• Out-of-the-box methodology applicable to experience sampling method (ESM) data
  • Multiple people measured multiple times in relatively short time span (several weeks)
• Up to three network structures can be obtained in ESM data:
  • Contemporaneous networks
  • Temporal networks
  • Between-subjects networks

• The Gausian graphical models the inverse variance-covariance matrix
  • \(\pmb{K} = \pmb{\Sigma}^{-1}\)
• Network of partial correlation coefficients:
  • \(\mathrm{Cor}\left(Y_i,Y_j \mid \pmb{Y}^{-(i,j)}\right) = - \frac{\kappa_{ij}}{\sqrt{\kappa_{ii}} \sqrt{\kappa_{jj}}}\)

The GGM model:

• Concentration \(-\) Fatigue \(-\) Insomnia

Is equivalent to three causal structures:

1. Concentration \(\rightarrow\) Fatigue \(\rightarrow\) Insomnia
2. Concentration \(\leftarrow\) Fatigue \(\rightarrow\) Insomnia
3. Concentration \(\leftarrow\) Fatigue \(\leftarrow\) Insomnia

Thus, the GGM highlights potential causal pathways. In addition, the partial correlations are proportional to multiple regression coefficients.

• We will use the lag-1 factorization

\[ \pmb{Y}_t \mid \pmb{y}_{t-1} \sim N\left( \pmb{\mu} + \pmb{B} \left(\pmb{y}_{t-1} - \pmb{\mu}\right), \pmb{\Theta} \right) \]

• \(\pmb{B}\) encodes the temporal network
  • Granger causality
• \(\pmb{\Theta}^{-1}\) encodes the contemporaneous network
  • GGM
• The sample means can be used as plugin to center the predictors

• Many causal effects likely faster than the time-window of measurement
  • Somatic arousal \(\rightarrow\) anticipation of panic attack \(\rightarrow\) anxiety
• These can be caught in a contemporaneous network of partial correlations
• Thus, the contemporaneous network can also be seen to highlight potential causal relationships
• As the contemporaneous network is the GGM, the temporal network can be seen as a correction for dependent measurements in estimating the GGM

Data collected by Date C. Van der Veen, in collaboration with Harriette Riese en Renske Kroeze.

• Patient suffering from panic disorder and depressive symptoms
  • Perfectionist
• Measured over a period of two weeks
• Five times per day
• Items were chosen after intake together with therapist

^{1: anxious, 2: stressed, 3: angry, 4: sad, 5: guilty, 6: weak, 7: worthless, 8: helpless, 9: full of energy, 10: fear panic attack, 11: fear to cry, 12: fear to appear angry, 13: 'had to do things', 14: bodily discomfort, 15: enjoying, 16: let something pass, 17: social env pleasurable, 18: physically active}

Adding superscript \(p\) for subject. Level 1 model: \[ \pmb{Y}^{(p)}_t \mid \pmb{y}^{(p)}_t = N\left(\pmb{\mu}^{(p)} + \pmb{B}^{(p)} \left(\pmb{y}_{t-1}^{(p)} - \pmb{\mu}^{(p)} \right), \pmb{\Theta}^{(p)} \right) \]

Level 2 model: \[ \begin{bmatrix} \pmb{R}_{\pmb{\mu}} \\ \pmb{R}_{\pmb{B}} \end{bmatrix} \sim N\left(\pmb{0}, \begin{bmatrix} \pmb{\Omega}_{\pmb{\mu}} & \pmb{\Omega}_{\pmb{\mu}\pmb{B}} \\ \pmb{\Omega}_{\pmb{B}\pmb{\mu}} & \pmb{\Omega}_{\pmb{B}} \end{bmatrix} \right). \]

• Block \(\pmb{\Omega}_{\pmb{\mu}}\) encodes the between-subject relationships between means
• These can be used to estimate a GGM
  • Between-subjects network of partial correlations

^{Example based on Hamaker, E. L. (2012). Why Researchers Should Think 'Within-Person': A Paradigmatic Rationale. Handbook of Research Methods for Studying Daily Life. The Guilford Press New York, NY, 43–61.}

Hypothetical example of networks based on two persons:

• Clinically depressed person constantly scoring high on both
• Healthy person constantly scoring low on both

• Two datasets
  • Original: 26 subjects, 51 measurements on average, 1323 total observations
  • Replication: 65 subjects, 35.5 measurements on average, 2309 total observations
• 16 indicators of neuroticism, extroversion, conscientiousness
• Orthogonal estimation of temporal and contemporaneous effects
• Only significant effects shown
  • Alpha = 0.05 and using the "or" rule
• Very preliminary results
  • I ran the analysis two days ago!

