The qgraph package for network visualizations of ......The qgraph package for network visualizations of psychometric data in R Sacha Epskamp, Angelique O. J. Cramer, Lourens J. Waldorp,

The qgraph package for network visualizations ofpsychometric data in R

Sacha Epskamp, Angelique O. J. Cramer, Lourens J. Waldorp,Verena D. Schmittmann and Denny Borsboom

University of AmsterdamDepartment of Psychological Methods

Psychoco 2011

All codes in these slides were run using R version 2.12.1(2010-12-16) and qgraph version 0.4.8 and were made onWindows 7 x64 x86-64 build 7600.

qgraph

I A R package (CRAN link)

I Can be used to plot various types of graphsI Different from other R packages (e.g. igraph Csardi &

Nepusz, 2006) in:I Focus on weighted graphsI Intended for visualization of data as graphsI Optimized for vector-type image files (e.g. PDF, SVG)

I Aims in qgraphI Simple inputI Summarize a large amount of statistics without needing data

reduction methods.I Visualize relations between variables

I Main idea: Show variables as nodes, relationships as edges

Graphs

I A graph is a network that consists of n nodes (or vertices)that are connected with m edges.

I Each edge has a weight indicating the strength of thatconnection

I An edge can be directed (have an arrow) or undirected

http://cran.r-project.org/web/packages/qgraph/index.html

Unweighted graph

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Weighted graph

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Weighted graph

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Directed graph

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The qgraph() function

I The main function in qgraph is qgraph()I Most other functions are either wrapping functions using

qgraph() or functions used in qgraph()

I The qgraph() function requires only one argument (adj)

I A lot of other arguments can be specified, but these are alloptional

Usage:

qgraph( adj, ... )

The adjacency matrix

I The adj argument is the input. This can be an adjacencymatrix

I An adjacency matrix is a square n by n matrix in which eachelement indicates the relationship between two variables

I Any relationship can be used as long as:I A 0 indicates no relationshipI Absolute negative values are similar in strength to positive

values

I Examples:I A 1 indicating a connection (unweighted graphs)I CorrelationsI Regression parametersI Factor loadings

I Adjacency matrices occur naturally in statistics!

[,1] [,2] [,3]

[1,] 0 1 1

[2,] 0 0 1

[3,] 0 0 0

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Weighted graphs

Y = ηΛT + Θ

> set.seed(2)

> eta <- matrix(rnorm(200 * 5), ncol = 5)

> lam <- matrix(rnorm(50 * 5, 0, 0.15), 50,

+ 5)

> lam[apply(diag(5) == 1, 1, rep, each = 10)] <- rnorm(50,

+ 0.7, 0.3)

> th <- matrix(rnorm(200 * 50), ncol = 50)

> Y <- eta %*% t(lam) + th

Weighted graphs

> cor(Y)[1:15, 1:3]

[,1] [,2] [,3]

[1,] 1.000000000 0.218987651 0.197325805

[2,] 0.218987651 1.000000000 0.231696634

[3,] 0.197325805 0.231696634 1.000000000

[4,] 0.464897112 0.346780772 0.279845144

[5,] 0.295912130 0.275030523 0.209220603

[6,] 0.235201044 0.272947122 0.197676521

[7,] 0.157314986 -0.001815960 -0.027551034

[8,] 0.234392422 0.212721700 0.192401237

[9,] 0.321680277 0.350685995 0.210808452

[10,] 0.204097076 0.277127339 0.148343574

[11,] -0.072734280 -0.000913891 -0.085440215

[12,] 0.052842181 0.105870583 -0.056247479

[13,] 0.001850306 0.025604291 0.081077133

[14,] -0.083391834 -0.088641129 -0.215127641

[15,] -0.055847218 0.007651554 0.004209916

Weighted graphs

> qgraph(cor(Y))

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Weighted graphs

> gr <- list(1:10, 11:20, 21:30, 31:40, 41:50)

> qgraph(cor(Y), groups = gr)

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Fruchterman-Reingold layout (500 iterations)

> qgraph(cor(Y), groups = gr, layout = "spring")

