Journal of Family Psychology1999, Vol. 13, No. 1,3-19

Copyright 1999 by the American Psychological Association, Inc.0893-32«V99/$3.00

The Mathematics of Marital Conflict: DynamicMathematical Nonlinear Modeling of Newly wed

Marital InteractionJohn Gottman, Catherine Swanson, and James Murray

University of Washington

This article extends a mathematical approach to modeling marital interaction usingnonlinear difference equations. Parameters of the model predicted divorce in asample of newlyweds. The parameters reflected uninfluenced husband and wifesteady states, emotional inertia, influenced husband and wife steady states, andinfluence functions. The model permits separation of uninfluenced parameters—thatis, what is initially brought to the interaction by each person's personality or therelationship's history—from where the interaction heads once influence begins. Inthe present model, a theoretical shape of the influence functions is proposed thatpermits estimation of negative and positive threshold parameters. Couples whoeventually divorced initially had more negative uninfluenced husband and wifesteady states, more negative influenced husband steady state, and lower negativethreshold in the influence function.

The application of mathematics to the studyof marriage was presaged by von Bertalanffy(1968), who wrote a classic and highly influen-tial book titled General System Theory. Thisbook was an attempt to view biological andother complex organizational units across awide variety of sciences in terms of theinteraction of these units. The work was anattempt to provide a holistic approach tocomplex systems. Von Bertalanffy's enterprisefit a general Zeitgeist, with Wiener's (1948)Cybernetics, Shannon and Weaver's (1949)information theory, and VonNeumann and Mor-genstern's (1947) game theory. Von Bertalanffyinspired many major thinkers of family systemsand family therapy. Unfortunately, the mathemat-ics of general systems theory was neverdeveloped, and theorists of family interactionkept these systems concepts only at the level ofmetaphor. Even at the level of metaphor these

John Gottman and Catherine Swanson, Departmentof Psychology, University of Washington; JamesMurray, Department of Applied Mathematics, Univer-sity of Washington.

Correspondence concerning this article should beaddressed to John Gottman, Department of Psychol-ogy, Box 351525, University of Washington, Seattle,Washington 98195.

concepts were tremendously influential in thefield of family therapy (see Rosenblatt, 1994).However, they were never really subjected toexperimental processes, and so they becamefrozen and reified. Systems concepts came to beconsidered true without evidence. This washighly unfortunate, and it was the directconsequence of not making the ideas testable orat least disconfirmable.

Von Bertalanffy (1968) viewed his theory asessentially mathematical. He believed that theinteraction of complex systems with many unitscould be characterized by a set of values thatchange over time, denoted Qu Q2, Q3, and so on.The Qs were variables each of which indexed aparticular unit in the system, such as mother,father, and child. He thought that the systemcould be best described by a set of ordinarydifferential equations of the following form:dg./dt = f,(fi,, Q2, ft,...), d<22/dt = f2(Q,, Q2,Q3,...), and so on. The terms on the left of theequal sign are time derivatives, that is, rates ofchange of the quantitative sets of values Qu Q2,and so on. The terms on the right of the equalsign are functions, fb f2,.... of the Qs. Therewas no suggestion of how to operationalize the"g-variables." Von Bertalanffy thought that thefunctions, the fs, might generally be nonlinear.

GOTTMAN, SWANSON, AND MURRAY

The equations that von Bertalanffy selected havea particular form, called "autonomous," mean-ing that the fs have no explicit function of timein them, except through the Qs, which arefunctions of time.

Von Bertalanffy thought that for practicalsolution, the equations had to be linear. Hepresented a table in which these nonlinearequations were classified as "impossible" (vonBertalanffy, 1968, p. 20). Rapoport (1972)suggested that investigators apply von Bertalan-ffy's linear equations to an analysis of marriage,but he presented no data, no suggestion of howto operationalize von Bertalanffy's Q-variables,and no explanation of how to apply theequations to real data. Unfortunately, linearequations are not generally stable, so they tendto give erroneous solutions, except as approxima-tions under very local conditions near a steadystate. Von Bertalanffy was not aware of themathematics that Poincare and others haddeveloped in the last quarter of the 19th centuryfor the study of nonlinear systems. The model-ing of complex deterministic (and stochastic)systems with a set of nonlinear difference ordifferential equations has now become a produc-tive enterprise across a wide set of phenomenaand across a wide range of sciences, includingthe biological sciences. The use of nonlinearequations forms the basis of our modeling ofmarital interaction.

Other than the possibility of stability, anadvantage of nonlinear equations is that byusing nonlinear terms in the equations of changesome very complex processes can be repre-sented with very few parameters. Unfortunately,unlike many linear equations, these nonlinearequations are generally not solvable in closedfunctional mathematical form. For this reason,the methods are often called "qualitative," andvisual graphical methods and numerical approxi-mation must be relied on. For this purpose,numerical and graphical methods have beendeveloped such as phase space plots. Thesevisual approaches to mathematical modelingcan be very appealing for engaging the intuitionof a scientist working in a field that has nomathematically stated theory. If the scientist hasan intuitive familiarity with the data of the field,our approach may suggest a way of buildingtheory using mathematics in an initially qualita-tive manner. The use of these graphical solutionsto nonlinear differential equations makes it

possible to talk about "qualitative" mathemati-cal modeling. In qualitative mathematical mod-eling, one searches for solutions that havesimilarly shaped phase space plots, whichprovide a good qualitative description of thesolution and how it varies with the parameters.

There are many excellent introductions to thisgeneral approach to qualitative nonlinear dy-namic modeling and its subtopics of chaos andcatastrophe theory (e.g., Baker & Gollub, 1996;Beltrami, 1993; Berge, Pomeau, & Vidal, 1984;Glass & Mackey, 1988; Gleick, 1987; Lorenz,1993; Morrison, 1991; Peters, 1991; Vallacher& Nowack, 1994; Winfree, 1990). An introduc-tion to catastrophe theory may be found inworks by Arnold (1986), Castrigiano and Hayes(1993), Gilmore (1981), and Saunders (1990).The currently vast and expanding area ofmathematical biology was introduced by Mur-ray's (1989) classic text.

Our modeling of marital interaction using themathematical methods of nonlinear differenceequations is an attempt to integrate the math-ematical insights of von Bertalanffy with thegeneral systems theorists of family systems(Bateson, Jackson, Haley, & Weakland, 1956)using nonlinear equations. In modeling maritalinteraction, Cook et al. (1995) developed anapproach that used both the data and themathematics of differential or difference equa-tions in conjunction with the creation ofqualitative mathematical representations of theforms of change. Our approach was uniquebecause the modeling itself generated theequations, and the objective of our mathematicalmodeling was to generate theory. We suggestedthat the data be used to guide the scientificintuition so that equations of change weretheoretically meaningful.

In finding a practical example of vonBertalanffy's g-variable, we used a report byGottman and Levenson (1992) that one variabledescriptive of specific interaction patterns of thebalance between negativity and positivity waspredictive of marital dissolution. Gottman andLevenson used a methodology for obtainingsynchronized physiological, behavioral, andself-report data in a sample of 73 couples whowere followed longitudinally between 1983 and1987. Applying observational coding of interac-tive behavior, they computed, for each conversa-tional turn, the number of positive minusnegative speaker codes and plotted the cumula-

MARITAL CONFLICT

tive running total for each spouse. An observa-tional system (the Rapid Couples InteractionScoring System [RCISS]) was used to code thesalient behaviors; this system had been devel-oped by reviewing literature on the behavioralcorrelates of marital satisfaction. A behaviorwas considered negative if it correlated nega-tively with marital happiness and positive if itcorrelated positively (Krokoff, Gottman, &Haas, 1989). The slopes of these plots deter-mined a risk variable, low risk if both husband'sand wife's graphs had a positive slope and highrisk if not. Computing the graph's slope wasguided by a balance theory of marriage, namely,that those processes most important in predict-ing dissolution would involve a balance, or aregulation, of positive and negative interaction.Four years after the initial assessment, theoriginal participants were recontacted. Thehigh-low risk distinction was able to predict thecascade toward divorce, which consisted ofmarital dissatisfaction, persistent thoughts aboutdivorce and separation, and actual separationand divorce.

