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

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).

In some ways, Gottman's ideas are not new. Gottman's idea of the initial state of the person as an important factor may be underlying the warning about marrying an "unloved woman" (Proverbs 30: 23a), someone who brings low self-esteem or even a background of parental abuse or neglect to a marriage. Concerns about negative start-up often leading to escalating negativity seem to be echoed in warnings such as "Do you see a man hasty in his words? There is more hope for a fool than for him," and in concerns about aversive, hasty, and thoughtless words that evoke further contention (Proverbs 12:15; 15:18, 15:28; 16:32; 17:14; 20:3; 29:20, 29:22). The idea of using soothing to calm an argument and using editing of one's words to avoid escalation of negative affect seem to be contained in the proverb, "a soft answer turns away wrath" (Proverbs 15:1; 17:27; 25:11). The greater influence of negative affect relative to positive may underlie the frequent discussions of the aversiveness of spousal hostility (Proverbs 21:9, 21:19; 25:24). The importance of some negative affect may underlie the need to rebuke a friend and the need for wise persons to heed such rebuke rather than reject it (Proverbs 17: I0; 19:20; 27:5; 28:23; 29:1). Acting so as to build up a reservoir of trust, positive sentiment, and friendship that leads to positive levels of the "influenced steady state" may be related to the exhortation to do good and not evil to others all the days of your life (Proverbs 31:11-12).

Toward the end of Chapter 9, Gottman et al. show how shifts in uninfluenced negativity by either spouse can create conditions in which there are only negative stable states, that the only possible outcome of interaction is a condition of an imbalance of greater negative affect.

Chapter 10 describes five new couple types in terms of their influence functions, concepts we found confusing. Nevertheless, the point appears to be that couples with matched influence functions appear to find stable, happy states, whereas those with mismatches tend toward stable, unhappy states (Glenn, 2003). Chapter 10 is also very confusing because the labels for tables in the narrative do not appear to match the labels on the tables themselves. Gottman concludes the chapter on the grim note that only a small percentage of couples (4-14%) were able to reverse initial negative conditions and establish a positive trend.

Chapter 11 begins with a critique of the limitations of general systems theory; Gottman argues that its "nonquantitative use has not led anywhere scientifically productive" (p. 168). He argues that it casts the therapist at odds with the family, assumes too few stable states in the marriage, overlooks the potential for catastrophic change, and sets couples up for disappointment by failing to ensure long lasting change. "To avoid this intolerable state of affairs, in which empirical findings are of no value and in which the ideas live on in spite of truth, we need mathematics" (p. 171).

In chapter 12, the models are applied to longitudinal studies of newlyweds. Gottman hypothesizes that unhappy couples may adapt too readily to negativity, ignoring it until it gets too much to bear, by which time it is much more difficult to resolve. As in chapter 10, the tables and their numbers in the narrative do not match, creating more confusion. Of the nearly 40 statistical tests conducted, only 8 were always highly significant (p < .001) predictors of marital outcomes. The results for the threshold hypothesis were not conclusive in our opinion.

Chapter 13 turns to how couples repair or exit dysfunctional interactions. Gottman admits that even happy, stable couples have some negative interactions. It turned out that positive sentiment override (PSO) and negative sentiment override (NSO) explained the success of repair attempts. PSO occurs when one spouse says something with negative affect as observed by others but the partner interprets it as neutral; NSO occurs when a neutral message is received as negative. The source of PSO is positive affect in nonconflict situations, what the authors call the emotional bank account model, "an index of the quality of the friendship between husband and wife" (p. 191). So far, so good. However, the rest of the chapter focuses on repair and dampening processes. Even though the empirical results did not appear to support the importance of those processes, the mathematical models are still developed in great detail.

Chapter 14 extends the model to gay and lesbian relationships. Gottman concludes that compared to heterosexuals, gay male and lesbian couples begin conflicts more positively and are more willing to accept influence, which keeps the conversation positive. We would interpret the results differently: ceteris paribus, it is easier to communicate with a person of the same gender, partly because of physiological similarities and partly because of greater socialization experience with members of the same gender.

