a general systems theory of marriage: nonlinear difference...

A General Systems Theory of Marriage: Nonlinear Difference EquationModeling of Marital Interaction

John Gottman, Catherine Swanson, & Kristin SwansonDepartment of PsychologyUniversity of Washington

This article describes a new mathematical approach for modeling the prediction of di-vorce or marital stability from marital interaction using nonlinear difference equa-tions. The approach is quite general for modeling social interaction, and can be appliedto any time series data generated over time for two individuals. We pursued a balancemodel in selecting the dependent variables of this modeling. Both the mathematicalmethods and the theoretical gains obtained when using this approach are reviewed.

What is the Modeling Trying toAccomplish?

In our laboratory’s multi-method research on mari-tal interaction we have shown, in four longitudinalstudies, that we can predict with over 90% accuracywhether a couple will divorce or stay married, and theirmarital satisfaction if they do stay married (Gottman,1994; 1999). Furthermore, we were able to predict inthree measurement domains: the couple’s interactivebehavior, their perception of their interaction, and theirphysiology during the interaction. The prediction inthe behavioral domain was based largely on codingpositive and negative emotions.

What we started doing nine years ago, in collabora-tion with the mathematical biologist James Murray andhis students, was to build a mathematical model forthese predictions. Unlike other problems in mathemati-cal biology, in which it was possible to write the differ-ential equations from existing theory, we had no lawsand no mathematical theory in the area of marriage. Forthat reason the goal of our efforts became the construc-tion of theory. We wound up developing a new languagefor social interaction and a theory that utilizes that lan-guage to attempt to understand our predictions.

The mathematically-based theory we developedmade it possible to simulate the couple’s behavior un-der conditions we had never observed them in, and so itled to the idea of conducting proximal change experi-ments. Instead of intervening to change the entire mar-riage, these proximal change experiments had as theirgoal changing a particular parameter of the mathemati-cal model for that couple. Thus, the study of marriagewas brought into the social psychology laboratory.

This article is an introduction to the work we did tobuild that mathematical theory. First, let us briefly re-view research on marriage and marital interaction.

Observational Research on MaritalRelationships: Brief Review

Psychology was a late comer to the study of mar-riage. Sociologists had been studying marriages for 35years before psychologists got wind of the idea, al-though the first published study on marriage was con-ducted by a psychologist, Louis Terman in 1938(Terman, Buttenweiser, Ferguson, Johnson, & Wilson,1938). Terman’s question was “What are the correlatesof marital happiness and misery?” What psychologistsinitially brought to the study of marriage in the 1970swas the use of observational methods, the design andevaluation of intervention programs, and an unbridledoptimism. These contributions have had an enormousimpact on the study of marriage. Psychology was ini-tially skeptical about studying marriage, in part be-cause personality theory was facing a severe challengein the 1970s from the work by Mischel (1968). In thatbook, Mischel reviewed research on personality andsuggested that personality theory had come far short ofbeing able to predict and understand behavior. He con-cluded that correlations were quite low on the whole,that the field was plagued with common method vari-ance (mostly self-reports predicting self-reports), andthat the best predictors of future behavior were past be-havior in similar situations. This book was a great stim-ulus to many researchers. It encouraged a new look atpersonality measurement, validity, and reliability(Wiggins, 1973). It stimulated new kinds of research inpersonality. However, it also contributed to a pessimis-tic view that research in interpersonal psychologywould have very little payoff. In hindsight, this pessi-mism was wrong. We are learning that, in fact, much of

Personality and Social Psychology Review2002, Vol. 6, No. 4, 326–340

Copyright © 2002 byLawrence Erlbaum Associates, Inc.

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Requests for reprints should be sent to John Gottman, JamesMifflin Professor, University of Washington, Box 351525, Depart-ment of Psychology, Seattle, WA 98195. E-mail: [email protected]

the order in individual personality exists at the inter-personal level. Patterson (1982), in his conclusion thatthere is a great deal of consistency across time and situ-ations in aggression, suggested that the aggressive traitought to be rethought in interpersonal terms as the ag-gressive boy’s recasting people in his social world toplay out dramatic coercive scenes shaped in his family.The same is true of gender differences: they appear toemerge primarily in the context of relationships(Maccoby, 1990).

The use of self-report measures, including person-ality measures, had initially dominated the field ofmarriage research. Unfortunately, even with the prob-lem of common method variance, the self-report pa-per-and-pencil personality measures yielded relativelyweak correlates of marital satisfaction (Burgess,Locke, & Thomes, 1971). Not until studies askedspouses to fill out questionnaires about their spouse’spersonality, were substantial correlations discoveredwith marital satisfaction. Unhappily married coupleswere found to endorse nearly every negative trait ascharacteristic of their spouses (the negative halo ef-fect), while happily married spouses were found to en-dorse nearly every positive trait as characteristic oftheir spouses (the positive halo effect; Nye, 1988).

A complete review of findings on marriage is be-yond the scope of this article. We will limit ourselves toa sampler, and some general conclusions. Because weare interested in temporal patterns of behavior, percep-tion, and physiology that unfold over time, we will re-strict ourselves to observational studies that have in-cluded some sequential analyses of the data. Thisrestricts us to the work of seven laboratories, Weiss’s(Oregon), Raush’s (Massachusetts), Gottman’s (Wash-ington), Schapp’s (Holland), Ting-Toomey’s (New Jer-sey), the Max Planck group in Munich (Revenstorf,Hahlweg, Schindler, Vogel), and Fitzpatrick’s (Wis-consin). What were the results of various laboratoriesthat investigated the Terman question, and in particularwhat were the results of sequential analyses? Of themany studies that have observed marital interaction,very few have employed sequential analyses. With theMarital Interaction Coding System (MICS), there areonly two such studies (Margolin & Wampold, 1981;Revenstorf, Vogel, Wegener, Hahlweg, & Schindler,1980). They collapsed the many codes of the MICSinto positive or negative, or into positive, negative, andneutral. They each defined negativity in their own way.Margolin and Wampold (1981) reported the results ofinteraction with 39 couples (combined from two stud-ies conducted in Eugene, Oregon and Santa Barbara,California). Codes were collapsed into 3 global catego-ries: Positive (problem-solving, verbal and nonverbalpositive), Negative (verbal and nonverbal negative),and Neutral.

What were the results? Margolin and Wampold’s(1981) results on negative reciprocity were that dis-

tressed couples showed negative reciprocity throughLag 2, whereas nondistressed couples do not demon-strate it to any significant extent. For positive reciproc-ity, they found that “whereas both groups evidencedpositive reciprocity through Lag 2, this pattern appearsto continue even into Lag 3 for distressed couples” (p.559). Thus, reciprocating positive acts was more likelyfor distressed than for nondistressed couples. Gottman(1979) had reported similar results, suggesting that dis-tressed couples showed greater rigidity in temporalinteractional structure than nondistressed couples.Margolin and Wampold also defined a sequence called“negative reactivity,”which involvesapositive responsetoanegativeantecedentbyone’s spouse.Theyproposedthat there is a suppression of positivity following a nega-tive antecedent in distressed couples. They found thisfor all four lags for distressed coupes, but they found noevidence for this suppression of positivity by negativityfor any lag for nondistressed couples.