The modeling began with a sequence ofGottman-Levenson scores for each couple overthe 15-min conversation: W,,Ht, Wt+i,Ht+u

and so on. These scores were numbers obtainedby giving a numerical weight to each categoricalobservational code, with the turn at speech asthe unit (ignoring vocal backchannels; Duncan& Fiske, 1977). The weights were determinedand were positive or negative by dependingentirely on their correlations with maritalsatisfaction in previous research. In the processof modeling, two parameters were obtained foreach spouse. One parameter was his or her"emotional inertia" (positive or negative),which is each person's tendency to remain in thesame state for a period of time, and the otherparameter was his or her natural uninfluencedsteady state, which was each spouse's averagelevel of positive minus negative when the otherspouse's score was zero, that is, equally positiveand negative.

The mathematical model involved breakingdown the rate of change of the Gottman-Levenson variable for each spouse into the sumof three terms, a constant term related to theuninfluenced value that each spouse brought tothe interaction, an autoregressive term related tothat person's past, and an influence functiondescribing the spouse's influence on the rate of

change of the partner. If, at time t, the husband'squantitative variable is represented by H, thewife's variable by Wt, and r represents theemotional inertia parameter, the differenceequations are as follows:

Wt+l =a + r.W, + IHW(H,), (1)

Hl+l = b + r2Ht + IWH(Wt). (2)

The parameters a and b and the rs wereestimated from the data. The constants a and bare related to the initial uninfluenced level ofpositivity minus negativity that each spousebrings to the interaction. [The uninfluencedsteady states are actually a/(l — rx) andbl{\ — r2), respectively.] These parameters werethought to reflect the past history of therelationship and aspects of each partner'spersonality (that is, modal affectivity). The rsthen reflect the influence of each person'simmediate past on that person; that is, they areautoregressive parameters.

The /s are the influence functions, and theyare the nonlinear part of the equations. I[fw> f° r

example, is the influence of the husband on thewife at time t. This dismantling of the Gottman-Levenson variable into "influenced" and "unin-fluenced" behavior represented a theory of howthis dependent variable may be decomposed intocomponents that suggest a mechanism for thesuccessful prediction of marital stability ordissolution. For example, for the wife, once therh r2, a, and b are estimated from the data (seeCook et al., 1995), at each time point, t,(a + rxWt) is subtracted from W;+1, and theremainder is the influence function, IHy(H,).These data points for the influence function arethen averaged across specific values of H, and aplot is made of H as the x-axis and Iip^H) as they-axis. A similar procedure holds for the otherinfluence function.1

The "qualitative" portion of the equationsinvolved writing down the mathematical form ofthe influence functions. Two influence functionswere used to describe the nature of the couple'sinfluence on one another (see Figure 1). Theycan potentially link power and affective parts ofthe couple's relationship in the following

1 We could model cycles by including a delayparameter between husband and wife responding (seeMurray, 1989, Section 1.3, p. 8).

GOTTMAN, SWANSON, AND MURRAY

A

NegativeThreshold

\HusbandNegative

WifePositive

WifeNegative

HusbandPositive

B WifePositive

HusbandPositive

WifeNegative

Figure 1. Two possible functional forms for theinfluence functions: For the influence function of thehusband on the wife, the jc-axis is the husband'sprevious score, H,, and the y-axis is the influencedcomponent, IHW (H,+i), of the wife's following score,W,+1. The wife's influence on the husband, IWH(W/+1), could be graphed in a similar way. In (A)O-Jive or sigmoidal shape, there is no influenceunless the partner's previous score lies outside somerange. Outside that range the influence takes either afixed positive value or a fixed negative value. In (B)bilinear shape, influence increases linearly with thevalue of the previous score, but negative scores canhave either a stronger or less strong influence thanpositive scores. In both graphs, a score of zero haszero influence on the partner's next score (one of theassumptions of the model).

manner: They present a precise function of howone partner's affect (e.g., the husband's) at time taffects the other's (the wife's) affect at time t +1, as a function of the entire range of affectdisplayed (by the husband). This influence of thehusband is determined by the equations control-ling for autocorrelation, that is the wife's priorinfluence on herself (rxWt), controlling for herbaseline level of affectivity (a). Hence, we

can use the influence function to compute whatthe average effect is of the husband on the nextaffective behavior of his wife when he ispositive at level +6 on our affect scale, or at—17, and so on. The mathematical form of theinfluence function is represented graphicallywith the x-axis as the range of values of thedependent variable (positive minus negative at aturn of speech) for one spouse and with they-axis as the average value of the dependentvariable for the other spouse immediatelyfollowing behavior, averaged across turns atspeech.

Negative Threshold and the "MaritalNegativity Detector"

In this study, we extended Cook et al.'s (1995)work by selecting a particular theoretical formof the influence functions. The form we selectedwas the O-Jive. We selected this functionbecause it permitted us to create four importantnew parameters in the model: the thresholds fornegativity and the thresholds for positivityfor each spouse. The threshold for negativityimplies that this is the point at which nega-tivity has an impact on the partner's immedi-ately following behavior. The threshold fornegativity in the influence function is thus afunction of both a couple's perception of therelationship and the partner's subsequent action.If the threshold for negativity is set lower (asmaller negative number on the x-axis) fornewlywed marriages that eventually wind upstable and happy, this could be called the maritalnegativity detector effect: Some spouses arehaving a negative response to lower levels ofnegativity from their partner; they are noticingand responding to negative behavior when it isless escalated. In other marriages, people areadapting to and trying to accept this negativity,setting their threshold for response at a muchhigher or more negative level. It's as if they havesaid to themselves, "Just ignore this negativity.Don't respond to it until it gets much worse (thatis, more negative)." It may be the case that thiskind of adaptation to negativity is dysfunctionalbecause having a lower threshold for negativityimplies that people would discuss issues beforethey escalate too much. People with lowerthresholds may follow the biblical principlefrom Ephesians (Eph. 4:27), "The sun must notgo down on your wrath." This advice hastypically been taken by many couples to detect

MARITAL CONFLICT

negativity quickly and repair the marriagebefore retiring for the evening.

The introduction of the negativity thresholdsrelates to an important effect recently discoveredby Baucom and his associates (Baucom, Ep-stein, Rankin, & Burnett, 1996). Lederer andJackson (1968) had originally suggested thatmarriages may become dysfunctional if spouseshave too high expectations for the marriage, butBaucom et al. reported that, on the contrary,couples who held the highest expectations fortheir marriage generally scored higher on everyindex of positive marital outcome that wasstudied. This may mean that the couples whosemarriages were doing well detected, respondedto, and attempted reasonably immediate repairof interaction before it became too negative.This may be contrasted with a style in whichcouples decide to adapt to levels of negativityexpressed by their partners. The direction ofcausation, however, is unclear.

This point of view is currently controversialin the marital therapy literature. It is not at allobvious that adapting to one's partner's negativ-ity is not functional. In fact, many therapistshave suggested that couples cultivate an empa-thetic response to their partner's negativity.Hendrix (1988) suggested a therapy that wouldtrain people to have an "X-ray vision" in whichthey see their partner's childhood wound behindthe hostility, and then they are able to respondempathetically even when they have beenattacked. This could lead to people setting theirnegative threshold at a more negative level.Many other therapies have proposed a similarmodel for effective conflict resolution (e.g.,Guerney, 1977). On the other hand, Wile (1993)disagreed. He recommended that people shouldhold high standards for their marriages and thatthey should feel "entitled" to their complaints:

It is the adjustments, accommodations, andsacrifices that partners make without telling eachother or fully recognizing it themselves—that is,their automatic and unverbalized attempts tocompromise and be reasonable—that may lie atthe root of the difficulty. What these partnersmay be needing is to stop compulsively compro-mising and to develop an ability to complain.(p. 73)

We set out to test these two alternative points ofview by selecting the O-Jive form of theinfluence functions. If Wile's point of view werecorrect, newlywed couples in marriages thathave a less negative threshold would be less

likely to divorce than couples in marriages thathave a more negative threshold. However, ifHendix's point of view were correct, newlywedcouples in marriages that have a less negativethreshold would be more likely to wind updivorced than couples in marriages that have amore negative threshold.

