^{ 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}

26 Feb 2016

^{ 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