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Dynamic Mathematical Nonlinear Modeling of Newly Wed Marital Interaction John Gottman, Catherine Swanson, and James Murray University of Washington

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Dynamic Mathematical Nonlinear Modeling of Newly Wed Marital Interaction John Gottman, Catherine Swanson, and James Murray University of Washington Journal of Family Psychology Copyright 1999 by the American Psychological Association, Inc. 1999, Vol. 13, No. 1,3-19 0893-32«V99/$3.00 The Mathematics of Marital Conflict: Dynamic Mathematical Nonlinear Modeling of Newly wed Marital Interaction John Gottman, Catherine Swanson, and James Murray University of Washington This article extends a mathematical approach to modeling marital interaction using nonlinear difference equations. Parameters of the model predicted divorce in a sample of newlyweds. The parameters reflected uninfluenced husband and wife steady states, emotional inertia, influenced husband and wife steady states, and influence functions. The model permits separation of uninfluenced parameters—that is, what is initially brought to the interaction by each person's personality or the relationship's history—from where the interaction heads once influence begins. In the present model, a theoretical shape of the influence functions is proposed that permits estimation of negative and positive threshold parameters. Couples who eventually divorced initially had more negative uninfluenced husband and wife steady states, more negative influenced husband steady state, and lower negative threshold in the influence function. The application of mathematics to the study concepts were tremendously influential in the of marriage was presaged by von Bertalanffy field of family therapy (see Rosenblatt, 1994). (1968), who wrote a classic and highly influen- However, they were never really subjected to tial book titled General System Theory. This experimental processes, and so they became book was an attempt to view biological and frozen and reified. Systems concepts came to be other complex organizational units across a considered true without evidence. This was wide variety of sciences in terms of the highly unfortunate, and it was the direct interaction of these units. The work was an consequence of not making the ideas testable or attempt to provide a holistic approach to at least disconfirmable. complex systems. Von Bertalanffy's enterprise Von Bertalanffy (1968) viewed his theory as fit a general Zeitgeist, with Wiener's (1948) essentially mathematical. He believed that the Cybernetics, Shannon and Weaver's (1949) interaction of complex systems with many units information theory, and VonNeumann and Mor- could be characterized by a set of values that genstern's (1947) game theory. Von Bertalanffy inspired many major thinkers of family systems change over time, denoted Qu Q2, Q3, and so on. and family therapy. Unfortunately, the mathemat- The Qs were variables each of which indexed a ics of general systems theory was never particular unit in the system, such as mother, developed, and theorists of family interaction father, and child. He thought that the system kept these systems concepts only at the level of could be best described by a set of ordinary metaphor. Even at the level of metaphor these differential equations of the following form: dg./dt = f,(fi,, Q2, ft,...), d<22/dt = f2(Q,, Q2, Q3,...), and so on. The terms on the left of the John Gottman and Catherine Swanson, Department equal sign are time derivatives, that is, rates of of Psychology, University of Washington; James change of the quantitative sets of values Qu Q2, Murray, Department of Applied Mathematics, Univer- and so on. The terms on the right of the equal sity of Washington. Correspondence concerning this article should be sign are functions, fb f2,.... of the Qs. There addressed to John Gottman, Department of Psychol- was no suggestion of how to operationalize the ogy, Box 351525, University of Washington, Seattle, "g-variables." Von Bertalanffy thought that the Washington 98195. functions, the fs, might generally be nonlinear. GOTTMAN, SWANSON, AND MURRAY The equations that von Bertalanffy selected have possible to talk about "qualitative" mathemati- a particular form, called "autonomous," mean- cal modeling. In qualitative mathematical mod- ing that the fs have no explicit function of time eling, one searches for solutions that have in them, except through the Qs, which are similarly shaped phase space plots, which functions of time. provide a good qualitative description of the Von Bertalanffy thought that for practical solution and how it varies with the parameters. solution, the equations had to be linear. He There are many excellent introductions to this presented a table in which these nonlinear general approach to qualitative nonlinear dy- equations were classified as "impossible" (von namic modeling and its subtopics of chaos and Bertalanffy, 1968, p. 20). Rapoport (1972) catastrophe theory (e.g., Baker & Gollub, 1996; suggested that investigators apply von Bertalan- Beltrami, 1993; Berge, Pomeau, & Vidal, 1984; ffy's linear equations to an analysis of marriage, Glass & Mackey, 1988; Gleick, 1987; Lorenz, but he presented no data, no suggestion of how 1993; Morrison, 1991; Peters, 1991; Vallacher to operationalize von Bertalanffy's Q-variables, & Nowack, 1994; Winfree, 1990). An introduc- and no explanation of how to apply the tion to catastrophe theory may be found in equations to real data. Unfortunately, linear works by Arnold (1986), Castrigiano and Hayes equations are not generally stable, so they tend (1993), Gilmore (1981), and Saunders (1990). to give erroneous solutions, except as approxima- The currently vast and expanding area of tions under very local conditions near a steady mathematical biology was introduced by Mur- state. Von Bertalanffy was not aware of the ray's (1989) classic text. mathematics that Poincare and others had Our modeling of marital interaction using the developed in the last quarter of the 19th century mathematical methods of nonlinear difference for the study of nonlinear systems. The model- equations is an attempt to integrate the math- ing of complex deterministic (and stochastic) ematical insights of von Bertalanffy with the systems with a set of nonlinear difference or general systems theorists of family systems differential equations has now become a produc- (Bateson, Jackson, Haley, & Weakland, 1956) tive enterprise across a wide set of phenomena using nonlinear equations. In modeling marital and across a wide range of sciences, including interaction, Cook et al. (1995) developed an the biological sciences. The use of nonlinear approach that used both the data and the equations forms the basis of our modeling of mathematics of differential or difference equa- marital interaction. tions in conjunction with the creation of Other than the possibility of stability, an qualitative mathematical representations of the advantage of nonlinear equations is that by forms of change. Our approach was unique using nonlinear terms in the equations of change because the modeling itself generated the some very complex processes can be repre- equations, and the objective of our mathematical sented with very few parameters. Unfortunately, modeling was to generate theory. We suggested unlike many linear equations, these nonlinear that the data be used to guide the scientific equations are generally not solvable in closed intuition so that equations of change were functional mathematical form. For this reason, theoretically meaningful. the methods are often called "qualitative," and In finding a practical example of von visual graphical methods and numerical approxi- Bertalanffy's g-variable, we used a report by mation must be relied on. For this purpose, Gottman and Levenson (1992) that one variable numerical and graphical methods have been descriptive of specific interaction patterns of the developed such as phase space plots. These balance between negativity and positivity was visual approaches to mathematical modeling predictive of marital dissolution. Gottman and can be very appealing for engaging the intuition Levenson used a methodology for obtaining of a scientist working in a field that has no synchronized physiological, behavioral, and mathematically stated theory. If the scientist has self-report data in a sample of 73 couples who an intuitive familiarity with the data of the field, were followed longitudinally between 1983 and our approach may suggest a way of building 1987. Applying observational coding of interac- theory using mathematics in an initially qualita- tive behavior, they computed, for each conversa- tive manner. The use of these graphical solutions tional turn, the number of positive minus to nonlinear differential equations makes it negative speaker codes and plotted the cumula- MARITAL CONFLICT tive running total for each spouse. An observa- change of the partner. If, at time t, the husband's tional system (the Rapid Couples Interaction quantitative variable is represented by H, the Scoring System [RCISS]) was used to code the wife's variable by Wt, and r represents the salient behaviors; this system had been devel- emotional inertia parameter, the difference oped by reviewing literature on the behavioral equations are as follows: correlates of marital satisfaction. A behavior was considered negative if it correlated nega- Wt+l =a + r.W, + IHW(H,), (1) tively with marital happiness and positive if it correlated positively (Krokoff, Gottman, & Hl+l = b + r2Ht + IWH(Wt). (2) Haas, 1989). The slopes of these plots deter- mined a risk variable, low risk if both husband's The parameters a and b and the rs were and wife's graphs had a positive slope and high estimated from the data. The constants a and b risk if not. Computing the graph's slope was are related to the initial uninfluenced level of guided
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