June 27, 2016


  • 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

Vector Auto-regression

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

  • \(\pmb{B}\) 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

Contemporaneous Causation

  • 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

Empirical Example

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

Multi-level VAR

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

Empirical Example

  • 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!