© 2006 society for chaos theory in psychology & life...

Nonlinear Dynamics, Psychology, and Life Sciences, Vol. 10, No. 3, pp. 365-399. © 2006 Society for Chaos Theory in Psychology & Life Sciences Electrodermal Arousal Between Participants in a Conversation: Nonlinear Dynamics and Linkage Effects Stephen J. Guastello1, Marquette University David Pincus, Chapman University Patrick R. Gunderson, Marquette University

Abstract: Physiological linkages between two people in conversation were investigated from a nonlinear dynamical systems theory perspec-tive. Thirty-seven pairs of college students engaged in a 20-minute di-scussion while connected to electrodes that measured their electrodermal response (ED). Greater incidents of linkage were detected through the nonlinear regression analysis compared to linear models. The nonlinear models were exponential structures that determined positive Lyapunov exponents (and thus the presence of chaos) for 73 out of 74 participants. Results support the use of nonlinear models for interpersonal linkage ef-fects of this type. For each participant, Social Sensitivity from the Social Skills Inventory was correlated with the nonlinear model R2 coefficient, but not with the size of the Lyapunov exponent. The latter results supported the conclusion that empathy acted as a moderator of the nonlinear process that underlies changes in ED levels over time. Key Words: physiological linkage, electrodermal response, chaos, self-organization, Lyapunov exponent, empathy, conflict

INTRODUCTION There is a growing interest in various types of synchronization

that can occur when two people engage in a problem-solving activity, conflict situation, or normal conversation. “Synchronization” is not particularly different from “coordination” in meaning, where two or more people take the same action, or compatible actions at the same time. “Synchronization” as described by Strogatz (2003) suggests a greater element of precision in the timing of events than “coordination.” “Coordination” emanates from game theory and human task performance research (e.g., Brannick, Prince, Prince, & Salas, 1993; Friedman, 1994;

1 Correspondence address: Stephen J. Guastello, Ph.D., Dept. Psychology, Marquette University, P. O. Box 1881, Milwaukee, WI 53201-1881, USA. E-mail: [email protected]

365

366 NDPLS, 10(3), Guastello, Pincus & Gunderson

Guastello & Guastello, 1998). “Physiological linkage” is an example of either synchronization or coordination, and it said to occur when the elevation levels of cardiovascular or electrodermal (ED, also known as the galvanic skin response) indicators of one person are affected closely in time by the same indicators of another person during an interaction (e.g. Gottman & Levinson, 1985, 1986; Levinson & Gottman, 1983).

Thus, the primary objective for the present study was to formu-late and test a viable nonlinear theory of physiological linkage that could be used as a platform for future research on this type of synchronization. The second objective was to determine the relationship between param-eters of the nonlinear model and social skills of the participants.

PSYCHOLOGICAL EXAMPLES OF SYNCHRONIZATION Synchronization phenomena can be observed in both living and

nonliving systems. The synchronization of clocks and flashing of South-east Asian fireflies are two poignant examples (Strogatz, 2003). Al-though the fireflies might begin their flashing for an evening with each firefly flashing at its own rate, all the fireflies eventually flash on and off in unison. The common elements that apparently underlie synchroniza-tion phenomena are a fundamental oscillating behavior, feedback loops among the oscillating entities, and a mechanism by which the feedback from one oscillator affects the speed of the next oscillator. The driving force of one oscillator on another often involves increasing the speed of oscillation.

In the case of human communication, synchronization can be observed as a patterning of verbal exchanges between a therapist and patient (Hartkamp & Schmitz, 1999), among members of a therapy group (Pincus & Guastello, 2005), among family members who may be in a state of conflict (Pincus, 2001), and among affable problem-solving team members (Fritz, Nagurney, & Helgeson, 2003; Guastello, 2000; Guastello, Hyde & Odak, 1998). A second, nonverbal type of synchronization is observed as the development of coordinated task sequences among work team members who cannot talk to each other (Guastello, Bock, Caldwell, & Bond, 2005; Guastello & Guastello, 1998), or the patterns by which two people shift their weight from the left foot to the right while in conversation (Shockley, 2003).

A third type of synchronization is observed as physiological arousal, which can be measured as cardiovascular activity (e.g. blood pressure) or ED; the latter is a more pure measure of sympathetic ner-vous system arousal (Levinson & Reuf, 1992). Physiological synchroni-zation as measured by ED is the central focus of the present investiga-

NDPLS, 10(3), Nonlinear Physiological Linkage 367

tion. Physiological synchronization can occur between a therapist and pa-tient when conflict is anticipated (Dimasco, Boyd, & Greenblatt, 1957; Malmo, Boag, & Smith, 1957), between two people in a distressed or vi-olent marriage (Gottman et al., 1995; Gottman & Levinson, 1986; Jacobson, Gottman, Waltz, Rushe, Babcock, & Holtzworth-Munroe, 1994; Levinson & Gottman, 1983; Margolin, John, & Gleberman 1988), in children as a result of their parents’ levels of conflict (Davies & Cummings, 1994). It can occur among friends during a problem disclo-sure (Fritz et al., 2003), or between two people who dislike each other (Kaplan, Burch, & Bloom, 1964; Mendes, Blascovitch-Lickel, & Hunter, 2002).

Nonlinear dynamical processes have been identified for all the foregoing synchronization processes except the particular group of physi-ological linkage phenomena (Guastello, 2000; Guastello et al., 2005; Guastello & Guastello, 1998; Guastello et al., 1998; Hartkamp & Schmitz, 1999; Pincus 2001; Pincus & Guastello, 2005; Shockley, 2003). Thus the primary objective of the present study was to pursue the non-linear dynamics of physiological linkage.

LINEAR DYNAMICS In their study of married couples, Levinson and Gottman (1983)

demonstrated that the degree of prediction in physiological arousal fluc-tuation, or physiological linkage, was significantly higher in the high-conflict conversation segments compared to the low-conflict segments. Furthermore, they found that physiological linkage was a strong predic-tor of self-reported marital dissatisfaction, accounting for 60 percent of the variance in a self-report measure. In later work, Gottman and Levinson (1986) characterized physiological linkages within distressed marriages by three consistent patterns: (a) higher degrees of negative affect, (b) more negative affect reciprocity, and (c) greater structure. In the latter, there is “more predictability of one spouse’s behaviors from those of the other, and less statistical independence than is found in the interactions of satisfied couples” (p. 39). Physiological linkage is believed to be a by-product of the behavioral and emotional entrainment that occurs between individuals engrossed in a heated discussion. In other words, greater behavioral responsivity across dissatisfied couples is reflected in greater physiological responsivity.

There was also a fourth common pattern of interaction within Gottman’s (1979a) earlier model that typified distressed marriages but was not used as support for physiological linkage. This pattern was one of dominance or non-egalitarianism in which there was “more


asymmetry in predictability.” (p. 73). In other words, one could predict the arousal level for one spouse (e.g. husband) from the arousal level of the other (e.g. wife), but not the other way around.

Levinson and Gottman (1983) used a combined index of physio-logical linkage for each couple in their study. This approach precluded their ability to explore the role of asymmetry during conflict. However, when using Levinson and Gottman’s (1983) methodology it is not neces-sary to use a combined index of physiological linkage.

