• ^{Hamaker, E. L., & Grasman, R. P. P. P. (2014). To center or not to center? Investigating inertia with a multilevel autoregressive model. Frontiers in Psychology, 5, 1492. http://doi.org/10.3389/fpsyg.2014.01492}

• 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

• Introduction Vector Auto-regression
• Problem 1: \(n=1\) and limited observations
  • Graphical VAR
• Problem 2: \(n>1\)
  • Multi-level VAR
• Problem 3: Measurement error
  • State-space models
• Even more problems

• Regress a vector of variables from a single subject on the previous time point
• Assume multivariate normality
• Assume errors are correlated
• This leads to three things to estimate:
  • A vector of intercepts
  • A matrix encoding a temporal network
  • A matrix encoding a contemporaneous network
• Can be estimated using lm() or any least squares regression method

For a single subject: \[ \begin{aligned} \pmb{y}_t &= \pmb{\tau} + \pmb{B} \pmb{y}_{t-1} + \pmb{\varepsilon}_t \\ \pmb{\varepsilon}_t &\sim N\left( \pmb{0}, \pmb{\Theta} \right). \end{aligned} \] \(\pmb{B}\) encodes a directed temporal network and \(\pmb{\Theta}\) and undirected contemporaneous network

With \(\pmb{K} = \pmb{\Sigma}^{-1}\) encoding partial correlations coefficients

• Measure a patient over a short time, estimate network structures and use these in clinical practice
• Naturally \(n=1\) problem
  • Different estimation periods
  • Different nodes
• Naturally a limited data problem
  • You can't measure a patient 10 times per day
  • You can't measure a patient for months

• Only model temporal effects between consecutive measurements
  • Lag-1
• Assume both the temporal and contemporaneous effects are sparse
  • Only a relatively little number of edges in both networks
• To do this, we use the graphical VAR model (Wild et al. 2010)
  • Estimation via LASSO regularization, using extended BIC to select optimal tuning parameter (Rothman, Levina, and Zhu 2010; Abegaz and Wit 2013).
• We implemented these methods in the R package graphicalVAR (cran.r-project.org/package=graphicalVAR)

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

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

• Graphical VAR can be used to estimate network structures on limited data
• Simulation study is still work in progress
  • Simulated networks might not resemble clinical patient networks
• Around 50 observations work well for 10 node networks

• Each subject is assumed to have their own temporal VAR model
  • Contemporaneous model equal across people
• VAR parameters come from distribution
  • Fixed effect
  • Random effect

Adding superscript \(p\) for subject. Level 1 model: \[ \begin{aligned} \pmb{y}^{(p)}_t &= \pmb{\tau}^{(p)} + \pmb{B}^{(p)} \pmb{y}_{t-1} + \pmb{\varepsilon}^{(p)}_t \\ \pmb{\varepsilon}^{(p)}_t &\sim N\left( \pmb{0}, \pmb{\Theta} \right). \end{aligned} \]

Level 2 model: \[ \begin{bmatrix} \pmb{\tau}^{(p)} \\ \mathrm{Vec}\left(\pmb{B}^{(p)}\right) \end{bmatrix} \sim N\left( \pmb{\gamma}, \pmb{\Omega} \right). \] \(\pmb{\gamma}\) encodes fixed effects and \(\pmb{\Omega}\) the distribution of random effects.

• Multi-variate multi-level regression estimation is complicated and not yet well implemented in open source software
• lme4 packages implements univariate multi-level regression
  • ^{Douglas Bates, Martin Maechler, Ben Bolker, Steve Walker (2015). Fitting Linear Mixed-Effects Models Using lme4. Journal of Statistical Software, 67(1), 1-48. doi:10.18637/jss.v067.i01}
  • lmer function
• A multi-level VAR model can be estimated by sequentially estimating univariate models
  • Estimate all incoming edges per node
  • Used by Bringmann et al. (2013)

• Sequential estimation:
  • Needs to integrate out a high-dimensional distribution over parameters
  • Only feasible for up to ~6 nodes
  • Does not estimate all parameter covariances
    • Not all parameters together in the same model
• Orthogonal estimation
  • Alternatively, parameter covariances can be fixed to zero
  • Fast, and works for high dimensions (e.g., 20 nodes)
  • But, does not return any parameter correlation

