A � � � � � � �D

A �������� �� R ������� I����F��

Adapted from:

https://cran.r-project.org/web/packages/IsingFit/IsingFit.pdf

���

APPENDIX D. A TUTORIAL ON R PACKAGE ISINGFIT

This tutorial provides and explanation of R package IsingFit (Van Borkulo, Epskamp, & Ro-bitzsch, ���6). IsingFit �ts Ising models (Ising, ����) using the eLasso method. eLassocombines `

1

-regularized logistic regression with model selection based on the ExtendedBayesian Information Criterion (EBIC). EBIC is a �t measure that identi�es relevant relationshipsbetween binary variables. The resulting network consists of variables as nodes and relevant re-lationships as edges. In this tutorial, we will provide a short introduction to eLasso — pleasesee Chapter � and Appendix A for more detailed information — and illustrate how IsingFit()

can be used, what options there are and how applying different options affect results. To il-lustrate this, we provide examples with real data and provide the accompanying R code. Thevarious arguments of IsingFit() are explained and the impact of adjusting the arguments isillustratedwith data of the Virginia Adult Twin Study of Psychiatric and Substance Use Disorders(VATSPUD; Kendler & Prescott, ���6; Prescott et al., ����). For this tutorial, we use the data ofpresence/absence of the �� disaggregated symptoms of MDD for at least � days during the pre-vious year of 8��� individuals from the population as used in Chapter � of this dissertation. ThisAppendix is an extended version of https://cran.r-project.org/web/packages/IsingFit/IsingFit.pdf.

D.� Introduction

The edges in a network represent conditional dependencies: if there is an edge between symp-tom X

1

and X2

, they are related even after conditioning on (or controlling for) all other variablesin the network. This means that the relationship between X

1

and X2

cannot be explained by anyof the other variables. It is not a spurious relationship that arose because the two are relatedthrough another (third) confounding variable in the dataset. Conversely, if two variables are notconnected, say X

2

and X3

, this means that they are not related. Not directly related, to be morespeci�c. They might be correlated, but only because they share connections to other variablesin the network. When conditioned on all other variables, the relationship between X

2

and X3

disappears — it is explained away by the other variables. The network representing conditionaldependencies is also called aMarkov Random Field (Kindermann & Snell, ��8�; Lauritzen, ���6).

For continuous data, conditional dependencies can be established by means of the covari-ance matrix of the observations of the variables. A zero entry in the inverse of the covariancematrix represents a conditional independence between the focal variables, given the other vari-ables (Speed & Kiiveri, ��86). The simplest model that can explain the data as adequately aspossible according to the principle of parsimony, can be found with different strategies to �nda sparse approximation of the inverse covariance matrix. One of the strategies to �nd such asparse approximation is to impose an `

1

-penalty (also called lasso) on the estimation of theinverse covariance matrix (Foygel & Drton, ����; Friedman et al., ���8; Ravikumar et al., ����).The `

1

-penalty guarantees shrinkage of partial correlations and puts others exactly to zero (Tib-

���

D.�. ARGUMENTS

shirani, ���6). Another strategy entails estimating the neighborhood of each variable individu-ally with `

1

-penalized regression (Meinshausen & Bühlmann, ���6), instead of imposing an `1

-penalty on the inverse covariance matrix and is, therefore, an approximation to the `

1

-penalizedinverse covariancematrix. This approximationmethod using `

1

-penalized regression is an inter-esting alternative — it is computationally ef�cient and asymptotically consistent (Meinshausen& Bühlmann, ���6) — that can also be applied to binary data and is implemented in IsingFit.