EMPATHY EXPLANATION The typical behavioral patterns of interaction in distressed mar-

riages that Gottman identified, which are characterized by negative affect reciprocity and predictability, have strong empirical support (1979a; Gottman, Murray, Swanson, Tyson, & Swanson, 2002). Those studies also serve as straightforward explanations for physiological linkage, with the increased patterning and predictability in the ebb and flow of physio-logical arousal in distressed couples mirroring the increased patterning and predictability in the ebb and flow of verbal antagonism. Yet, they do not address the role of cognition in the phenomenon of physiological linkage between individuals. Malmo et al. (1957) studied the role of per-ceived threat in the form of a therapist’s criticism or praise of a client’s TAT story. The role of perception was also mentioned by Kaplan et al. (1964), who wrote, “[T]he simultaneous variation in physiological responses of the participants might provide clues to the specific stimuli (such as verbal content) that have common (or similar) tension-inducing or tension-reducing significance” (p. 93).

To take matters further, nonverbal linkages of other sorts have been identified between people who are engaged in non-hostile interac-tions (e.g. Guastello & Guastello, 1998; Guastello et al., 2005; Shockley, 2003). It would stand to reason that physiological linkage might in fact exist in non-hostile interactions, which would in turn suggest that other mechanisms beside threat could be responsible for shaping linkages. If that were the case, greater skill or greater empathy would predict stronger linkage.

Empathy contains both cognitive and emotional components. The cognitive aspect is such that one person can see another’s point of view and reasoning. The emotional aspect is such that one person can experience another person’s feelings. In this study, we investigated the role of social skills, as defined by Riggio (1989). His Social Skills Inventory (SSI) contains six measurements: Emotional Expressivity, Emotional Sensitivity, Emotional Control, Social Expressivity, Social


Sensitivity, and Social Control. The former three variables are forms of receptivity, and the latter three are forms of social behavior production. The SSI scales, on the one hand, are related to empathy (Rieke, Guastello, & Conn, 1994; Riggio, Tucker, & Coffaro, 1989). In fact, they are more specific than empathy and appear to be more proximally related to the type of interactions investigated in the experiment that follows.

NONLINEAR DYNAMICS The physiological linkage between two people that has been ob-

served in previous studies is the basis of self-organizing phenomena, wherein information flows between two subsystems, each affecting the other (Guastello, 2002; Haken, 1988). This form of connection reduces the entropy (i.e., unpredictability) in the behavior of either subsystem, and typically occurs at times of chaotic behavior. Chaotic behavior of systems is very unpredictable in the short term, although in the longer term it is deterministic and predictable in other respects. Like other ex-amples of self-organization in natural and social phenomena, self-organi-zation results from the spontaneous emergence of order from a chaotic system. In living systems, the chaotic states are precursors to successful adaptations. The phase transition between the state of adaptive readiness and an adaptive result occurs after there has been some local interaction among system elements.

Oscillations appear to underlie synchronization in human com-munication. According to Gottman (1979b), oscillations can be observed as patterns of gaze and gaze aversion, synchronization of respiration and heartbeat rates, and patterns of positive and negative affect in an im-provised conflict situation. The oscillations in these measures for two people in conversation appear to coincide over time in the same manner as a cosine function would coincide with a sine function. The main point of that paper (Gottman, 1979b), however, was that nonlinear forms of analysis, spectral analysis in that case, might be used to identify patterns. The formal constructs from nonlinear dynamics, other than oscillations, had not been brought to bear on these phenomena yet.

Two parties in a conversation might not have equal influences on each other. Coupling or synchronization between people can be strong or weak, equally balanced or one-sided (Gottman, 1979b; Nowak & Vallacher, 1998). Different dynamics can be expected in any combina-tion of conditions. One might find low-dimensional chaotic dynamics (dimensionalities between 1.0 and 2.0) in loosely coupled bilateral dyads. Tightly coupled bilateral dyads may end up producing strong oscilla-tions, as one “pulls up” while another “pushes down” as with a see-saw.


A similar type of coupling is occasionally observed in other situations (Balduzzo, Ferro Milone, Minelli, Pittaro-Cadore, & Turricchia, 2003).

Strong predictability of one partner’s responses using the other’s preceding responses without a corresponding degree of predictability in the other direction is, in essence, a driver-slave relationship (Haken, 1984). Psychologically, it is a form of dominance (Gottman, 1979b). Driver-slave dynamics are typical in high-conflict marriages (Gottman, 1979a; Gottman et al., 2002) and may be meaningful within other types of conflictual relationships. Also relevant here is a study of verbal ex-changes between a therapist and client (Hartkamp & Schmitz, 1999). Ac-cording to their operating theory, the concordance between the therapist’s behaviors and the client’s would be linear if they were in-fluencing each other equally, but nonlinear if there was an imbalance.

The principle nonlinear dynamic for this application is the Lyapunov exponent, which is a measure of turbulence (Ruelle, 1991). The Lyapunov exponent can be estimated statistically through nonlinear regression using a simple model:

z2 = e(θz1) (1)

(Guastello, 1995, 2002). In Eq. 1, the nonlinear regression weight θ is the Lyapunov exponent, and zi is the time series variable such that

zi = (yi - λ)/ σs. (2)

In Eq. 2, the raw score of the dependent measure y has been corrected for location and scale; an elaboration appears in the method section of this article. There is a convenient relationship between the fractal dimension DL and θ such that

DL = eθ (3)

(Guastello, 1995, 2002; Ott, Sauer, & Yorke, 1994). Note that Eq. 1 is essentially a nonlinear autocorrelation model,

and it is the first necessary step in testing for nonlinear dynamics in psy-chological linkage. In order to ascertain the linkage itself, Eq. 4 must also be tested:

z2 = e(θz1) + e(θ2p1) . (4)

The second part of Eq. 4 contains a second nonlinear regression weight that is associated with the partner’s ED measure at a given lag length. The partner’s contribution is also known a transfer effect in time series analysis.


Synchronization phenomena that have been studied from a non-linear dynamics perspective illustrate a process of self-organization, where information flowing from one person (or element) in the system affects the behavior of another person or element. The system then ex-hibits a transition from chaotic behavior to an ordering of behavior that occurs as a natural course of system events without any deliberate exter-nal cause (Strogatz, 2003). Spontaneous ordering reduces internal entro-py and makes the system’s operation more efficient, and thus adaptive to a situation. The relative amount of chaotic and self-organized behavior in a system varies with the situation. The toggling between chaotic and self-organized behavior that is observed in some systems is characteristic of a complex adaptive system (Dooley, 1997, 2004; Dooley & Van de Ven, 1999; Guastello, 2002). The chaotic behavior signifies a readiness to re-spond in many possible ways to incoming novel stimuli. The self-organi-zation signifies that the system has locked onto a suitable behavior pat-tern and is operating more efficiently, that is, with less entropy in its behavior.

In this application, we would expect to find positive Lyapunov exponents for most, if not all, participants, as well as a nonlinear transfer effect. The latter would be evidence that a basic amount (only one link-age is possible between two people) of self-organizing behavior has occurred.

HYPOTHESES The first hypothesis of the study was that greater amounts of

physiological linkage would be found in conversations between strangers in a contrived high-conflict situation, compared to a low-conflict or no-conflict situation. Physiological linkages were assessed by ED measures.

The second hypothesis was that nonlinear time series analysis would uncover greater numbers of physiological linkages compared to linear time series analysis. Concomitantly, the nonlinear approach would account for more variance in the ED measures than linear models. The nonlinear model in Eqs. 1-5 were based on the exponential model series from Guastello (1995, 2002) and are detailed in the Method section of this article. Importantly, the current study did not combine indices of physiological linkage across dyad members, thus allowing for the identi-fication and investigation of patterns of asymmetry in predictability.

