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

Published online by Cambridge University Press:  01 January 2025

Sy-Miin Chow Open the ORCID record for Sy-Miin Chow [Opens in a new window] ,
Lu Ou ,
Arridhana Ciptadi ,
Emily B. Prince ,
Dongjun You ,
Michael D. Hunter ,
James M. Rehg ,
Agata Rozga  and
Daniel S. Messinger
Sy-Miin Chow*
Affiliation:
Pennsylvania State University
Lu Ou
Affiliation:
Pennsylvania State University
Arridhana Ciptadi
Affiliation:
Georgia Institute of Technology
Emily B. Prince
Affiliation:
University of Miami
Dongjun You
Affiliation:
Pennsylvania State University
Michael D. Hunter
Affiliation:
University of Oklahoma Health Sciences Center
James M. Rehg
Affiliation:
Georgia Institute of Technology
Agata Rozga
Affiliation:
Georgia Institute of Technology
Daniel S. Messinger
Affiliation:
University of Miami
*
Correspondence should be made to Sy-Miin Chow, Pennsylvania State University, 413 Biobehavioral Health Building, University Park, PA16802, USA. Email: [email protected]
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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\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n= 29$$\end{document} 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.

Keywords

differential equationregime switchinghidden Markovdynamicdyadicstrange situation
Type
Original Paper
Information
Psychometrika , Volume 83 , Issue 2 , June 2018 , pp. 476 - 510
Copyright
Copyright © 2018 The Psychometric Society

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Footnotes

Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11336-018-9605-1) contains supplementary material, which is available to authorized users.

References

Ainsworth, M.D.S., Blehar, M.C., Waters, E., Wall, S.Patterns of attachment: A psychological study of the strange situation 1978 Oxford:Lawrence Erlbaum.Google Scholar
Akaike, H., (1973). Information theory and an extension of the maximum likelihood principle. In Petrov, B.N., Csaki, F.(Eds.), Second international symposium on information theory. (pp 267–281). Budapest:Akademiai Kiado.Google Scholar
Aptech Systems Inc. (2009). GAUSS (Version 10). WA: Black Diamond..Google Scholar
Arnold, L., (1974). Stochastic differential equations. New York:Wiley.Google Scholar
Bar-Shalom, Y., Li, X.R., Kirubarajan, T., (2001). Estimation with applications to tracking and navigation: Theory algorithms and software. New York:Wiley.Google Scholar
Behrens, K.Y., Parker, A.C., Haltigan, J.D., (2011). Maternal sensitivity assessed during the strange situation procedure predicts child’s attachment quality and reunion behaviors, Infant Behavior and Development, 34(2), 378–381.CrossRefGoogle ScholarPubMed
Beskos, A., Papaspiliopoulos, O., Roberts, G., (2009). Monte carlo maximum likelihood estimation for discretely observed diffusion processes, The Annals of Statistics, 37(1), 223–245.CrossRefGoogle Scholar
Boker, S.M., Deboeck, P.R., Edler, C., Keel, P.K., (2010). Generalized local linear approximation of derivatives from time series. In Chow, S., Ferrer, E., Hsieh, F.(Eds.), Statistical methods for modeling human dynamics: An interdisciplinary dialogue. (pp 161–178). New York:Taylor & Francis.Google Scholar
Boker, S.M., Graham, J., (1998). A dynamical systems analysis of adolescent substance abuse, Multivariate Behavioral Research, 33, 479–507.CrossRefGoogle ScholarPubMed
Boker, S.M., Neale, H., Maes, H., Wilde, M., Spiegel, M., Brick, T., Spies, J., Estabrook, R., Kenny, S., Bates, T., Mehta, P., Fox, J., (2011). Openmx: An open source extended structural equation modeling framework, Psychometrika, 76(2), 306–317.CrossRefGoogle ScholarPubMed