Revenstorf, et al. (1980), studying 20 German cou-ples, collapsed the MICS categories into six rather thanthree summary codes. These codes were Positive Reac-tion, Negative Reaction, Problem Solution, ProblemDescription, Neutral Reaction, and Filler. Interrupts,Disagrees, Negative Solution, and Commands wereconsidered Negative. They employed both lag sequen-tial analyses that allowed them to examine sequencesout for four lags, as well as the multivariate informa-tion theory that Raush, Barry, Hertel and Swain (1974)had employed with couples undergoing the transitionto parenthood. From the multivariate information anal-ysis, they concluded

In problem discussions distressed couples responddifferently from non-distressed couples....In particu-lar [distressed couples] are more negative and lesspositive following positive (+) and negative (-) reac-tions. At the same time they are more negative andmore positive, that is more emotional, following prob-lem descriptions (P) of the spouse. Above all dis-tressed couples are more negative and less positive ingeneral that non-distressed couples. (p.103)

They also found 17 sequences that differentiated thetwo groups. There is some inconsistency in the group dif-ferences for sequences with similar names (e.g., “recon-ciliation”), here only their clearest results will be summa-rized. For what might be called “constructive” interactionsequences, they found that nondistressed couples en-gaged in more “validation” sequences (problem descrip-tion followed by positivity), and positive reciprocity se-quences (positive followed by positive). On the“destructive side,” they found that distressed couples en-gaged in more “devaluation” sequences (negative followspositive), negative continuance sequences (which theycalled “fighting on” or “fighting back” in 3-chain se-quences) and negative startup sequences (which they

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called “yes-butting,” meaning that somewhere in the4-chain sequence, negative follows positive) thannondistressed couples. After an analysis of the sequencesfollowing a problem description, they concluded

It appears as if the distressed couples would interactlike non-distressed—had they only higher positive re-sponse rates following a problem description of thespouse. And vice versa. The non-distressed would re-act equally detrimentally as the distressed—were theyto respond more negatively to problem description oftheir spouse. The way they handle problems [problemdescription statements] seems to be the critical issue—not the sheer number of problems stated. (p.107)

Revenstorf et al. (1980) also continued their se-quential analyses for five lags and found that these rec-iprocity differences held across lags. They wrote:

In summary, different patterns of response tendenciesemerge for distressed and non-distressed couples. Af-ter a positive statement the partner continues to recipro-cate itpositively innon-distressed,whereasnoimmedi-ate response is likely in distressed couples. After anegative statement no immediate response is mostlikely in non-distressed, whereas in distressed couplesboth partners continue to reciprocate negatively. Aproblem description finally is repeatedly followed by apositive response in non-distressed. In distressed cou-ples, negative statements follow repeatedly. (p. 109)

Revenstorf et al. then described four types of sequences.The first type of sequence is continued negativity (theycalled it “distancing”). This sequence measures the ex-tent towhichnegativitybecomesanabsorbingstate.Thesecond sequence type was positive reciprocity (whichthey called “attraction”). This sequence measures theextent to which positivity becomes an absorbing state.The third sequence consisted of alternating problem de-scriptions and negativity (they called it “problem esca-lation”). The fourth type of sequence consists of Valida-tion sequences, sequences of alternating problemdescriptions and positive responses to it (they called it“problem acceptance”). In most of their graphs (e.g., forpositive reciprocity), the differences between thegroups were not very great. However, the evidence wasvery clear that negativity represented an “absorbingstate” for distressed couples, but not for non-distressedcouples. By Lag-2, non-distressed couples begin to es-cape from the negativity, but distressed couples can notescape. These graphs provide dramatic information ofgroup differences reflected in sequential patterning ofMICS codes.

The consistent findings in these two and other stud-ies that have employed sequential analysis(Fitzpatrick, 1988; Gottman, 1979; Raush et al., 1974;Schaap, 1982; Ting-Toomey, 1982; for a review, seeGottman, 1994) is that: (1) unhappily married couples

appear to engage in long chains of reciprocatednegativity, and (2) there is a climate of agreement cre-ated in the interaction of happily married couples.

These findings using observational research onmarital interaction are interesting and important. How-ever, they are also frustrating because of the absence oftheory either in generating or in summarizing the em-pirical findings. Just where is all this empirical re-search heading? What is it saying about dysfunctionaland functional marriages?

Another dust-bowl empirical approach was takenby Gottman and Levenson (1992). They used a graphi-cal method for combining categorical observationaldata to produce husband and wife time-series data overa 15-minute conflict interaction. They developed amethodology for obtaining synchronized physiologi-cal, behavioral, and self-report data in a sample of 73couples who were followed longitudinally, initially be-tween 1983 and 1987. Using observational coding ofinteractive behavior with the Rapid Couples Interac-tion Scoring System (Gottman, 1996), couples were di-vided into two groups, which we here call “high risk,”and “low risk” for divorce. This classification wasbased on a graphical method originally proposed byGottman (1979) for use with the Couples InteractionScoring System, a predecessor of the RCISS, for cod-ing how couples attempt to resolve a marital conflict.This observational system used as its unit the “inter-act,” which is everything each of two people say in twoconversational turns at speech during a 15-minute taskin which they attempt to resolve a major area of contin-uing disagreement in their marriage. This unit pro-vided an uneven number of interacts for each couple.The coding also took about 6 hr, and an additional 10 hrto create a necessary verbatim transcript. We have nowbeen able to derive essentially the same data using aweighting of the codes of the Specific Affect CodingSystem (SPAFF; Gottman, 1996), which can be codedon-line almost in real time, and, with a numericalweighting scheme for each coding category, averagedover –second blocks, which provides 150 observationsfor each person in each couple. SPAFF also has the ad-vantage that it can be coded for any conversation, notjust for conflict resolution conversations.

In the initial data, on each conversational turn thetotal number of positive RCISS speaker codes minusthe total number of negative speaker codes was com-puted for each spouse. The classifications of “positive”or “negative” were based on previous studies that hadsought to discriminate happy from unhappy couplesfrom a scoring of the behavior they exhibited duringconflictual marital interaction. Based on this review ofthe literature, the RCISS system was devised. Then thecumulative total of these points was plotted for eachspouse. The slopes of these plots, which were thoughtto provide a stable estimate of the difference betweenpositive and negative codes over time, were deter-

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mined using linear regression analysis. If both husbandand wife graphs had a positive slope, they were called“low risk;” if not, they were called “high risk.” Thisclassification is referred to as the couple’s “affectivebalance.” All couples, even happily married ones, havesome amount of negative interaction; similarly, all cou-ples, even unhappily married, have some degree ofpositive interaction.

There was some implicit theory in this dust-bowlempirical approach. Computing the graph’s slope wasguided by a balance theory of marriage, namely thatthose processes most important in predicting dissolu-tion would involve a balance, or a regulation, of posi-tive and negative interaction. Low risk couples weredefined as those for whom both husband and wifespeaker slopes were significantly positive; high riskcouples had at least one of the speaker slopes that wasnot significantly positive. By definition, low risk cou-ples were those who showed, more or less consistently,that they displayed more positive than negative RCISScodes. Classifying couples in the current sample in thismanner produced two groups consisting of 42 low riskcouples and 31 high risk couples. We can easily imag-ine two cumulative graphs, one from the interaction ofa low risk and one the interaction of a high risk couple.

In 1987, four years after the initial assessment, theoriginal subjects were recontacted and at least onespouse (70 husbands, 72 wives) from 73 of the original79 couples (92.4%) agreed to participate in the fol-low-up. Marital status information was obtained.Gottman and Levenson then used their longitudinaldata to examine the results for the dissolution variablesof their “dissolution cascade.” The dissolution cascadeis a Guttman scale in which precursors of separationand divorce were identified as continued marital un-happiness and serious thoughts of dissolution. Basedonly on the Time-1 graphs, they found that after fouryears low risk couples were indeed less likely to be un-happy, to have persistent thoughts of divorce, lesslikely to be lonely in the marriage, less likely to lead“parallel lives” (avoiding one another), and less likelyto separate and divorce than high risk couples. We havenow followed the original sample for 14 years.

