Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-01-08T10:02:14.876Z Has data issue: false hasContentIssue false
  • English
  • Français

Using Threshold Autoregressive Models to Study Dyadic Interactions

Published online by Cambridge University Press:  01 January 2025

Ellen L. Hamaker ,
Zhiyong Zhang  and
Han L. J. van der Maas
Ellen L. Hamaker*
Affiliation:
Methodology and Statistics, Faculty of Social Sciences, Utrecht University
Zhiyong Zhang
Affiliation:
Quantitative Psychology, University of Notre Dame
Han L. J. van der Maas
Affiliation:
Psychological Methodology, University of Amsterdam
*
Requests for reprints should be sent to Ellen L. Hamaker, Methods and Statistics, Faculty of Social Sciences, Utrecht University, P.O. Box 80140, 3508 TC, Utrecht, The Netherlands. E-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Considering a dyad as a dynamic system whose current state depends on its past state has allowed researchers to investigate whether and how partners influence each other. Some researchers have also focused on how differences between dyads in their interaction patterns are related to other differences between them. A promising approach in this area is the model that was proposed by Gottman and Murray, which is based on nonlinear coupled difference equations. In this paper, it is shown that their model is a special case of the threshold autoregressive (TAR) model. As a consequence, we can make use of existing knowledge about TAR models with respect to parameter estimation, model alternatives and model selection. We propose a new estimation procedure and perform a simulation study to compare it to the estimation procedure developed by Gottman and Murray. In addition, we include an empirical example based on interaction data of three dyads.

Keywords

dynamic systemTAR modelautoregressivedyadic interaction
Type
Theory and Methods
Information
Psychometrika , Volume 74 , Issue 4 , December 2009 , pp. 727 - 745
Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NC
This article distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Copyright
Copyright © 2009 The Psychometric Society

Footnotes

This study was supported by the National Institute on Aging (Grant 5T32AG020500), and the Dutch Organization for Scientific Research (NWO; VENI Grant 451-05-012).

