estimation of (logistic) vector-autoregression models · 2013. 7. 25. · fit var and logistic var...
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Estimation of (Logistic) Vector-autoregressionModels
Using generalized linear modelling
Sacha Epskamp
University of AmsterdamDepartment of Psychological Methods
IMPS 2013
MDInsomnia
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MDInsomnia
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MDInsomnia
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MDInsomnia
Fatigue
Concentration
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MDInsomnia
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MDInsomnia
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MDInsomnia
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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
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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
β13
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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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Variable:
V1
V2
V3
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0 100 200 300Time
Sum
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.