Let us take a step back from these empirical find-ings and think about how dynamical systems theorymight be employed to head our research in a more the-oretical direction.

The General Systems Theory of vonBertalanffy

The application of applied mathematics to thestudy of marriage was presaged by von Bertalanffy(1968), who wrote a classic and highly influentialbook called General System Theory. This book wasan attempt to view biological and other complex or-ganizational units across a wide variety of sciences in

terms of the interaction of these units. The work wasan attempt to provide a holistic approach to complexsystems. The work fit a general Zeitgeist. As vonBertalanffy noted, his work fit with Wiener’s (1948)Cybernetics, Shannon & Weaver’s information the-ory, and VonNeumann and Morgenstern’s (1947)game theory. The concepts of homeostasis (derivedfrom the physiologist Walter Cannon), feedback, andinformation provided the basis for a new approach tothe study of complex interacting systems.

The mathematics of general system theory was lostto most of the people in the social sciences who wereinspired by von Bertalanffy’s work. Our work is, there-fore, a return to von Bertalanffy’s original dream. VonBertalanffy’s dream was that the interaction of com-plex systems with many units could be characterizedby a set of values that changed over time, denoted Q1,Q2, Q3, and so on. We can presume that each Q variableindexed a particular unit in the “system,” such asmother, father, and child, and, furthermore that thesevariables measured some relevant characteristic of aperson that changes over time, such as the number ofangry facial expressions per unit time. Actually, theQ’s were quantitative variables that von Bertalanffynever specified.

However, he thought that the system could be bestdescribed by a set of ordinary differential equations ofthe form:

dQ1/dt = f1(Q1, Q2,Q3,…)dQ2/dt = f2(Q1, Q2,Q3,…)and so on.

The terms on the left of the equal sign are time deriva-tives, that is, rates of change of the quantitative sets ofvaluesQ1,Q2,Q3, andsoon.The termson the rightof theequal sign are functions, f1, f2,…, of the Q’s. VonBertalanffy thought that these functions, the f’s, wouldgenerally be nonlinear. The equations he selected have aparticular form, called “autonomous,” meaning that thef’s have no explicit function of time in them, exceptthrough the Q’s, which are functions of time. These are,in fact, the type of equations we worked with. However,von Bertalanffy presented a table in which these nonlin-ear equations were classified as “impossible” (vonBertalanffy, 1968, p.20). He was referring to a very pop-ular mathematical method of approximating nonlinearfunctions with a linear approximation.

However, it was not the case that these systems were“impossible”; von Bertalanffy was unaware of the ex-tensivemathematicalworkbeginning in the19th centurywith Poincaré, on nonlinear differential equations,chaos, and fractal theory, which was only to becomeknownto thepopularpress in the1980s. In fact, in recenttimes the modeling of complex deterministic (and sto-chastic) systemswithasetofnonlineardifferenceordif-ferential equations has become a very productive enter-

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prise across a wide set of phenomena, across a widerange of sciences, including the biological sciences.

Nonlinear Dynamic Modeling WithMore Than One Equation

We thus applied a relatively old approach to the newproblem of modeling social interaction using the mathe-matics of difference and differential equations. Theseequations express, in mathematical form, a proposedmechanism of change over time. They do not represent astatistical approach to modeling but rather they are de-signed to suggest a precise mechanism of change. Thismethod has been employed with great success in thephysical and the biological sciences (e.g., see Murray,1989). It is a quantitative approach that requires themodeler to be able to write down, in mathematical formand on the basis of some theory, the causes of change inthe dependent variables. For example, in the classicpredator-prey problem, one writes down that the rate ofchange in the population densities is some function ofthe current densities. The equations are designed towritedowntheprecise formof ratesofchangeover time.The ideal mathematical technique for describingchange is the area of differential equations. These equa-tions usually used linear terms or linear approximationsof nonlinear terms, and they often gave very good re-sults. In fact, most of the statistics psychology uses arebased upon linear models. In the area of differentialequations, linear equations simply assume that rates ofchange follow generalized straight-line functions of thevariables rather than curved line functions.1 Unfortu-nately, linear models are generally unstable.

In recent years it has become clear that most sys-tems are complex and must be described by nonlinearterms. Interestingly, by employing nonlinear terms inthe equations of change, some very complex processescan be represented with very few parameters. Unfortu-nately, unlike many linear equations, these nonlinearequations are generally not solvable in closed func-tional mathematical form. For this reason the methodshave been called “qualitative,” and visual methodsmust be relied upon. For this purpose, numerical andgraphical methods have been developed such as “phasespace plots.” These visual approaches to mathematicalmodeling can be very appealing because they can en-gage the intuition of a scientist working in a field thathas no mathematically stated theory. If the scientist hasan intuitive familiarity with the data of the field, our ap-proach may suggest a way of building theory usingmathematics in an initially qualitative manner. The useof these graphical solutions to nonlinear differentialequations makes it possible to talk about “qualitative”

mathematical modeling. In qualitative mathematicalmodeling, one searches for solutions that have simi-larly shaped phase space plots. We will now describethese methods of mathematical modeling in detail.

The First Question: What are theSteady States of the System?

Once we write down the equations of marital interac-tion, the first question is “Toward what values is the sys-tem drawn?” To answer the question we define a “steadystate” as one for which the derivatives (on the left side ofthe von Bertalanffy equations) are zero. This means thatthe system at a steady state does not change.

The Second Question: Which SteadyStates are Stable?

What does “stability” mean? It means that if youperturb the system slightly from a stable steady state, itwill return to that steady state. It is as if the steady stateis an attractor that pulls the system back to the steadystate. This is like a rubber band snapping back oncepulled and released. If the steady state is unstable, onthe other hand, and you perturb the system slightly atthat steady state, it will move away from that steadystate. If we have the more general equation N´ = dN/dt= f(N), it is easy to show that N* is a stable steady stateif f´ = df/dt is less than zero at N* and unstable if f´ =df/dt is positive at N*. Graphically we can look at theslope of f(N) where it crosses the x-axis; each point ofnegative slope would imply a stable steady state, whileeach point of positive slope would imply an unstablesteady state. Recall that the function f(N) is given byour equation, once we know how to write it down. Inour work with steady states, we will refer to the stablesteady states as “set points,” a term borrowed from re-search on the management of body weight. In this area,it has been noted that the body defends a particularweight as a set point, adjusting metabolism to maintainthe body’s weight in homeostatic fashion. It is interest-ing that the existence of homeostasis does not, by it-self, imply that the system is regulated in a functionalmanner. Recently a man who weighed 1,000 poundsdied in his early thirties; he had to be lifted out of hisapartment with a crane. Clearly this set point was dys-functional for this man’s health, despite the fact that hisbody defended this set point.

