Abstract
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.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
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.
Carver, C. S., & Scheier, M. F. (1998). On the self-regulation of behavior. New York: Cambridge University Press.
Chan, K. S., & Tong, H. (1990). On likelihood ratio tests for threshold autoregression. Journal of the Royal Statistical Society, 52, 469–476.
Chen, C. W. S. (1998). A Bayesian analysis of generalized threshold autoregressive models. Statistical and Probability Letters, 40, 15–22.
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.
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.
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.
De Gooijer, J. G. (2001). Cross-validation criteria for SETAR model selection. Journal of Time Series Analysis, 22, 267–281.
Fan, J., & Yao, Q. (2003). Nonlinear time series: Nonparametric and parametric methods. New York: Springer.
Gonzalo, J., & Pitarakis, J.-Y. (2002). Estimation and model selection based inference in single and multiple threshold models. Journal of Econometrics, 110, 319–352.
Gonzalo, J., & Wolf, M. (2005). Subsampling inference in threshold autoregressive models. Journal of Econometrics, 127, 201–224.
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.
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.
Gottman, J. M., McCoy, K., Coan, J., & Collier, H. (1996). The specific affect coding system SPAFF. In J. M. Gottman (Ed.), What predicts divorce? The measures (pp. 1–169). Hillsdale: Lawrence Erlbaum Associates.
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.
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.
Granger, C. W. J., & Andersen, A. P. (1978). An introduction to bilinear time series models. Göttingen: Vandenhoeck und Ruprecht.
Granic, I., & Hollenstein, T. (2003). Dynamic system methods for models of developmental psychopathology. Development and Psychopathology, 15, 641–669.
Guastello, S. J. (1997). Science evolves: An introduction to nonlinear dynamics, psychology, and life sciences. Nonlinear Dynamics, Psychology, and Life Sciences, 1, 1–6.
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.
Hansen, B. E. (1997). Inference in TAR models. Studies in Nonlinear Dynamics and Econometrics, 2, 1–14.
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.
Murray, J. D. (2002). Mathematical biology: I. An introduction (3 ed.). New York: Springer.
Normand, S.-L. (1999). Tutorial in biostatistics. Meta-analysis: Formulating, evaluating, combining and reporting. Statistics in Medicine, 18, 321–359.
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.
Politis, D. N. (2003). The impact of bootstrap methods on time series analysis. Statistical Science, 18, 219–230.
Politis, D. N., & Romano, J. P. (1994). Large sample confidence regions based on subsamples under minimal assumptions. The Annals of Statistics, 22, 2031–2050.
Schiepek, G. (2003). A dynamic system approach to clinical case formulation. European Journal of Psychological Assessment, 19, 175–184.
Schwartz Gottman, J. (2004). The marriage clinic casebook. New York: W.W. Norton and Co.
Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics, 6, 461–464.
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.
Stijnen, T. (2000). Tutorial in biostatistics. meta-analysis: Formulating, evaluating, combining and reporting. Statistics in Medicine, 19, 753–761.
Strikholm, B., & Teräsvirta, T. (2006). A sequential procedure for determining the number of regimes in a threshold autoregressive model. Econometrics Journal, 9, 472–491.
Thelen, E., & Smith, L. B. (1994). A dynamic system approach to the development of cognition and action. Cambridge: MIT Press.
Tong, H., & Lim, K. S. (1980). Threshold autoregression, limit cycles and cyclical data. Journal of the Royal Statistical Society, B, 42, 245–292.
Tsay, R. S. (1998). Testing and modeling multivariate threshold models. Journal of the American Statistical Association, 93, 1188–1202.
Vallacher, R. R., & Nowak, A. (1997). The emergence of dynamical social psychology. Psychological Inquiry, 8, 73–99.
Vallacher, R. R., Read, S. J., & Nowak, A. (2002). The dynamical perspective in personality and social psychology. Personality and Social Psychology, 4, 264–273.
Van der Maas, H. L. J., & Molenaar, P. C. M. (1992). Stagewise cognitive development: An application of catastrophe theory. Psychological Review, 99, 395–417.
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.
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.
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.
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.
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.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
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).
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Hamaker, E.L., Zhang, Z. & van der Maas, H.L.J. Using Threshold Autoregressive Models to Study Dyadic Interactions. Psychometrika 74, 727–745 (2009). https://doi.org/10.1007/s11336-009-9113-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11336-009-9113-4