoidance o...

ideas of reference

A change in cognit...

development of a p...

Recent ingestion o...

autonomic hyperact...

increased hand tre...

grand mal seizures

feelings of hopele...

dissociative amnesia

behavior or appear...

frantic efforts to...

a pa

ttern

of u

nsta

...

impulsivity in at ...

recurrent suicidal...

affective instabil...

chronic fee lings ...

failure to conform...deceitfulness

Per

sist

ent a

void

an...

often loses temperoften argues with ...

ofte

n de

liber

atel

y...

often blames other...

is often angry and...

is often spiteful ...

hypervigilance

exaggerated startl...

impulsivity or fai...

reckless disregard...

consistent irrespo...

lack of remorse

often actively def... Dysphoric mood

drowsiness

neither desires no...

has little, if any...

takes pleasure in ...

Pupillary constric...

catalepsy

almost always choo...

appears indifferen...

conjunctival injec...

dry

mou

th

Relative unrespons...

No detailed dream ...

Rec

urre

nt e

piso

des.

..


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Closeness

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insomnia / difficu...psychomotor agitat...

increased appetitedepressed mood

psychomotor retard...

nausua

transient visual, ...

difficulty concent...

fatigue / fatigue ...

Weight loss

weight gain

sweating / perspir...is often touchy or...

anxiety

tach

ycar

dia

/ acc

e...

vomiting

often easily distr...

pupillary dilation

delusions

disorganized speech

diso

rgan

ized

beh

av...

extreme negativism

Hypersomnia

feel

ings

of w

orth

l...

echolalia

echopraxia

mutism

com

a

palpitations

incoordination

mus

cle

ache

s

diarrheayawning

fever chills

mus

cula

r w

eakn

ess,

...

tremors

excitement

flush

ed fa

ce

stupor

leth

argy

ataxia

unexpected travel ...


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Small world

The famous paper of Watts and Strogatz (1998)—alreadycited 22691 times—describes the “small world” principlethat frequently occurs in natural graphs.

I “Six degrees of separation”I High clustering and low average path length


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mouseover for friend details

Hbased on data from 190 of 203 friendsL


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Small world

A graph exhibits a small world if it has both highclustering and a short average path length. Thisinformation can be summarized in a small-world index,which should be higher than 3.


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Network Analysis with qgraph


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What is R?

I R is a statistical programming languageI Statistical analysisI Data visualizationI Data miningI General programming

I It is open-sourceI Free as in “free beer” and “free speech”I Large active community around RI Many contributed packages


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What is R?

R can be downloaded fromhttp://cran.r-project.org/. A good environmentfor R is RStudio which can be downloaded fromhttps://www.rstudio.com/. Some good links tostart learning R are:

I http://cran.r-project.org/doc/contrib/Torfs+Brauer-Short-R-Intro.pdf

I https://www.codeschool.com/courses/try-r

http://cran.r-project.org/

https://www.rstudio.com/

http://cran.r-project.org/doc/contrib/Torfs+Brauer-Short-R-Intro.pdf

http://cran.r-project.org/doc/contrib/Torfs+Brauer-Short-R-Intro.pdf

https://www.codeschool.com/courses/try-r

https://www.codeschool.com/courses/try-r


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R uses functions, such as rnorm(), which take somearguments between the parantheses. All these functionsare documented using ?. For example, rnorm generatesrandom numbers:

# rnorm samples random normal numbers:rnorm(3)

## [1] 1.59291 0.04501 -0.71513


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Multiple arguments can be used. They can also benamed. See the documentation for a list of arguments!?rnorm

# Using arguments we can change its behavior. e.g.,rnorm(3, mean = 100, sd = 15)

## [1] 113.0 116.1 128.4


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The <- symbol can be used to store results and reusethem later.

# We can use '<-' to store the result:result <- rnorm(3)result

## [1] -0.6030 -0.3909 -0.4162

# Reuse:mean(result)

## [1] -0.47


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Packages

I Packages are extensions contributed to R containingextra functions

I They can be installed using install.packages()

I Afterwards they can be loaded using library()


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Packages

# Install packageinstall.packages("qgraph", dep = TRUE)

# Load packagelibrary("qgraph")


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Load data into R

I There are many ways to load data into RI The most common way is to read a plain text file

(which can be exported from excel for instance)using read.table or read.csv (see their helpfiles for how to do this)

I Because psychologists often use SPSS, it is usefulto directly import data from SPSS

I This can be done using read.spss() in theforeign package.