Describing the Behavior of the Model

The next step in modeling is to describe the behav-ior of the model near each steady state and as the pa-rameters of the model vary. We want to know what themodel tells us qualitatively about how the system issupposed to act. Then, if the model isn’t acting the waywe think it ought to (given the phenomena we are try-

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1The strength of an attractor is one over the smallest eigenvaluefor that state. Eigenvalues are solutions to two equations whenlinearizing around a steady state.

ing to model), we alter the model by changing the func-tion “f” . This process can be represented with a flowchart for dynamic modeling of any process over time.We begin with identifying the phenomenon or phe-nomena we wish to model. This is given to us by our in-sights and intuitions about the science. Then we writethe equations for building the model. This is a difficulttask, one that requires some knowledge of mathemat-ics. Next, we find the steady states of the model, thosepoints where the derivatives are zero, and determine ifthese steady states are stable or unstable. Then westudy the qualitative behavior of the model near thesteady states. Next, we study how the model behaves aswe vary the parameters of the model. Finally, returningto the science, we ask whether this model is doing thejob we want, and, if not, we modify the model. There issometimes an additional step in the model building,namely, non-dimensionalizing the model. This step re-duces the number of parameters in the model.

Writing the Equations of MaritalInteraction

In modeling marital interaction there are two equa-tions, one for the husband, and one for the wife. Ourdependent variable was the positive minus negative ateach turn of speech. Because we could not come upwith any theory we knew of to write down the equa-tions of change (linear or nonlinear) in marital interac-tion over time, we developed an approach that usesboth the data and the mathematics of differential or dif-ference equations in conjunction with the creation ofqualitative mathematical representations of the formsof change. The expressions we wrote down were thenused with the data to “test” our qualitative forms. Whatwe discovered was different about this approach wasthat we needed to use the modeling approaches to gen-erate the equations themselves. Thus, the objectives ofthe mathematical modeling in our case became to gen-erate theory. We think that our experience is quite gen-eral and may also be useful for other social psychologi-cal problems.

The Details of our DynamicMathematical Modeling

The goal of our modeling was to dismantle theunaccumulated RCISS point graphs of positive minusnegative behaviors at each turn into components thathad some theoretical meaning. This is an attempt at un-derstanding the ability of these data to predict maritaldissolution via the interactional dynamics. We thus be-gan with the Gottman-Levenson dependent variablefor husband and for wife and dismantled it into compo-nents that represent: (1) a function of interpersonal in-fluence from spouse to spouse, and (2) terms contain-

ing parameters related to an individual’s own dynam-ics. This dismantling of the Gottman-Levenson de-pendent variable into “influenced” and “uninfluenced”behavior represents part of our theory of how the de-pendent variable may be decomposed into componentsthat suggest a mechanism for the successful predictionof marital stability or dissolution.

The qualitative and theoretical portion of writingour equations lies in writing down the mathematicalform of the influence functions. An influence functionis used to describe the couple’s interaction. The mathe-matical form is represented graphically, with the x-axisas the range of values of the dependent variable (posi-tive minus negative at a turn of speech) for one spouseand the y-axis the average value of the dependent vari-able for the other spouse’s immediately following be-havior, averaged across turns at speech. This latterpoint is critical, and it may be unfamiliar to social sci-entists: The influence functions represent averagesacross the whole interaction.

Now what did we know about marital interactionthat could help us write down the mathematical form ofthe influence functions? Gottman (1994) suggestedthat one consistent result that had been obtained bymany laboratories studying marital conflict interactionwith observational methods was that negative affectwas a better correlate of marital satisfaction and pre-dictor of longitudinal course than positive affect. Thismeans that we could expect that the theoretical form ofthe influence functions would probably be bilinear,with a steeper slope in the negative affect ranges than inthe positive affect ranges (this is also reminiscent ofAlexander’s defensive/supportive cycle in family inter-action; e.g., Alexander, 1973). Thus, we would expectthe influence functions to be somewhat asymmetric, asshown in Figure 1.

The Parameters

Notice in Figure 1 that the slopes of the influencefunctions in the two regions are the important parame-ters for the bilinear form of the influence functions. If

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Figure 1. Bilinear form of the influence functions. This form isbased on the frequent result that negative affect during conflict ismorestronglyrelatedtomarital satisfactionthanpositiveaffect.

we had selected as our theoretical influence functionsan ojive form (step functions instead of straight lines)we could have had two parameters indicating theheights of each part of the O-Jive; furthermore, wewould have also added two parameters, the thresholdsof positivity and negativity. These thresholds representthe values of how negative the husband’s negativity hasto get before it starts having an impact on the wife, andhow positive the husband’s behavior has to get before itstarts having a positive impact on the wife. But in thebilinear model only two parameters are obtained foreach spouse’s influence on their partner. The choice ofthe influence function thus determines the nature of thetheory we will build. We have experimented with theojive form of the influence function (see Gottman,Murray, Swanson, Tyson, & Swanson, in press).

Using the existing research on marital interaction,another parameter we decided that it was important toinclude is the emotional inertia (positive or negative)of each spouse, which is each person’s tendency toremain in the same state for a period of time. Thegreater the emotional inertia, the more likely the per-son is to stay in the same state for a longer period oftime. It has been consistently found, in marital inter-action research, that the reciprocation of negativity ismore characteristic of unhappy than of happy cou-ples. This finding has held cross-nationally as well aswithin the United States (for a review, see Gottman,1994). Surprisingly, the tendency to reciprocate posi-tive affect is also greater in unhappy than in happycouples (see Gottman, 1979). There is generally moretime linkage or temporal structure in the interactionof distressed marriages and families. Another way ofstating this finding is that there is more new informa-tion in every behavior in well-functioning family sys-tems. The system is also more flexible because it isless time-locked. A high inertia spouse is also lessopen to being influenced by the partner. Emotionalinertia came from including the autocorrelation com-ponent of human behavior.

Another parameter we added after four years ofworking on the model was a constant that representedthe initial starting values of the conversation. A derivedparameter from knowing both this starting value andthe emotional inertias of both people (one that emergedfrom solving the equations) was the couple’s uninflu-enced set point, which is their average level of positiveminus negative when their spouse did not influencethem (the influence function was zero). We decidedthat this state of affairs was most likely when the affectwas most neutral or equally positive and negative. Wethink of this parameter as what each spouse brings intothe interaction, before being influenced by the partner.Clearly this uninfluenced set point may be a functionof the past history of the couple’s interactions. It mayalso be a function of individual characteristics of eachspouse, such as a tendency to dysphoria, or optimism.

Another derived parameter was the influenced setpoint of the interaction, which is a steady state of thesystem. One way of thinking about the influenced setpoint is that it is a sequence of two scores (one for eachpartner) that would be repeated ad infinitum if the theo-retical model exactly described the time series; if sucha steady state is stable, then sequences of scores willapproach the point over time. In a loose sense it repre-sents the average score the theoretical model wouldpredict for each partner. We also thought it might be in-teresting to examine the difference between influencedand uninfluenced set points. We expect that the influ-enced set point will be more positive than the uninflu-enced set point in marriages that are stable and happy;that is, we asked the question, Did the marital interac-tion pull the individual in a more positive or a morenegative direction? This was an additional derived pa-rameter in our modeling.

We think it will be interesting to discover whethersome interventions will alter either the fundamentalshape of the influence function and whether other inter-ventions will alter the influenced or uninfluenced setpoints. Changing the influence functions seems to us torepresent much more fundamental change of the relation-ship itself than changing the uninfluenced set points.