The negative threshold parameter could beimportant because it might explain two myster-ies in the area of marriage. The first effect wecall here the delay effect. In many areas ofpublic health, one concern is decreasing the"delay time" to getting competent medical helpfrom the time a patient notices a symptom, suchas chest pains or a lump in the breast. Notariusand Buongiorno (1995) found that the averageamount of time a married couple waits to getprofessional help from the time one of themdetects serious problems in the marriage is 6years! The second mystery we call here therelapse effect. This refers to a pervasive findingthat initial gains in marital therapy relapse after1 to 2 years (e.g., see Jacobson & Addis, 1993).How could the mathematical model explainthese two effects? We proposed that thisparticular parameter of our mathematical modelof marriage, the negative threshold, couldexplain both of these effects. A finding with thenegative threshold parameter would suggest thatpeople delay getting help for their marriagebecause they have inadvertently raised thenegativity threshold. In marriages that work,spouses do not do this adapting to negativity.The negative threshold effect could also explainthe relapse effect in marital therapy. It suggeststhat relapse occurs because people adapt toincreasingly higher levels of negativity, insteadof repairing the relationship. Relapse and delaybecome parts of the same phenomenon, namelyadaptation to increasingly higher levels ofnegativity.

We might find that spouses who do not adaptto negativity will fairly quickly bring up an issueabout which they are unhappy, within days ofthings becoming negative. This would be anexample of longer term repair. So, this findingwould suggest that the key to avoiding decay inmarriages is for the therapy to reset what couldbe called "the marital negativity detector" to alower level of negativity and build in someformal mechanisms for the couple to do repairon a continual basis. The finding about thresholdfor negativity could also suggest an hypothesis

GOTTMAN, SWANSON, AND MURRAY

for the pervasive marital therapy relapse effect.It would suggest that people relapse after initialtherapeutic gains because the therapy hasaffected the overall levels of positivity andnegativity in the marriage, but it has not reset thenegative threshold. This then could become oneof the goals of therapy: to reset the maritalnegativity detector so that couples notice thenegativity, do not adapt to it, and instead makecontinual repairs in the marriage. It is alsoconceivable that couples who put off makingrepairs to their marriage also erode the overallamount of positivity in the marriage through theprocess of carrying grudges that are notaddressed.

Steady States and Stability

As Cook et al. (1995) noted, for each couplewe plot a phase plane containing the model'snull dines (see Figure 2). The phase plane referssimply to the plane with the husband's and thewife's scores as coordinates. Hence, a point inthis plane is a pair representing the husband'sand the wife's scores at a particular time block inthe interaction; using the RCISS data, this unitwas a particular interact (a two-turn unit). Astime progresses, this point moves, and charts atrajectory in phase space. In phase space thereare sometimes points called stable steady states.These are points that the trajectories are drawntoward, and if the system is perturbed awayfrom these states, it will be drawn back.Unstable steady states are the opposite: Ifperturbed, the system will drift away from thesepoints. Hence, it is of considerable importanceto find the steady states of the phase plane. Thisis accomplished mathematically by plotting thenull dines of the marital system. Null clinesinvolve searching for steady states in the phaseplane; they are theoretical curves where thingsstay the same over time, where the derivativesare zero (see Cook et al., 1995). Cook et al.(1995) derived the functional form of the nullclines in terms of the influence functions, hiFigure 2 the filled black dots indicate stablesteady states and the white dots indicateunstable steady states.

We call the set of points that approaches astable steady state the "basin of attraction" forthat steady state. Consider a sequence of scoresapproaching a more positive steady state. A"theoretical conversation" could be constructedby simply applying Equations 1 and 2 iterativelyfrom some initial pair of scores. Cook et al.

(1995) showed that the final outcome (positiveor negative trend) of a conversation coulddepend critically on the opening scores of eachpartner. Where one begins in the phase space isdetermined by the couple's actual initial condi-tions. We have generally found that the endpoints can indeed depend critically on startingvalues, hi addressing the issue of stability of thesteady states, we asked whether the mathemati-cal equations implied that the reconstructedseries would approach a given steady state.Analytically, we asked the question of where asteady state would move once it was slightlyperturbed from its position. The theoretical(stable or unstable) behavior of the model inresponse to perturbations of the steady states isonly possible once we assume a functional formfor the influence functions. For example, as wehave noted, for the sigmoidal influence function,we can have one, three, or five steady states.From the null-cline plot, we can see that thereare three stable and two unstable states, hi mostof our data, there was only one influenced steadystate per couple.

Is the "Uninfluenced" Steady StateInfluenced by Prior Relationship History,by Lasting Characteristics of Individuals,

or by Both?

Kelly and Conley (1987), in a 35-yearlongitudinal study, reported that neuroticismpredicts divorce. Kurdek (1991a, 1991b, 1993)obtained similar results in his longitudinal study.Karney and Bradbury's (1995) review of 115longitudinal studies also identified what theycalled "enduring vulnerabilities" within indi-viduals as risk variables for predicting negativemarital outcomes. However, personality dimen-sions, when they have predicted marital out-comes, have tended to be weak predictors, and,as Karney and Bradbury noted, the field hasshifted "from predicting outcomes to explainingchains and patterns of events at differentoutcomes" (p. 28). There also is a trend towardgreater precision in the predictor variables(Gottman, 1994). We raised the question ofwhether it was possible to use the parameters ofour mathematical model to tease out the separatecontributions of individual characteristics thatboth spouses bring to the marital interactionfrom the interaction and influence the processitself.

The parameters that we call the "uninflu-

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(W3 H3)

(W2 H2)

Curve separating two basinsof attraction (the "Separatrix")

Figure 2. The null-clines to graphically determine the steady states and separatethe phase space into two basins of attraction (the separatrix). Stable steady states(large filled circles) are separated by unstable steady states (large open circles). Thepairs of husband and wife scores (Wb H^ W2, H2; W3, H3) gradually approach oneof the stable steady states, the exact trajectory depending on the basin of attraction inwhich the initial pair of scores lies. H = husband; W = wife.

enced" steady states might actually containvarying degrees of information about therelationship's prior history, perhaps due either toremaining good feelings, to unresolved conflict,or to both. In fact, this "spillover effect" of priorinteractions can be assessed by dividing theinteraction into two halves and attempting topredict the second half's uninfluenced param-eters from the first half's parameters. If there is aspillover effect of negativity, predictabilitywould be greatest for couples who eventuallywind up in unhappy marriages or who wind updivorced. If it is a spillover effect of goodfeelings, predictability would be greatest forcouples who wind up in happy stable marriages.To assess these possibilities, we divided theinteraction into two equal halves and investi-gated the predictors in the first half of theinteraction of the second half's uninfluencedsteady state parameters. In this way, what wehave been calling the "uninfluenced" steadystate might be explainable, at least in part, interms of the prior interaction.

There were two parts of the analysis. In thefirst part, we assessed whether the first half'sinfluence function parameters (influenced steadystates and influence function thresholds) couldpredict the second half's uninfluenced steady

states, controlling for the first half's uninflu-enced parameters. This analysis assessed whetherthe prior relationship's interaction and influencehistory alone could predict the immediatelyfollowing uninfluenced steady states. This mustinclude a statistical control for the prioruninfluenced parameters. This was intendedsimply as an assessment of prior relationshipspillover. In the second part of the analysis, wefocused on whether there was predictabilityfrom the uninfluenced steady states in the firsthalf directly to the uninfluenced steady states inthe second half, controlling for interaction andinfluence in the first half of the interaction. Thisanalysis attempted to assess to what extent theuninfluenced steady states in the second half ofthe interaction were a function of enduringqualities in the individual, independent ofinteraction and influence. We further assessedhow these issues might vary with the eventualfate of the marriage, dividing the entire sampleinto two groups: (a) those who were divorced orstable but unhappy and (b) everyone else. Thiscomparison should provide adequate power forthe analysis in both groups. We expected that thesecond half's uninfluenced steady state wouldhave components of both interaction and theenduring qualities of the individual. Further-

10 GOTTMAN, SWANSON, AND MURRAY

more, we expected both the interaction's influ-ence and the enduring qualities of the individualto matter more in troubled marriages than inhappy marriages.