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Saving arguments

> Q <- qgraph(cor(Y), groups = gr, layout = "spring",

+ minimum = 0.2)

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Saving arguments

> Q <- qgraph(Q)

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Graphical arguments

> Q <- qgraph(Q, esize = 10)

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Graphical arguments

> Q <- qgraph(Q, vsize = 4)

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Graphical arguments

> Q <- qgraph(Q, borders = FALSE)

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Graphical arguments

> Q <- qgraph(Q, shape = "square")

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Graphical arguments

> Q <- qgraph(Q, shape = "circle")

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Graphical arguments

> Q <- qgraph(Q, vTrans = 150)

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Graphical arguments

> qgraph(Q, transparency = T, bg = T, bgcontrol = 5,

+ filetype = "png", filename = "bg", res = 144,

+ width = 7, height = 7)

[1] "Output stored in C:/Users/Sacha/Documents/Work/qgraph/Psychoco2011/bg.png"

Graphical arguments Correlations

> qgraph(cor(Y), layout = "spring", groups = gr,

+ cut = 0.3, minimum = 0.1, maximum = 1,

+ graph = "association")

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Partial correlations


+ cut = 0.3, minimum = 0.1, maximum = 1,

+ graph = "concentration")

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Factorial graph


+ cut = 0.2, vsize = 2, esize = 1, borders = F,

+ graph = "factorial")

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Factor loadings

I A factor loadings matrix can be visualized usingqgraph.loadings()

I There are two wrapper functions that perform an analysis andsend the results to qgraph.loadings():

I qgraph.efa() performs an exploratory factor analysis (EFA)using stats:::factanal

I qgraph.pca() performs a principal component analysis (PCA)using psych:::principal (Revelle, 2010)

I These functions use a correlation or covariance matrix as input

Factor loadings: EFA

> qgraph.efa(cor(Y), 5, rotation = "promax",

+ layout = "tree", vsize = c(3, 10), groups = gr)

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Factor loadings: EFA crossloadings

> qgraph.efa(cor(Y), 5, rotation = "promax",

+ layout = "tree", crossloadings = TRUE,

+ vsize = c(3, 10), groups = gr, cut = 0.2)

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Factor loadings: PCA

> qgraph.pca(cor(Y), 5, rotation = "promax",

+ vsize = c(4, 10), groups = gr, vTrans = 200)

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Confirmatory Factor Analysis

I qgraph.cfa() can be used to fit a simple confirmatory factormodel

I Each variable loads on only one factorI Factors are correlatedI Scaling by fixing first loading of each factor to 1

I This is done with the sem (Fox, 2010) package

I Returns a "sem" object

I Results can be send to qgraph.sem() for a full report


> res <- qgraph.cfa(cov(Y), N = 200, groups = gr,

+ vsize = c(2, 10))

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> qgraph.semModel(res, edge.label.cex = 0.6)

λ2 λ3 λ4λ5

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Specified model


> qgraph(res, edge.label.cex = 0.6)

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> qgraph.sem(res, filename = "sem%03d", onefile = F,

+ panels = 2, legend = FALSE, groups = gr,

+ edge.label.cex = 0.6)


Model statistics

Model:Observed variables: 50 / 55

Number of free parameters: 110 Number of observations: 200

Number of fixed exogenous variables: 0

Iterations: 49

Goodness of Fit:Chisq: 1769.10048 , df: 1165 , p= 0

RMSEA: 0.05105 ( 90 % CI: NA − NA )Goodness−of−fit index: 0.74863

Adjusted goodness−of−fit index: 0.7249 Bentler−Bonnett NFI: 0.58411 Tucker−Lewis NNFI: 0.79028

Bentler CFI: 0.80055 SRMR: 0.09508

BIC: −4403.43926

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Correlation differences

Structural Equation Modelling

I qgraph comes with a function that extends output from sem(Fox, 2010) with path diagrams and graphs visualizing theparameter estimates

