- 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
lm()
or any least squares regression methodFor 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
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
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.
lme4
packages implements univariate multi-level regression
lmer
functionSchuurman, 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.
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>
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 \(\left\{ \varepsilon_4, \varepsilon_5, \varepsilon_6 \right\}\).
Abegaz, Fentaw, and Ernst Wit. 2013. “Sparse Time Series Chain Graphical Models for Reconstructing Genetic Networks.” Biostatistics. Biometrika Trust, kxt005.
Rothman, Adam J, Elizaveta Levina, and Ji Zhu. 2010. “Sparse Multivariate Regression with Covariance Estimation.” Journal of Computational and Graphical Statistics 19 (4). Taylor & Francis: 947–62.
Wild, Beate, Michael Eichler, Hans-Christoph Friederich, Mechthild Hartmann, Stephan Zipfel, and Wolfgang Herzog. 2010. “A Graphical Vector Autoregressive Modelling Approach to the Analysis of Electronic Diary Data.” BMC Medical Research Methodology 10 (1). BioMed Central Ltd: 28.