Method

Participants

Between 1989 and 1992, a two-stage samplingprocedure was used to draw a sample of newly wedcouples from the Puget Sound area of Washington.Couples were initially recruited with newspaperadvertisements. To be eligible for the study, thecouples had to meet requirements (a) that it was a firstmarriage for both partners, (b) that they were marriedwithin 6 months of participating in the study, and (c)that they were childless. Couples were contacted byphone, were administered the telephone version of theMarital Adjustment Test (MAT; Krokoff, 1987; Locke& Wallace, 1959), and were surveyed to determinetheir eligibility on the other subject selection criteria.The MAT measures marital satisfaction. Higherscores on the MAT represent higher marital satisfac-tion. There were 179 newlywed couples who met theresearch criteria and who participated in the initialsurvey phase of the study. In the survey phase of thestudy, both the husbands and wives received sepa-rately mailed sets of questionnaires to fill out. Thequestionnaire included measures of demographiccharacteristics and indices about the couples' mar-riage, well-being, and health.

In the second phase of the study, 130 newlywedcouples, who represented an even distribution ofmarital satisfaction, were invited to participate in amarital interaction laboratory session and to completeadditional questionnaires. These couples fit thedemographic characteristics of the major ethnic andracial groups in the greater Seattle area, using theSeattle City Metropolitan Planning CommissionReport. The demographic characteristics for thesenewly married couples were (a) wife's age, M = 25.4years (.SD = 3.5); (b) husband's age, M = 26.5 years(SD = 4.2); (c) wife's marital satisfaction, M = 120.4(SD = 19.7); (d) husband's marital satisfaction, M =115.9 (SD = 18.4). Couples were seen in threecohorts of approximately 40 couples per cohort andwere followed for 6 years, so that the follow-upperiod varied from 3 to 6 years.

3-6-Year Marital Status and theCriterion Groups

Once each year the marital status and satisfactionof the 130 couples in the study were assessed. At theend of the 6-year period (Time 2), there had been 17divorces: 6, 6, and 5 in the first, second, and thirdcohorts, respectively. Only 13% of the couples had

divorced after 3-6 years. The divorce rate in thissample may be somewhat lower than one wouldexpect at this point in the marriage trajectory; onewould expect about 22 divorces by this point(Cherlin, 1981; U.S. National Center for HealthStatistics, 1986, 1988).2 Some of the analyses wereconcerned only with marital stability, in which casethe entire sample of 130 couples was used (with 17divorces). However, we were also interested in thecurrent marital satisfaction of the stable (i.e.,nondivorced) couples. To create similar-sized crite-rion groups of stable couples, we used the lowest ofeach couple's Time 2 Locke-Wallace marital satisfac-tion scores to form two criterion groups of 20 stablecouples each, those 20 most happily married andthose 20 most unhappily married. The mean Time 2marital satisfaction score of the stable, happilymarried group was 128.30 (SD = 27.65) and themean Time 2 marital satisfaction score of the stable,unhappily married group was 90.70 (SD = 16.08).

Missing data were created in two ways: (a) bycouples for whom we could not estimate themathematical model for either half of the data set,which occurred when there were fewer than 10 zerosin the seventy-five 6-s time blocks of data forestimating a, b, r\, and ri- Using this criterion, therewere 125 couples for the divorce prediction analyses,(b) For some couples, the null clines did not intersectat all, and these couples had no influenced steadystates.

Questionnaire

As described earner, the MAT (Locke & Wallace,1959) was administered during the initial telephone

2 An exploratory analysis of the correlations of thenegative threshold parameter with the questionnaireand interview data we collected in the newlywedstudy revealed the following. The negative thresholdwas set significantly lower if the husband rated thecouple's marital problems as severe. Using ourcoding of our oral history interview, we found that thenegative threshold was set significantly more nega-tive (a) if the couple reported that their lives were"chaotic"; that is, if they reported that unexpectedthings kept happening to them that they had tocontinually adjust to rather than feeling in control oftheir lives; (b) if during the interview the husband orthe wife spontaneously expressed negative affecttoward the partner; (c) if the husband whined a lotduring the marital conflict discussion; (d) if thehusband or the wife was low on "we-ness." If acouple received counseling, a more negative thresh-old was associated with more months spent incounseling (r = .35, p < .001). Hence, overall, amore negative threshold was associated with agenerally poorer marital quality as well as a poorerprognosis.

MARITAL CONFLICT 11

interview. The MAT is used to assess maritalsatisfaction and is frequently used in marital researchbecause it reliably and validly distinguishes betweenhappily and unhappily married couples (Cronbach'sa = .92). The telephone version of the MAT hasstrong psychometric properties as well (Krokoff,1987). Lower scores on the MAT represent lowermarital satisfaction.

Marital Interaction Laboratory Procedures

Behavioral observation. Two remotely con-trolled, high-resolution cameras filmed frontal viewsof both spouses and their upper torsos during theinteraction sessions. The images from the twocameras were combined in a split-screen imagethrough the use of a video special-effects generator.VHS videorecorders were used to record the behav-ioral data. Two lavaliere microphones were used torecord the couple's audio interactions. The computersynchronized the physiological data with the videodata by using the elapse time codes imposed on thevideo recordings. The Specific Affect Coding System(SPAFF; Gottman, McCoy, Coan, & Collier, 1996)was used to code the couples' conflict interactions.SPAFF was used to index the specific affectsexpressed during the marital problem resolutionsession. SPAFF focuses solely on the affects ex-pressed. The SPAFF system draws on facial expres-sion (based on Ekman and Friesen's Facial ActionCoding System; Ekman & Friesen, 1978), vocal tone,and speech content to characterize the emotionsdisplayed. Coders categorized the affects displayedusing 5 positive codes (interest, validation, affection,humor, and joy), 10 negative affect codes (disgust,contempt, belligerence, domineering, anger, fear/tension, defensiveness, whining, sadness, and stone-walling), and a neutral affect code. Every videotapewas coded in its entirety by two independentobservers using a computer-assisted coding systemthat automated the collection of timing information;each coder noted only the onsets of each code. Thecoders were paid staff in our laboratory, or they wereundergraduate volunteers who worked in the labora-tory for 3 quarters; the 1st quarter was training, andthen they coded for 2 quarters; a seminar of readings,discussion, and a required paper were also part of thevolunteer's activity. The time to code a videotapeindependently twice was three times real time. First,the two independent observers (who could not seeeach other's coding) watched the interaction through,then they both coded 1 spouse and then the otherspouse. Reliability was computed on all tapes, not asubsample of the tapes, and on 100% of each tape. Atime-locked confusion matrix for the entire videotapewas then computed using a 1-s window for determin-ing agreement of each code in one observer's

coding against all of the other observers' coding (seeBakeman & Gottman, 1986). The diagonal versus thediagonal-plus-off-diagonal entries in these matriceswere then entered into a repeated measures analysisof variance (ANOVA) using the method specified byWiggins (1977): The Cronbach's alpha generalizabil-ity coefficients were then computed for each code asthe ratio of the mean square for observers minus theerror mean square and the mean square for observersplus the error mean square (see also Bakeman &Gottman, 1986). The Cronbach's alpha generalizabil-ity coefficients ranged between .651 and .992 andaveraged .907 for the entire SPAFF coding of all 130videotapes. Kappas ranged between .84 and .72 andaveraged .80.

The marital interaction assessment consisted of adiscussion by the husband and wife of a problem areathat was a source of ongoing disagreement in theirmarriage and two recall sessions in which the couplesviewed their marital disagreement discussion. Thecouples were asked to complete the Couple's ProblemInventory (Gottman, Markman, & Notarius, 1977), anindex of marital problems. One of the experimentersthen reviewed with the couple those issues they ratedas most problematic and helped them to chooseseveral issues to use as the basis for the problem-areadiscussion. After choosing the topics for the discus-sion, couples were asked to sit quietly and not tointeract with each other during a 2-min baseline. Thecouples then discussed their chosen topics for 15 min.