D.� Arguments

Themain function of package IsingFit is function IsingFit(). To run IsingFit(), the only inputthat is required is the data set. This results in output according to default settings of severalarguments:

IsingFit(x, AND = TRUE, gamma = 0.25, plot = TRUE, progressbar = TRUE,

lowerbound.lambda = NA, ...)

x

The �rst argument (x) — and the only one you have to enter to run the default function — is thedata. Thismust be amatrix in which each variable is represented by a column and each observa-tion by a row. Observations should be independent. An example of a data set with independentobservations is one in which rows contain scores on items of a questionnaire of different indi-viduals. Whether person i (on row i ) has a set of scores on items does not depend on the scoresof person j (on row j ).

The following code is used to estimate a default network with our example data, in whichData is the name of the VATSPUD dataset and the output of IsingFit is stored under the nameRes.

Res <- IsingFit(Data)

The estimated network structure is shown in Figure D.�, which is simular to the empiricalnetwork in Figure �.�, Chapter �.

AND

This is a logical indicating whether the AND-rule is used or not. When AND = TRUE, the AND-rule is applied, requiring both regression coef�cients Ø j k and Øk j to be nonzero to result in anedge between node j and k (see Chapter � and Appendix A for an extensive explanation ofthe estimation procedure). A more lenient option is to set the AND argument to AND = FALSE,thereby applying the OR-rule, requiring only one of Ø j k and Øk j to be nonzero. For our dataset

���

APPENDIX D. A TUTORIAL ON R PACKAGE ISINGFIT

depint

los

gai

dap

iap

iso

hso

agi

ret

fat

wor

con

dea

Figure D.�: The estimated network structure based on the VATSPUD data, with the default set-ting of IsingFit(). dep - depressed mood; int - loss of interest; los - weight loss; gai - weightgain; dap - decreased appetite; iap - increased appetite; iso - insomnia; hso - hypersomnia; ret -psychomotor retardation; agi - psychomotor agitation; fat - fatigue; wor - feelings of worthless-ness; con - concentration problems; dea - thoughts of death.

there is no observable difference in the resulting network structure when applying the AND- orOR-rule. This is because of the high power due to high sample size (n = 8973). When the poweris not high enough to estimate a reasonable network structure — due to an unfavorable ratio ofnumber of nodes and sample size — one might use the OR-rule. Note that this might result inmore spurious connections (false positives).

In Figure D.�, we show the result of applying the AND and OR-rule when power is low. As canbe seen in the code below, we used a subsample of our original VATSPUD dataset in which onlythe �rst ��� individuals are included.

Res.150.AND <- IsingFit(Data[1:150,], AND=TRUE)

���

D.�. ARGUMENTS

Res.150.OR <- IsingFit(Data[1:150,], AND=FALSE)

As you can see in Figure D.�, there are a fewmore edges retrievedwhen the OR rule is applied(right panel), compared to when the AND rule is applied (left panel).

depint

los

gai

dap

iap

isohso

agi

ret

fat

wor

con

dea

AND−rule

depint

los

gai

dap

iap

isohso

agi

ret

fat

wor

con

dea

OR−rule

Figure D.�: Network estimation with AND- (left panel) and OR-rule (right panel). Only the �rst ���observations are included in the network estimation procedure.

gamma

With this argument, you can adjust the value of hyperparameter ∞. Hyperparameter ∞ plays arole in Goodness-of-Fit measure EBIC (equation A.��) with which the optimal tuning parameter(i.e., Ω which represents the best set of neighbors of the focal node) is selected (again, see Chap-ter � and Appendix A for an extensive explanation of the estimation procedure of IsingFit). Inessence, ∞ determines the strength of prior information on the size of the model space. It penal-izes the number of nodes in neighborhood selection and puts an extra penalty on the numberof neighbors (see equation A.��). Note that — and this is often misunderstood — when ∞ is setto zero, there will still be regularization and the number of neighbors is still penalized. However,instead of an extra penalty on the size of the model space, the optimal tuning parameter is nowselected with the ordinary BIC instead of the extended BIC.

