Using Threshold Autoregressive Models to Study Dyadic Interactions

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.Key words: dynamic system, TAR model, autoregressive, dyadic interaction.The notion of a dynamic system has become increasingly popular in psychology, both as a metaphor and as a quantitative approach. Recently, there has been a burst of applications, for instance in the study of development (Thelen & Smith, 1994;Van der Maas & Molenaar, 1992;Van Geert & Van Dijk, 2002), self-regulation of behavior (Carver & Scheier, 1998), personality (Shoda, Tiernan, & Mischel, 2002), addiction (Warren, Hawkins, & Sprott, 2003;Witkiewitz, Van der Maas, Hufford, & Marlatt, 2007), grief (Bisconti, Bergeman, & Boker, 2004), psychopathology (Granic & Hollenstein, 2003), and psychotherapy (Schiepek, 2003). In addition, several leading journals have devoted special issues to the topic (Vallacher & Nowak, 1997;Vallacher, Read, & Nowak, 2002), and in 1997 a new journal called Nonlinear Dynamics, Psychology, and Life Sciences was started (Guastello, 1997).Mathematically, a dynamic (or dynamical) system is a set of equations that expresses how the state of a system (represented by one or more variables) changes as a function of its previous state. These equations may be deterministic or stochastic (Clayton, 1997), and they can be defined in continuous time as differential equations, or in discrete time as difference equations. A particularly useful application of dynamic system theory in psychological research concerns the study of dyadic interactions: Representing each partner by one or more variables and measuring them repeatedly over time allows the researcher to determine whether and how the partners influence each other.

Modeling Nonstationary Emotion Dynamics in Dyads using a Time-Varying Vector-Autoregressive Model

“…Examples of such new models include extensions of the dynamic factor model with time-varying parameters using a state space approach (Chow, Zu, Shifren, & Zhang, 2011;Molenaar, De Gooijer, & Schmitz, 1992), an extension of the (multilevel) vector-autoregressive (VAR) model using threshold parameters representing, for example, emotion dynamics under decreased and increased negative affect (threshold autoregressive models; Haan-Rietdijk, Gottman, Bergeman, & Hamaker, 2014;Hamaker, Zhang, & Maas, 2009;Madhyastha, Hamaker, & Gottman, 2011), and regime switching models, in which different states of emotion dynamics can be specified (Frühwirth-Schnatter, 2006;Hamaker, Grasman, & Kamphuis, 2010;Stifter & Rovine, 2015). Additionally, exploratory tools have been developed to discover which aspects or periods of dyadic interactions show similar patterns (Boker, Rotondo, Xu, & King, 2002;Ferrer, Steele, & Hsieh, 2012;Hsieh, Ferrer, Chen, & Chow, 2010).…”

Investigating spousal influence using moment-to-moment affect data from marital conflict.

“…However, the novel estimation method that was recently developed by Hamaker, Zhang, and Van der Maas (2009), is based on maximum likelihood estimation, which implies it is possible to compute the Bayesian Information Criterion (BIC; Schwarz, 1978). The BIC is a well-known criterion that can be used to compare multiple nested and nonnested models simultaneously.…”

Capturing Context-Related Change in Emotional Dynamics via Fixed Moderated Time Series Analysis

“…(Chow & Zhang, 2013;Dolan et al, 2004;Hamaker & Grasman, 2012;Hamaker, Zhang, & van der Maas, 2009;Hamilton, 2010;Hunter, 2014a;Kim & Nelson, 1999). These evoke a small number of distinct parameter sets (i.e., dynamic regimes) between which the outcome process switches over time according to a discrete-valued parameter process, for instance a Markov chain.…”