When the couple completed their discussion, theywere asked to view the videorecording of theinteraction. In counterbalanced order, husbands andwives were asked to first view and rate their ownaffect during the discussion and then to view and ratetheir spouse's affect. Both husbands and wives usedrating dials that provided continuous self-report data.Continuous physiological measures and videorecord-ings were made during all of the interaction sessions,and data were averaged over 1-s intervals.

Weighting the SPAFF codes. We developed aweighting scheme for using the on-line observationalsystem, the SPAFF (Gottman et al., 1996). Theweighting scheme for the SPAFF was based on thedifferential ability of the specific codes to predictdivorce in previous studies in our laboratory. Thus,for example, the codes of criticism, contempt,belligerence, defensiveness, and stonewalling re-ceived more negative weights than codes such asanger and sadness because of their greater ability topredict divorce. In the weighting scheme, positiveaffects such as affection and humor also receivedhighly positive weights. The weighting schemeresulted in a potential range of —24 to +24 for thedata. The observational coding for the SPAFF wasalso computer assisted, so that the coding wassynchronized with the video time code and thephysiological data, as well as later with the couple's

12 GOTTMAN, SWANSON, AND MURRAY

subjective recall of their own affects. The computerprogram that acquired the data scanned the electronicsignals from the observational coding station 32 timesa second and then determined the dominant code foreach second. This code received the appropriateweight, and then these weights were summed over 6-stime blocks. Because 6 s is the average length of aturn at speech during the conflict discussion, we usedit as the time block for summing. Interestingly, 6 s isalso the unit used in observational coding ofclassroom interaction (Amidon & Hough, 1967), andit is also the unit used by the Oregon Social LearningCenter for analyzing family interactions at home(Patterson, 1982). For the 900 s of the 15-minconversation, this gave us 150 data points for husbandand for wife. The weights were as follows: anger, - 1 ;sadness, — 1; fear, 0; defensiveness, —2; contempt,—4; belligerence, —2; domineering, —1; stonewall-ing, - 2 ; disgust, - 3 ; whining, - 1 ; neutral, 0.1;interest, +2; affection, +4; humor, +4; listenerbackchannels (or any validation), +4; and joy, +4.

The advantages of the SPAFF over the older RCISSobservational system are that it codes specificemotional behaviors of husband and wife and that itcan code as the couple interacts in the laboratory,essentially in real time. The SPAFF coding schemewas designed to be usable for coding any maritalconversation, whereas the RCISS was designed tocode only conflict resolution discussions. The SPAFFalso specifically discriminates among the positiveaffects, making fine distinctions between neutral andinterest, affection, humor, and validation. Positiveaffect models have received scant attention indescribing marital interaction. An exception is workby Birchler, Weiss, and Vincent (1975), who used aself-report diary measure of "pleases" and "dis-pleases," a precursor of the Spouse ObservationChecklist, and a version of the Marital InteractionCoding System (MICS) to code either generalconversation when they were supposedly setting upthe equipment or the Inventory of Marital Conflictdiscussion (IMC; Olson & Ryder, 1970). In thesummary MICS code, the positives were agreement,approval, humor, assent, laugh, positive physicalcontact, and smile. According to this scheme, in theinteraction distressed couples produced an average of1.49 positive codes per minute, whereas nondis-tressed couples produced an average of 1.93 positivecodes per minute, a significant difference. In the homeenvironment, distressed partners recorded signifi-cantly fewer pleasing and significantly greaterdispleasing events than was the case for nondistressedpartners. The previous RCISS system required averbatim transcript of the couple's speech, usuallytaking about 10 hr of a transcriber's time, and then ittook 6 hr for an observer to code the conversationfrom tape and transcript. The unit of analysis for the

RCISS was interact, which is two "turns" at speech,one for the wife and one for the husband, and thenumber of interacts at speech varied a great deal fromcouple to couple; on average, there were 75 interactsper 15-min conflict discussion. The average length ofa turn across our studies was approximately 6 s. The"turn" unit was everything one person said until theother began speaking, ignoring what are called"listener tracking backchannels" (Duncan & Fiske,1977). These backchannels are messages such as briefvocalizations (e.g., "Uh huh," "Yeah," "I see"), eyecontact, head nods, and facial movements that tell thespeaker that the listener is tracking the conversation.These signals also regulate taking turns at speech. Thetransformation from RCISS to SPAFF was designedto be a technical breakthrough in our laboratory. First,it made it possible to code on-line in three times realtime, instead of waiting 16 hr to obtain the coded data,which was the time for RCISS coding. Thisstreamlined the lab's ability to conduct experiments;we were able to see the effects of an experiment in themath model immediately after doing the experimentwith a particular couple. Second, because of SPAFF'sdesign, which applies universal codes suitable for anymarital interaction, it now made it possible to codeany conversation the couple had, such as talkingabout the events of the day, talking about an enjoyabletopic, or everyday conversations such as the 12 hr oftime couples spend in our apartment laboratory.Third, we could now obtain a larger number of pointsper couple, and a uniform number across couples, andthis moved us toward more reliable data within eachcouple, as well as toward the differential equationsform of the math model.

Computational Algorithms

As suggested by Cook et al. (1995), we assume thatthe influence functions are zero at zero values; at thesubset of the values for which H = 0, using leastsquares fitting, we fit the reduced linear equationW,+i = r{W, + a, and at the subset of values forwhich W = 0, we fit the reduced linear equationH,+\ = r2H, + b. Then, using these estimates of theparameters ru r2, a, and b, we compute the values ofthe influence functions, which is IHW = W,+1 — r\W, — a, and IWH = Ht+i — r2H, — b. We then fit afunctional form, the ogive, to these estimates of theinfluence functions, finding, by least squares fitting,the husband influence function parameters AH, BH,CH, DH, EH, and FH, and the wife influence functionparameters, Aw, Bw, Cw, Dw, Ew, and Fw. Thecomputer program, written by Catherine Swanson,can be supplied on request. All other computations—for example, the wife's uninfluenced steady state,which is a/(l — r,), and the husband's uninfluenced

MARITAL CONFLICT 13

steady state, which is bl{\ — r2)—follow from thesecomputations. The influenced steady states aredetermined graphically, from the intersection of thenull clines.

Results

Divorce Prediction

For these analyses, we compared the entiresample of couples who stayed married for whomwe had no missing data (N = 108) with those 17couples who eventually divorced.

Uninfluenced steady states. Table 1 showsthe predictive validity of the parameters of themathematical model using our weighting scheme.As can be seen from the table, the wife'suninfluenced steady state, al(\ - /•]), was muchmore negative in the first few months of thenewlywed phase for couples who eventuallydivorced. The same was true for the husband'suninfluenced steady state.

Influenced steady states. In the early monthsof newlywed marriage, for husbands, those whoeventually divorced had an average influencedsteady state of —1.89, whereas those whosemarriages turned out to be stable had aninfluenced steady state of 0.72. This parameterwas thus a significant predictor of divorce.However, the wife's average influenced steadystate was —1.95 for the group of wives whoeventually divorced and - .65 for the groupwhose marriages turned out to be stable, and

these were not significantly different. Even afterthe influence process, the husbands were stillvery different across the two groups.

Negative threshold. The husband's negativethreshold was a significant predictor of divorce,F(l , 123) = 8.25, p < .01, in the predicteddirection. In the first few months of marriage,couples who eventually divorced initially had anegativity threshold more negative than coupleswhose marriages turned out to be stable. A"more negative" negativity threshold meansthat, for example, a husband has to becomemore hostile before he gets a response from hiswife. The findings mean that it took morenegativity on the husband's part to get aresponse from his wife among couples whoeventually divorced than for couples whosemarriages turned out to be stable. In otherwords, the wife in a marriage that turned out tobe stable at Time 1 was responding to lessintense negative affect in her husband than wasthe case for wives in marriages that ended indivorce. For example, a wife in a marriage thatlater turned out to be stable had initially reactedto her husband's defensiveness, whereas in amarriage that dissolved, the wife did not reactuntil he became contemptuous, a more nega-tively weighted behavior. This was a spouseeffect rather than a couple effect.