I This is done with the qgraph.sem() function

I The output of qgraph.sem() is a multi-page pdf fileI We can use sem as usual and pass the output to

qgraph.sem() with only two things to note:I It is best to limit variable names in the model to three

charactersI qgraph supports Greek letters, by adding an asterisk a label is

printed in the symbol font


> library('sem')> R.thur <- read.moments(diag=FALSE, names=c('Sen','Voc',+ 'SC','FL','4LW','Suf','LS','Ped','LG'))1: .828

2: .776 .779

4: .439 .493 .46

7: .432 .464 .425 .674

11: .447 .489 .443 .59 .541

16: .447 .432 .401 .381 .402 .288

22: .541 .537 .534 .35 .367 .32 .555

29: .38 .358 .359 .424 .446 .325 .598 .452

37:

Read 36 items


> model.thur <- specify.model()

1: F1 -> Sen, *l11, NA

2: F1 -> Voc, *l21, NA

3: F1 -> SC, *l31, NA

4: F2 -> FL, *l41, NA

5: F2 -> 4LW, *l52, NA

6: F2 -> Suf, *l62, NA

7: F3 -> LS, *l73, NA

8: F3 -> Ped, *l83, NA

9: F3 -> LG, *l93, NA

10: F4 -> F1, *g1, NA

11: F4 -> F2, *g2, NA

12: F4 -> F3, *g3, NA

13: Sen <-> Sen, q*1, NA

14: Voc<-> Voc, q*2, NA

15: SC <-> SC, q*3, NA


16: FL <-> FL, q*4, NA

17: 4LW <-> 4LW, q*5, NA

18: Suf<-> Suf, q*6, NA

19: LS <-> LS, q*7, NA

20: Ped<-> Ped, q*8, NA

21: LG <-> LG, q*9, NA

22: F1 <-> F1, NA, 1

23: F2 <-> F2, NA, 1

24: F3 <-> F3, NA, 1

25: F4 <-> F4, NA, 1

26:

Read 25 records

> sem.thur <- sem(model.thur, R.thur, 213)


> qgraph(model.thur)

λ11

λ21

λ31

λ41

λ52

λ62

λ73λ83λ93

γ1γ2

γ3

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Sen

Voc

SC

F2

FL

4LW

Suf

F3

LSPed

LG

F4

Specified model


> qgraph(model.thur, manifest = rownames(R.thur),

+ layout = "tree")

λ11 λ21 λ31 λ41 λ52 λ62 λ73 λ83 λ93

γ1γ2

γ3

θ1 θ2 θ3 θ4 θ5 θ6 θ7 θ8 θ9

● ● ● ● ● ● ● ● ●

● ● ● ●

F1

Sen Voc SC

F2

FL 4LW Suf

F3

LS Ped LG

F4

Specified model


> qgraph(sem.thur, layout = "tree", curve = 0.4)

0.9 0.91 0.86 0.84 0.8 0.7 0.78 0.72 0.7

0.82

0.78

0.82

0.18 0.16 0.27 0.3 0.36 0.51 0.39 0.48 0.51

0.32 0.39 0.34 1

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● ● ● ●

F1

Sen Voc SC

F2

FL 4LW Suf

F3

LS Ped LG

F4

Standardized model


> qgraph(sem.thur, layout = "spring", residuals = FALSE)

0.90.91

0.86

0.84

0.8

0.7

0.78

0.72

0.7

0.82

0.78

0.82

F1Sen

Voc

SC

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Suf

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Ped

LG

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Standardized model

The big 5

> library(qgraph)

> data(big5)

> str(big5)

num [1:500, 1:240] 2 3 4 4 5 2 2 1 4 2 ...

- attr(*, "dimnames")=List of 2

..$ : NULL

..$ : chr [1:240] "N1" "E2" "O3" "A4" ...

> data(big5groups)

> str(big5groups)

List of 5

$ Neuroticism : num [1:48] 1 6 11 16 21 26 31 36 41 46 ...

$ Extraversion : num [1:48] 2 7 12 17 22 27 32 37 42 47 ...

$ Openness : num [1:48] 3 8 13 18 23 28 33 38 43 48 ...

$ Agreeableness : num [1:48] 4 9 14 19 24 29 34 39 44 49 ...