The third hypothesis was that the Lyapunov exponents would be positive, signifying chaos.


METHOD Participants

Participants in this study were 76 (62 females, 14 males) intro-ductory psychology students at a private Midwestern university. The eth-nic distribution of the student pool was 90% Caucasian, and 10% ethnic minorities. Participants were administered informed consent forms, where they were asked to record their names and phone numbers. Each form contained an identification number, which was used after initial screening. Participants were then asked to complete a brief questionnaire, rating their opinions on a variety of emotionally-arousing topics (e.g. legalized abortion, prayer in public schools, the death penalty). Opinion ratings were measured on 9-point Likert scales, with “1” representing “strongly disagree” and “9” representing “strongly agree.” The question-naire also asked participants to report their gender and times of avail-ability to help facilitate pairing.

Participants were matched on gender (same-sex pairings), availability for testing, and their ratings of one of the opinion statements within the opposite extremes of the 9-point scale (1-3 vs. 6-9). These dyads were then randomly assigned to one of the three following conditions: competition, shared understanding, and control. The competi-tion and shared understanding groups each contained 26 participants (13 dyads) and the control group contained 24 participants (12 dyads). Two participants were deleted from analyses, due to equipment malfunction. Thus Dyad 13 does not appear in any data tables.

Procedure Prior to the discussion segments, participants were seated on

comfortable furniture at roughly a 90-degree angle, approximately 0.5 meters away from one another. Electrodes were attached to the first and second fingers of their non-dominant hands to enable the recording of their electrodermal responses (ED) during the interactions. All questions about the equipment and procedures were answered at this time. Data from each member of a dyad were recorded separately in time-indexed files; linkage was determined analytically. Testing began with the record-ing of a five-minute ED baseline on each participant, during which the dyads were instructed to keep their eyes closed and relax. Subsequently, participants received the competition, shared-understanding, or control instructions.

Participants in the competition group were told that their partner possessed the opposite opinion on a given topic and were instructed to


convince the partner that his or her view point on the topic was correct. Each participant was told to think of the discussion as a contest where he or she has 10 minutes to get his or her point across, while not allowing the partner to change his or her mind. Participants in the shared under-standing group were also told that their partners possessed the opposite opinion on a given topic. However, these participants were instructed to try to listen to their partner and understand their reasoning. They were also told that they had 10 minutes, but that the goal was to leave the dis-cussion with the best possible understanding of the other person’s opin-ion. Participant dyads in the control group were simply told that they had 10 minutes to get to know each other. They were instructed to talk about anything they wanted and were given some recommendations (e.g., interests, a recent event). After the instructions were given, the partici-pants in the two experimental conditions were told to begin their dis-cussion, while video-recording commenced through a one-way mirror.

The Social Skills Inventory (SSI; Riggio, 1989), a self-report in-strument, was used to assess interpersonal personality traits. The SSI is a 90-item instrument used to measure basic communication skills. The items consist of descriptive self statements (e.g., “I am usually wary of strangers.”) with responses presented in a five-point Likert-type format ranging from “not at all like me” (1) to “exactly like me” (5). It is design-ed to provide a total score, which is a measure of global social competen-ce as well as six subtest scores measuring distinct subdomains of social ability. The six subdomains are as follows: emotional expressivity, emo-tional sensitivity, emotional control, social expressivity, social sensitivi-ty, and social control. This instrument was chosen because of its brevity, comprehensiveness in generating several indices, and because of its relia-bility and validity, particularly when used with college samples (Riggio, 1989).

Overview of Analyses The objective of the present analyses was to determine whether

ED trends over time were a result of autocorrelation, transfer, or both. Autocorrelation occurs when a person’s score at any given time point is significantly predicted by his or her score at an earlier time point. Trans-fer refers to the prediction of a person’s score by his or her partner’s score at an earlier time point. In order to evaluate these kinds of patterns, the first step is to produce a lag function. In each dataset (dyad of two in-teracting participants), a variable was created indicating, at every time point, a participant’s ED reading 20 seconds prior. This variable – the lag function – was used in one set of linear and two sets of nonlinear multiple regression analyses.


A set of linear multiple regressions was conducted for each parti-cipant within each dyad, with ED reading as the dependent variable. The two predictors in this model were the participant’s own lag function (ED reading 20 seconds prior) and the lag function of the partner. Signifi-cance of a participant’s own lag function as a predictor indicates autocor-relation, while significance of a partner’s lag function indicates linkage.

Two sets of nonlinear regressions were then conducted in order to further evaluate autocorrelation and linkage effects. To begin, raw ob-served ED readings were transformed into z scores as defined in Eq. 2, where yi was the raw observation of ED, λ was location parameter (mean of time points before 5 minutes) and σs was the scale parameter (standard deviation of time points after 5 minutes). Two equations were run for each participant within each dyad. In the first equation, which was an amplification of Eq. 1

z2 = A*exp(B*z1)+C (5) A, B, and, C were regression parameters and z1 was the transformed score of a participant’s own relative lag function (his or her ED reading 20 seconds earlier than a given point in time). This equation captures autocorrelation effects. The constant C is an expendable parameter. If significance is not obtained for parameter B, which is far more critical in meaning, then C would be dropped from the analysis. Similarly, parame-ter A would be more expendable than parameter B (Guastello, 2002).

In the second equation, which was an amplification of Eq. 4, z2 = A*exp(B*z1)+C*exp(D*zp)+E (6)

A, B, C, D, and E were regression parameters, z1 again represented a participant’s own relative lag function, and zp was the transformed score of the partner’s relative lag function (ED reading of a participant’s partner 20 seconds earlier than a given point in time). This equation examines any possible effects of linkage. Parameters A, C, and E are expendable parameters in Eq. 6.

RESULTS Linear Linkage Effects

Autocorrelation and linkage were assessed for each participant in several steps. A series of linear, lag-sequential multiple regression analyses were conducted for each of the 37 dyads using a 20-second lag. Autocorrelation was found in all cases except two, indicating that ED ratings at each time point, for each participant, were significantly predicted by ratings 20 seconds prior.


Autocorrelation Linkage location scale Dataset R² t (ß) t (ß) (x < 5 min) (sd > 5 min)

1 Person 1 .65 6.39** (.69) 1.33 (.14) 5.61 2.42 Person 2 .83 13.48** (1.03) -2.15* (-.16) 9.14 2.55

7 Person 1 .85 9.73** (.60) 6.76** (.42) 6.76 4.33 Person 2 .76 13.67** (1.04) -4.53** (-.35) 5.39 1.10

11 Person 1 .48 7.57** (.76) -2.72* (-.23) 11.73 2.77 Person 2 .45 3.77** (.39) 3.90** (.40) 6.19 1.86

14 Person 1 .32 2.29* (.35) 1.66 (.25) 4.7 3.52 Person 2 .29 5.06** (.78) -2.90* (-.45) 4.06 2.95

21 Person 1 .92 9.43** (.88) .92 (.09) 4.03 1.76 Person 2 .94 10.14** (.84) 1.60 (.13) 4.15 2.49

22 Person 1 88 12.16** (1.03) -1.29 (-.11) 5.66 3.34 Person 2 .85 8.49** (.79) 1.59 (.15) 7.83 3.33

25 Person 1 .72 5.08** (.68) 1.43 (.19) 7.84 3.22 Person 2 .80 5.25** (.58) 3.02** (.34) 18.11 3.69

26 Person 1 .20 3.59** (.44) .25 (.03) 5.68 1.12 Person 2 .94 23.67** (.82) 7.28** (.25) 6.56 2.34

29 Person 1 .69 3.83** (.89) -.28 (.07) 12.09 2.54 Person 2 .58 .66 (.18) 2.19* (.59) 7.03 1.13

32 Person 1 .66 12.17** (.77) 1.46 (.09) 4.86 3.19 Person 2 .80 18.44** (.88) .84 (.04) 5.81 2.09

33 Person 1 26 4.80** (.62) -2.46** (-.32) 7.43 1.01 Person 2 .69 9.49** (.80) .55 (.05) 9.93 2.16

Note. ** indicates significance at the .01 level; * indicates significance at the .05 level.