Inertia. Neither the husband's nor the wife'sinertia parameter was a significant predictor,

Table 1Predicting Divorce Using Parameters of the Mathematical Model

Variable

WifeInertiaUninfluenced steady stateInfluence functions

Positive thresholdNegative threshold

Influenced steady stateHusband

InertiaUninfluenced steady stateInfluence functions

Positive thresholdNegative threshold

Influenced steady state

F ratio

0.2014.86***

0.710.031.29

3.07t18.40***

2.498.25**4.36*

df

1,1231,123

1,1231,1231,106

1,1231,123

1,1231,1231, 106

Stable

0.390.51

8.25-6.23-0.65

0.400.72

8.45-5.36

0.71

M

Unstable

0.42-2.26

7.24-6.06-1.95

0.30-1.89

6.65-7.71-1.49

14 GOTTMAN, SWANSON, AND MURRAY

unlike the results with the RCISS data (Cooketal., 1995).

Results for the Three Criterion Groups

Table 2 summarizes these results. There wereonly three significant results, two for thehusband and wife uninfluenced steady states,and one for the husband's influenced steadystate. According to least significant differencetests subsequent to the ANOVA, the husbandand wife uninfluenced steady states were able topredictively discriminate all three groups fromone another. For both husbands and wives, thedivorced group's uninfluenced steady state wassignificantly lower than the unhappy stablegroup's uninfluenced steady state, and theunhappy stable group's steady state was signifi-cantly lower than the happy stable group'suninfluenced steady state. The husband's influ-enced steady state also predictively discrimi-nated the three criterion groups from oneanother. Least significant difference tests subse-quent to the ANOVA showed that the husbandinfluenced steady states were able to predic-tively discriminate all three groups from oneanother. The happy stable group's mean wassignificantly different from and greater than theunhappy stable group's mean, which, in turn,was significantly different from and greater than

the divorced group's mean. In their influencedsteady states, the group means of the wives of allthree groups were negative, although that of thehappy stable group was near zero. To furtherinvestigate this puzzling finding, we conductedANOVAs on the actual wives' maxima andminima of the data for the three groups.Although the minima were not significantlydifferent, F(2,48) = 0.72; happy group mean =-10.94, unhappy group mean = -12.94,divorced group mean = -12.94, the maximawere marginally significant, F(2, 48) = 2.94,p = .062, and the least significant differencetests subsequent to the ANOVA showed that thehappy stable mean of 20.06 was significantlygreater than the divorced group mean of 14.88(the unhappy stable mean was 17.76). Hence,wives in the stable happy group were beinginfluenced by their husbands to increased rangesof positivity, compared with the divorced group.

Understanding the UninfluencedSteady State

As noted in our discussion of spillover in theintroduction, we wished to assess whether theparameters we have been calling the "uninflu-enced" steady states may have actually con-tained some information about the relationship'sprior history and about enduring qualities of the

Table 2Tests of Significance of the Three Criterion Groups Using ParametersFrom the Mathematical Model

Variable

WifeInertiaUninfluenced steady stateInfluence functions

Positive thresholdNegative threshold

Influenced steady stateHusband

InertiaUninfluenced steady stateInfluence functions

Positive thresholdNegative threshold

Influenced steady state

F ratio

0.2812.61***

0.730.281.60

0.639.10***

0.542.221.15

df

2,482,48

2,482,482,41

2,482,48

2,482,482,41

Happy,stable

0.371.07

9.12-5.76-0.05

0.381.06

7.94-5.94-0.33

M

Unhappy,stable

0.41-0.03

8.53-6.76-1.88

0.35-0.41

7.94-5.29

0.31

Divorced

0.42-2.26

7.24-6.06-1.95

0.30-1.89

6.65-7.71-1.49

***p < .001.

MARITAL CONFLICT 15

individual, as well as how these influencesmight vary with the eventual fate of themarriage (happy vs. unhappy-stable or di-vorced). To understand the possible effect ofimmediate prior relationship history on theuninfluenced steady state parameters, we di-vided the interaction into two equal halves andinvestigated the predictors in the first half of theinteraction of the second half's uninfluencedsteady state parameters. We did this predictionseparately for those couples who were eventu-ally either divorced or stable but unhappy,compared with everyone else. There were twoparts to answering this question. The first partwas to assess whether there was a spillover ofuninfluenced steady states from the first to thesecond half of the interaction (controlling thefirst half's influence parameters, i.e., influencedsteady state and threshold parameters), and thesecond part was to assess whether the nature ofthe influence process itself in the first halfaffected the uninfluenced steady states of thesecond half (controlling the first half's uninflu-enced steady state).

To test these hypotheses, we conducted aregression analysis with group as the maineffects variable (group = 2 for die divorced andunhappy couples and group = 1 for every othercouple, so that increases in the grouping variableimplied the declining health of the marriage)and with Group X Predictors as the interaction.The variables to be predicted were the secondhalf's uninfluenced steady states. The regressionhad three steps. In the first step, we entered therelevant control variables. In the second step,the main effect variables were then stepped intothe equation. In the third step, the interactionswere stepped in (assessed by multiplying thegroup variable by the predictors, after firsthaving stepped in the predictors themselves).

Assessing the influence of enduring qualitiesof the individual on the uninfluenced steadystate, by controlling first-half influence param-eters. When the wife's second-half uninflu-enced steady state was predicted, controlling thesocial interaction variables, the F-ratio-for-change was F(7,108) = 8.62, p = .0041. In thenext step, the interaction with group was notsignificant, F-for-change F(8,107) = .79. Whenthe husband's second-half uninfluenced steadystate was predicted, controlling the socialinteraction variables, the F-ratio-for-change wasF(7, 107) = 17.72, p = .0001. In the next step,

the interaction with group was significant,F-for-change F(8, 106) = 6.87, p < .05. Theinteraction partial correlation was —.25 (p <.05), so that the degree of the stability of thehusband's enduring quality was negativelyrelated to marital health. Hence, the analysisshowed that the uninfluenced steady state didcontain a significant component that reflectedthe enduring qualities of the individual, and, forthe husband, this is related to negative maritaloutcomes.

The effects of the influence process itself.When the wife's second-half uninfluencedsteady state was predicted, the F-ratio-for-change was F(7,108) = 3.76, p = .0019. In thenext step, the interaction with group was notsignificant, F-for-change F(13, 102) = 0.93.When the husband's second-half uninfluencedsteady state was predicted, the F-ratio-for-change was F(7, 107) = 5.95, p = .0000.Hence, the influence process itself made signifi-cant contributions. In the next step, the interac-tion with group was not significant, F-for-change F(13,101) = 1.67.

Discussion

The purpose of the dynamic mathematicalmodeling proposed in this article was to extendto a newlywed sample the mathematical model-ing and the theory that previously sought toexplain the ability of the RCISS point graphs topredict the longitudinal course of marriages. Wemade a number of changes. First, we used thepreviously recommended ogive sigmoidal influ-ence function. We used this function, in part,because the negative threshold could be used toexplain the delay and relapse effects if theresults supported this.

A weighting of a new observational codingscheme, the SPAFF, was substituted, whichdescribed the couple's affective behavior. TheSPAFF system makes it possible to code in realtime, on-line, and it will make it possible to codenot only conflict-resolution interactions but anycouple interactions. Using the weighted SPAFFdata taken from a conflict-resolution conversa-tion the couple had in the first few months ofmarriage, we found that the uninfluenced steadystates were enough to accomplish the task ofpredicting divorce in a sample of newlyweds.This alone was an interesting result because itwas a replication of Cook et al.'s (1995) result.In using three criterion groups, we found that

16 GOTTMAN, SWANSON, AND MURRAY

these parameters not only predicted divorce butalso predictively discriminated happy fromunhappy stable marriages.

As noted, we also extended the model in thisarticle to use the ogive form of the influencefunction because this form of the influencefunction had a parameter, the negative thresh-old, that theoretically represented that point atwhich the husband's negativity has a majorimpact on the wife's behavior. We hypothesizedthat if this threshold were more negative indivorcing couples, it would offer one possibleexplanation for the delay effect that Notariusand Buongiorno (1995) had detected andpotentially for the marital therapy relapse effect.Indeed, we found evidence that this was the case.