By increasing ∞, the strength of the extra penalty on the size of the model space increases.A lower value of ∞ — maybe even zero — could be advantageous when the power is low andapplying EBIC results in very few edges or none at all. A higher value of ∞ could be favorablewhen you have high power (a good number of nodes to observations ratio) and want the leastnumber of false positives.

���

APPENDIX D. A TUTORIAL ON R PACKAGE ISINGFIT

To illustrate the effect of different values of ∞, we used the subsample of ��� individualsand estimated network structures with ∞ = 0, .3, .6,1. As you can see in Figure D.�, the networkbased on ∞= 0 (most left panel) is least sparse (most connections) and ∞= 1 (most right panel)is most sparse. The connections in the latter network have the highest probability of being truepositives: they are still present, while being estimated with a high value of ∞ (i.e., a high extrapenalty on the number of neighbors).

12

3

4

5

6

78

9

10

11

12

13

14

gamma = 0

12

3

4

5

6

78

9

10

11

12

13

14

gamma = 0.3

12

3

4

5

6

78

9

10

11

12

13

14

gamma = 0.6

12

3

4

5

6

78

9

10

11

12

13

14

gamma = 1

Figure D.�: Networks estimation based on the �rst ��� observations of the VATSPUD data withincreasing values of ∞ (from left to right).

plot

A logical indicating whether the estimated network should be plotted.

progressbar

Logical. Should the progressbar be plotted in order to see the progress of the estimation proce-dure? Especially with a large data set, it may take a while before all node-wise regressions areperformed. With the progressbar, you can keep track of the progress of the estimation proce-dure. Defaults to TRUE.

lowerbound.lambda

The minimum value of tuning parameter lambda (regularization parameter). Can be used tocompare networks that are based on different sample sizes. The lowerbound.lambda is basedon the number of observations n in the smallest group:

qlog p

n , where p is the number of vari-ables, that should be the same in both groups. When both networks are estimated with thesame lowerbound for lambda (based on the smallest group), the two networks can be visuallycompared. For a statistical comparison, NCT can be used (see Chapter � and Appendix E for atutorial on the NCT package).

��6

D.�. OUTPUT

T���� D.�. Thresholds of the symptoms based on the VATSPUD data with default set-ting of IsingFit. See Figure D.� for abbreviations.

symptom thresholddep -�.��int -�.��los -�.��gai -�.8�dap -�.��iap -�.�iso -�.��hso -�.��agi -�.�8ret -�.��fat -�.8�wor -�.��con -�.��dea -�.8�

D.� Output

Function IsingFit returns an IsingFit object and contains several items. Below follows a de-scription of each item.

weiadj

The weighted adjacency matrix. This contains the edge weights. You can use this item to plotthe network with qgraph and adjust the plot as you like. Below you can �nd the code to use theoutput of IsingFit to make your own plot.

qgraph(Res$weiadj, layout=’circular’)

thresholds

A vector that contains the thresholds of the variables. The threshold of a variable representsthe autonomous disposition to be present. For the network in Figure D.� based on the wholesample, the thresholds are all negative (see Table D.�). This means that all symptoms have anautonomeous disposition to be absent. When zooming in on speci�c symptoms, “depressedmood” (dep) has the highest threshold (-�.��). This means that this symptom has the highestprobability of being present in the sample compared to the other symptoms. “Thoughts of death”(dea) has the lowest threshold and, consequently, has the lowest probability of being present inthe sample.

���

APPENDIX D. A TUTORIAL ON R PACKAGE ISINGFIT

q

The object that is returned by qgraph (class qgraph).

gamma

The value of ∞ that was applied.

AND

A logical indicating whether the AND-rule is applied. If FALSE, the OR-rule was applied.

time

The time it took to estimate the network.

asymm.weights

The asymmetrical weigthed adjacency matrix before applying the AND/OR-rule. These are theoriginal regression coef�cients from the nodewise regressions (see Section �.�.�).

lambda.values

The values of the tuning parameter per node that ensured the best �tting set of neighbors.

��8

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