Dynamical Analysis of Therapist-Client Interactions

  • Paul R. PelusoEmail author
  • Andrew Z. Baker
  • Ashley Sauer
  • Jennifer P. Peluso
Part of the Computational Social Sciences book series (CSS)


Dynamic nonlinear mathematical equations allow investigators to deeply understand complex systems that are apt to change. They can be used to determine the stable steady states or points of homeostasis within the system (i.e., the relationship). These steady states function as an anchor that brings the system back to homeostasis if the system is perturbed or if it has been moved away from homeostasis by a force. Liebovitch and his associates (Liebovitch et al., 2008) modified Gottman’s dynamic nonlinear equations to study the dynamics of conflicts between two parties (which could be individuals or groups). They determined how the dynamics of such conflicts and their emergent properties depended on the actions of each person. Our modeling of the therapeutic relationship constitutes a concrete extension of these results to a different social interaction. In this chapter, we will outline an ongoing research project (described in some detail in the previous chapter) that employs mathematical modeling to discover what underlies a successful therapeutic relationship. We will then discuss, through the use of case studies, the issues related to the parameters generated, the graphical displays of the derived models, and the indices of model fit. Specifically, we present three therapy sessions from two separate case studies to illustrate the mathematical modeling, the difference in the model parameters, and a graphical representation of each.


  1. Acquah, H. G. (2010). Comparison of Akaike information criterion (AIC) and Bayesian information criterion (BIC) in selection of an asymmetric price relationship. Journal of Development and Agricultural Economics, 2(1), 1–6.Google Scholar
  2. Burnham, K. P., & Anderson, D. R. (2004). Multimodel inference: Understanding AIC and BIC in model selection. Sociological Methods & Research, 33, 261–304. Scholar
  3. Gardner, B. C., Burr, B. K., & Wiedower, S. E. (2006). Reconceptualizing strategic family therapy: Insights from a dynamic systems perspective. Contemporary Family Therapy, 28, 339–352. Scholar
  4. Gottman, J. M., & Notarius, C. (2000). Decade review: Observing marital interaction. Journal of Marriage and the Family, 62, 927–947. Scholar
  5. Gottman, J. M., & Notarius, C. (2002). Marital research in the 20th century and a research agenda for the 21st century. Family Process, 41(2), 159–197. Scholar
  6. Gottman, J., Murray, J., Swanson, C., Tyson, R., & Swanson, K. (2002). The mathematics of marriage: Dynamic nonlinear models. Cambridge, MA: MIT Press.Google Scholar
  7. Granic, I., & Hollenstein, T. (2003). Dynamic systems methods for models of developmental psychopathology. Development and Psychopathology, 15, 641–669.
  8. Granic, I., & Lamey, A. V. (2002). Combining dynamic systems and multivariate analyses to compare the mother-child interactions of externalizing subtypes. Journal of Abnormal Child Psychology, 30, 265–283. Scholar
  9. Hamaker, E. L., Zhang, Z., & Van der Maas, H. L. J. (2009). Using threshold autoregressive models to study dyadic interactions. Psychometrika, 74, 727–745. Scholar
  10. Kass, R. E., & Raftery, A. E. (1995). Bayes factors. Journal of the American Statistical Association, 90(430), 773–795. Scholar
  11. Kass, R. E., & Wasserman, L. (1995). A reference Bayesian test for nested hypotheses and its relationship to the Schwarz criterion. Journal of the American Statistical Association, 90, 928–934. Scholar
  12. Liebovitch, L. S., Naudot, V., Vallacher, R., Nowak, A., Bui-Wrzosinska, L., & Coleman, P. (2008). Dynamics of two-actor cooperation-competition conflict models. Physica A, 387, 6360–6378. Scholar
  13. Liebovitch, L. S., Peluso, P. R., Norman, M. D., Su, J., & Gottman, J. M. (2011). Mathematical model of the dynamics of psychotherapy. Cognitive Neurodynamics, 5(3), 265–275. Scholar
  14. Madhyastha, T. M., Hamaker, E. L., & Gottman, J. M. (2011). Investigating spousal influence using moment to moment affect data from marital conflict. Journal of Family Psychology, 25(2), 292–300. Scholar
  15. Norcross, J. C. (2011). Psychotherapy realtionships that work: Evidence based responsiveness (2nd ed.pp. 3–21). New York: Oxford University Press.Google Scholar
  16. Nowak, A. (2004). Dynamical minimalism: Why less is more in psychology. Personality and Social Psychology Review, 8(2), 183–192. Scholar
  17. Peluso, P. R., Liebovitch, L. S., Gottman, J. M., Norman, M. D., & Su, J. (2012). A mathematical model of psychotherapy: An investigation using dynamic non-linear equations to model the therapeutic relationship. Psychotherapy Research, 22(1), 40–55. Scholar
  18. Vallacher, R. R., Coleman, P. T., Nowak, A., & Bui-Wrzosinska, L. (2010). Rethinking intractable conflict: The perspective of dynamical systems. American Psychologist, 65(4), 262–278. Scholar
  19. Vallacher, R. R., Coleman, P. T., Nowak, A., Bui-Wrzosinska, L., Liebovitch, L., Kugler, K., & Bartoli, A. (2013). Attracted to conflict: Dynamic foundations of destructive social relations dynamic foundations of destructive social relations. Berlin, Heidelberg: Springer.CrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Paul R. Peluso
    • 1
    Email author
  • Andrew Z. Baker
    • 2
  • Ashley Sauer
    • 2
  • Jennifer P. Peluso
    • 3
  1. 1.Department of Counselor EducationFlorida Atlantic UniversityBoca RatonUSA
  2. 2.The Alliance Lab, Department of Counselor EducationFlorida Atlantic UniversityBoca RatonUSA
  3. 3.Department of PsychologyKeiser UniversityFort LauderdaleUSA

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