References

Bisconti, T., Bergeman, C. S., Boker, S. M. (2004). Emotional well-being in recently bereaved widows: A dynamical system approach. Journal of Gerontology, Series B: Psychological Sciences and Social Sciences, 59, 158–167.CrossRefGoogle Scholar
Carver, C. S., Scheier, M. F. (1998). On the self-regulation of behavior, New York: Cambridge University Press.CrossRefGoogle Scholar
Chan, K. S., Tong, H. (1990). On likelihood ratio tests for threshold autoregression. Journal of the Royal Statistical Society, 52, 469–476.CrossRefGoogle Scholar
Chen, C. W. S. (1998). A Bayesian analysis of generalized threshold autoregressive models. Statistical and Probability Letters, 40, 15–22.CrossRefGoogle Scholar
Chow, S.-M., Ferrer, E., Nesselroade, J. R. (2007). An unscented Kalman filter approach to the estimation of nonlinear dynamic system models. Multivariate Behavioral Research, 42, 283–321.CrossRefGoogle Scholar
Clayton, K. (1997). Basic concepts in nonlinear dynamics and chaos. Workshop presented at the society for chaos theory in psychology and the life sciences meeting.Google Scholar
Cook, J., Tyson, R., White, R. R., Gottman, J. M., Murray, J. (1995). Mathematics of marital conflict: Qualitative dynamic mathematical modeling of marital interaction. Journal of Family Psychology, 9, 110–130.CrossRefGoogle Scholar
De Gooijer, J. G. (2001). Cross-validation criteria for SETAR model selection. Journal of Time Series Analysis, 22, 267–281.CrossRefGoogle Scholar
Fan, J., Yao, Q. (2003). Nonlinear time series: Nonparametric and parametric methods, New York: Springer.CrossRefGoogle Scholar
Gonzalo, J., Pitarakis, J.-Y. (2002). Estimation and model selection based inference in single and multiple threshold models. Journal of Econometrics, 110, 319–352.CrossRefGoogle Scholar
Gonzalo, J., Wolf, M. (2005). Subsampling inference in threshold autoregressive models. Journal of Econometrics, 127, 201–224.CrossRefGoogle Scholar
Gottman, J. M., Coan, J., Carrere, S., Swanson, C. (1998). Predicting marital happiness and stability from newlywed interactions. Journal of Marriage and the Family, 60, 5–22.CrossRefGoogle Scholar
Gottman, J. M., Levenson, R. W., Swanson, C., Swanson, K., Tyson, R., Yoshimoto, D. (2003). Observing gay, lesbian, and heterosexual couple’s relationships: Mathematical modeling of conflict interaction. Journal of Homosexuality, 45, 65–91.CrossRefGoogle ScholarPubMed
Gottman, J. M., McCoy, K., Coan, J., Collier, H. (1996). The specific affect coding system SPAFF. In Gottman, J. M. (Eds.), What predicts divorce? The measures (pp. 1–169). Hillsdale: Lawrence Erlbaum Associates.Google Scholar
Gottman, J. M., Murray, J. D., Swanson, C. C., Tyson, R., Swanson, K. R. (2002). The mathematics of marriage: Dynamic nonlinear models, Cambridge: MIT Press.CrossRefGoogle Scholar
Gottman, J. M., Swanson, C. C., Murray, J. D. (1999). The mathematics of marital conflict: Dynamic mathematical nonlinear modeling of newlywed marital interaction. Journal of Family Psychology, 13, 3–19.CrossRefGoogle Scholar
Granger, C. W. J., Andersen, A. P. (1978). An introduction to bilinear time series models, Göttingen: Vandenhoeck und Ruprecht.Google Scholar
Granic, I., Hollenstein, T. (2003). Dynamic system methods for models of developmental psychopathology. Development and Psychopathology, 15, 641–669.CrossRefGoogle ScholarPubMed
Guastello, S. J. (1997). Science evolves: An introduction to nonlinear dynamics, psychology, and life sciences. Nonlinear Dynamics, Psychology, and Life Sciences, 1, 1–6.CrossRefGoogle Scholar
Hamaker, E. L., Dolan, C. V., Molenaar, P. C. M. (2003). ARMA-based SEM when the number of time points T exceeds the number of cases N: Raw data maximum likelihood. Structural Equation Modeling, 10, 352–379.CrossRefGoogle Scholar
Hansen, B. E. (1997). Inference in TAR models. Studies in Nonlinear Dynamics and Econometrics, 2, 1–14.Google Scholar
Kapetanios, G. (2003). Using extraneous information and GMM to estimate threshold parameters in TAR models (U of London Queen Mary Economics Working Paper No. 494). Available at SSRN: http://ssrn.com/abstract=425380 or doi: 10.2139/ssrn.425380.CrossRefGoogle Scholar
Murray, J. D. (2002). Mathematical biology: I. An introduction, (3rd ed.). New York: Springer.CrossRefGoogle Scholar
Normand, S.-L. (1999). Tutorial in biostatistics. Meta-analysis: Formulating, evaluating, combining and reporting. Statistics in Medicine, 18, 321–359.3.0.CO;2-P>CrossRefGoogle Scholar
Olthof, T., Kunnen, E. S., Boom, J. (2000). Simulating mother-child interaction: Exploring two varieties of a non-linear dynamic system approach. Infant and Child Development, 9, 33–60.3.0.CO;2-6>CrossRefGoogle Scholar
Politis, D. N. (2003). The impact of bootstrap methods on time series analysis. Statistical Science, 18, 219–230.CrossRefGoogle Scholar
Politis, D. N., Romano, J. P. (1994). Large sample confidence regions based on subsamples under minimal assumptions. The Annals of Statistics, 22, 2031–2050.CrossRefGoogle Scholar
Schiepek, G. (2003). A dynamic system approach to clinical case formulation. European Journal of Psychological Assessment, 19, 175–184.CrossRefGoogle Scholar
Schwartz Gottman, J. (2004). The marriage clinic casebook, New York: W.W. Norton and Co.Google Scholar
Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics, 6, 461–464.CrossRefGoogle Scholar
Shoda, Y., Tiernan, S. L., Mischel, W. (2002). Personality as a dynamic system: Emergence of stability and distinctiveness from intra- and interpersonal interactions. Personality and Social Psychology Review, 6, 316–325.CrossRefGoogle Scholar
Stijnen, T. (2000). Tutorial in biostatistics. meta-analysis: Formulating, evaluating, combining and reporting. Statistics in Medicine, 19, 753–761.3.0.CO;2-V>CrossRefGoogle ScholarPubMed
Strikholm, B., Teräsvirta, T. (2006). A sequential procedure for determining the number of regimes in a threshold autoregressive model. Econometrics Journal, 9, 472–491.CrossRefGoogle Scholar
Thelen, E., Smith, L. B. (1994). A dynamic system approach to the development of cognition and action, Cambridge: MIT Press.CrossRefGoogle Scholar
Tong, H., Lim, K. S. (1980). Threshold autoregression, limit cycles and cyclical data. Journal of the Royal Statistical Society, B, 42, 245–292.CrossRefGoogle Scholar
Tsay, R. S. (1998). Testing and modeling multivariate threshold models. Journal of the American Statistical Association, 93, 1188–1202.CrossRefGoogle Scholar
Vallacher, R. R., Nowak, A. (1997). The emergence of dynamical social psychology. Psychological Inquiry, 8, 73–99.CrossRefGoogle Scholar
Vallacher, R. R., Read, S. J., Nowak, A. (2002). The dynamical perspective in personality and social psychology. Personality and Social Psychology, 4, 264–273.Google Scholar
Van der Maas, H. L. J., Molenaar, P. C. M. (1992). Stagewise cognitive development: An application of catastrophe theory. Psychological Review, 99, 395–417.CrossRefGoogle ScholarPubMed
Van der Maas, H. L. J., Raijmakers, M. E. J. (2000). A phase transition model for mother-child interaction: Comment on Olthof et al., 2000. Infant and Child Development, 9, 75–83.3.0.CO;2-Q>CrossRefGoogle Scholar
Van Geert, P., Van Dijk, M. (2002). Focus on variability: New tools to study intra-individual variability in developmental data. Infant Behavior and Development, 25, 340–374.CrossRefGoogle Scholar
Wang, L., McArdle, J. J. (2008). A simulation study comparison of Bayesian estimate with conventional methods for estimating unknown change points. Structural Equation Modeling, 15, 52–74.CrossRefGoogle Scholar
Warren, K. (2002). Thresholds and the abstinence violation effect: A nonlinear dynamic model of the behaviors of intellectually disabled sex offenders. Journal of Interpersonal Violence, 17, 1198–1217.CrossRefGoogle Scholar
Warren, K., Hawkins, R. C., Sprott, J. C. (2003). Substance abuse as a dynamical disease: Evidence and clinical implications of nonlinearity in a time series of daily alcohol consumption. Addictive Behaviors, 28, 369–374.CrossRefGoogle Scholar
Witkiewitz, K., Van der Maas, J. L., Hufford, M. R., Marlatt, G. A. (2007). Nonnormality and divergence in posttreatment alcohol use: Reexamining the Project MATCH data another way. Journal of Abnormal Psychology, 116, 378–394.CrossRefGoogle Scholar