June 27, 2016

- 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:

- Concentration \(\rightarrow\) Fatigue \(\rightarrow\) Insomnia
- Concentration \(\leftarrow\) Fatigue \(\rightarrow\) Insomnia
- Concentration \(\leftarrow\) Fatigue \(\leftarrow\) Insomnia

Thus, the GGM highlights potential causal pathways

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

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