Using the fact that the new approach producesfar more data points than the old RCISS method(which relies on the turn at speech as the unit),we were able to estimate model parameters forboth first and second halves of the conversation.This allowed us to attempt to understand moreabout the uninfluenced steady state. We wereable to assess the extent to which the uninflu-enced steady state represents enduring qualitiesof the individual, as opposed to the immediatepast history of interactive influence processes inthe relationship. The analyses showed that bothare the case. There was also evidence that theenduring qualities of the husband were predic-tive of negative marital outcomes.

The modeling transforms an empirical rela-tionship between the point graphs and eventualmarital outcomes into concepts that can explainthe prediction. The new language includesparameters of uninfluenced steady state, influ-enced steady state, and the influence functions.These equations can help suggest a mechanismfor the observed effect, as opposed to astatistical model. A statistical model suggeststhat variables are related, but it does not proposea mechanism for understanding this relation-ship. For example, we may find that socioeco-nomic status is related to divorce prediction, butwe will have no idea from this fact how thiseffect may operate as a mechanism to explainmarital dissolution. The nonlinear differenceequation model approach suggests a theoreticaland mathematical language for such a theory ofmechanism. The mathematical model differsfrom the statistical model in presenting anequation linking a particular husband and wifeover time, instead of a representation of

husbands and wives, aggregated across couplesas well as time.

By varying parameters, we can make predic-tions of what would happen to this couple if wecould change specific aspects of their interac-tion, which is a quantitative thought experimentof what is possible for this particular couple. Weare currently using this approach in a series ofspecific intervention experiments designed tochange a couple's second interaction about aparticular issue. The model is to be derived fromthe couple's first interaction in the laboratory;the intervention is designed to change a modelparameter (whether it does or not is assessed).Thus, the model can be tested and expanded byan interplay of modeling and experimentation.

The qualitative assumptions that form theunderpinnings of this effort are also laid bare bythe process. For example, the choice of theshape of the influence functions can be modifiedwith considerable effect on the model. Inaccordance with this qualitative approach, subse-quent correlational data can quantitatively testthe theory. We might use a bilinear form of theinfluence function, in which there are twodifferent slopes, one for positive and one fornegative ranges. This bilinear form of theinfluence function proposes that every marriagehas a negative and a positive stable steady state.In effect, this mathematical form of the influ-ence function is positing that every marriagepotentially has a "dark side attractor," and a"bright side attractor." How the marriageproceeds then depends only on the startingvalues (uninfluenced steady states) and therelative strength of the two attractors. This is anappealing theoretical formulation for any bal-ance theory.

There are several approaches we can take todeveloping these equations further. One is toexplore whether the parameters are functions ofother theoretical variables, such as the couple'sphysiology or the couple's perception of theinteraction (derived from our video recallprocedure). We expect that physiological arousalmay be related to greater emotional inertia andthat a negative perception of the interactionwould relate to feeling flooded by one'spartner's negative affect (see Gottman, 1993)and negative attributions (see Fincham, Brad-bury, & Scott, 1991). A second approach is tomodify the model. The model is, in some ways,rather grim. Depending on the parameters, the

MARITAL CONFLICT 17

initial conditions determine the eventual slopeof the cumulated curves. Unfortunately, this isessentially true of most of our data. However,some couples began their interaction by startingnegatively but then changed the nature of theirinteraction to a positively sloping cumulativepoint graph; their cumulative graph lookedsomewhat like a check mark. This was quite rare(characterizing only 3.6% of the sample), but itdid characterize about 14.0% of the couples atleast for part of their interaction. Also, mostcurves have local ups following downs. Thismore optimistic type of curve suggests adding tothe model the possibility of repair of theinteraction once it has passed some threshold ofnegativity. This could be incorporated bychanging the influence function so that its basicsigmoidal shape had the possibility of a repairfunction in the very negative parts of the x-axisof Figure 1. The size of the repair jolts and thethreshold at which the jolt took effect would addother parameters to the model, each of whichwould have to be estimated from the data.

Another way that the model could bedeveloped is by inquiring about what we mightcall "strength of the attractors in phase space."Each influenced steady state is like a gravita-tional force point, and it has a mass or a strengthof attraction. This strength is the speed of returnof a point perturbed away from the steady state.This notion could have some theoretical appeal.For example, one consequence of our briefinterventions could be that they increase thestrength of attraction of the positive stablesteady state and decrease the strength ofattraction of the negative stable steady state.This might be an adequate therapeutic outcome.To operationalize this measure, we are planningto use a standard mathematical parameter calledthe eigenvalue obtained from linearizing themodel using Taylor series in the region near anattractor (see Murray, 1989, Appendix 1 and pp.65-68). We will have to see if this measure hasvalidity.

We may ask the question, How permanentwould these changes be that might be suggestedby the models? For example, is changing only ahusband's positive affect enough to completelychange a dysfunctional marriage? Potentially,according to this model, this is adequate,assuming that the influence functions and theother parameters were such that there wasindeed a positive influenced steady state for the

couple. The model suggests that as long as thisstate exists, starting on the right side of theseparatrix (i.e., husband and wife startingpositively), they will inevitably drift toward thispositive attractor. The model also suggests akind of "second-order" change in which theinfluence functions themselves change. In thiscase, the marriage can provide a buffer againstnegativity. The concepts of "first-order" and"second-order" change were appealing to theoriginal systems theorists, and they wrote aboutthem, but they had little notion of how tooperationalize them.

The mathematical model made it possible toseparate the influence of enduring qualities ofthe individual on the uninfluenced steady state,by controlling first-half influence parameters,and then to estimate the effects of the influenceprocess itself. The analysis showed that theuninfluenced steady state does contain a signifi-cant component that reflects the enduringqualities of the individual, and, for the husband,this is related to negative marital outcomes.Hence, both the history of the individual and theprior history of the relationship affect the"starting" places of the marital interaction.Analyses also showed that the first-half influ-ence process itself makes a significant contribu-tion to the second half's uninfluenced steadystate.

In clinical recommendations, a number ofwriters have suggested the proposition that forhealthy marriages, people ought to lower theirexpectations (Lederer & Jackson, 1968). In asimilar vein, Jacobson and Christensen (1996),in their recent book on couple's therapy,suggested moderating the demand for changewith acceptance of the partner and also recom-mended that people should learn how to takebetter care of themselves and meet their ownneeds. Although these are extreme views of verybalanced positions, the implications for mainte-nance of change in our results is generally quitethe opposite of these recommendations, namely,our results suggest that couples fix problemssoon and detect even small issues. Our recom-mendation, based on the negativity thresholdresults, would be: "Don't let things ride andhave a chance to build up." Baucom's system-atic research on expectations and standards inmarriage (e.g., Baucom et al., 1996) wasintended as a direct test of Lederer and Jackson'sproposition. His findings were quite clear that

18 GOTTMAN, SWANSON, AND MURRAY

the people with the highest expectations havethe best marital outcomes. His findings supportour recommendations. Despite the predictivenature of the relationship, it is unclear in thefindings about the relationship between thenegative threshold parameter and marital out-comes which is cause and which is effect.However, the negative threshold parameter issomething that a couple may establish incourtship as they progress toward commitment.We hypothesize that it seems logical that alowered threshold for negativity implies thatnegativity does not become escalated, becauselower intensity negativity is responded to anddealt with before it escalates. In courtship, acouple may first establish a lower negativitythreshold to deal with problems before theybecome too escalated. This would act tominimize the degree of reciprocity of negativitythat leads to escalated conflicts and could bebeneficial in the long-term stability and happi-ness of the relationship. Some support for thisnotion comes from a 5-year longitudinal studyby Filsinger and Thoma (1988) with premaritalcouples. Dissolution of the relationship andrelationship satisfaction were predicted bynegative reciprocity, assessed from sequentialanalysis of observational data.

References

Amidon, E., & Hough, J. B. (Eds.). (1967).Interaction analysis: Theory, research, and applica-tion. New York: Addison-Wesley.

Arnold, V. I. (1986). Catastrophe theory. Berlin,Germany: Springer-Verlag.

Bakeman, R., & Gottman, J. (1986). Observinginteraction: An introduction to sequential analysis.New York: Cambridge University Press.