Estimating the Parameters

We needed a plan for estimating these and other sa-lient parameters. We begin with a sequence ofGottman-Levenson scores: Wt, Ht, Wt+1, Ht+1,… Forpurposes of estimation we assumed that zero scoreshave no influence on the partner’s subsequent score.We then subtract the uninfluenced effects from the en-tire time series to reveal the influence function, whichsummarizes the partner’s influence. We then can plotthe influence function as a function of values of thedata (positive minus negative affect), and use the theo-retical form of the influence function to estimate thetwo slopes. There are two influence functions, the in-fluence of the husband on the wife and the influence ofthe wife on the husband. We can also estimate the cou-ple’s relative power by subtracting the two sets ofslopes from one another, comparing the wife’s and thehusband’s influence. Notice we have complicated theusual discussions of relative power, because relativepower here depends on the affect.

Description of the Model

The simplifying assumption that each person’sscore is determined solely by their own and their part-ner’s previous score restricts us to a particular class ofmathematical models. It we denote Wt and Ht as thehusband’s and wife’s scores respectively at turn t, thenthe sequence of scores is given by an alternating pair ofcoupled difference equations:

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The functions f and g remain to be determined. Theasymmetry in the indexes is due to the fact that we areassuming, without loss of generality, that the wifespeaks first. We therefore label the turns of speech W1,H1, W2, H2,… To select a reasonable f and g, we makesome simplifying assumptions. First we assume thatthe past two scores contribute separately and that theeffects can be added together. Hence, a person’s scoreis regarded as the sum of two components, one ofwhich depends on their previous score only and theother on the score for their partner’s last turn of speech.We term these the “uninfluenced” and the “influenced”components, respectively.

Consider the uninfluenced component of behaviorfirst. This is the behavior one would exhibit if not influ-enced by one’s partner. As we noted, it could primarilybe a function of the individual, rather than the couple,or it could be a cumulative effect of previous interac-tions, or both. It seems reasonable to assume that somepeople would tend to be more negative when left tothemselves while others would naturally be more posi-tive in the same situation. This “baseline temperament”we term the individual’s “uninfluenced set point.” Wesuppose that each individual would eventually ap-proach that set point after some time regardless of howhappy or how sad they were made by a previous inter-action. The simplest way to model the sequence of un-influenced scores is to assume that uninfluenced be-havior can be modeled by a simple linear relationship.This leads us to the linear relationship:

where Pt is the score at turn t, ri determines the rate atwhich the individual returns to the uninfluenced setpoint and ai is a constant. The constant ri will hence-forth be referred to as the “emotional inertia,” parame-ter, or more simply, just the “inertia.”

The uninfluenced set point is the steady state of thisequation and is found by solving Pt+1 = Pt = P = ai /(1-ri). This is the uninfluenced steady state , that is, theattractor of the system if there were no influence. Notethat the stability of the attractor in Equation 2 is gov-erned by the value of ri. If the absolute value of ri is lessthan 1.0, then the system will tend toward the steadystate regardless of the initial conditions, while if the ab-solute value of ri is greater than 1.0, the system will al-ways evolve away from a steady state. Clearly we re-quire the uninfluenced steady state to be stable, and sowe are only interested in the case in which the absolutevalue of ri is less than 1.0. The magnitude of ri deter-mines how quickly the uninfluenced state is reachedfrom some other state, or how easily a person changestheir frame of mind, hence the use of the word “iner-

tia.” For selecting the form of the influenced compo-nent of behavior, we can take several approaches. Theinfluence function is a plot of one person’s behavior atturn, t, on the x-axis, and the subsequent turn, t+1, be-havior of the spouse on the y-axis. Averages are plottedacross the whole interaction. Our approach was towrite down a theoretical form for these influence func-tions (recall Figure 1). As we noted, we posited atwo-slope function: We have two straight lines goingthrough the origin, with two different slopes, one forthe positive range and one for the negative range. Otherforms of the influence function are also reasonable.

The Full Equations

We denote the influence functions by IAB (At), theinfluence of person A’s state at turn t on person B’sstate. With these assumptions the complete model is:

Again, the asymmetry in the indexes is due to the factthat we are assuming that the wife speaks first. Theproblem now facing us is estimation of our four param-eters, r1, a , r2, and b, and the empirical determinationof the two unknown influence functions.

Proximal Change Experiments:Changing Marriages Through ModelSimulation and Intervention

It isvery important that themodelbederived insuchamanner that the parameters and functions of the modelhave interpretable physical meaning. For example, themodel parameters a and b can be interpreted as startupvaluesbeforeautocorrelation (self-influence)orpartnerinfluence begins. Once we have a model, with equa-tions, and we have estimates of the model’s parameters,we can simulate the couple’s interaction under condi-tions different from those that were used in the estima-tion. This means that we can change the model’s param-eters in ways that are meaningful in the sense that theycan be translated into behavior. For example, supposewe have a couple whose conflict interaction we model,andwefind that thehusband’s startupparameter isnega-tive, and, furthermore, we find that the model has only anegative attractor (stable steady state). Now let us simu-late the model’s behavior if the husband’s startup pa-rameter, b, were far more positive. This is done simplybybeginningwith the sameequations (withonlyonepa-rameter changed) and the same initial values, and run-ning off the predicted values. We calculate the new in-fluence functions, and we once again compute the newstable steady states. Suppose we find that now the modelhas a positive stable steady state. That means that the

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1

1 1

( , )( , ) (1)

t t t

t t t

W f W HH g W H

�

� �

�

�

1 (2)t i t iP r P a� � �

1 1( ) (3)t HW t tW I H r W a� � � �

1 1 2( ) (4)t WH t tH I W r H b� � ��

model has predicted that changing the husband’s startvalues will have a major positive effect on the relation-ship. What we now need to do is to create an interventionthat has the desired effect on the husband. Then the cou-ple actually has the post-intervention conversation, andthe model is refit. The question then is, does the newmodel look like the simulatedmodel? In thatwaywecanuse the modeling and the simulation to test the model.For the past six years we have pilot tested these proximalchange experiments and they have greatly enriched ourunderstanding of how to change marriages. The mathe-matical modeling has been an essential ingredient inthese experiments.

Estimation of Parameters and theUnknown Influence Functions

The algorithms and computer program for using ourmodeling will soon be available upon request. We willnow describe the general methods we have developedfor estimation of model parameters and the influencefunctions. We begin by examining the model for thatsubset of data points for which we can safely assumethat there was no partner influence. We assumed thatthese points were those for which the partner’s scorewas near zero.

To isolate and estimate the uninfluenced behaviorwe look only at pairs of scores for one person forwhich the intervening score of their partner was zero(about 15% of the data). That is we are assuming thatat such points, IHW = 0 and IWH = 0, and then Equa-tions 3 and 4 collapse to Equation 2, and we can useleast squares on this subset of the data to estimate thetwo unknown autocorrelation constants for each per-son. Note that we can now compute the uninfluencedstates and inertia of each partner. Now here is an im-portant part of the estimation, the derivation of theinfluence functions. Once we have estimated the un-influenced component of the scores we subtract itfrom the scores at turn t + 1 to find the observed in-fluenced component. We can plot the influencedcomponent of the wife’s score against her husband’sprevious score. This is used to derive one of the in-fluence functions. For each value of the husband’sscore during the conversation there is likely to be arange of observed values of the influence componentdue to noise in the data. To convert these into esti-mates for the influence functions of the model (IHW

and IWH), we simply average the observations foreach partner score. Both the raw influence data andthe averaged influence function are then plotted foreach member of each couple.

Steady States and Stability

For each couple, we also plot a phase plane contain-ing the model’s null clines. The phase plane refers sim-

ply to the plane with the husband’s and the wife’sscores as coordinates. Hence, a point in this plane is apair representing the husband’s and the wife’s scoresfor a particular interact (a two-turn unit). As time pro-gresses, this point moves, and charts a trajectory inphase space.

