Abstract
Dynamic systems modeling techniques provide a convenient platform for representing multidimensional and multidirectional change processes over time. Central to dynamic systems models is the notion that a system may show emergent properties that allow the system to self-organize into qualitatively distinct states through temporal fluctuations in selected key parameters of interest. Using computer vision-based measurement of smiling in one infant-mother dyad’s interactions during a face-to-face interaction, we illustrate the use of generalized additive modeling techniques to fit multivariate dynamic systems models with self-organizing, time-moderated dynamic parameters. We found evidence for systematic over-time changes in the infant → mother cross-regression effect, which provided a glimpse into how the dyad self-organized into distinct states over the course of the interaction, including periods where the mother’s positivity was reinforced and strengthened by the infant’s positivity, as well as periods where the mother’s positivity was inversely related to the infant’s past positivity levels.
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Notes
- 1.
Heterogeneities may stem from (a) between-person differences in the population due, for example, to the presence of subgroups/subpopulations or other individual difference characteristics; and (b) within-person variations in change characteristics during portions of the individual’s repeated assessments.
- 2.
Strict stationarity refers to the property that the probability distribution of a stochastic process is assumed to be constant over time, whereas weak stationarity only requires the mean and variance of a probability distribution to be time invariant (Chatfield, 2004).
- 3.
GAMM provides a collection of procedures for approximating these functions and the resultant curves using different smoothers. \( f_{1,k} (.) - f_{3,s,q} \) are typically referred to as smooth functions, and the curves or lines produced by these functions are denoted as smooths (Hastie & Tibshirani, 1990; McKeown & Sneddon, 2014). Note that the first subscript in \( x_{1,ki} - x_{3,si}^{*} \) is used to distinguish the kind of smoothing function with which a specific covariate is associated, and the second subscript distinguishes among the covariates that are subjected to that particular kind of smoothing function. The superscript * in \( x_{2,o,i}^{*} \) and \( x_{3,si }^{*} \) is used to distinguish between the two sets of covariates that appear in the second and third types of smoothing functions. For instance, \( x_{2,ri} \) denotes the rth covariate that is subjected to smoothing in \( f_{2,o} \) whereas \( x_{2,o,i}^{*} \) denotes the oth (unsmoothed) covariate that moderates the effect of \( f_{2,o} \,(x_{2,ri} ) \) on \( \eta_{i} \).
- 4.
Edfs are inversely related to the smoothing parameter used in the penalized basis functions to smooth out “wiggliness” in the data. Roughly speaking, they may be viewed as weights that map the penalized smoothed coefficient of a covariate to the unpenalized linear parametric coefficient associated with the covariate. An edf value that is close to zero implies that a particular covariate does not have statistically significant effect on the dependent variable whereas an edf value close to 1.0 suggests insufficient evidence for the effect of the covariate to be nonlinear.
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Acknowledgements
The authors thank the families participating in the research and acknowledge support from NIH grant R01GM105004, NSF grant BCS-0826844, Penn State Quantitative Social Sciences Initiative and UL TR000127 from the National Center for Advancing Translational Sciences.
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Chow, SM., Ou, L., Cohn, J.F., Messinger, D.S. (2017). Representing Self-organization and Nonstationarities in Dyadic Interaction Processes Using Dynamic Systems Modeling Techniques. In: von Davier, A., Zhu, M., Kyllonen, P. (eds) Innovative Assessment of Collaboration. Methodology of Educational Measurement and Assessment. Springer, Cham. https://doi.org/10.1007/978-3-319-33261-1_17
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