Baker, G. L., & Gollub, J. P. (1996). Chaoticdynamics: An introduction. New York: CambridgeUniversity Press.

Bateson, G., Jackson, D. D., Haley, J., & Weakland, J.(1956). Toward a theory of schizophrenia. Behav-ioral Science, 1, 251-264.

Baucom, D. H., Epstein, N., Rankin, L. A., & Burnett,C. K. (1996). Assessing relationship standards: Theinventory of specific standards. Journal of FamilyPsychology, 10, 72-88.

Beltrami, E. (1993). Mathematical models in thesocial and biological sciences. Boston: Jones andBartlett.

Berge, P., Pomeau, Y., & Vidal, C. (1984). Orderwithin chaos: Toward a deterministic approach toturbulence. New York: Wiley Interscience.

Birchler, G. R., Weiss, R. L., & Vincent, J. P. (1975).

Multimethod analysis of social reinforcement ex-change between maritally distressed and nondis-tressed spouse and stranger dyads. Journal ofPersonality and Social Psychology, 31, 349-360.

Castrigiano, D. P. L., & Hayes, S. A. (1993).Catastrophe theory. New York: Addison-Wesley.

Cherlin, A. (1981). Marriage, divorce, remarriage.Cambridge, MA: Harvard University Press.

Cook, J., Tyson, R., White, J., Rushe, R., Gottman, J.,& Murray, J. (1995). The mathematics of maritalconflict: Qualitative dynamic mathematical model-ing of marital interaction. Journal of FamilyPsychology, 9, 110-130.

Duncan, S. D., Jr., & Fiske, D. W. (1977). Face-to-face interaction: Research, methods, and theory.Hillsdale, NJ: Erlbaum.

Ekman, P., & Friesen, W. V. (1978). Facial ActionCoding System. Palo Alto, CA: Consulting Psycholo-gists Press.

Filsinger, E. E., & Thoma, S. J. (1988). Behavioralantecedents of relationship stability and adjustment:A five-year longitudinal study. Journal of Marriageand the Family, 50, 785-795.

Fincham, F. D., Bradbury, T. N., & Scott, C. K.(1990). Cognition in marriage. In F. D. Fincham &T. N. Bradbury (Eds.), The psychology of marriage(pp. 118-149). New York: Guilford Press.

Gilmore, R. (1981). Catastrophe theory for scientistsand engineers. New York: Dover.

Glass, L., & Mackey, M. C. (1988). From clocks tochaos: The rhythms of life. Princeton, NJ: PrincetonUniversity Press.

Gleick, J. (1987). Chaos: Making a new science. NewYork: Viking Press.

Gottman, J., Markman, J., & Notarius, C. (1977). Thetopography of marital conflict: A sequential analysisof verbal and nonverbal behavior. Journal ofMarriage and the Family, 39, 461—477.

Gottman, J. M. (1993). The roles of conflictengagement, escalation, or avoidance in maritalinteraction: A longitudinal view of five types ofcouples. Journal of Consulting and Clinical Psychol-ogy, 61, 6-15.

Gottman, J. M. (1994). What predicts divorce?Hillsdale, NJ: Erlbaum.

Gottman, J. M., & Levenson, R. W. (1992). Maritalprocesses predictive of later dissolution: Behavior,physiology, and health. Journal of Personality andSocial Psychology, 63, 221-233.

Gottman, J. M., McCoy, K., Coan, J., & Collier, H.(1996). The Specific Affect Coding System (SPAFF).In J. M. Gottman (Ed.), What predicts divorce? Themeasures (pp. 1-169). Hillsdale, NJ: Erlbaum.

Guemey, B. (1977). Relationship enhancement. SanFrancisco: Jossey-Bass.

Hendrix, H. (1988). Getting the love you want. NewYork: Holt Rinehart, & Winston.

MARITAL CONFLICT 19

Jacobson, N. S., & Addis, M. E. (1993). Research oncouples and couple therapy: What do we know?Where are we going? Journal of Consulting andClinical Psychology, 61, 85-93.

Jacobson, N. S., & Christensen, A. (1996). Integrativecouples therapy. New York: Norton.

Karney, B. R., & Bradbury, T. N. (1995). Thelongitudinal course of marital quality and stability:A review of theory, method, and research. Psycho-logical Bulletin, 118, 3-34.

Kelly, E. L., & Conley, J. J. (1987). Personality andcompatibility: A prospective analysis of maritalstability and marital satisfaction. Journal of Person-ality and Social Psychology, 52, 27-40.

Krokoff, L. J. (1987). Anatomy of negative affect inworking class marriages. Dissertation AbstractsInternational, 45, 7A. (University Microfilms No.84-22, 109)

Krokoff, L. J., Gottman, J. M., & Haas, S. D. (1989).Validation of a global Rapid Couples InteractionScoring System. Behavioral Assessment, 11, 65-79.

Kurdek, L. A. (1991a). Marital instability andchanges in marital quality of newly wed couples: Atest of the contextual model. Journal of Social andPersonal Relationships, 8, 27-48.

Kurdek, L. A. (1991b). Predictors of increases inmarital distress in newly wed couples: A three-yearlongitudinal study. Developmental Psychology, 27,627-636.

Kurdek, L. A. (1993). Predicting marital dissolution:A five-year prospective longitudinal study ofnewly wed couples. Journal of Personality andSocial Psychology, 64, 221-242.

Lederer, W. J., & Jackson, D. D. (1968). The miragesof marriage. New York: Norton.

Locke, H. J., & Wallace, K. M. (1959). Short maritaladjustment and prediction tests: Their reliability andvalidity. Marriage and Family Living, 21, 251-255.

Lorenz, E. (1993). The essence of chaos. Seattle:University of Washington Press.

Morrison, F. (1991). The art of modeling dynamicalsystems: Forecasting for chaos, randomness, &determinism. New York: Wiley Interscience.

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

Notarius, C , & Buongiorno, J. (1995). Wait times formarital therapy. Unpublished paper, Catholic Uni-versity of America, Washington, DC.

Olson, D. H., & Ryder, R. G. (1970). Inventory of

Marital Conflicts (IMC): An experimental interac-tion procedure. Journal of Marriage and the Family,32, 443-448.

Patterson, G. R. (1982). Coercive family process.Eugene, OR: Castalia Press.

Peters, E. E. (1991). Chaos and order in the capitalmarkets. New York: Wiley.

Rapoport, A. (1972). The uses of mathematicalisomorphism in general systems theory. In G. J. Klir(Ed.), Trends in general systems theory (pp. 42-77).New York: Wiley Interscience.

Rosenblatt, P. C. (1994). Metaphors of family systemstheory. New York: Guilford Press.

Saunders, P. T. (1990). An introduction to catastrophetheory. Cambridge, England: Cambridge UniversityPress.

Shannon, C , & Weaver, W. (1949). The mathematicaltheory of communication. Urbana: University ofIllinois Press.

U.S. National Center for Health Statistics (1986,September 25). Percent distribution of divorces andannulments by duration of marriage to decree, andmedian and mean duration of marriage to decree:Divorce-registration area 1984 (Table 11). MonthlyVital Statistics Report (Vol. 35, No. 6, p. 14).Hyattsville, MD: Author.

U.S. National Center for Health Statistics (1988).Percent of all divorces by duration of marriage.Washington, DC: U. S. Government Printing Office.

Vallacher, R. R., & Nowack, A. (1994). Dynamicalsystems in social psychology. New York: AcademicPress,

von Bertalanffy, L. (1968). General system theory.New York: Braziller.

VonNeumann, J., & Morgenstern, O. (1947). Theoryof games and economic behavior. Princeton, NJ:University of Princeton Press.

Wiener, N. (1948). Cybernetics. New York: Wiley.Wiggins, J. (1977). Personality and prediction. New

York: Addison- Wesley.Wile, D. B. (1993). Couples therapy: A nontradi-

tional approach. New York: Wiley Interscience.Winfree, A. T. (1990). The geometry of biological

time. Berlin, Germany: Springer-Verlag.

Received August 4, 1997Revision received July 30, 1998

Accepted October 19, 1998 •