$ Conscientiousness: num [1:48] 5 10 15 20 25 30 35 40 45 50 ...

The big 5

> Q <- qgraph(cor(big5), minimum = 0.25, cut = 0.4,

+ vsize = 2, groups = big5groups, legend = T,

+ borders = F, vTrans = 200)

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> qgraph(cor(big5), Q, layout = "spring")

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> qgraph.efa(cor(big5), factors = 5, Q, layout = "circle",

+ vsize = c(1, 15), borders = F, asize = 0.07,

+ esize = 4, rotation = "promax", residSize = 0.1)

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Concluding comments

I qgraph is still work in progressI Plans for the future:

I More wrapper functions for different statistics (e.g. IRT)I More layout modesI Estimating and fitting causal models

I Some things I couldn’t describe...

Layout constraints Grayscale colors

> Q <- qgraph(cor(big5), minimum = 0.25, cut = 0.4,

+ vsize = 2, groups = big5groups, legend = T,

+ borders = F, vTrans = 200, gray = TRUE)

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americas

enduring

strength

httpfbmesashrd

httpfbmepfjwduip

tragedy

arizona

httpfbmefcgtvdg

price

gold

ounce

hour

finale

sarah

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Cutoff: 0.3Minimum: 0.1 Maximum: 1

Concluding comments

Thank you for your attention!

ReferencesButts, C. T. (2010). sna: Tools for social network analysis

[Computer software manual]. Available fromhttp://CRAN.R-project.org/package=sna (R packageversion 2.2-0)

Csardi, G., & Nepusz, T. (2006). The igraph software package forcomplex network research. InterJournal, Complex Systems,1695. Available from http://igraph.sf.net

Fox, J. (2010). sem: Structural equation models [Computersoftware manual]. Available fromhttp://CRAN.R-project.org/package=sem (R packageversion 0.9-21)

Fruchterman, T., & Reingold, E. (1991). Graph drawing byforce-directed placement. Software: Practice and Experience,21(11), 1129–1164.

Revelle, W. (2010). psych: Procedures for psychological,psychometric, and personality research [Computer softwaremanual]. Evanston, Illinois. Available from http://

personality-project.org/r/psych.manual.pdf (Rpackage version 1.0-93)

Layout modes

I The placement of the nodes is specified with the layout

argument in qgraph()

I This can be a n by 2 matrix indicating the x and y position ofeach node

I layout can also be given a character indicating one of thetwo default layouts

I If layout="circular" the nodes are placed in circles pergroup (if the groups list is specified)

I If layout="spring" a force-embedded algorithm(Fruchterman & Reingold, 1991) is used for the placement

I This is an iterative algorithm that clusters the nodes so thatthe length of the edges correspond to the absolute strength ofthe edges

Fruchterman-Reingold layout (20 iterations) Fruchterman-Reingold layout (500 iterations)

> qgraph(cor(Y), groups = gr, layout = "spring")

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

http://igraph.sf.net

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

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

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

Grid layout

I A final option is to specify a grid as layout

I This can be done by specifying a matrix to layout with morethan two columns

I This matrix contains zeros and a number for each node

> dat.3 <- matrix(c(1:15 * 2 - 1, 1:15 * 2),

+ , 2)

> dat.3 <- cbind(dat.3, round(seq(-0.7, 0.7,

+ length = 15), 1))

> L.3 <- matrix(1:30, nrow = 2)

> L.3

[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]

[1,] 1 3 5 7 9 11 13 15 17

[2,] 2 4 6 8 10 12 14 16 18

[,10] [,11] [,12] [,13] [,14] [,15]

[1,] 19 21 23 25 27 29

[2,] 20 22 24 26 28 30

Grid layout

> qgraph(dat.3, layout = L.3, directed = FALSE)

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Fruchterman-Reingold layout

I layout="spring" uses a force-embedded algorithm that wasproposed by Fruchterman and Reingold (1991)

I This layout was ported from the sna package (Butts, 2010)I A solution for weighted graphs was taken from igraph (Csardi

& Nepusz, 2006)

I This is an iterative algorithm.

