June 27, 2016

The Dynamics of Psychology

  • 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

Goal

  • 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

Outline

  • 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

Experience Sampling Method

  • Rows are termed the cases

The Gaussian Graphical Model

Cross-sectional Data

  • 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
    • \(\pmb{Y} \sim N\left(\pmb{\mu}, \pmb{\Sigma}\right)\)
    • \(f\left( \pmb{y} \mid \pmb{\mu}, \pmb{\Sigma} \right) = \prod_{p=1}^N f\left( \pmb{y}^{(p)} \mid \pmb{\mu}, \pmb{\Sigma} \right)\)

The Gaussian Graphical Model

  • \(\pmb{\Sigma}\), 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

GGM and Multiple Regressions

GGM and Multiple Regressions

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

GGM and Multiple Regressions

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

GGM and Multiple Regressions