The 20-second lag was used because the equipment could only generate readings at intervals of 10 seconds. A preliminary analysis of data produced at 10-second intervals showed no benefit in accuracy associated with the nonlinear model and no linkages with the linear model. The time interval was thus widened by one increment because research of other types indicated that widening the prediction intervals can favor a globally nonlinear model. Globally nonlinear models are asymptotically linear locally when the time interval is too short (Guastello, 2002; Thieler & Eubank, 1993; Wiggins, 1988).

At the second step of the analysis, the ED measure for the con-versation partner was entered as a possible predictor, again using a 20- Table 1. Results of Linear Analysis for Control Condition.



2 Person 1 .25 2.21* (.27) 2.39* (.29) 7.72 2.25 Person 2 .62 10.91** (.94) -3.43* (.-.30) 17.17 3.83

3 Person 1 .87 7.36** (.69) 2.79** (.26) 6.02 1.60 Person 2 .71 5.68** (.81) .24 (.04) 2.79 3.40

5 Person 1 .34 4.21** (.56) .27 (.04) 7.63 2.01 Person 2 .61 6.16** (.64) 1.81 (.19) 11.01 1.40

6 Person 1 .69 7.49** (.77) .80 (.08) 5.05 3.05 Person 2 .67 6.32** (.67) 1.75 (.19) 8.05 1.63

8 Person 1 .30 5.74** (.60) -1.74 (-.19) 4.95 1.77 Person 2 .86 18.14** (.85) 3.59** (.17) 4.24 2.74

15 Person 1 .42 5.42** (.81) -1.51 (-.23) 8.09 4.72 Person 2 .55 3.19** (.42) 2.74** (.36) 7.85 1.24

20 Person 1 .73 6.34** (.63) 2.55* (.26) 8.75 1.19 Person 2 .48 5.55** (.77) -.67 (-.09) 13.04 3.51

24 Person 1 .72 4.28** (.65) 1.42 (.21) 6.47 1.47 Person 2 .82 6.52** (.78) 1.08 (.13) 14.17 1.26

27 Person 1 .58 4.24** (.60) 1.31 (.18) 5.58 0.76 Person 2 .79 5.99** (.60) 3.22** (.32) 15.07 2.68

30 Person 1 .78 11.22** (.72) 3.63** (.23) 8.91 1.32 Person 2 .64 11.29** (.92) -2.67** (-.22) 22.99 1.85

34 Person 1 .05 2.05* (.20) .86 (.09) 1.83 0.68 Person 2 .85 23.00** (.90) 3.17** (.12) 10.42 2.94

37 Person 1 .65 12.86** (.87) -4.47** (-.30) 10.83 2.62 Person 2 .57 10.72** (.80) -1.80 (-.13) 17.43 2.46

38 Person 1 .74 8.90** (.67) 3.34** (.25) 10.88 3.06 Person 2 .30 5.67** (.70) -2.56* (-.32) 13.81 1.36


second lag. Individual R2 and t values for the autocorrelation and linkage effects appear in Tables 1, 2, and 3 for control (no-conflict) low conflict and high conflict groups, respectively. Linkage was assessed as single, double, or none at all. For single linkage, only one participant’s ED rating at each point in time was significantly related to the partner’s relative ED rating 20 seconds prior. Double linkage occurred when both Table 2. Results of Linear Regression Analysis for Shared Under-standing Condition.



4 Person 1 .58 7.31** (.92) -1.62 (-.20) 6.48 1.59 Person 2 .86 6.58** (.48) 6.73** (.49) 4.64 3.62

9 Person 1 .46 5.98** (.80) -1.23 (-.16) 2.91 2.86 Person 2 .53 4.21** (.52) 2.05* (.25) 6.92 2.96

10 Person 1 .83 10.10** (.93) -.26 (-.02) 6.62 4.68 Person 2 .86 10.99** (.92) .08 (.01) 7.05 1.29

12 Person 1 .34 5.25** (.71) -1.45 (-.20) 12.82 1.16 Person 2 .44 4.42** (.55) 1.22 (.15) 8.36 2.48

16 Person 1 .66 7.33** (.98) -1.56 (-.21) 11.05 4.93 Person 2 .57 3.47** (.52) 1.76 (.26) 11.93 2.44

17 Person 1 .84 17.17** (.93) -.20 (-.01) 9.49 1.13 Person 2 .31 -.92 (-.11) 5.37** (.61) 9.58 2.06

18 Person 1 .79 12.92** (1.11) -3.86** (-.33) 4.47 3.51 Person 2 .89 12.18** (.75) 3.86** (.24) 5.53 2.04

19 Person 1 .76 13.34** (.83) 1.75 (.11) 5.86 5.66 Person 2 .38 6.65** (.67) -2.55* (-.26) 12.86 1.37

23 Person 1 .92 14.84** (.79) 3.61** (.19) 6.4 2.84 Person 2 .97 27.74** (1.00) -.50 (-.02) 6.72 1.87

28 Person 1 .52 9.14** (.69) 2.12* (.16) 9.73 0.43 Person 2 .56 10.25** (.74) -3.09** (-.22) 12.23 1.28

31 Person 1 .75 11.89** (.99) -2.22* (-.18) 14.01 1.46 Person 2 .93 19.91** (.88) 2.48* (.11) 7.56 2.87

35 Person 1 .41 5.31** (.55) 1.46 (.15) 8.29 0.76 Person 2 .34 6.27** (.68) -2.44* (-.26) 16.94 1.74

36 Person 1 .90 18.46** (.89) 1.89 (.09) 12.71 4.03 Person 2 .34 3.30** (.40) 1.90 (.23) 17.22 2.88


partners’ ED ratings were significantly related to each other’s ED ratings 20 seconds prior. For the control group, lack of linkage occurred in 3 cases, single linkage in 6 cases, and double linkage in 2 cases. For the shared understanding group, lack of linkage occurred in 3 cases, single linkage in 7 cases, and double linkage in 3 cases. For the competition group, lack of linkage occurred in 4 cases, single linkage in 6 cases, and double linkage in 3 cases. In the two cases without autocorrelation the Table 3. Results of Linear Regression Analysis for Competition Condition.


target person’s ED time series was predicted from the partner’s time series; those two people were apparently highly driven by their partner’s behavior.