Chapter 15 expands the models to include physiological and perceptual measures. Forgiveness--which was discussed by Broderick and Smith (1979), has an important role in some marital therapy (DiBlasio, 2000), and may be involved with Gottman's "repair" mechanisms--receives surprisingly little attention as part of the authors' differential equations. Another possibility is that very high levels of affective commitment to marriage, including sexual intimacy (Schumm, Bollman, Jurich, & Hatch, 2001), may help boost uninfluenced steady states and sustain positive marital interaction (Ripley, Worthington, Bromley, & Kemper, 2000).

Chapter 16 is an implications chapter about what needs to be done in therapy. At the most global level, what needs to be done is to increase positive affect and reduce negative affect during and before conflict. Creating positive sentiment override (as an outcome of a strong, positive marital friendship) is a primary goal. Gottman believes that positive sentiment override is the main way in which couples can regulate conflict; that without it, negative affect invariably leads to a downward spiral of negative paybacks. Gottman concludes, "Unfortunately, positive affect during conflict resolution is generally not programmable by intervention" (p. 300). However, three other intermediate goals are useful.

First, therapists should reduce negative affect reciprocity by teaching more effective repair techniques, reducing harsh starting comments in discussions, reducing defensiveness, and increasing psychological soothing. Second, they should increase the positivity of the uninfluenced steady state (making life more positive before the couple interacts extensively) by reducing harsh start-up and by replacing contempt with affection and admiration. Third, they should make the influenced steady state more positive than the uninfluenced steady state (i.e., each spouse is happier after interacting with the other than each was before) by teaching how to respond to anger constructively, enhancing emotional connections, and helping couples to understand and value the meaning of each other's goals and dreams.

Stanley, Bradbury, and Markman (2000) have previously commented on some of the methodological limitations of Gottman's research. In this book, the authors conduct so many statistical tests that some statistically "significant" results are probably products of chance. A further limitation is that most of the comparison groups are created by nonrandom assignment. There is some question as to whether Gottman's models would generalize to cross-cultural situations. Nevertheless, this work is a magnificent contribution to the field of family psychology and family studies.

REFERENCES

Broderick, C., & Smith, J. (1979). The general systems approach to the family. In W. R. Burr, R. Hill, F. I. Nye, & I. L. Reiss (Eds.), Contemporary theories about the family (Vol. 2, pp. 112-129). New York: Free Press.

DiBlasio, F. (2000). Decision-based forgiveness treatment in cases of marital infidelity. Psychotherapy: Theory, Research, Practice, Training, 37, 149-158.

Glenn, D. (2003, April 25). Bilinear influence function: Researchers propose a mathematical model of marriage. The Chronicle of Higher Education, A14-A15.

Gottman, J. M., & Notarius, C. I. (2000). Decade review: Observing marital interaction. Journal of Marriage and the Family, 62, 927-947.

Gottman, J. M., & Notarius, C. I. (2002). Marital research in the 20th century and a research agenda for the 21st century. Family Process, 41, 159-197.

Murray, J. D. (1989). Mathematical biology. Berlin: Springer-Verlag.

Ripley, J. S., Worthington, E. L., Bromley, D., & Kemper, S. (2000). Values Orientation towards Wedlock or Self-interest (VOWS) Scale: Convenantal and contractual values in marriage. Paper presented at the conference of the American Psychological Association, Washington, DC

Schumm, W. R. (1999). Satisfaction. In D. Levinson, J. J. Ponzetti, Jr., & P. F. Jorgensen (Eds.), Encyclopedia of human emotions (Vol. 2, pp. 583 590). New York: MacMillan.

Schumm, W. R. (2003). Comments on marriage in contemporary culture: Five models that might help families. Journal of Psychology and Theology, 31,213-223.

Schumm, W. R., Bollman, S. R., Jurich, A. P., & Hatch, R. C. (2001). Family strengths and the Kansas Marital Satisfaction Scale: A factor analytic study. Psychological Reports, 88, 965-973.

Stanley, S. M., Bradbury, T. M., & Markman, H. J. (2000). Structural flaws in the bridge from basic research on marriage to interventions for couples. Journal of Marriage and the Family, 62, 256-264.

Walter R. Schumm, Ph.D., 5900 Lake Elbo Road, Manhattan, KS 66502-8996; e-mail: [e-mail]; and Jennifer Ripley, Ph.D., Regent University Psy.D. Program, 1000 Regent University Drive, Virginia Beach, VA 23464; e-mail: [e-mail].