I file.choose() can be used to select a file (itmight be opened in the background of RStudio)


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Load SPSS data into R

# Load package "foreign":library("foreign")

# Select a SPSS file:file <- file.choose()

# Read data:Data <- read.spss(file,to.data.frame=TRUE)


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Rather than using a file as data, we will use a datasetalready included in R via the psysch package (Revelle,2010):

library("psych")data(bfi)bfi <- bfi[,1:25]

str(bfi)

## 'data.frame': 2800 obs. of 25 variables:## $ A1: int 2 2 5 4 2 6 2 4 4 2 ...## $ A2: int 4 4 4 4 3 6 5 3 3 5 ...## $ A3: int 3 5 5 6 3 5 5 1 6 6 ...## $ A4: int 4 2 4 5 4 6 3 5 3 6 ...## $ A5: int 4 5 4 5 5 5 5 1 3 5 ...## $ C1: int 2 5 4 4 4 6 5 3 6 6 ...## $ C2: int 3 4 5 4 4 6 4 2 6 5 ...## $ C3: int 3 4 4 3 5 6 4 4 3 6 ...## $ C4: int 4 3 2 5 3 1 2 2 4 2 ...## $ C5: int 4 4 5 5 2 3 3 4 5 1 ...## $ E1: int 3 1 2 5 2 2 4 3 5 2 ...## $ E2: int 3 1 4 3 2 1 3 6 3 2 ...## $ E3: int 3 6 4 4 5 6 4 4 NA 4 ...## $ E4: int 4 4 4 4 4 5 5 2 4 5 ...## $ E5: int 4 3 5 4 5 6 5 1 3 5 ...## $ N1: int 3 3 4 2 2 3 1 6 5 5 ...## $ N2: int 4 3 5 5 3 5 2 3 5 5 ...## $ N3: int 2 3 4 2 4 2 2 2 2 5 ...## $ N4: int 2 5 2 4 4 2 1 6 3 2 ...## $ N5: int 3 5 3 1 3 3 1 4 3 4 ...## $ O1: int 3 4 4 3 3 4 5 3 6 5 ...## $ O2: int 6 2 2 3 3 3 2 2 6 1 ...## $ O3: int 3 4 5 4 4 5 5 4 6 5 ...## $ O4: int 4 3 5 3 3 6 6 5 6 5 ...## $ O5: int 3 3 2 5 3 1 1 3 1 2 ...


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This is a dataset measuring the big 5 personality traitsusing an ordinal scale. The qgraph functioncor_auto() can be used to compute an appropriatecorrelation matrix. In this case, polychoric correlations:

cor_bfi <- cor_auto(bfi)

## Variables detected as ordinal: A1; A2; A3; A4;A5; C1; C2; C3; C4; C5; E1; E2; E3; E4; E5; N1; N2;N3; N4; N5; O1; O2; O3; O4; O5


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We can send this correlation matrix to qgraph() to plotthe association network. To use a force-embeddedalgorithm we set the layout argument to "spring"

graph_cor <- qgraph(cor_bfi, layout = "spring")

A1

A2

A3

A4

A5

C1

C2

C3

C4

C5

E1

E2

E3

E4

E5

N1N2

N3

N4

N5

O1O2

O3

O4

O5


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The partial correlation graph can be plotted by setting thegraph argument to "concentration"

graph_pcor <- qgraph(cor_bfi, graph = "concentration", layout = "spring")

A1

A2A3A4

A5

C1

C2

C3

C4

C5

E1E2

E3

E4

E5

N1

N2

N3

N4

N5

O1

O2

O3O4

O5


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To use the graphical lasso to estimate a sparse structureset graph to "glasso" and assign the sample size:

graph_glas <- qgraph(cor_bfi, graph = "glasso", sampleSize = nrow(bfi), layout = "spring")

A1

A2 A3A4

A5

C1C2

C3

C4C5

E1

E2

E3

E4

E5

N1

N2

N3

N4 N5 O1

O2

O3O4

O5


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With legend:

Names <- scan("http://sachaepskamp.com/files/BFIitems.txt",what = "character", sep = "\n")

Groups <- rep(c('A','C','E','N','O'),each=5)


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With legend:

qgraph(cor_bfi, graph = "glasso", sampleSize = nrow(bfi),layout = "spring", nodeNames = Names,legend.cex =0.6, groups = Groups)

