From the issue dated April 25,
2003
http://chronicle.com/weekly/v49/i33/33a01401.htm
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Every Unhappy Family Has Its Own Bilinear Influence
FunctionResearchers propose a mathematical model of
marriage
By DAVID GLENN
A few years ago, an unhappily
married couple identified as "Angie" and "Dave" 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).
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Copyright ©
2003 by The Chronicle of Higher Education