It was possible to compute an indicator average effect size asso-ciated with linkage, when it was detected in the linear analysis. This val- Table 4. Results of Nonlinear Regression Analysis Detecting Autocorrelation in Control Condition.

final final fractal Dataset R² A parameter B parameter dimension

1 Person 1 .82 .957 .334* 1.39 Person 2 .84 .895 .360* 1.43

7 Person 1 .73 .557 .558* 1.75 Person 2 .77 .973 .335* 1.40

11 Person 1 .73 1.073 .304* 1.36 Person 2 .68 1.426 .236* 1.27

14 Person 1 .67 1.072 .305* 1.35 Person 2 .58 1.108 .207* 1.23

21 Person 1 .83 .522 .603* 1.83 Person 2 .86 .511 .604* 1.83

22 Person 1 .86 .431 .644* 1.90 Person 2 .81 .440 .682* 1.98

25 Person 1 .79 .713 .448* 1.57 Person 2 .76 .518 .595* 1.81

26 Person 1 .40 .944 .303* 1.35 Person 2 .82 .386 .742* 2.10

29 Person 1 .86 .631 .453* 1.57 Person 2 .77 .629 .488* 1.63

32 Person 1 .46 .286 .947* 2.58 Person 2 .86 .749 .417* 1.52

33 Person 1 .34 .802 .337* 1.40 Person 2 .80 .704 .443* 1.56

Note. * indicates significance at the .05 level.



2 Person 1 .59 1.204 .274* 1.32Person 2 .72 .960 .346* 1.41

3 Person 1 .86 .659 .461* 1.59Person 2 .83 .819 .387* 1.47

5 Person 1 .50 1.008 .303* 1.35Person 2 .78 1.126 .294* 1.34

6 Person 1 .80 .731 .423* 1.53Person 2 .79 1.061 .308* 1.36

8 Person 1 .61 1.053 .312* 1.37Person 2 .77 .651 .485* 1.62

15 Person 1 .63 .892 .359* 1.43Person 2 .76 .978 .334* 1.40

20 Person 1 .86 .823 .372* 1.45Person 2 .71 .899 .363* 1.44

24 Person 1 .81 .560 .543* 1.72Person 2 .86 .618 .482* 1.62

27 Person 1 .68 .814 .399* 1.49Person 2 .86 .760 .416* 1.52

30 Person 1 .78 .671 .477* 1.61Person 2 .62 .839 .373* 1.45

34 Person 1 .07 .261 .369* 1.45Person 2 .78 .529 .597* 1.82

37 Person 1 .73 .859 .374* 1.45Person 2 .71 .724 .452* 1.57

38 Person 1 .81 .467 .585* 1.79Person 2 .66 1.312 .255* 1.29


ue was obtained by taking the ratio of the beta weight for the linkage ef-fect to the sum of the two weights (autocorrelation and linkage) in the linear model. The average value was .45, suggesting that a little less than half of the prediction of a person’s ED trend was associated with transfer Table 5. Results of Nonlinear Regression Analysis Detecting Autocorre-lation in Shared Understanding Group.



4 Person 1 .78 .686 .446* 1.56 Person 2 .81 .606 .500* 1.65

9 Person 1 .70 .962 .340* 1.40 Person 2 .66 .991 .333* 1.40

10 Person 1 .86 .669 .475* 1.61 Person 2 .92 .928 .331* 1.39

12 Person 1 .63 1.194 .275* 1.32 Person 2 .71 1.057 .314* 1.37

16 Person 1 .78 .832 .386* 1.47 Person 2 .78 .899 .361* 1.43

17 Person 1 .88 .992 .324* 1.38 Person 2 .56 1.533 .218* 1.24

18 Person 1 .82 .874 .362* 1.44 Person 2 .82 .436 .650* 1.92

19 Person 1 .82 .740 .422* 1.53 Person 2 .67 1.278 .262* 1.30

23 Person 1 .77 .579 .534* 1.71 Person 2 .88 .262 1.01* 2.75

28 Person 1 .81 1.497 .225* 1.25 Person 2 .34 .509 -1.094* 2.99

31 Person 1 .66 .873 .378* 1.46 Person 2 .84 .408 .698* 2.01

35 Person 1 .68 1.531 .222* 1.25 Person 2 .63 1.043 .313* 1.37

36 Person 1 .86 .592 .519* 1.68 Person 2 .63 .977 .337* 1.40


from the other member of the dyad, if there was any linkage detection at all. Table 6. Results of Nonlinear Regression Analysis Detecting Autocorre-lation in Competition Group.


final final final fractal Dataset R² A parameter B parameter D parameter dimension

1 Person 1 .79 .224 .470* .226* 2.85 Person 2 .80 .142 .589* .186* 3.00

7 Person 1 .58 .014 1.313 .168* 2.18 Person 2 .69 .295 .483* .209* 2.85

11 Person 1 .69 .301 .444* .132* 2.70 Person 2 .69 .407 .300* .258* 2.64

14 Person 1 .64 .365 .378* .221* 2.71 Person 2 .57 .133 .481* .201* 2.84

21 Person 1 .71 .003 1.743 .290* 2.34 Person 2 .72 .015 1.304 .289* 2.34

22 Person 1 .70 .049 1.080* .226* 4.19 Person 2 .61 .011 1.569 .198* 2.22

25 Person 1 .74 .060 .786* .333* 3.59 Person 2 .64 .008 1.442 .238* 2.27

26 Person 1 .41 .025 .809 .304* 2.36 Person 2 .59 .039 1.326* .127* 4.91

29 Person 1 .80 .096 .676* .279* 3.29 Person 2 .74 .001 1.378 .274* 2.32

32 Person 1 .61 -1.151 -.531 .127* 2.14 Person 2 .76 .287 .581* -.033 2.79

33 Person 1 .25 .052 .721* .117* 3.18 Person 2 .71 .185 .664* .174* 3.13


Nonlinear Effects A series of nonlinear regression analyses were then conducted on

each dyad to investigate the presence of chaotic autocorrelation, as signified by a significant and positive Lyapunov exponent (parameter B in Eq. 1). Nonlinear autocorrelation occurred in all cases. Fractal dimensions (Eq. 3) were calculated also. Lyapunov exponents and fractal dimensions are given in Tables 4, 5, and 6. In the course of the analyses, Table 7. Results of Nonlinear Regression Analysis Detecting Autocorre-lation and Transfer in Control Condition.


the constant parameter C was dropped from the model as it made no contribution to the explanation of the ED series.

A second set of nonlinear regression analyses were conducted in order to evaluate physiological linkage effects. In each condition, lack of linkage never occurred. For the control group, single linkage occurred in Table 8. Results of Nonlinear Regression Analysis Detecting Autocorre-lation and Transfer in Shared Understanding Group.


