August 2015

Psychometrics

Psychometrics

Item scores correlate positively with each-other

Mostly item scores are stable over time within a person

Phase transitions can occur

Modeling Psychology

  • Item responses often modeled to be causal result of some latent factor
    • Intelligence
    • Depression
    • Neuroticism
  • The correlation between any pair of items is modeled as being solely due to the common influence of the latent trait
    • Local independence

  • Local independence is not plausible; psychological variables interact with each other
  • Allowing these interactions, do we still need latent variables to explain the macroscopic behavior?
  • Correlated structures could be emergent behavior in a network of cellular automata

Mutualism

Van der Maas et al. (2006)

Psychopathology as a virus

Video

The Ising Model

Network Psychometrics

  • What is the structure of psychology?

Weighted network

The Ising Model

Stationary distribution:

\[ \Pr\left(\pmb{X} = \pmb{x}\right) \propto \exp\left( \pmb{\tau}^\top \pmb{x} + \frac{1}{2} \pmb{x}^\top\pmb{\Omega}\pmb{x} \right) \]

  • \(\pmb{x}\) is a vector of binary variables (\(-1\) or \(1\))
  • \(\pmb{\tau}\) is a vector containing threshold parameters
  • \(\pmb{\Omega}\) is a matrix containing the network parameters and an arbitrary diagonal

Gaussian Graphical Model

\[ f\left(\pmb{x}\right) \propto \exp\left( \pmb{\tau}^\top \pmb{x} + \frac{1}{2} \pmb{x}^\top\pmb{\Omega}\pmb{x} \right) \]

  • Now, \(\pmb{x}\) is assumed Gaussian distributed
  • Same as Ising model; the inverse covariance matrix encodes a network
  • Edges can be standardized to partial correlation coefficients between nodes conditioned on all other nodes
    • \(\rho_{ij} = \omega_{ij} / \sqrt{\omega_{ii}\omega_{jj}} = \mathrm{Cor}\left( X_i, X_j \mid \pmb{X}^{-(i,j)} = \pmb{x}^{-(i,j)} \right)\)

Model Selection

  • A well accepted approach for jointly model selection and parameter estimation is the least absolute shrinkage and selection operator (LASSO)
    • Penalized maximum likelihood estimation
  • LASSO utilizes a penalty parameter, which can be chosen to optimize some information criterion
    • Extended Bayesian information criterion (EBIC)

Empirical example: personality

  • Responses to questions such as "Am indifferent to the feelings of others" and "Make friends easily" are traditionally modeled to be the results of five stable personality traits:
    • Neurotocism
    • Extraversion
    • Agreeableness
    • Conscientiousness
    • Openess to Experience
  • I will analyze 2800 responses on 25 such items
    • Data from "psych" package in R

Longitudinal Analysis

Clinical Setting

Graphical VAR

\[ \begin{aligned} \pmb{y}_t &= \pmb{B} \pmb{y}_{t-1} + \pmb{\varepsilon}_t \\ \pmb{\varepsilon}_t &\sim N(\pmb{0}, \pmb{K}^{-1}) \\ \mathrm{Cov}\left( \pmb{\varepsilon}_t, \pmb{\varepsilon}_{t+i} \right) &= \pmb{O} \, \forall i \not= 0 \end{aligned} \]

  • This model forms both a temporal and a contemporaneous network
    • Estimation via LASSO regularization, using BIC to select optimal tuning parameter (Rothman, Levina, and Zhu 2010; Abegaz and Wit 2013).
  • LASSO is used to regularize both \(\pmb{B}\) and \(\pmb{K}\)
  • Afterwards, both matrices are standardized

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

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

Multi-level VAR

  • When multiple persons are measured over a period of time, multi-level VAR can be used
  • Estimates a fixed effects network, that describes the average structure across persons
  • As well as random effects, which describe individual differences
  • Explained by Bringmann et al. (2013)

Multi-level VAR

Lag-1 model

Level 1: \[ \pmb{y}_t^{(p)} = \pmb{B}^{(p)} \pmb{y}_ {t-1}^{(p)} + \pmb{\varepsilon}_t^{(p)} \]

Level 2: \[ \begin{aligned} \pmb{\beta}_ {ij}^{(p)} &= b_{ij} + u^{(p)}_{ij} \\ u^{(p)}_{ij} &\sim N(0, \sigma_{ij}) \end{aligned} \]

26 subjects; 10-100 measurements

Software Implementation (1/2)

  • qgraph (github.com/SachaEpskamp/qgraph)
    • Network visualization
    • Gaussian graphical model estimation
      • Unregularized and regularized using glasso/EBIC
    • Epskamp et al. (2012)
    • Costantini et al. (2015)
  • IsingSampler (github.com/SachaEpskamp/IsingSampler)
    • Sample from stationary Ising distribution
    • Unregularized Ising model estimation
  • IsingFit (github.com/cvborkulo/IsingFit)
    • Regularized Ising model estimation using LASSO/EBIC
    • Borkulo et al. (2014)

Software Implementation (2/2)

  • graphicalVAR (github.com/SachaEpskamp/graphicalVAR)
    • LASSO estimation of lag-1 VAR models
  • mlVAR (github.com/SachaEpskamp/mlVAR)
    • Multi-level VAR estimation
    • Bringmann et al. (2013)

Network Psychometrics Ecosystem

References

Abegaz, Fentaw, and Ernst Wit. 2013. “Sparse Time Series Chain Graphical Models for Reconstructing Genetic Networks.” Biostatistics. Biometrika Trust, kxt005.

Borkulo, Claudia D van, Denny Borsboom, Sacha Epskamp, Tessa F Blanken, Lynn Boschloo, Robert A Schoevers, and Lourens J Waldorp. 2014. “A New Method for Constructing Networks from Binary Data.” Scientific Reports 4. Nature Publishing Group: Article number: 5918.

Bringmann, Laura F, Nathalie Vissers, Marieke Wichers, Nicole Geschwind, Peter Kuppens, Frenk Peeters, Denny Borsboom, and Francis Tuerlinckx. 2013. “A Network Approach to Psychopathology: New Insights into Clinical Longitudinal Data.” PloS One 8 (4). Public Library of Science: e60188.

Costantini, Giulio, Sacha Epskamp, Denny Borsboom, Marco Perugini, René Mõttus, Lourens J. Waldorp, and Angélique O.J. Cramer. 2015. “State of the ARt Personality Research: A Tutorial on Network Analysis of Personality Data in R.” Journal of Research in Personality 54: 13–29.

Epskamp, Sacha, Ang élique OJ Cramer, Lourens J Waldorp, Verena D Schmittmann, and Denny Borsboom. 2012. “Qgraph: Network Visualizations of Relationships in Psychometric Data.” Journal of Statistical Software 48 (4): 1–18.

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.

Van der Maas, Han LJ, Conor V Dolan, Raoul PPP Grasman, Jelte M Wicherts, Hilde M Huizenga, and Maartje EJ Raijmakers. 2006. “A Dynamical Model of General Intelligence: The Positive Manifold of Intelligence by Mutualism.” Psychological Review 113. American Psychological Association: 842–61.