We summarize our definitions here. Recall that thenull clines are the curves in the phase plane for whichthe derivatives are zero, or the values of theGottman-Levenson variable stay constant. Once themodel has arrived at a stable steady state, it will remainthere, hence the term “attractor;” the attractor is similarto a gravitational attractor, it “draws” the values of themodel back to it if it is perturbed slightly from the sta-ble steady state. In phase space there are sometimespoints called “stable steady states.” These are pointsthat the trajectories are drawn toward, and if the systemis perturbed away from these states, it will be drawnback. Unstable steady states are the opposite: if per-turbed, the system will drift away from these points.

Finding the Null Clines and the Influenced SetPoints. It is of considerable importance to find thesteady states of the phase plane, the influenced setpoints, and this is accomplished by finding thosepoints were the null clines intersect. Remember thatthe null clines are determined by the equations. Find-ing the null clines is accomplished mathematically byplotting them. Null clines involve searching forsteady states in the phase plane; they are theoreticalcurves where things stay the same over time. A per-son’s null cline is a function of their partner’s lastscore and gives the value of their own score when thisis unchanged over one iteration. Mathematically, thisis written as:

This last equation says that, for the wife’s behavior,things stay the same over time, and that is preciselyhow we find the shapes of the null clines. Plotting nullclines and finding their intersections provides a graphi-cal means of determining steady states. First we beginwith simple algebra in which we substitute W for allthe wife terms. This process gives us:

That last equation is the wife’s null cline. It’s the curvewhere she doesn’t change. When we do the same anal-ysis for the husband’s null cline, and recall that thesteady states are the intersection of the null clines, thisthen gives the final form of our null clines as:

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� � �( 1) ( )W t W t W

1

1

1

1

( ),( ) ,

(1 ) ( ) ,( ( ) ) /(1 )

HW t

HW t

HW t

HW t

W r W a I H orW r W I H a or

r W I H a orW I H a r

� � �

� � �

� � �

� � �

Therefore, we have discovered by simple algebrathat our null clines are simply the influence functions,scaled (by 1-r1, or 1-r2) and moved (by a or b). In otherwords, we have shown that the null clines have thesame shape as the influence functions, they are movedover (translated) by a constant, and they are scaled byanother constant. Null clines play an important role inmathematical analysis since they give a visual indica-tion of the dynamics of the system.

As we noted, the equilibria or steady states are de-termined by looking for intersections of the nullclines, since, by definition, if the system started atthis point then it would stay there. Of course, the sta-bility of these steady states to perturbations is yet tobe determined. Since we have not specified the func-tional form of the influence functions, we can onlyproceed qualitatively.

To derive the influenced steady states of a maritalsystem, the pair of equations (Equation 5) can be solvedgraphically. The method is similar to solving two simul-taneous linearequations (ax+by=c;dx+ey=f)byfind-ing intersection points. If these two lines are plotted onthe same graph, the point where they intersect gives thesolution value (x,y) that satisfies both equations. In ourcase these functions are not straight lines; they are prob-ably nonlinear (depending on our theory of marital in-fluence). Therefore, if we plot the two curves fromEquation 5, their solution will be given by any pointswhere the curves intersect. Now we need to think aboutwhat we know about couples’ interaction to generate afunctional form for the influence functions.

Becoming familiar with influence functions andnull clines in the bilinear case. One thing thatseems to have emerged from marital research is that,during conflict, negativity has a bigger impact on one’spartner’s immediately subsequent behavior thanpositivity. In Figure 2, we depict a graphical summaryof this idea and plot both of the bilinear influence func-tions. Here the husband’s influence on his wife is

drawn as the dotted line. The solid line is the reverse,the wife’s influence function on her husband. This lat-ter function is drawn on the same graph by mentally ro-tating axes. The positive part of the wife axis, which isvertical, now gets viewed as an abcissa (x-axis), andthe first half of the bilinear influence function is thendrawn in the positive-wife/positive-husband quadrant.A similar line is drawn in the negative-wife/negativehusband quadrant. These influence functions will thenbe translated and stretched to become the null clines,whose intersection determines the marital system’ssteady states, given these parameter estimates. Noticethat we are plotting two null clines, and doing thisgraphically is a little tricky. This is illustrated in Figure3 for the bilinear form of the influence function.

Figure 4 shows how the shape of the null clines attheir intersection determines the stability of thesteady state. We are plotting two functions: The valueof Wt for which Wt+1 = Wt for any given interveningHt, and the converse for the husband. Intersectionpoints are, by definition, points for which both thewife’s and the husband’s score remain constant onconsecutive turns of speech. These are the points wecall the “influenced steady states.” If a couple were toreach one of these states during a conversation, theywould theoretically remain there with each partnerscoring the same on each of their future turns ofspeech. If they were perturbed away from one ofthese steady states, they would be drawn back to it.

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Figure 2. Bilinear form of the influence functions. To draw thewife’s influence on the husband (solid line) it is necessary to men-tally flip axes, considering the vertical axis as the “wife’sabcissa” and the horizontal axis as the “wife’s ordinate.”

Figure 3. Null clines using the bilinear form of the influencefunctions. Notice that the null cline for the husband (dotted line)is the husband’s influence function on his wife moved to the left(and stretched). The null cline for the wife (solid line) is the wife’sinfluence function on her husband moved down (and stretched).The black dots are the stable steady states.

1

1

2

( ) ( ( ) ) /1 )( 1) ( ) /(1 ) (5)t

t HW t

t WH

W H I H a rH W I W b r�

� � �

� � � �

This perturbation could happen by assuming thatthere is some random error that also affects people’sbehavior. In phase space there are paths that each per-turbed point will take back to a stable steady state, oraway from an unstable steady state. These potentialflow lines can be used to map potential trajectories,or solutions to the equations in phase space. Note thattheoretically there are two steady states for everycouple with the bilinear influence function.

If a couple begins somewhere in state space, allthings being equal, it will generally be drawn to thesteady state it is closest to. Thus, if a couple beginsnegatively they are most likely to be drawn to theirnegative steady state; if a couple begins positivelythey are most likely to be draw to their positivesteady state. However, Murray has shown (Gottman,Murray, Swanson, Tyson, & Swanson, in press) thatevery attractor in phase space has a strength muchlike some gravitational fields, which vary with themass of an object. This strength will determine whichattractor is most influential in predicting a couple’sconversational trajectory.

Steady States and Trajectories in PhaseSpace

As we have noted, there are two types of steadystates, stable and unstable. Theoretically, if a conversa-tion were continued for a very long time, then pairs ofscores would approach a stable steady state and moveaway from an unstable one. Mathematicians call the setof points that approach a stable steady state (we ignorethepossibilityofcycles) the“basinofattraction” for thatsteady state. Theoretically, this very long conversationwould be constructed by simply applying Equations 3and4 iteratively fromsome initialpairof scores.Thepo-tential existence of multiple stable steady states eachwith its own basin of attraction has practical implica-tions. The model suggests that the final outcome (posi-tive or negative trend) of a conversation could dependcritically on the opening scores of each partner and thestrength of each attractor. If the attractors were of equalstrength, where one ends up in the phase space is deter-mined by the couple’s actual initial conditions. In ourexperience, we have generally found that the end pointscan depend critically on starting values.