• Multi-level models can naturally be formulated as a Bayesian model
• Estimable in open-source software such as OpenBUGS, Jags and Stan
• Requires prior distributions to be specified
  • Normal priors for fixed effects
  • Inverse-Wishart priors for residual covariance structures and parameter covariances
• Flat priors are preferred to model prior ignorance

• Schuurman, Grasman and Hamaker (in press) show that an Inverse-Wishart prior can be highly informative and thus problematic
• They propose a two-step procedure of specifying the scale parameter
  • Fit model using lmer
  • Obtain prior guess of the parameter standard deviations
  • Put these in a diagonal matrix to use as scale matrix
• The degrees of freedom can be set to the number of rows or columns in the covariance matrix

^{Schuurman, N. K., Grasman, R. P. P. P., & Hamaker, E.l. (in press). A Comparison of InverseWishart Prior Specifications for Covariance Matrices in Multilevel Autoregressive Models. Multivariate Behavioral Research.}

• The number of correlations to estimate grows fast with the number of nodes in the network
  • For \(P\) nodes, the number of random parameters equals \(P^4 + 2P^3 + P^2\)
• While in theory estimable, practically high-dimensional models will take a long time
• Again, models can be estimated sequentially or orthogonal

• So far, we have assumed no measurement error on the VAR process
• However, in psychology measurement error is a dominant problem
• Is within-person variability due to genuine within-person dynamics or simply due to white noise?

^{Schuurman, N. K., Houtveen, J. H., & Hamaker, E. L. (2015). Incorporating measurement error in n = 1 psychological autoregressive modeling. Frontiers in Psychology, 6, 1038. <.sup>}

• Schuurman, Houtveen and Hamaker (2015) suggest to to use Bayesian estimation of a white noise or state space model
• In a state-space model, we assume a latent underlying VAR process measured through some measurement model plus (correlated) measurement error
• Each observed can be represented by a latent or observed variables can be seen as indicators of a few latents (e.g., personality traits)
• Added complexity: we need to explicitly model days

Adding superscript \(d\) for days. Level 1 model for the observed variables: \[ \begin{aligned} \pmb{y}^{(p,d)}_{t} &= \pmb{\tau} + \pmb{\Lambda} \pmb{\eta}_t^{(p,d)} + \pmb{\varepsilon}_{t}^{(p,d)} \\ \pmb{\varepsilon}_{t}^{(p,d)} &\sim N(\pmb{0}, \pmb{\Theta}) \end{aligned} \] \(\pmb{\Lambda}\) encodes the measurement model and \(\pmb{\Theta}\) now encodes the variance-covariance of the measurement error. At the latent level we model a VAR process: \[ \begin{aligned} \pmb{\eta}_t^{(p,d)} &= \pmb{\alpha}^{(p)} + \pmb{B}^{(p)} \pmb{\eta}_{t-1}^{(p,d)} + \pmb{\zeta}_t^{(p,d)} \\ \pmb{\zeta}_{t}^{(p,d)} &\sim N(\pmb{0}, \pmb{\Psi}). \end{aligned} \] \(\pmb{\Psi}\) encodes the contemporaneous relationships.

Level 2 model: \[ \begin{bmatrix} \pmb{\alpha}^{(p)} \\ \mathrm{Vec}\left(\pmb{B}^{(p)}\right) \end{bmatrix} \sim N\left( \pmb{\gamma}, \pmb{\Omega} \right). \] \(\pmb{\gamma}\) encodes fixed effects and \(\pmb{\Omega}\) the distribution of random effects.

Not shown: correlations between \(\left\{ \varepsilon_1, \varepsilon_2, \varepsilon_3 \right\}\) and \(\left\{ \varepsilon_4, \varepsilon_5, \varepsilon_6 \right\}\).

• 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 Bayesian estimation (uncorrelated random effects)
• Very preliminary results
  • I ran the analysis yesterday!

• Multi-level estimation of contemporaneous effects
• High-dimensional estimation of random effect correlations
• Day-processes
• …

• mlVAR
  • Multi-level vector autoregression
  • Will contain state space soon
  • https://github.com/SachaEpskamp/mlVAR
• murmur
  • Multivariate multi-level regression
  • Also does VAR
  • Contains Bayesian and lmer estimation
  • https://github.com/SachaEpskamp/murmur