Representing Sudden Shifts in Intensive Dyadic Interaction Data Using Differential Equation Models with Regime Switching

Abstract

A growing number of social scientists have turned to differential equations as a tool for capturing the dynamic interdependence among a system of variables. Current tools for fitting differential equation models do not provide a straightforward mechanism for diagnosing evidence for qualitative shifts in dynamics, nor do they provide ways of identifying the timing and possible determinants of such shifts. In this paper, we discuss regime-switching differential equation models, a novel modeling framework for representing abrupt changes in a system of differential equation models. Estimation was performed by combining the Kim filter (Kim and Nelson State-space models with regime switching: classical and Gibbs-sampling approaches with applications, MIT Press, Cambridge, 1999) and a numerical differential equation solver that can handle both ordinary and stochastic differential equations. The proposed approach was motivated by the need to represent discrete shifts in the movement dynamics of \(n= 29\) mother–infant dyads during the Strange Situation Procedure (SSP), a behavioral assessment where the infant is separated from and reunited with the mother twice. We illustrate the utility of a novel regime-switching differential equation model in representing children’s tendency to exhibit shifts between the goal of staying close to their mothers and intermittent interest in moving away from their mothers to explore the room during the SSP. Results from empirical model fitting were supplemented with a Monte Carlo simulation study to evaluate the use of information criterion measures to diagnose sudden shifts in dynamics.

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Notes

  1. 1.

    This was accomplished by optimizing the log transformations of these parameters on an unconstrained scale, and upon convergence, obtaining the standard error estimates of the final, transformed (constrained) parameters via the delta method (Casella & Berger, 2001).

  2. 2.

    An alternative form of Eq.  4 that may be easier to understand is one where all components of the equation are divided by dt, yielding the first derivatives of elements in \(\varvec{\eta }_i(t)\), namely, \(\frac{d\varvec{\eta }_i(t)}{dt}\), on the left-hand side of Eq. (4). The expression shown in 4 is, in the so-called Ito form, often adopted to highlight explicitly that the Wiener process is not differentiable with respect to time (i.e., \(\frac{d\varvec{w}_i(t)}{dt}\) is not defined as dt approaches zero; Arnold, 1974, Molenaar & Newell, 2003).

  3. 3.

    These means and variance parameters could also be estimated as modeling parameters (see, e.g., Chow et al., 2016b). However, because these parameters were not of great substantive interest and with the large numbers of time points within participants in the current data set, slight misspecifications of the values and structures of these specific initial conditions were likely to have minimal impact on the overall estimation. Thus, they were fixed at heuristic, empirically determined values.

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Correspondence to Sy-Miin Chow.

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Funding for this study was provided by NSF Grant SES-1357666, NIH Grant R01GM105004, Penn State Quantitative Social Sciences Initiative and UL TR000127 from the National Center for Advancing Translational Sciences.

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Appendices

Appendix

Sample dynr Code

The R package, dynr, can be downloaded from CRAN. Here, we include some sample codes for fitting the empirically motivated model in Eqs. 13 to simulated data.

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Chow, SM., Ou, L., Ciptadi, A. et al. Representing Sudden Shifts in Intensive Dyadic Interaction Data Using Differential Equation Models with Regime Switching. Psychometrika 83, 476–510 (2018). https://doi.org/10.1007/s11336-018-9605-1

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Keywords

  • differential equation
  • regime switching
  • hidden Markov
  • dynamic
  • dyadic
  • strange situation