A1

A2 A3A4

A5

C1C2

C3

C4C5

E1

E2E3

E4

E5

N1

N2

N3

N4 N5 O1

O2

O3O4

O5

A1: 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.C1: 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.E1: 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.N1: Get angry easily.N2: Get irritated easily.N3: Have frequent mood swings.N4: Often feel blue.N5: Panic easily.O1: 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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A graph can be saved using filetype and filename.PDF files produce the best results:

qgraph(cor_bfi, graph = "glasso",sampleSize = nrow(bfi), layout = "spring",filetype = "pdf", filename = "glasso")

## Output stored in/home/sacha/[email protected]/Documents/Work/Talks/APS/Sheets/glasso.pdf


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The centrality_auto() function automaticallycomputes centrality measures and shortest path lengths:

cent <- centrality_auto(graph_glas)names(cent)

## [1] "node.centrality"## [2] "edge.betweenness.centrality"## [3] "ShortestPathLengths"


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centralityPlot(graph_glas, labels=Names)

Betweenness Closeness Strength

Am exacting in my work.

Am full of ideas.

Am indifferent to the feelings of others.

Avoid difficult reading material.

Carry the conversation to a higher level.

Continue until everything is perfect.

Don't talk a lot.

Do things according to a plan.

Do things in a half−way manner.

Find it difficult to approach others.

Get angry easily.

Get irritated easily.

Have frequent mood swings.

Inquire about others' well−being.

Know how to captivate people.

Know how to comfort others.

Love children.

Make friends easily.

Make people feel at ease.

Often feel blue.

Panic easily.

Spend time reflecting on things.

Take charge.

Waste my time.

Will not probe deeply into a subject.

0.00 0.25 0.50 0.75 1.00 0.8 0.9 1.00.6 0.7 0.8 0.9 1.0


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centralityPlot(list(cor = graph_cor, pcor = graph_pcor, glasso = graph_glas), labels=Names)

Betweenness Closeness Strength

Am exacting in my work.

Am full of ideas.

Am indifferent to the feelings of others.

Avoid difficult reading material.

Carry the conversation to a higher level.

Continue until everything is perfect.

Don't talk a lot.

Do things according to a plan.

Do things in a half−way manner.

Find it difficult to approach others.

Get angry easily.

Get irritated easily.

Have frequent mood swings.

Inquire about others' well−being.

Know how to captivate people.

Know how to comfort others.

Love children.

Make friends easily.

Make people feel at ease.

Often feel blue.

Panic easily.

Spend time reflecting on things.

Take charge.

Waste my time.

Will not probe deeply into a subject.

0.000.250.500.751.00 0.7 0.8 0.9 1.0 0.6 0.8 1.0

type

cor

glasso

pcor


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The smallworldness() function computes thesmallworldness index:

smallworldness(graph_glas)

## smallworldness## 1.0105## trans_target## 0.6358## averagelength_target## 1.3833## trans_rnd_M## 0.6292## trans_rnd_lo## 0.6131## trans_rnd_up## 0.6438## averagelength_rnd_M## 1.3833## averagelength_rnd_lo## 1.3833## averagelength_rnd_up## 1.3833


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Concluding comments

More information is on our website:http://psychosystems.org/ and the developmentalversion of qgraph is on GitHubhttps://github.com/SachaEpskamp/qgraph

http://psychosystems.org/

https://github.com/SachaEpskamp/qgraph


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Thank you for your attention!


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References I

Borsboom, D., Cramer, A. O., Schmittmann, V. D.,Epskamp, S., & Waldorp, L. J. (2011). The smallworld of psychopathology. PloS one, 6(11),e27407.

Costa, P. T., & McCrae, R. R. (1992). Revised NEOPersonality Inventory and NEO Five-FactorInventory. Odessa, FL: PsychologicalAssessment Services.

Cramer, A. O., Waldorp, L. J., van der Maas, H. L.,Borsboom, D., et al. (2010). Comorbidity: Anetwork perspective. Behavioral and BrainSciences, 33(2-3), 137–150.


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References II

Dolan, C., Oort, F., Stoel, R., & Wicherts, J. (2009).Testing Measurement Invariance in the TargetRotated Multigroup Exploratory Factor Model.Structural Equation Modeling: A MultidisciplinaryJournal, 16(2), 20.

Doosje, B., Loseman, A., & Bos, K. (2013). Determinantsof radicalization of islamic youth in thenetherlands: Personal uncertainty, perceivedinjustice, and perceived group threat. Journal ofSocial Issues, 69(3), 586–604.

Foygel, R., & Drton, M. (2010). Extended bayesianinformation criteria for gaussian graphical models.In Nips (pp. 604–612).

