• Psychological constructs can be conceptualized as dynamical systems, featuring complex emergent behavior:
  • Correlated responses
  • Stable "traits"
  • Phase transitions
  • Individual Differences
• These systems can be portrayed as networks
• As network structures are unknown in psychology, they need to be estimated
  • Network Psychometrics

• Up to three network structures can be obtained in time-series data:
  • Contemporaneous networks
  • Temporal networks
  • Between-subjects networks
• All three structures can be interpreted in several ways:
  • Highlighting potential causal pathways
  • Showing predictive effects and mediation
  • Under very strict assumptions: a causal model

• When cases are independent
  • The Gaussian Graphical model
  • Interpreting network structures
• When cases are not independent: \(N = 1\)
  • The VAR model
  • Temporal and contemporaneous networks and causation
• When cases are not independent: \(N > 1\)
  • The multi-level VAR model
  • Between-subjects networks and causation
• Conclusion

• Rows are termed the cases

• Every person measured only once
• Cases can reasonably be assumed to be independent
  • Given IQ has a mean of 100 and SD of 15, does knowing that Peter has an IQ of 90 help us predict better that Sarah had an IQ of 110?
• Because of this assumption, likelihood reduces to a product
  • \(\pmb{Y} \sim N\left(\pmb{\mu}, \pmb{\Sigma}\right)\)
  • \(f\left( \pmb{y} \mid \pmb{\mu}, \pmb{\Sigma} \right) = \prod_{p=1}^N f\left( \pmb{y}^{(p)} \mid \pmb{\mu}, \pmb{\Sigma} \right)\)

• \(\pmb{\Sigma}\), the variance-covariance matrix, encodes all information how variables relate to one-another
• Because of the Schur complement, it also encodes all conditional relationships
• We will focus on its inverse, \(\pmb{K}\):
  • \(\pmb{K} = \pmb{\Sigma}^{-1}\)
• The inverse variance-covariance matrix is called a Gaussian graphical model (GGM)
  • Encodes an undirected network

• GGM is a 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

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

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