2 Person 1 .62 .233 .390* .260* 2.78 Person 2 .64 .274 .520* .106* 2.79

3 Person 1 .81 .040 .789* .268* 3.51 Person 2 .79 .081 .632* .283* 3.21

5 Person 1 .50 .031 .695 .175* 2.19 Person 2 .73 .472 .374* .196* 2.67

6 Person 1 .77 .060 .792* .196* 3.43 Person 2 .80 .135 .500* .300* 3.00

8 Person 1 .57 .249 .496* .243* 2.92 Person 2 .64 .045 1.037* .133* 4.24

15 Person 1 .60 .059 .676* .210* 3.20 Person 2 .72 .276 .451* .235* 2.83

20 Person 1 .82 .200 .527* .120* 2.82 Person 2 .69 .066 .690* .217* 3.23

24 Person 1 .73 .016 1.175 .239* 2.27 Person 2 .80 .047 .826* .317* 3.65

27 Person 1 .68 .06 1.108 .256* 2.29 Person 2 .79 .187 .606* .196* 3.05

30 Person 1 .66 .117 .747* .185* 3.31 Person 2 .48 .140 .627* .152* 3.03

34 Person 1 .08 -.990 -.130 .129* 2.14 Person 2 .58 .119 .943* .105* 3.68

37 Person 1 .63 .209 .603* .094 2.83 Person 2 .60 .163 .720* .107* 3.16

38 Person 1 .70 .066 .936* .142* 3.70 Person 2 .64 .236 .420* .278* 2.84



1 case and double linkage in 10 cases. For the shared understanding group, single linkage occurred in 1 case and double linkage in 12 cases. For the competition group, single linkage occurred in 2 cases, and double linkage in 11 cases. Individual R2 values, significance tests on the confidence intervals for each nonlinear regression parameter (at p < .05), and fractal dimension values appear in Tables 7, 8, and 9. In the course of the analyses, the constant parameters C and E were dropped from the models as they made no contribution to the explanation of the ED series. Table 9. Results of Nonlinear Regression Analysis Detecting Autocorre-lation and Transfer in Competition Group.


4 Person 1 .70 .130 .663* .226* 3.19 Person 2 .78 .01 1.140 .267* 2.31

9 Person 1 .64 .205 .501* .212* 2.89 Person 2 .66 .139 .494* .248* 2.92

10 Person 1 .79 .028 1.060* .152* 4.05 Person 2 .89 .302 .465* .184* 2.79

12 Person 1 .58 .276 .393* .199* 2.70 Person 2 .67 .296 .414* .204* 2.74

16 Person 1 .74 .066 .704* .235* 3.28 Person 2 .75 .144 .529* .269* 3.01

17 Person 1 .85 .244 .485* .169* 2.80 Person 2 .67 .174 .314* .274* 2.69

18 Person 1 .76 .166 .565* .286* 3.09 Person 2 .68 .013 1.413 .174* 2.19

19 Person 1 .77 .113 .712* .164* 3.22 Person 2 .64 .321 .395* .223* 2.73

23 Person 1 .62 .007 1.421* .333* 5.54 Person 2 .43 .010 2.119* .084 9.32

28 Person 1 .52 -1.094 -.500* .049* 2.70 Person 2 .76 .948 .267* -.091 2.31

31 Person 1 .57 .133 .599* .283* 3.15 Person 2 .67 .015 1.544* .163* 5.86

35 Person 1 .68 .567 .271* .236* 2.58 Person 2 .61 .222 .486* .149* 2.79

36 Person 1 .74 .048 1.017* .158* 3.93 Person 2 .64 .106 .582* .309* 3.15



The fractal dimension for the nonlinear linkage analyses was calculated as

DL = eB + eD (8)

which is a co-dimension composed of two partial functions (cf. Guastello, 1995). If the confidence interval for parameters B or D did not indicate statistical significance, the nonsignificant parameter was treated as 0. Thus if D were 0, then eD simplified to 1.0.

Phase Portraits

The phase portrait is a plot of velocity as a function of position. In both cases the dots are connected to represent the serial ordering of points as they occurred in the process. In the case of a one-variable plot, the coordinates of each point would be ∆Z as a function of Z (our measure of ED level for one person in this study) for each point in time. In the case of a two-variable plot, the coordinates of each point would be would be ∆Z for Person 1 as a function of ∆Z for Person 2.

Phase portraits of real data typically do not produce the pretty pictures that are associated with the classic chaotic attractors, such as the Lorenz, Rossler, or Henon-Heiles systems. Portraits of real data are often tainted with noise, and information is often blurred because the data were not projected in a sufficient number of dimensions (Abarbanel, 1996; Abraham, 1997; Guastello & Bock, 2001). Thus we have taken steps to extricate noise and to project the data for three typical dyads in both two and four dimensions. The four-dimensional plot required the use of two three-dimensional sectionings, where each sectioning showed DZ for one member of the dyad as a function of the initial positions of both members of the dyad.

Figure 1 shows the time series for the two members of a Dyad 37 where single linkage occurred. In this dyad, Person 1 affected Person 2, but not the other way around. Figures 2 and 3 show their phase portraits. They are plotted in three dimensions in Fig. 3 so that, in the upper panel, the three axes are Person 1 at Time 1, Person 2 at Time 1, and the predicted value of Person 1 at Time 2. In the lower panel, the axes are Person 1 at Time 1, Person 2 at Time 1, and the predicted value of Person 2 at Time 2. The five-minute resting period was omitted from the return maps and phase portraits. Person 1 was placed on the ordinate of Fig. 2 because Person 1 appeared to be the “driver” in that conversation.


Fig. 1. Time series plots for Dyad 37.

Fig. 2. Two-dimensional phase portrait for Dyad 37 showing single linkage with Person 1 affecting Person 2, but not vice versa.


Fig. 3. Four dimensional phase portrait for Dyad 37.

The diagrams in Figs. 2 and 3 and in others shown subsequently

were produced using the z values for the Time 1 scores as defined in Eq. 2. The Time 2 scores were produced using the predicted values of z based on the model parameters obtained for Eq. 5, as given in Table 5. These transformations of raw data extricated the noise (variance unac-counted for by the best available model) from the plots; the presence of noise often obfuscates the underlying visual patterns (Guastello & Bock, 2001).

The Lyapunov dimension for Person 1’s time series taken singly was 1.45, and the dimension for Person 2’s time series taken singly was 1.82 (based on values shown in Table 5). The co-dimension for the time



Fig. 5. Two-dimensional phase portrait for Dyad 23 showing single linkage with Person 2 affecting Person 1 but not vice versa.


Fig. 6. Four-dimensional phase portraits for Dyad 23. series for Person 1, when considered with Person 2 as a possible linkage agent was 2.83. The dimension for Person 2, when considered with Per-son 1 as a possible linkage agent was 3.16 (based on values given in Table 8).

The phase portraits showed that the arousal of both conversants increased sharply at first with a slowing rate. Person 1 remained relatively stable as shown by the cluster of points in the upper portion of Figs 2 and 3. Person 2’s ED measurements were much more volatile than Person 1’s measurements.

Figure 4 shows the time series for Dyad 23 where single linkage also occurred by the dimensionality was higher. In this dyad, Person 2 affected Person 1, but not the other way around. Figures 5 and 6 show their phase portraits. The Lyapunov dimension for Person 1’s time series



Fig. 8. Two-dimensional phase portrait for Dyad 31 showing double linkage.


Fig. 9. Four-dimensional phase portrait for Dyad 31.

taken singly was 1.71, and the dimension for Person 2’s time series taken singly was 2.75 (see Table 6). The co-dimension for the time series for Person 1, when considered with Person 2 as a possible linkage agent was 5.54. The dimension for Person 2, when considered with Person 1 as a possible linkage agent was 9.32 (see Table 9).

Compared to Dyad 37, the two members of Dyad 23 escalated in ED arousal more gradually and steadily in the earlier phase of the conversation. Person 2, who was driving Person 1, became much more volatile than Person 1. Person 1 seemed to react more to Person 2’s overall arousal level, and less to Person 2’s local ups and downs. After a period of stabilization, however, Person 1 displayed a sharp increase in ED arousal.