Friedman, J., Hastie, T., & Tibshirani, R. (2008). Sparseinverse covariance estimation with the graphicallasso. Biostatistics, 9(3), 432–441.


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References III

Friedman, J., Hastie, T., & Tibshirani, R. (2011). glasso:Graphical lasso- estimation of gaussian graphicalmodels [Computer software manual]. Retrievedfrom http://CRAN.R-project.org/package=glasso (R package version 1.7)

Hoekstra, H., de Fruyt, F., & Ormel, J. (2003).Neo-persoonlijkheidsvragenlijsten: NEO-PI-R,NEO-FFI. Lisse: Swets Test Services.

Liu, H., Lafferty, J., & Wasserman, L. (2009). Thenonparanormal: Semiparametric estimation ofhigh dimensional undirected graphs. The Journalof Machine Learning Research, 10, 2295–2328.

http://CRAN.R-project.org/package=glasso

http://CRAN.R-project.org/package=glasso


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References IVRevelle, W. (2010). psych: Procedures for psychological,

psychometric, and personality research[Computer software manual]. Evanston, Illinois.Retrieved fromhttp://personality-project.org/r/psych.manual.pdf (R package version 1.0-93)

van Borkulo, C., Epskamp, S., & with contributions fromAlexander Robitzsch. (2014). Isingfit: Fitting isingmodels using the elasso method [Computersoftware manual]. Retrieved from http://CRAN.R-project.org/package=IsingFit(R package version 0.2.0)

Watts, D., & Strogatz, S. (1998). Collective dynamics ofsmall-world networks. Nature, 393(6684),440–442.

http://personality-project.org/r/psych.manual.pdf

http://personality-project.org/r/psych.manual.pdf

http://CRAN.R-project.org/package=IsingFit

http://CRAN.R-project.org/package=IsingFit


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Doosje, Loseman, and Bos (2013)


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Doosje et al. (2013)


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Correlation network

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1: In−group Identification2: Individual Deprivation3: Collective Deprivation4: Intergroup Anxiety5: Symbolic Threat6: Realistic Threat7: Personal Emotional Uncertainty8: Perceived Injustice9: Perceived Illegitimacy authorities10: Perceived In−group superiority11: Distance to Other People12: Societal Disconnected13: Attitude towards Muslim Violence14: Own Violent Intentions


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Partial correlation network

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Partial correlation networkAfter glasso:

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Partial correlation networkAfter glasso (Friedman et al., 2011):

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Muslim Violence Violent IntentionsIn-group Identification 16.12 19.89Individual Deprivation 20.10 23.87Collective Deprivation 17.05 20.82

Intergroup Anxiety Inf InfSymbolic Threat 11.44 15.21Realistic Threat 14.81 18.58

Personal Emotional Uncertainty 10.73 14.50Perceived Injustice 24.83 28.60

Perceived Illegitimacy authorities Inf InfPerceived In-group superiority 3.16 6.93

Distance to Other People 5.70 8.81Societal Disconnected 9.15 12.92

Attitude towards Muslim Violence 0.00 3.77Own Violent Intentions 3.77 0.00


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Muslim Violence Violent IntentionsIn-group Identification 16.12 19.89Individual Deprivation 20.10 23.87Collective Deprivation 17.05 20.82

Intergroup Anxiety Inf InfSymbolic Threat 11.44 15.21Realistic Threat 14.81 18.58

Personal Emotional Uncertainty 10.73 14.50Perceived Injustice 24.83 28.60

Perceived Illegitimacy authorities Inf InfPerceived In-group superiority 3.16 6.93

Distance to Other People 5.70 8.81Societal Disconnected 9.15 12.92

Attitude towards Muslim Violence 0.00 3.77Own Violent Intentions 3.77 0.00


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Extra: Interpreting qgraph

I Under the default coloring scheme, positive edgeweights (here correlations) are shown as greenedges and negative edge weights as red.

I An edge weight of 0 is omitted. The wider and morecolorful an edge the stronger the edge weight.


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Interpreting qgraph

To interpret qgraph networks, three values need to beknown:

Minimum Edges with absolute weights under thisvalue are omitted

Cut If specified, splits scaling of width and colorMaximum If set, edge width and color scale such that

an edge with this value would be the widestand most colorful


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Interpreting qgraph

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Interpreting qgraph

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Interpreting qgraph

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Interpreting qgraph

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Interpreting qgraph

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Interpreting qgraph

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Interpreting qgraph

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