Estimation of (Logistic) Vector-autoregressionModels

Using generalized linear modelling

Sacha Epskamp

University of AmsterdamDepartment of Psychological Methods

IMPS 2013

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Friday 11.15 in Concertzaal: Network Psychometricssymposium

Goal

Suppose we measure a patient several times per day, can wethen estimate the structure of his or her disorder?

Vector autoregression (VAR)

Regression on the previous time point (lag 1):

y t |y t−1 ∼ N (µ,θ)

µ = δ + By t−1

For a single variable:

yt ,i = δi +n∑

j=1

βijyt−1,j + εt ,i

Network representationβ11

β12

β13β21

β22β23

β31

β32

β33

1

23

Estimation

Many methods of estimating VAR models exist (including morelags and more advanced models such as VARIMA). Thesimplest way to do this is by using linear regression using thegeneral linear model or the generalized linear model withidentity link function.

Due to independence incoming edges can be estimatedseparately using univariate analyses.

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Variable:

V1

V2

V3

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Sum

scor

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β21

β22β23

β31

β32

β33

β11

β12

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Res <- glm(Y[-1, 1] ~ Y[-nrow(Y), ])coef(Res)

## (Intercept) Y[-nrow(Y), ]1 Y[-nrow(Y), ]2## 0.05264 0.40306 0.86754## Y[-nrow(Y), ]3## 0.08734

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Original network

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Estimated network

The logistic VAR model

I What if nodes are not continuous, but binary?I “on” or “off”

I What if some symptoms require more “input”I e.g., “suicidal thoughts”

I A logistic model can be helpful here, and has very niceproperties

I Very similar to Ising modelI Reduces to Rasch and Birnbaum models under special

conditionsI Allows for “phase transitions’

The logistic VAR model

yt ,i |yt−1 ∼ Bernoulli(pt ,i)

(1)

πt ,i =eδi+

∑nj=1 βij yt−1,j

1 + eδi+∑n

j=1 βij yt−1,j(2)

Network representationβ11

β12

β13β21

β22β23

β31

β32

β33

1

23

Parameter interpretation

yt ,i |yt−1 ∼ Bernoulli(pt ,i)

(3)

πt ,i =eδi+

∑nj=1 βij yt−1,j

1 + eδi+∑n

j=1 βij yt−1,j(4)

I βij can be seen as the effect of node j being on at timet − 1 on the probability that node i is on at time t

I −δi can be seen as a threshold, the amount of activationneeded to get a 0.5 probability of node i switching on attime t

The logistic VAR model

If all incoming edges are equal per node:

βi1 = βi2 = . . . = βin = βi

Then the model reduces to 2PL form, with the sumscore onprevious time point as latent trait:

πt ,i =eδi+βj yt−1,+

1 + eδi+βi yt−1,+

Correspondence to Birnbaum model

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Correspondence to Rasch model

If all edges are equal:

β11 = . . . = βn1 = β12 = . . . = βnn = β

Then the model reduces to Rasch form, with the sumscore onprevious time point as latent trait:

πt ,i =eδi+βyt−1,+

1 + eδi+βyt−1,+

Correspondence to Rasch model

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Phase transitions

Estimation

L(y t |δ B,y t−1) =n∏

i=1

πyt,it ,i

(1− πt ,i

)1−yt,i

πt ,i =eδi+

∑nj=1 βij yt−1,j

1 + eδi+∑n

j=1 βij yt−1,j

Estimation

The logistic VAR model is equal to a logistic regression on theprevious time point, and so generalized linear modelling can beused using a the logit link function.

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scor

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Res <- glm(Y[-1, 1] ~ Y[-nrow(Y), ], family = binomial)coef(Res)

## (Intercept) Y[-nrow(Y), ]1 Y[-nrow(Y), ]2## -1.84437 1.16282 1.23745## Y[-nrow(Y), ]3 Y[-nrow(Y), ]4## -0.22106 -0.05827

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Concluding comments

I Generalized linear modelling (GLM) can be used to easilyfit VAR and logistic VAR models in a single subject

I Can be done in many software packagesI Does however require quite some measures for reliable

estimatesI For multiple subjects, any software package that supports

multi-level generalized linear models can be used.I e.g., see Bringmann et al. (2013)

I A function to estimate VAR networks using GLM can befound on my website:

www.sachaepskamp.com

Thank you for your attention!

References

Bringmann, L. F., Vissers, N., Wichers, M., Geschwind, N.,Kuppens, P., Peeters, F., . . . Tuerlinckx, F. (2013). Anetwork approach to psychopathology: New insights intoclinical longitudinal data. PloS one, 8(4), e60188.