June 27, 2016
The GGM model:
Is equivalent to three causal structures:
Thus, the GGM highlights potential causal pathways
\[ y_1 = \tau_1 + \gamma_{12} y_2 + \gamma_{13} y_3 + \gamma_{14} y_4 + \varepsilon_1 \]
\[ y_2 = \tau_2 + \gamma_{21} y_1 + \gamma_{23} y_3 + \gamma_{24} y_4 + \varepsilon_2 \]
\[ y_3 = \tau_3 + \gamma_{31} y_1 + \gamma_{32} y_2 + \gamma_{34} y_4 + \varepsilon_3 \]
\[ y_4 = \tau_4 + \gamma_{41} y_1 + \gamma_{42} y_2 + \gamma_{43} y_3 + \varepsilon_4 \]
\[ \rho_{ij} = \frac{\gamma_{ij} \mathrm{Var}\left(\varepsilon_j\right)}{\mathrm{Var}\left(\varepsilon_i\right)} = \frac{\gamma_{ji} \mathrm{Var}\left(\varepsilon_i\right)}{\mathrm{Var}\left(\varepsilon_j\right)} \]
I will analyze the BFI dataset from the pych package:
library("psych") ?bfi
\[ \pmb{Y}_t \mid \pmb{y}_{t-1} \sim N\left( \pmb{\mu} + \pmb{B} \left(\pmb{y}_{t-1} - \pmb{\mu}\right), \pmb{\Theta} \right) \]
graphicalVAR
sparseTSCGM
Data collected by Date C. Van der Veen, in collaboration with Harriette Riese en Renske Kroeze.
Feeling worthless interacts with feeling helpless
Feeling stressed interacts with feeling the need to do things
Central node: Feeling sad
Cycle of enjoyment, feeling sad, feeling worthless and being active
Having to had to do things leads to letting important things pass
Fear of panic attack is not connected
Adding superscript \(p\) for subject. Level 1 model: \[ \pmb{Y}^{(p)}_t \mid \pmb{y}^{(p)}_t = N\left(\pmb{\mu}^{(p)} + \pmb{B}^{(p)} \pmb{y}_{t-1}^{(p)}, \pmb{\Theta}^{(p)} \right) \]
Level 2 model: \[ \begin{bmatrix} \pmb{\mu}^{(p)} \\ \mathrm{Vec}\left(\pmb{B}^{(p)}\right) \end{bmatrix} \sim N\left( \pmb{f}, \pmb{\Omega} \right). \] \(\pmb{f}\) encodes fixed effects and \(\pmb{\Omega}\) the distribution of random effects.
The random effects variance-covariance matrix can be divided in four blocks:
\[ \begin{bmatrix} \pmb{R}_{\pmb{\mu}} \\ \pmb{R}_{\pmb{B}} \end{bmatrix} \sim N\left(\pmb{0}, \begin{bmatrix} \pmb{\Omega}_{\pmb{\mu}} & \pmb{\Omega}_{\pmb{\mu}\pmb{B}} \\ \pmb{\Omega}_{\pmb{B}\pmb{\mu}} & \pmb{\Omega}_{\pmb{B}} \end{bmatrix} \right). \]
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:
lme4
packages implements univariate multi-level regression
lmer
functionclusterGeneration
R package with "onion" method1: 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