Figure 7 shows the time series for the two members of Dyad 31


where double linkage occurred. Figures 8 and 9 show their phase portraits. The Lyapunov dimension for Person 1’s time series taken singly was 1.45, and the dimension for Person 2’s time series taken singly was 2.01 (see Table 6). The time series for Person 1, when considered with Person 2 as a possible linkage agent was 3.15. The dimension for Person 2, when considered with Person 1 as a possible linkage agent was 5.96 (see Table 9).

Both members of the Dyad increased ED arousal gradually and reached a stability point indicated by the cluster of points. Both shared a momentary increase in arousal toward the end of the conversation. Neither was visibly more volatile than the other, although the numerical analysis indicated a difference of about 0.5 dimensions.

Experimental Conflict Manipulation Two split-plot ANOVAs were conducted using R2 and fractal di-

mension values as dependent measures. The fixed effect was experimen-tal condition differences (control vs. shared understanding vs. competition). Type of analysis (linear vs. nonlinear without linkage vs. nonlinear with linkage) was the repeated measure. There were three levels of the repeated factor in the analysis of R2 coefficients, which showed that R2 values generated by the nonlinear analyses testing for autocorrelation without linkage (M = .73, SD = .15) were significantly greater than those generated by the nonlinear analyses testing for linkage (M = .67, SD = .13) and the linear analyses (M = .63, SD = .22, F(2, 70) = 36.50, p<.01). No significant effect was found for experimental condition, nor was there any interaction.

There were two levels of the repeated measure in the analysis of fractal dimension (nonlinear without linkage vs. nonlinear with linkage) wherein fractal dimension values generated by the nonlinear linkage analyses (M = 3.10, SD = 1.01) were significantly greater than those gen-erated by the nonlinear analyses without the linkage (M = 1.57, SD = .32, F(1, 71) = 201.28, p<.01). These findings indicate that a greater degree of turbulence occurred in a time series in which nonlinear analyses took into account both autocorrelation and linkage effects than those which took into account only autocorrelation effects. This was an unsurprising result based on the method by which co-dimension is calculated. Again, no experimental condition or interaction effects were detected.

Two maximum likelihood chi-square analyses were conducted to evaluate the association between experimental condition and rates of linkage, i.e., how often single, double, and no linkage occurred in each experimental condition. Observed frequencies were not significantly


different than expected frequencies for control, shared differences, and competition groups in both the linear analysis, χ ² (df = 4, N = 37) = 0.40, n.s., and nonlinear analysis, χ² (df = 4, N = 37) = 0.23, n.s.

A third maximum likelihood chi-square analysis was conducted to evaluate the association between rates of linkage and nonlinear vs. linear modes of analysis. In this case, the observed frequencies were significantly different than the expected frequencies, χ ² (df = 2, N = 74) = 40.86, p<.001. In the linear analysis, lack of linkage and single linkage occurred more frequently than expected, while double linkage occurred less frequently than expected. Conversely, in the nonlinear analysis, lack of linkage and single linkage occurred less frequently than expected, while double linkage occurred more frequently than expected. In other words, the nonlinear analysis detected physiological linkages far more often than the linear analysis.

Social Skills A linear regression was conducted using the stepwise method,

with R² values obtained in the nonlinear autocorrelation analyses as the dependent variable. Scores for each SSI subscale were used as predictors. Only Social Sensitivity was found to be significant, (β = .24, t = 2.08, p < .05, R² = .06). A second stepwise linear regression was conducted, with R² values obtained in the nonlinear linkage analyses as the dependent variable and scores for each SSI scale as predictors. Again, only Social Sensitivity significantly predicted R² values, (β = .26, t = 2.27, p<.05, R² = .07). The remaining variables of emotional expressivity, emotional sensitivity, emotional control, social expressivity, and social control were not found to be significant predictors in either of these analyses. None of the SSI scales predicted fractal dimension from either of the nonlinear analyses.

DISCUSSION The first noteworthy finding is that the contrived conflict condi-

tions did not produce more frequent linkages than the control group of dyads. In retrospect, there were some fundamental differences between these experimental dyads and the distressed marital relationships or ther-apeutic relationships in terms of history, expectations, intimacy, and norms. Because the participants in each dyad had no previous relation-ship history or future expectations for interaction, the potential for hos-tility and emotional vulnerability were probably minimal. It also seems likely that demand characteristics of the contrived situation created a norm for politeness and tolerance in these student volunteers.


On the contrary, it is possible that covert conflict existed across the experimental groups (not easily observable or noticeable to participants based on positive rating for “liking” their partners even in the high-conflict condition). The experimental groups were designed to assess three levels of conflict with other pairing factors (i.e., opposing attitudes) controlled. The possibility of unconscious conflict underlying linkage across conditions could have been more fully explored with an attitude-similar control condition.

Linkage Detection Given the lack of expressed conflict overall, it was all the more

surprising that physiological linkages were detected in all 37 dyads, and in 70 out of 74 people using the nonlinear analysis. The linear analysis, in contrast, detected physiological linkage in 26 of out 37 dyads, and in only 35 out of 74 people. Not only did the nonlinear models produce higher R² values than the linear models, they detected linkage twice as often. These results strongly suggest that linkage phenomena in social psychology should be studied from the nonlinear theoretical viewpoint and its accompanying methodology. They also strongly suggest, given the outcome of the experimental manipulation here, that conflict is not a necessary condition for physiological linkage. Thus the empathy explanation is a stronger explanation for the origin of physiological linkage; this point is revisited below.

Unfortunately, it was not possible to detect directly how much ED variance the linkage effects accounted for by themselves in the non-linear analyses. R² values actually dropped somewhat when the linkage parameter was added to the nonlinear model. This is an idiosyncrasy of nonlinear regression that does not correspond with linear regression. In nonlinear regression, the size of R² and the final model parameters are not as closely related as they are in linear regression. The introduction of a new variable could actually introduce another source of error, whereas in linear regression an additional variable would have a trivial positive impact on R² in the worst case. Thus the interpretation of a nonlinear regression model relies more strongly on the significance tests on the individual parameters. If the linear results were of any indicator here, however, a little less than half of the prediction of a person’s ED series was associated with physiological linkage with the partner.

Beyond the linear versus nonlinear question, the fact that non-linear linkages can occur in the absence of linear linkages (and in more rare cases the opposite result) suggests that these phenomena are some-what distinct, and that different types of linkages each may provide dif-


ferent information about small group coordination. It may be helpful to think of linear linkage as a first order phenomenon, whereby individuals may drive one another up or down physiologically over time with the lag-function allowing for driver-slave relationships. In the case of non-linear linkages, members are driving one another’s levels of deterministic turbulence, a second order level of influence. Beyond the finding that se-cond-order effects appear to be more common and robust, the current study suggests that first and second-order linkages can occur indepen-dently of one another. The differing putative interpersonal functions of first and second-order linkages may be a fruitful avenue for further investigation.

Nonlinear Theory Linkage was detected with the nonlinear analysis twice as often

as it was with the linear analysis, and the overall model accuracy was significantly larger for the nonlinear analyses. Thus, three important points follow. First, the results supported the underlying theory of com-plex adaptive systems and self-organizing behavior as an explanation or description of the process by which any physiological or other linkage might occur in a dyad or larger social unit. For each person, there was a low-dimensional chaos as signified by a positive Lyapunov exponent with an average fractal dimension of 1.57. Fractal dimensions between 1.0 and 2.0 are usually regarded as the range of “pink noise” where self-organizing behavior is thought to be taking place (Bak, 1996; Dooley & Van de Ven, 1999).

Second, any research paradigm that is concerned with physiological linkage may be more likely to detect a linkage effect, by using nonlinear analysis and the exponential model series (Eqs. 2-3, and 5-7). It is reasonable to expect that deeper understanding of factors that influence physiological linkage could be obtained from the nonlinear perspective. In fact self-organized behavior among three or more people can be studied as extrapolations of dyadic interactions.

Third, the physiological linkage, as detected by ED, can be interpreted further as one member of class of synchronization phenomena that includes synchronized EEGs within a person (Balduzzo et al., 2003), weight-shifting in a dyadic conversation (Shockley, 2003), and coordinated nonverbal task performance (Guastello, 2002; Guastello et al., 2005; Guastello & Guastello, 1998). The foregoing phenomena all exhibit the shift from independent and chaotic behavior to synchronized and self-organized behavior. Inasmuch as weight-shifting, physiological linkage, and coordinated task performance all involve communicative


interactions among individuals, it would follow that many other aspects of social interaction would follow the same dynamics.

Social Skills Social Sensitivity was correlated with the degree to which the

nonlinear models fit the data. As such it acted as a moderator of the entire nonlinear model. There were greater amounts of determinism and less stochastic noise in the ED time series taken from people with higher levels of social Sensitivity. Social Sensitivity was not related to the amount of turbulence in the time series, however, nor was any other SSI variable. The exclusive relationship of social sensitivity and no other SSI variable could in part be related the fact that the SSI is a single, static, self-report measure. Future analyses may find more robust relationships among linkages, turbulence, and numerous interpersonal factors if obser-vation-based measures are included. It might also be worthwhile to assess perceptions of the relationship under investigation in addition to people’s general interpersonal styles.

Nevertheless, the positive results for Social Sensitivity suggest that empathy and attunement would be important central constructs in future investigations. Perhaps Fritz et al. (2003) were on the right track by examining the impact of a listener’s reactions to a friend’s disclosure of a conflict with a third party. The discloser’s cardiovascular arousal level fluctuated with the supportiveness of the listener.

The size of the Lyapunov exponents themselves are another matter. They represent relative amounts of turbulence or volatility in time series. Their psychological interpretation in this context is not yet known and should be a topic of future research.

Future Research There are several avenues for future research and theory build-

ing. Given that it now appears that physiological linkage is more frequent than the conflict-centered literature seemed to indicate, it may be worth-while to examine physiological linkage in conversations where the dyads had a greater vested interest in the topic of conversation or in the contin-uing relationship with the conversation partner. We can question whether the same nonlinear dynamics are produced and whether they are more or less turbulent or predictable from psychological variables.

An obvious limitation of the present study is that the attempt to manipulate levels of conflict did not work. On the one hand, this result allowed us to interpret the results as meaning that the intended conflict dyads were no different from dyads in the control condition. On the other


hand, there remains the question as to how linkage from artificially created stranger-dyads would compare with linkages from truly distressed relationships.

Nowak and Vallacher (1998) indicated that tight versus loose coupling can produce different dynamics over time. Tight or loose coupling may be more adaptive, depending on the social context. We have yet to see the effect of coupling strength in the nonlinear dynamics in the behavior of real people (as opposed to numerical simulations). We could venture a guess, however, that the dyads in the present study were all loosely coupled by virtue of their strategic pairing for opposite views on controversial topics and their apparent unwillingness to pick a verbal fight with each other during the experimental sessions.

The nonlinear dynamics of linkage in hostile versus non-hostile interactions of marriage partners would be an important area of further investigation. In addition to the coupling issue, such an investigation could examine the possible role of nonlinear linkage in a context where linear linkage has been found to have a robust connection with marital satisfaction (Levinson & Gottman, 1983).

It would be valuable to know what preconditions of a relation-ship would lead to single versus double linkage. Single linkage, which is characteristic of a driver-slave relationship, might be likely if the part-ners have very different dominance scores. Double linkage might be likely if their dominance scores were similar. In light of the large number of double linkage dyads observed in this study relative to single linkage dyads, a fresh study should be conducted where the dyads are delib-erately composed on the basis of dominance scores. Again, the possible differences between linear versus nonlinear dominance might prove interesting as well.

Although dominance and empathy might foster linkage effect, there is little reason to believe that those two variables alone would be responsible for the amount of turbulence in ED measurements; the turbu-lence is what is indicated by the Lyapunov exponent. Personality vari-ables that are related to emotional stability might be responsible here instead.

REFERENCES Abarbanel, H. D. I. (1996). Analysis of chaotic data. New York: Springer-

Verlag. Abraham, F. D. (1997). Nonlinear coherence in multivariate research: Invariants

and the reconstruction of attractors. Nonlinear Dynamics, Psychology, and Life Sciences, 1, 7-34.

Bak, P. (1996). How nature works. New York: Springer-Verlag.


Balduzzo, M., Ferro Milone, F., Minelli, T. A., Pitarro-Cadore, I., & Turicchia, L. (2003). Mathematical phenomenology of neural stimulation by periodic fields. Nonlinear Dynamics, Psychology, and Life Sciences, 7, 115-138.

Brannick, M. T., Prince, A., Prince, P., & Salas, E. (1995). The measurement of team process. Human Factors, 37, 641-651.

Davies, P. T., & Cummings, E. M. (1994). Martial conflict and child adjustment: An emotional security hypothesis. Psychological Bulletin, 116, 387-411.

Dimascio, A., Boyd, R. W., & Greenblatt, M. (1957). Physiological correlates of tension and antagonism during psychotherapy: A study of “interpersonal physiology.” Psychosomatic Medicine, 19, 99-104.

Dooley, K. J. (1997). A complex adaptive systems model of organization change. Nonlinear Dynamics, Psychology, and Life Sciences, 1, 69-97.

Dooley, K. J., (2004). Complexity science models of organizational change and innovation. In M. S. Poole & A. H. Van de Ven (Eds.). Handbook of organizational change and innovation (pp. 354-373). New York: Oxford.

Dooley, K. J., & Van de Ven, A. H. (1999). Explaining complex organizational dynamics. Organization Science, 10, 358-372.

Friedman, J. W. (Ed.). (1994). Problems of coordination in economic activity. Boston: Kluwer.

Fritz, H. L., Nagurney, A. J., & Helgeson, V. S. (2003). Social interactions and cardiovascular reactivity during problem disclosure among friends. Personality and Social Psychology Bulletin, 29, 713-725.

Gottman, J. M. (1979a). Marital interaction: Experimental Investigations. New York: Academic Press.

Gottman, J. M. (1979b). Detecting cyclicity in social interaction. Psychological Bulletin, 86, 338-348.

Gottman, J. M., Jacobson, N. S., Rushe, R. H., Shortt, J. W., Babcock, J., La Taillade, J. J., & Waltz, J. (1995). The relationship between heart rate reactivity, emotionally aggressive behavior, and general violence in batterers. Journal of Family Psychology, 9, 227-248.

Gottman, J. M., & Levinson, R. W. (1985). A valid procedure for obtaining self-report of affect in marital interaction. Journal of Consulting and Clinical Psychology, 53, 151-160.

Gottman, J. M., & