The Chronicle: 4/25/2003: Every Unhappy Family Has Its Own Bilinear Influence Function

Researchers propose a mathematical model of marriage

By DAVID GLENN

A few years ago, an unhappily married couple identified as "Angie" and "Dave"

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Quantifying Conversations

visited the University of Washington at Seattle's Family Research Laboratory, better known on that campus as the Love Lab. Using techniques he first developed in 1979, John M. Gottman, a professor emeritus of psychology there, recorded the couple as they had a difficult conversation, carefully noting their emotional signals each time they spoke.

Dave: "We got to keep the communication lines open, we got to try to figure out how we can work on it. Some bills we got we can't pay off."

Angie: "The bills aren't the issue."

Dave: "Usually it's how can I try to please you and how you"

Angie: "No, I can please myself. I get happiness from my job, I get happiness from work, I get happiness from you, but I feel stifled ..."

In the last 24 years, Mr. Gottman and his colleagues have recorded thousands of such conversations, using careful techniques to measure and notate the participants' emotions each step of the way. After Angie and Dave's talk, Mr. Gottman says, his assistants reviewed a videotape, scoring each sentence and facial expression on such measures as disgust (-3), affection (+4), whining (-1), and contempt (-4). (Angie's grimace as she said "The bills aren't the issue" was scored as contempt.)

During the 1980s, Mr. Gottman gained prominence by mining such laboratory data for insights into the quandaries of marriage. He has argued, for example, that a crucial predictor of divorce is a husband's inability or unwillingness to be influenced by his wife's suggestions and emotional expressions. This husbandly tone-deafness, he says, augurs divorce much more strongly than, say, the general level of anger expressed in the marriage. In longitudinal studies of couples who have passed through the Love Lab, Mr. Gottman and his colleagues have successfully predicted which ones will divorce with greater than 90-percent accuracy.

Despite those successes, in the early 1990s Mr. Gottman found himself feeling anxious about his scholarly legacy. "My wife was pregnant, and we had one of those heart-to-heart talks," he recalls. "I said, I'm just not going to feel complete unless I can get back to my mathematical roots. I didn't really have respect for my work unless I could put it on a mathematical footing." (Mr. Gottman intended to earn a Ph.D. in mathematics at the Massachusetts Institute of Technology in the late 1960s, but switched fields because he found his psychologist roommate's textbooks to be much more interesting than his own.)

A decade later, Mr. Gottman believes he has found that footing. In The Mathematics of Marriage: Dynamic Nonlinear Models (MIT Press), which he wrote in collaboration with four mathematicians, Mr. Gottman uses the tools of calculus to describe the interactions of couples like Angie and Dave. The models presented in the book, he says, offer insights into the heaven and hell of couplehood that he would never have found by sifting through his data with standard linear statistical tools. He has already begun to apply those insights in his therapeutic work -- including with Angie and Dave themselves, whose conversations are transcribed in the book.

The Stars Were Aligned

The germ of The Mathematics of Marriage was a remarkable piece of luck. Around the time of the heart-to-heart conversation with his wife, Mr. Gottman forgot to send in a reply card to a scientific book-of-the-month club, and therefore received a book he'd never heard of: Mathematical Biology (Springer-Verlag, 1989), by James D. Murray, a professor of mathematics at the University of Oxford. The book explains how to use nonlinear equations to illuminate the mechanics of complex dynamic systems, such as the growth of brain tumors.

"The book was so different from all the other books I'd been reading in applied mathematics," Mr. Gottman says. "Finally, concepts like catastrophe theory were very clear. I understood what it all meant." Mr. Gottman sent a letter to Mr. Murray in Oxford -- but the reply came from just five blocks away. Mr. Murray had retired early from Oxford and moved to Seattle; he was teaching at Mr. Gottman's own university.

The two men met for lunch. "I thought John Gottman's ideas about having mathematics involved [in his marriage studies] were ridiculous, and I told him that," recalls Mr. Murray. "But by the end of the lunch, when I saw what he had in mind, I was totally hooked."

A similar process ensued when Mr. Murray recruited some of his graduate students to work on the project. "They raised their eyebrows," he says. Six months later, however, "you couldn't go into the lounge in applied mathematics without getting drawn into arguments about how to write these marriage equations. It sort of took over the department for a while." (Three of those students are, with Mr. Gottman and Mr. Murray, the book's authors: Kristin R. Swanson, now a research assistant professor of mathematics in the department of pathology at the University of Washington Medical Center; Catherine C. Swanson, now a software engineer at the university; and Rebecca Tyson, now an assistant professor of mathematics at Okanagan University College, in Kelowna, British Columbia.)

What the students were modeling in that lounge was not "marriage" per se, but the dynamics of marital conversations. Before looking at data from any real-world couples, they began with some very simple hypotheses: the idea, for example, that spouses will react emotionally to the most recent comment made by their partners. At this early stage they sketched crude "influence functions" -- calculus equations that described a dynamic system in which a snarky comment by one spouse would result in negative emotions in the partner, sometimes resulting in a downward spiral. When they tested those first equations against the Love Lab's data, however, they did not match at all.

The scholars soon realized that they needed to add a constant that represented each partner's "uninfluenced steady state" -- that is, the person's general level of cheerfulness or gloom, independent of the spouse's behavior on a particular day. "In retrospect, we should have thought of that at the very beginning," says Mr. Murray. "But once we added that constant, everything fell in just beautifully."

Mood-Altering Powers

The researchers now had equations that fit Mr. Gottman's data quite closely. They used these to develop bilinear influence functions, which describe a person's ability to affect his or her spouse's mood. Some couples, which Mr. Gottman calls "volatile," routinely unleash anger at one another -- but offset that anger with even larger doses of warm feelings. Those couples tend to be stable and successful, Mr. Gottman says, because they are able to influence one another with both anger and affection. (The influence functions of "conflict avoiding" couples look very different.)

The scholars also constructed a more sophisticated variant, using equations known as ojive functions. This second model incorporates tipping points and threshold effects. "The idea here," says Ms. Tyson, "is that you have to be negative enough before you make your partner negative, and positive enough before you make your partner positive." Ojive functions fit the Love Lab's data better still. "All of a sudden we had a language for describing what was going on that appeared to be quite universal," says Mr. Gottman.

A final innovation came in 1998, when the team added the concepts of "repair" and "damping" to their models. Those terms refer to spouses' conscious attempts to turn difficult conversations in positive directions (repair) or negative ones (damping). Repair attempts typically include jokes, soothing comments, or changes of subject; damping generally involves making hurtful comments even when one's partner is clearly trying to be positive. "Without repair and damping, we knew that the bilinear influence functions weren't going to work," says Kristin Swanson, "because the equations allowed people to just spontaneously combust with happiness or spontaneously combust with unhappiness. The repair and damping concepts don't completely fix that, but they mediate it."

So what's the cash value of all this mathematics? What insights has it offered Mr. Gottman that he couldn't have derived through ordinary intuition, or by analyzing his data with humdrum statistical techniques? "I believe it was Lord Rutherford who said, If you need to use statistics, then you should design a better experiment," says Mr. Murray. "Statistics will just give you the bare facts. If you want to understand why a dynamic system behaves as it does, then you need to use nonlinear tools."

Perhaps the most important insight generated by these marriage models is that particular couples tend to have more than one style of emotional connection. In 1968, the theoretical biologist Ludwig von Bertalanffy suggested that couples might have a homeostatic emotional set point, just as our metabolism regulates our weight around a particular set point. That metaphor has since been taken up by many family theorists.

Mr. Gottman and his colleagues believe, however, that the metaphor of homeostasis is misleading, because couples tend to have multiple set points, not just one. "Volatile" couples, for example, tend to shift between highly positive and highly negative exchanges, without stopping in between on some neutral ground. "The concept of homeostasis doesn't easily describe the potential existence of sudden, catastrophic change," the authors write, "or any other 'bifurcations' in which the system has two possible paths it can take, depending on the critical threshold values of the parameters. So again we need mathematics."

Moreover, says Mr. Gottman, "The mathematical model draws you to doing certain types of experiments that you wouldn't have thought of doing without the model." The nonlinear equations suggest particular ways for therapists to teach distressed couples to snap out of it when they find themselves in a negative steady state. Mr. Gottman and his colleagues have recently tried several such techniques. In one of these experiments, the scholars trained people to short-circuit their feelings of contempt by thinking and talking about their partners' admirable qualities before beginning a difficult conversation.

With Angie and Dave, Mr. Gottman and his colleagues used a method they term "dreams within conflict." In this technique, the therapist encourages the couple to discuss the "hidden life dreams" that often lie behind conflicts over prosaic subjects such as money. Rather than encouraging conflict resolution or "active listening," which more orthodox schools of marital therapy emphasize, Mr. Gottman wants couples to find ways of validating each other's deepest interests and anxieties. Angie, for example, worries that marriage is eroding her earlier sense of herself as a "free spirit," while Dave maintains a self-image as a level-headed family provider. Teaching couples to talk to each other about such anxieties, the authors write, can help "change the direction of [their] influence functions," by breaking the downward cycle of difficult conversations: contempt followed by defensiveness followed by withdrawal.

The goal of these therapeutic interventions is quite limited, according to Mr. Gottman. "We want to see if we can make their next conversation on the same topic better than their last one." He hopes to learn how to quickly give couples cognitive and behavior skills that will prevent their conversations from cascading into negativity.

Whether any of this leads to lasting innovations in real-world marriage therapy remains to be seen. Robert L. Weiss, a professor emeritus of psychology at the University of Oregon and a pioneer of observational studies of marriage, says that Mr. Gottman has a reputation for moving quickly from one enthusiasm to another. "He finds gems, but then sometimes builds articles around a small amount of initial evidence." Mr. Weiss believes, however, that the new book will be an enduring contribution. "My gut feeling is that John will turn out to be more right than wrong ... he's got some very interesting constructs here."

Psychological applications of dynamic nonlinear models are quite rare, according to Steven H. Strogatz, a professor of theoretical and applied mechanics in the College of Engineering at Cornell University and the author of the new book Sync: The Emerging Science of Spontaneous Order (Theia).

"There's definitely a community of people out there who are interested in the interaction between two people as a dynamical system," he says, citing a recent mathematical study of psychotherapists and their patients. But on the whole, he continues, such studies still face skepticism. "Biologists, who by definition don't like math -- they're people who like science but didn't want to study math, right? -- have been resistant to mathematical biology, and often wonder whether the equations give you anything more than a redescription of what you already knew from common sense. And so far, there are really only perhaps 10 clear-cut cases where the mathematical modeling has provided fundamentally new insights."

But even if Mr. Gottman's nonlinear models never transform marital therapy, Mr. Strogatz says, they are likely to prove a useful exercise. "The mere fact of writing equations forces you to be clear about your hypotheses. With nonlinear modeling there's nothing hidden."

Mr. Gottman retired from the university last fall and has established an independent laboratory in Seattle called the Relationship Research Institute. For his next project, which he calls "bringing baby home," Mr. Gottman is studying the effects of newborns on married couples' emotional equilibriums. Because these models will involve three actors rather than two, he will be able to experiment with new mathematical tools. In a declaration that will probably come as no surprise to the families in his study, he says, "Adding the baby allows you to get into chaos theory."

QUANTIFYING CONVERSATIONS

John M. Gottman and his colleagues have developed "influence functions" to illustrate how the partners in various types of couples influence each other's moods over the course of a difficult conversation. The horizontal axis represents a range of verbal and facial expressions -- from highly critical and contemptuous (on the left) to highly supportive and affectionate (on the right).

The vertical axis represents the degree to which a person is influenced positively (up) or negatively (down) by his or her spouse's behavior. For example, the husband in a "validating" couple tends to be strongly influenced (in a negative direction) by his wife's negative behavior: That line has a slope of 0.31. The husband in a "conflict avoiding" couple, on the other hand, actually tends to behave slightly more positively when his wife is negative: That line has a slope of 0.05.

Both "validating" couples and "conflict avoiding" couples tend to have marriages that are stable and long lasting. Couples are in trouble, though, when they have mismatched influence functions. Consider the "hostile detached" couple, where the slopes for the husband's and the wife's influences look very different. When the husband behaves positively, he has almost no impact (slope of 0.02) on his detached wife.

SOURCE: The Mathematics of Marriage: Dynamic Linear Models (MIT Press); graphic adapted By Dave Allen

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Volume 49, Issue 33, Page A14

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