1: Worried, 2: Organized, 3: Ambitious, 4: Depressed, 5: Outgoing, 6: Selfconscious, 7: Selfdisciplined, 8: Energetic, 9: Frustrated, 10: Focused, 11: Guilty, 12: Adventurous, 13: Happy, 14: Control, 15: Achieved, 16: Angry

• Network structures are useful in discovering potential causal relationships
• Cross-sectional data:
  • Gaussian graphical model (GGM)
• ESM data:
  • Contemporaneous network (GGM)
  • Temporal network (VAR)
  • Between-subjects network (GGM)

• A lot of potential problems with multi-level estimation
  • Multivariate estimation
  • Modeling random contemporaneous effects
  • Parameter variance-covariances
  • Model selection
• Possibly move away from multi-level
  • LASSO variants?
• Lag-interval

• Network structures are only hypothesis generating
  • Highlighting potential causal pathways
• Observational data can never confirm causality
  • Mixture of experimental and observational data needed
• We need to completely rethink the modeling framework to do so

• Estimation methods
  • Sequential estimation using lme4
  • Between-subject effects as level 2 predictors
  • Contemporaneous effects estimated post-hoc
• Simulation studies
  • 8 nodes
  • 20 nodes

\[ y_1 = \tau_1 + \gamma_{12} y_2 + \gamma_{13} y_3 + \gamma_{14} y_4 + \varepsilon_1 \]

\[ y_2 = \tau_2 + \gamma_{21} y_1 + \gamma_{23} y_3 + \gamma_{24} y_4 + \varepsilon_2 \]

\[ y_3 = \tau_3 + \gamma_{31} y_1 + \gamma_{32} y_2 + \gamma_{34} y_4 + \varepsilon_3 \]

\[ y_4 = \tau_4 + \gamma_{41} y_1 + \gamma_{42} y_2 + \gamma_{43} y_3 + \varepsilon_4 \]

\[ \rho_{ij} = \frac{\gamma_{ij} \mathrm{Var}\left(\varepsilon_j\right)}{\mathrm{Var}\left(\varepsilon_i\right)} = \frac{\gamma_{ji} \mathrm{Var}\left(\varepsilon_i\right)}{\mathrm{Var}\left(\varepsilon_j\right)} \]

• Between subject effects can be obtained by centering predictors and adding the person-means as level 2 predictors
  • ^{Hamaker, E. L., & Grasman, R. P. (2015). To center or not to center? Investigating inertia with a multilevel autoregressive model. Frontiers in psychology, 5, 1492.}
• This can be seen as node-wise estimation of a GGM
• Thus, an estimate for the between-subjects GGM can be obtained by averaging the level-2 predictive effects standardized with the residual variances

• Contemporaneous networks need to be estimated post-hoc by investigating the residuals
• Either inverting the sample variance-covariance matrix of residuals:
  • Fixed
  • Unique
• Or as a second multi-level model using nodewise estimation of a GGM:
  • Correlated
  • Orthogonal

• 8 and 20 nodes
• Random between-subjects covariance matrix for means and temporal effects
  • clusterGeneration R package with "onion" method
  • No correlations between means and temporal effects
• Temporal effects scaled to enforce stationarity
• Random fixed contemporaneous covariance matrix
• Contemporaneous person-specific covariances drawn from Wishart distribution with 2P DF
• Performance checked with temporal effects and partial correlations
• Each condition (# persons, # time, temporal estimation method and contemporaneous estimation method) replicated 100 times