An observed or a “reconstructed” conversation canbe represented in the phase plane as a series of con-nected points. In addressing the issue of stability of thesteady states, we are asking whether the mathematicalequations imply that the reconstructed series will ap-proach a given steady state. Analytically, we ask thequestion of where a couple will move once they areslightly perturbed from their position away from asteady state. The theoretical (stable or unstable) behav-ior of the model in response to perturbations is onlypossible once we assume a functional form for the in-fluence functions. For example, for an influence func-tion that has a sigmoidal shape, we can have 1, 3, or 5steady states, rather than 2.

What does it mean for there to be multiple steadystates? It means that these are all possible states for aparticular couple. Even if we only observe the couplenear one of them in our study, all are possible for thiscouple, given the equations. Each stable steady statewill have a “basin of attraction.” This is the set of start-ing points from which a reconstructed time series willapproach the steady state in question. If there is a sin-gle steady state, then its basin of attraction is the wholeplane; i.e., no matter what the initial scores were, thesequence would approach this one steady state. If, onthe other hand, there are two stable steady states (and,necessarily, one unstable one) generally the plane willbe divided into two regions (the basins of attraction). Ifthe scores start in the first stable steady state’s basin ofattraction, then, in time, the sequence of scores will ap-proach that steady state. The same goes for the secondsteady state and its basin of attraction. The couple be-gins at the point (W1, H1) in phase space, next moves tothe point (W2, H2), and next moves to the point (W3,

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Figure 4. Null clines and the stability of steady states for themarriage model. Steady states correspond to points at which thenull clines intersect. The stability of a steady state can be deter-mined graphically (see text for details): when the null clines in-tersect as is shown in (a), the steady state is unstable; when theyintersect as is shown in (b), the steady state is stable.

H3), and so on, heading for the large black dot that rep-resents the stable steady state intersection of the twonull clines.

Notice that this implies that the eventual trend thatthe conversation follows can be highly dependent onthe initial conditions. Thus, high inertia, high influencecouples (who are more likely to have multiple steadystates) could potentially exhibit a positive conversationon one day and yet not be able to resolve conflict on an-other. The only difference could be the way the conver-sation began (their initial scores). The influence func-tions and uninfluenced parameters would be identicalon each day. This discussion makes concrete the gen-eral systems theory notion of “first-order” (or more su-perficial, surface structure) change and “second-order”(or more meaningful, deeper structure) change. In ourmodel, first-order change means that the steady statesmay change but not the influence functions; second-or-der change would imply a change in the influencefunctions as well.

Catastrophes Can Be RepresentedWith This Model

In dynamical theory, “catastrophe” means that amodel parameter can change continuously, but, oncethe parameter crosses a critical threshold, a qualitativechange occurs and the same laws no longer govern thesystem. The classic example is the straw that broke thecamel’s back. For example, it is possible to change pa-rameters of the marriage model continuously so thatthe couple loses a positive steady state. This can hap-pen by slowly changing the slopes of the bilinear influ-ence functions, or by slowly changing the parameters aand b. If that happened, suddenly the marital systemwill have lost its positive influenced steady state. Thenno matter where on the basin of attraction the couplestarted the conversation, they would be inexorablydrawn to the negative stable steady state. That wouldbe all that were left to them in phase space! Now, inex-plicably all the couple’s conflict resolution discussionswould degenerate into very aversive and highly nega-tive experiences. This would be a literal catastrophe,and we would predict divorce as inevitable for this cou-ple. This is consistent with Gottman’s (1994) reportthat when that happens, the couple enters a series ofcascades that results in increasing flooding, diffusephysiological arousal, arranging their lives in parallelso that they have less interaction, and becoming in-creasingly lonely and vulnerable to other relationships.To the couple the change is inexplicable. They haveweathered many stresses in the past and succeeded instaying together. But now every disagreement headssouth toward the negative attractor. That is becausethere has been a qualitative change: There is no longera positive attractor.

Here, then, is a model for a very gradual trend dur-ing which the couple often thinks that they are simplyadapting to increasing stresses in their lives, gettingused to seeing less of each other and more fighting, butfully expecting that things will get better eventually.However, they are vulnerable for losing their positivestable influenced steady states, and then the modelwould predict a real marital catastrophe. The gradualchanges would suddenly change the marriage sud-denly, and then it would qualitatively become an en-tirely different relationship.

Although the model predicts this sudden cata-strophic change under these conditions, it also predictswhat is called “hysterisis” in catastrophe theory, whichmeans that this state of affairs is reversible. Clinical ex-perience suggests that, for longer time spans, this re-versibility is probably not the case when the marriagehas been neglected for long enough. For example,Buongiorno (1992) reported that the average wait timefor couples to obtain professional help for their mar-riage after they have noticed serious marital problemsis about six years. This problem of high delay in seek-ing help for an ailing marriage is one of the great mys-teries in this field of inquiry. It may very well be relatedto another great mystery, which is the nearly universal“relapse phenomenon” in marital therapy. Recentlysome of our best scholars (e.g., Jacobson & Addis,1993) have contended that marital therapy has relapserates so high (30 to 50 percent within a year after mari-tal therapy ends) that the entire enterprise of maritaltreatment may be in a state of crisis. Consistent withthese conclusions, the recent Consumer Reports studyof psychotherapy (Seligman, 1995) reported that mari-tal therapy received the lowest marks from psychother-apy consumers. Marital therapy may be at an impassebecause it is not based on a process model derived fromprospective longitudinal studies of what real couplesdo that predicts that their marriages will wind up happyand stable, unhappy and stable, or end in divorce.

After so long a delay before getting help, it makessome sense to propose that a positive hysterisis journeymay be less likely than a negative one. Also, some keylife transitions may make going back to the more posi-tive way things were less likely. This is particularlytrue for the transition to parenthood. Half of all the di-vorces occur in the first seven years of marriage, and agreat deal of stress is associated with the transition toparenthood. There are other vulnerable transitionpoints for marriages in the life course. The low pointcross-nationally for marital satisfaction is when thefirst child reaches the age of 14, although this phenom-enon is not well understood. Retirement is also such adelicate transition point. If these speculations are true,the model would have to be altered to accommodatethese asymmetrical phenomenon.

It does seem likely that there is something like a sec-ond law of thermodynamics for marital relationships,

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that things fall apart unless energy is supplied to keepmaking the relationship alive and well. At this time inthe history of Western civilization, marriages seemmore likely to fall apart than to stay together (Martin &Bumpass, 1989). Hence the hysterisis property of themodel may turn out to be incorrect. However, our re-cent research with long-term first marriages(Carstensen, Gottman, & Levenson, 1995) paints a farmore optimistic picture, one that suggests that somemarriages mellow with age and get better and better.

It should be pointed out that the model is designed toobtainparameters fromjusta15-minute interaction,andone useful way of extending the model is to attempt tomodel two sequential interactions, in which the parame-ters of the second interaction are affected by the first in-teraction. What is very interesting about the cata-strophic aspects of the model is that it does tend to fit agreat deal of our experience, in which we have observedthat many marriages suddenly fall apart, often after hav-ing successfully endured a period of high stress.

Implications of the model

One of the interesting implications of the mathe-matical model is that even in the best of marriages, it ispossible that there will be both a positive and a negativestable steady state. This means that, depending entirelyon starting values, there will be times that the couplewill be drawn toward a very negative interaction. Thismay not happen very often in a satisfying and stablemarriage, but the model predicts that it will happen. Tosome extent, these events are minimized if the strengthof the negative steady state (or “attractor”) is muchsmaller than the strength of the positive steady state.This means that the old concept of family homeostasishas to be modified: there are usually at least twohomeostatic set points in a family, one more positivethan the other. This concept may do a great deal towardending what Wile (1993) has called the “adversarial”approach of family systems therapy. Here the therapiststruggles gallantly against great odds as the family’shomeostasis holds on to dysfunctional interaction pat-terns. However, if there are two homeostatic set points,one more positive and one more negative, then the ther-apist can align with a family toward making their occu-pation time greater in the more positive homeostatic setpoint than in the more negative homeostatic set point.

