Using dynamic nonlinear mathematical models to explain patterns of marital interaction and the failure of marital therapy

Journal of Sex Research, August, 2004 by Walter R. Schumm, Jennifer Ripley

The Mathematics of Marriage: Dynamic Nonlinear Models. By John M. Gottman, James D. Murray, Catherine C. Swanson, Rebecca Tyson, and Kristin R. Swanson. Cambridge, MA: The MIT Press, 2002, 403 pages. Hardcover, $42.95.

This book shows how dynamic nonlinear modeling can be used to describe intimate relationship interaction in a variety of situations. The idea of using differential equations to model dyadic interaction is not new--Broderick and Smith (1979) developed the concept many years ago--but Gottman and his colleagues have taken it to a new depth in The Mathematics of Marriage. The complexity of this book limits the ability of any reviewer to do an adequate job of responding; our first review was three times as long as permitted! Here we greatly abbreviate our summary of the book to focus on its limitations.

Gottman and his colleagues begin by arguing that we have learned little about what makes marriages work and that the benefits, if any, of marital therapy are usually short-lived. They argue that the "field of marriage research is in desperate need of theory" (p. xi). Their answer is the use of mathematical models inspired by recent works from Murray (1989) and others. They believe that their work "represents the missing step necessary to complete the seminal thinking that the family general systems theorists started in the 1950s" (p. xiii). However, recent reviews have indicated moderate effect sizes for treatment (Denton, Burleson, Clark, Rodriquez, & Hobbs, 2000; Dunn & Schwebel, 1995). A primary contribution of this book, though, is more specific identification of how interventions could be targeted to improve relationship processes.

Chapter 2 is a solid review of the literature on marital interaction (also see Gottman & Notarius, 2000, 2002). Chapters 3 through 9 explain the development and application of dynamic nonlinear models. On page 37 the authors state, "The amount of mathematics needed is relatively minimal." However, even though I (Schumm) had six semesters of calculus in college, I was on the verge of being lost after chapter 7. Chapter 4 does include a basic review of calculus and differential equations. Chapter 5 concludes with an illustration of how a hyperbolic tangent function could be used to model regulation of affect within a marriage, based on three control parameters.

Chapter 6 deals with models of catastrophic change. The chapter begins with the spruce budworm problem, which models the impact of a predator, the spruce budworm, on the population development of its food, the leaves of the balsam fir tree. The authors carry this theme over into marriage with respect to the regulation of negative affect: You want neither the elimination of negative affect nor its runaway escalation. They compare courtship (no budworms), which features an impossibly low state of negative affect coupled with high positive affect that is unstable (and we know that limerance fades in all types of relationships within a year or two), a balanced condition (some budworms) with more positive affect than negative, and a runaway condition of escalating negative affect (budworms take over) that usually predicts divorce. The authors believe that repair attempts are the key to controlling outbreaks "of runaway negative affect" (p. 89). The key to repair attempts is seen as the quality of the marital friendship, probably positive sentiment override.

Chapter 7 deals with Poincare's phase space plots, based on two-equation models. From this perspective, therapy becomes the challenge of moving a couple from stable negative conditions to stable positive conditions. Chapter 8 develops more complex two-equation models concluding with models of competition and cooperation.

In Chapter 9, the authors present several new variables to help describe marital interaction. Influenced behavior is behavior influenced by one's spouse, whereas uninfluenced behavior is behavior caused by a person's own characteristics. Baseline temperament, also called the uninfluenced set point, refers to each person's emotional steady state independent of other factors, such as ongoing interaction. Some spouses are happier or sadder than others, everything else being equal. Emotional inertia refers to a person's tendency to remain in the same emotional state for a period of time in spite of current interactions. The equations used by the authors predict the Gottman-Levenson variable (GLV) for one spouse at Time 1 as a function of the other's influence plus inertia times the GLV at Time 0 plus a constant (initial emotional status). In plain English, a person's affective state is a function of his or her spouse's affect, his or her own previous affect dampened by his or her tendency to cool off or heat up, and his or her general affective tendency.

One error associated with Gottman's work so far (it seems to us) is that he focused strictly on interaction without looking at the context of higher order aspects of the marital system, including how the system revises interaction rules, how it creates totally new rules, how it changes values or goals, and how it develops totally new values and/or goals, as discussed by Broderick and Smith (1979). Another limitation is that Gottman relies exclusively on marital happiness or satisfaction as outcome variables while overlooking joy as an outcome variable, where we would hypothesize that joy is related more to how the couple shares transcendent values or goals in life (see Schumm, 1999).