I The initial layout is a circleI Then in each iteration:

I Each node is repulsed by all other nodesI Connected nodes are also attracted to each otherI The maximum displacement weakens each iteration

I After this process the layout is rescaled to fit the −1 to 1xy -plane

I The unscaled layout is returned as layout.orig

Fruchterman-Reingold layout

I The Fruchterman-Reingold algorithm can be controlled withthe layout.par argument

I This must be a list containing other arguments:

niter Number of iterations, default is 500max.delta Maximum displacement, default is n

area The area of the plot, default is n2

cool.exp Cooling exponent, default is 1.5repulse.rad Repulse radius, default is n · area

init Matrix indicating initial layout

Constraints in the Fruchterman-Reingold layout

I The Fruchterman-Reingold algorithm behaves like a chaoticsystem

I A small difference in initial setup can result in a completelydifferent layout

I In qgraph the layout can be constrained to compensate this

I Hard constraintsI Hard constraints can be used to fix the x and y position of

certain nodesI This can be done with the constraints argument in

layout.par

I Soft constraintsI Soft constraints can be used to limit the displacement of

certain nodesI This can be done with the max.delta and init arguments in

layout.par

No constraints

> dat.3 <- matrix(c(1, 2, 1, 3, 2, 3), 3, 2,

+ byrow = T)

> L <- matrix(c(1:3, 1, 3, 1), 3, 2)

> Q <- qgraph(dat.3, layout = L, vsize = 3,

+ esize = 1)

> par <- list(init = L)

> set.seed(1)

> for (i in 3:20) {

+ dat.3 <- rbind(dat.3, c(sample(c(i, sample(1:i,

+ 1)), 1), i + 1))

+ L <- rbind(L, 0)

+ par$init <- L

+ L <- qgraph(dat.3, Q, layout.par = par,

+ layout = "spring")$layout.orig

+ }

No constraints Hard constraints

> dat.3 <- matrix(c(1, 2, 1, 3, 2, 3), 3, 2,

+ byrow = T)

> L <- matrix(c(1:3, 1, 3, 1), 3, 2)

> par <- list(max.delta = 10, area = 10^2,

+ repulse.rad = 10^3)


+ esize = 1, layout.par = par)

> set.seed(1)

> for (i in 3:20) {


+ 1)), 1), i + 1))

+ par$init <- rbind(L, 0)

+ L <- rbind(L, NA)

+ par$constraints <- L



+ }

Hard constraints Soft constraints

> dat.3 <- matrix(c(1, 2, 1, 3, 2, 3), 3, 2,

+ byrow = T)

> L <- matrix(c(1:3, 1, 3, 1), 3, 2)

> par <- list(max.delta = 10, area = 10^2,

+ repulse.rad = 10^3)


+ esize = 1, layout.par = par)

> set.seed(1)

> for (i in 3:20) {


+ 1)), 1), i + 1))

+ par$init <- rbind(L, 0)

+ L <- rbind(L, NA)

+ par$max.delta <- 10/(i + 1):1



+ }

Soft constraints Interpreting weighted graphs

> qgraph(dat.3, layout = L.3, directed = FALSE,

+ edge.labels = T)

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Interpreting weighted graphs


+ edge.labels = T, esize = 14)

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> qgraph(dat.3[-c(1:3, 13:15), ], layout = L.3,

+ nNodes = 30, directed = FALSE, edge.labels = T,

+ esize = 14)

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+ edge.labels = T, esize = 14, maximum = 1)

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> qgraph(dat.3[-c(1:3, 13:15), ], layout = L.3,

+ nNodes = 30, directed = FALSE, edge.labels = T,

+ esize = 14, maximum = 1)

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maximum must be set to be able to compare multiple graphs!



+ edge.labels = T, esize = 14, minimum = 0.1)

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+ edge.labels = T, esize = 14, cut = 0.4)

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+ edge.labels = T, esize = 14, minimum = 0.1,

+ maximum = 1, cut = 0.4, details = TRUE)

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30Cutoff: 0.4Minimum: 0.1 Maximum: 1


Graphs can not be interpreted without knowing minimum, cutand maximum!