Another implicationof this twohomeostaticsetpointtheory is that a negative steady state may have some pos-itive functions in a relationship; the therapist ought notto make war on negative affect, for example. Negativitymight be useful in a relationship for a variety of reasons.Oneis that inanyrealcloserelationship,our legacy is thefull repertoire of emotions (they are controlled in moreformal and more casual relationships). It would not bevery intimate if some emotions were expurgated from

the full repertoire of emotions that is our legacy as homosapiens; in fact, a relationship with only positive emo-tionsmightactuallybea livinghell.Second,negativeaf-fectsmayserve the functionofcullingoutbehaviors thatdo not work in the relationship, continually fine tuningthe relationship over time so that there is a better andbetter fitbetweenpartners.Third,negativitymightservethe function of continually renewing courtship over thecourse of a long relationship; after the fight there isgreater emotional distance, which needs to be healedwitha re-courtship.Themodelhasaccomplishedagreatdeal just by dismantling the Gottman-Levenson vari-able into components and parameters. This has created anew theoretical language for describing interaction. In-steadofhaving just avariable thatpredicts the longitudi-nal course of marriages, we now can speak theoreticallyabout the mechanism of this prediction. We can expectthat compared to happy, stable marriages, what happensin marriages headed for divorce is that:

• there is more emotional inertia;• even before being influenced, the uninfluenced

set point is more negative;• when interaction begins, the couple influences

one another to become even more negative, ratherthan more positive;

• over time, as these negative interactions continueand become characteristic of the marriage, thecouple may catastrophically lose its positive sta-ble steady state.

The model suggests one possible integration of theconcepts of affect and power in relationships, whichhas haunted the field since its inception. The integra-tion is that power or influence is defined as one per-son’s affect having an influence over the other person’simmediately following affect. The integration alsosuggests a greater order of complexity to the concept ofpower. Who is more powerful in the relationship maybe a function of the level of affect, and how positive ornegative it is. In one relationship, for example, a wifemight be more powerful than her husband only withextreme negative affect, while her husband might bemore powerful only with mild positive affect. We mayalso discover that the very shape of the influence func-tions are different for couples heading for divorce,compared to happy, stable couples.

Therefore, unlike prior general systems writings,which remained at the level of metaphor, the mathe-matical model has also given birth to a new theoreticallanguage about the mechanism of change. In the mari-tal research area we did not have such a language be-fore the model was successfully constructed. Themodel provides the language of set point theory, inwhich a number of quantities, or parameters, may beregulated and protected by the marital interaction. Italso provides a precise mechanism for change. The

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model itself suggests variables that can be targeted forchange using interventions. In short, the model leadssomewhere. It helps us raise questions, helps us won-der what the parameters may be related to, and why. Itraises questions of etiology. Why might a couple beginan interaction with a negative uninfluenced set point?Why and how would they then influence one another tobe even more negative?

Thus, a major contribution of the model is the the-oretical language and the mathematical tools it pro-vides. It will give us a way of thinking about maritalinteraction that we never had before. Previous experi-ence in the biological and physical sciences suggeststhat any model that accomplishes these things willprobably be useful.

The model also gives us insight about its own in-adequacy. It is a very grim model. Where a couple be-gins the interaction in phase space will determine itsoutcome. Does it make sense to have a model inwhich there is no possibility of repair? Wouldn’t itmake more sense to include “switch” terms, whichare multiplied by parameters so that when the interac-tion becomes too negative, the switch is turned onand the interaction gets a jolt of positivity? The pa-rameter could vary for each individual. When we ex-amined our data we found only 4% of the coupleswho began the interaction negatively but were able tosignificantly turn the interaction around so it eventu-ally became more positive than negative! Hence, eventhough major repair is a rare phenomenon, wethought that the model should have such a switchterm in it. Couples would then vary in the extent towhich they were able to repair the interaction. Per-haps in marital therapy this repair term becomesstrengthened. The repair effectiveness parametercould be estimated from small, or “local” turnaboutsin the overall direction of the interaction. Also itmakes sense to think about the threshold at which re-pair begins, and whether the repair term is deter-mined by one’s own negativity (slope or level) orone’s partner’s. We have extended the model in thismanner. Similarly, once one imagines a term for thedown-regulation of negativity, it is possible to have aswitch term that down regulates positivity as well asnegativity. We have also experimented with this“damping” term in the model. The addition of both arepair and a damping term makes it possible to havemany more than two set points for the interaction.Thus, the nature of the nonlinearity determines thecomplexity of the marital system.

We have also discovered new things with the model,things we never expected to find. In the Cook et al.(1995) article we reported the discovery that differenttypes of marriages have different types of influencefunctions, and that divorce is only predicted by funda-mental mismatches in influence function shape. Cookalso showed (Cook et al., 1995) that there is an optimal

balance between influence and inertia. Cook’s analysisof the stability of the steady states of the marriagemodel shows that there is a dialectic between theamount of influence each spouse should have on hispartner and the level of emotional inertia in eachspouse’s uninfluenced behavior for the steady state tobe stable. Steady states tend to be stable when theyhave a lower level of influence and a lower level of in-ertia. Another way of saying this is that if a marriage isgoing to have high levels of mutual influence, for sta-bility of (say) the positive steady state, there needs tobe lowered inertia.

The model continues to develop. For example, it ispossible to model not only interactive behavior, butalso any time series synchronized to behavior. Hence,we can also model our perceptual (rating dial video re-call) data, and the couple’s physiology. Indeed, mixedmodels are possible across measurement domains, andthis has led us to propose that in functional marriagesthere is a “core triad of balance” that is regulated by thecouple toward greater positivity in behavior and per-ception, and greater calm physiologically. In the courseof these analyses we discovered that if the wife’s be-havior drives the husband’s physiology, the marriage islikely to end in divorce.

Conclusions

Our work has demonstrated that it is possible toplace the study of personal relationships on a solidmathematical footing. We have only begun to explorethe utility of this mathematics as we have begun to doproximal change experiments. After pilot testing theseproximal change experiments for three years, we havebegun a monthly column with the Reader’s Digestmagazine in which once a month a couple comes to ourlaboratory, we do an assessment (with a pre-interven-tion conversation that we model) and then perform anintervention, and evaluate the intervention with ourmathematical modeling of a post-intervention conver-sation. At the time of this writing we have successfullyworked with 10 couples. These experiments are slowlybuilding a library and a technology of proximal rela-tionship change interventions.

This dream of a mathematics for social relation-ships is not new. The work we have done is reminis-cent of Isaac Asimov’s science fiction classic seriesof books, called the Foundation series. In that seriesof books, a fictional mathematician named HariSeldon creates a set of equations for predicting thefuture of the entire human species, a new branch ofstudy he calls “psychohistory.” Considered muchharder to accomplish, later in the books anothermathematician creates a new set of equations for pre-dicting the future of smaller social units, which hecalls “micro-psychohistory.” We believe that it is that

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latter field that we have created. In our view, puttingthe study of social relationships on a mathematicalfooting is a major advance in our ability to under-stand and perhaps to regulate these relationships forthe betterment of all mankind.

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