Boker, S.M., Neale, M.C., Rausch, J., (2008). Latent differential equation modeling with multivariate multi-occasion indicators. In van Montfort, K., Oud, H., Satorra, A.(Eds.), Recent developments on structural equation models: Theory and applications. (pp 151–174). Amsterdam:Kluwer.Google Scholar
Bolger, N., Davis, A., Rafaeli, E., (2003). Diary methods: Capturing life as it is lived, Annual Review of Psychology, 54, 579–616.CrossRefGoogle ScholarPubMed
Bowlby, J., (1973). Separation: Anxiety & anger. Attachment and Loss. (International psycho-analytical library no. 95). London:Hogarth Press.Google Scholar
Bowlby, J., (1982). Attachment. 2 New York:Basic Books.Google ScholarPubMed
Casella, G., Berger, R.L., (2001). Statistical inference. 2 Pacific Grove:Duxbury Press.Google Scholar
Chow, S-M, Bendezú, J.J., Cole, P.M., Ram, N., (2016). A comparison of two- stage approaches for fitting nonlinear ordinary differential equation (ode) models with mixed effects, Multivariate Behavioral Research, 51(2–3), 154–184.CrossRefGoogle ScholarPubMed
Chow, S-M, Ferrer, E., Nesselroade, J.R., (2007). An unscented kalman filter approach to the estimation of nonlinear dynamical systems models, Multivariate Behavioral Research, 42(2), 283–321.CrossRefGoogle Scholar
Chow, S-M, Ho, M-HR, Hamaker, E.J., Dolan, C.V., (2010). Equivalences and differences between structural equation and state-space modeling frameworks, Structural Equation Modeling, 17, 303–332.CrossRefGoogle Scholar
Chow, S-M, Lu, Z., Sherwood, A., Zhu, H., (2016). Fitting nonlinear ordinary differential equation models with random effects and unknown initial conditions using the stochastic approximation expectation maximization (SAEM) algorithm, Psychometrika, 81, 102–134.CrossRefGoogle ScholarPubMed
Chow, S-M, Zhang, G., (2013). Nonlinear regime-switching state-space (RSSS) models, Psychometrika: Application Reviews and Case Studies, 78(4), 740–768.CrossRefGoogle ScholarPubMed
Chow, S-M, Zu, J., Shifren, K., Zhang, G., (2011). Dynamic factor analysis models with time-varying parameters, Multivariate Behavioral Research, 46(2), 303–339.CrossRefGoogle ScholarPubMed
Collins, L.M., Wugalter, S.E., (1992). Latent class models for stage-sequential dynamic latent variables, Multivariate Behavioral Research, 28, 131–157.CrossRefGoogle Scholar
Deboeck, P.R., (2010). Estimating dynamical systems: Derivative estimation hints from Sir Ronald A Fisher, Multivariate Behavioral Research, 45, 725–745.CrossRefGoogle Scholar
Dijkstra, T.K., (1992). On statistical inference with parameter estimates on the boundary of the parameter space, British Journal of Mathematical and Statistical Psychology, 45(2), 289–309.CrossRefGoogle Scholar
Dolan, C.V., (2009). Structural equation mixture modeling. In Millsap, R.E., Maydeu-Olivares, A. (Eds.), The SAGE handbook of quantitative methods in psychology. (pp 568–592). Thousand Oaks:Sage.CrossRefGoogle Scholar
Dolan, C.V., Molenaar, P.C.M., (1991). A note on the calculation of latent trajectories in the quasi Markov simplex model by means of regression method and the discrete kalman filter, Kwantitatieve Methoden, 38, 29–44.Google Scholar
Dolan, C.V., Schmittmann, V.D., Lubke, G.H., Neale, M.C., (2005). Regime switching in the latent growth curve mixture model, Structural Equation Modeling, 12(1), 94–119.CrossRefGoogle Scholar
Doornik, J.A., (1998). Object-oriented matrix programming using Ox 2.0. London:Timberlake Consultants Press.Google Scholar
Durbin, J., Koopman, S.J., (2001). Time series analysis by state space methods. New York:Oxford University Press.Google Scholar
Elliott, R.J., Aggoun, L., Moore, J., (1995). Hidden markov models: Estimation and control. New York:Springer.Google Scholar
Enders, C., Tofighi, D., (2008). The impact of misspecifying class-specific residual variances in growth mixture models, Structural Equation Modeling: A Multidisciplinary Journal, 15, 75–95.CrossRefGoogle Scholar
Fahrmeir, L., Tutz, G., (2001). Multivariate statistical modelling based on generalized linear models. Berlin:Springer.CrossRefGoogle Scholar
Fukuda, K., Ishihara, K., (1997). Development of human sleep and wakefulness rhythm during the first six months of life: Discontinuous changes at the 7th and 12th week after birth, Biological Rhythm Research, 28, 94–103.CrossRefGoogle Scholar
Gates, K.M., Molenaar, P.C.M., (2012). Group search algorithm recovers effective connectivity maps for individuals in homogeneous and heterogeneous samples, Neuroimage, 63, 310–319.CrossRefGoogle ScholarPubMed
Hamilton, J.D., (1989). A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica, 57, 357–384.CrossRefGoogle Scholar
Harvey, A.C., (2001). Forecasting, structural time series models and the Kalman filter. Cambridge:Cambridge University Press.Google Scholar
Jackson, C.H., (2011). Multi-state models for panel data: The msm package for R, Journal of Statistical Software, 38(8), 1–29.CrossRefGoogle Scholar
Jazwinski, A.H., (1970). Stochastic processes and filtering theory. New York:Academic Press.Google Scholar
Jones, R.H., (1993). Longitudinal data with serial correlation: A state-space approach. Boca Raton:Chapman & Hall/CRC.CrossRefGoogle Scholar
Kalman, R. E., & Bucy, R. S., (1961). New results in linear filtering and prediction theory. In Trans. ASME, Ser. D, J. Basic Eng, p 109..CrossRefGoogle Scholar
Kim, C-J, Nelson, C.R., (1999). State-space models with regime switching: Classical and Gibbs-sampling approaches with applications. Cambridge:MIT Press.CrossRefGoogle Scholar
Kulikov, G., Kulikova, M., (2014). Accurate numerical implementation of the continuous-discrete extended Kalman filter, IEEE Transactions on Automatic Control, 59(1), 273–279.CrossRefGoogle Scholar
Kulikova, M.V., Kulikov, G.Y., (2014). Adaptive ODE solvers in extended Kalman filtering algorithms, Journal of Computational and Applied Mathematics, 262, 205–216.CrossRefGoogle Scholar
Lanza, S.T., Collins, L.M., (2008). A new SAS procedure for latent transition analysis: Transitions in dating and sexual risk behavior, Developmental Psychology, 44(2), 446–456.CrossRefGoogle ScholarPubMed
Lawley, D.N., Maxwell, M.A., (1971). Factor analysis as a statistical method. 2 London:Butterworths.Google Scholar
Lu, Z-H, Chow, S-M, Sherwood, A., Zhu, H., (2015). Bayesian analysis of ambulatory cardiovascular dynamics with application to irregularly spaced sparse data, Annals of Applied Statistics, 9, 1601–16204773035.CrossRefGoogle ScholarPubMed
Mbalawata, I.S., Särkkä, S., Haario, H., (2013). Parameter estimation in stochastic differential equations with markov chain monte carlo and non-linear Kalman filtering, Computational Statistics, 28(3), 1195–1223.CrossRefGoogle Scholar
Miao, H., Xin, X., Perelson, A.S., Wu, H., (2011). On identifiability of nonlinear ODE models and applications in viral dynamics, SIAM Review, 53(1), 3–393140286.CrossRefGoogle Scholar
Molenaar, P.C.M., (2004). A manifesto on psychology as idiographic science: Bringing the person back into scientific pyschology-this time forever, Measurement: Interdisciplinary Research and Perspectives, 2, 201–218.Google Scholar
Molenaar, P.C.M., Newell, K.M., (2003). Direct fit of a theoretical model of phase transition in oscillatory finger motions, British Journal of Mathematical and Statistical Psychology, 56, 199–214.CrossRefGoogle ScholarPubMed
Muthén, B. O., & Asparouhov, T., (2011). LTA in Mplus: Transition probabilities influenced by covariates. Mplus web notes: No. 13. Available at http://www.statmodel.com/examples/LTAwebnote.pdf. Accessed 23 July 2016.Google Scholar
Muthén, L.K., Muthén, B.O., (2001). Mplus: The comprehensive modeling program for applied researchers: User’s guide (1998–2001). Los Angeles:Muthén & Muthén.Google Scholar
Nylund, K.L., Asparouhov, T., Muthén, B.O., (2007). Deciding on the number of classes in latent class analysis and growth mixture modeling: A Monte Carlo simulation study, Structural Equation Modeling, 14(4), 535–569.CrossRefGoogle Scholar
Nylund-Gibson, K., Muthén, B. O., Nishina, A., Bellmore, A., & Graham, S., (2013). Stability and instability of peer victimization during middle school: Using latent transition analysis with covariates, distal outcomes, and modeling extensions. Manuscript submitted for publication.Google Scholar
Oravecz, Z., Tuerlinckx, F., Vandekerckhove, J., (2009). A hierarchical Ornstein-Uhlenbeck model for continuous repeated measurement data, Psychometrika, 74, 395–418.CrossRefGoogle Scholar
Oravecz, Z., Tuerlinckx, F., Vandekerckhove, J., (2011). A hierarchical latent stochastic differential equation model for affective dynamics, Psychological Methods, 16, 468–490.CrossRefGoogle ScholarPubMed
Ou, L., Hunter, M. D., & Chow, S.-M., (2016). dynr: Dynamic Modeling in R. R package version 0.1.7-22..Google Scholar
Oud, J.H.L., Jansen, RARG, (2000). Continuous time state space modeling of panel data by means of SEM, Psychometrika, 65(2), 199–215.CrossRefGoogle Scholar
Oud, J. H. L., & Singer, H., (Eds.) (2008). Special issue: Continuous time modeling of panel data, Vol. 62..Google Scholar
Oud, J.H.L., van den Bercken, J.H., Essers, R.J., (1990). Longitudinal factor score estimation using the Kalman filter, Applied psychological measurement, 14, 395–418.CrossRefGoogle Scholar
Piaget, J., Inhelder, B., (1969). The psychology of the child. New York:Basic Books.Google Scholar
Pincus, S., Gladstone, I., Ehrenkranz, R., (1991). A regularity statistic for medical data analysis, Journal of Clinical Monitoring, 7, 335–345.CrossRefGoogle ScholarPubMed
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., (2002). Numerical recipes in C. Cambridge:Cambridge University Press.Google Scholar
R Development Core Team (2009). R: A language and environment for statistical computing. Vienna: Austria. ISBN: 3-900051-07-0.Google Scholar
Ramsay, J.O., Hooker, G., Campbell, D., Cao, J., (2007). Parameter estimation for differential equations: a generalized smoothing approach (with discussion), Journal of Royal Statistical Society: Series B, 69(5), 741–796.CrossRefGoogle Scholar
Särkkä, S., (2013). Bayesian Filtering and Smoothing. Hillsdale:Cambridge University Press.CrossRefGoogle Scholar
SAS Institute Inc. (2008). SAS 9.2 help and documentation. Cary, NC..Google Scholar
Schmittmann, V.D., Dolan, C.V., van der Maas, H., Neale, M.C., (2005). Discrete latent Markov models for normally distributed response data, Multivariate Behavioral Research, 40(2), 207–233.CrossRefGoogle ScholarPubMed
Schwartz, J.E., Stone, A.A., (1998). Strategies for analyzing ecological momentary assessment data, Health Psychology, 17, 6–16.CrossRefGoogle ScholarPubMed
Schwarz, G., (1978). Estimating the dimension of a model, The Annals of Statistics, 6, 461–464.CrossRefGoogle Scholar
Schweppe, F., (1965). Evaluation of likelihood functions for Gaussian signals, IEEE Transactions on Information Theory, 11, 61–70.CrossRefGoogle Scholar
Shapiro, A., (1985). Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints, Biometrics, 72(1), 133–44.CrossRefGoogle Scholar
Shapiro, A., (1986). Asymptotic theory or overparameterized structural models, American Statistical Association Journal, 81(393), 142–149.CrossRefGoogle Scholar
Sherwood, A., Thurston, R., Steffen, P., Blumenthal, J.A., Waugh, R.A., Hinderliter, A.L., (2001). Blunted nighttime blood pressure dipping in postmenopausal women, American Journal of Hypertension, 14, 749–754.CrossRefGoogle ScholarPubMed
Singer, H., (2002). Parameter estimation of nonlinear stochastic differential equations: Simulated maximum likelihood vs. extended Kaman filter and itô-Taylor expansion, Journal of Computational and Graphical Statistics, 11, 972–995.CrossRefGoogle Scholar
Singer, H., (2003). Nonlinear continuous discrete filtering using kernel density estimates and functional integrals, Journal of Mathematical Sociology, 27, 1–28.CrossRefGoogle Scholar
Singer, H., (2010). Sem modeling with singular moment matrices part i: Ml-estimation of time series, The Journal of Mathematical Sociology, 34(4), 301–320.CrossRefGoogle Scholar
Singer, H., (2012). Sem modeling with singular moment matrices part ii: Ml-estimation of sampled stochastic differential equations, The Journal of Mathematical Sociology, 36(1), 22–43.CrossRefGoogle Scholar
Sivalingam, R., Cherian, A., Fasching, J., Walczak, N., Bird, N., Morellas, V., Murphy, B., Cullen, K., Lim, K., Sapiro, G., & Papanikolopoulos, N., (2012). A multi-sensor visual tracking system for behavior monitoring of at-risk children. In IEEE international conference on robotics and automation (ICRA)..CrossRefGoogle Scholar
Tofghi, D., Enders, C.K., (2007). Identifying the correct number of classes in mixture models. In Hancock, G.R., Samulelsen, K.M.(Eds.), Advances in latent variable mixture models. (pp 317–341). Greenwich:Information Age .Google Scholar
Trail, J.B., Collins, L.M., Rivera, D.E., Li, R., Piper, M.E., Baker, T.B., (2013). Functional data analysis for dynamical system identification of behavioral processes, Psychological Methods, 19, 175–1824098896.CrossRefGoogle ScholarPubMed
van der Maas, H.L.J., Molenaar, P.C.M., (1992). Stagewise cognitive development: An application of catastrophe theory, Psychological Review, 99(3), 395–417.CrossRefGoogle ScholarPubMed
Van Geert, P., (2000). The dynamics of general developmental mechanisms: From Piaget and Vygotsky to dynamic systems models, Current Directions in Psychological Science, 9(2), 64–68.CrossRefGoogle Scholar
Vecchiato, W., (1997). New models for irregularly spaced time series analysis with applications to high frequency financial data. In Proceedings of the IEEE/IAFE 1997 computational intelligence for financial engineering (CIFEr), pp. 144–149..CrossRefGoogle Scholar
Visser, I., (2007). Depmix: An R-package for fitting mixture models on mixed multivariate data with markov dependencies. Technical report, University of Amsterdam.Google Scholar
Voelkle, M.C., Oud, J.H.L., Davidov, E., Schmidt, P., (2012). An sem approach to continuous time modeling of panel data: relating authoritarianism and anomia, Psychological Methods, 17, 176–192.CrossRefGoogle ScholarPubMed
Waters, E., (2002). Comments on strange situation classification. Retrieved on May 1, 2012 from http://www.johnbowlby.com..Google Scholar
Yang, M., Chow, S-M, (2010). Using state-space model with regime switching to represent the dynamics of facial electromyography (EMG) data, Psychometrika: Application and Case Studies, 74(4), 744–771.CrossRefGoogle Scholar
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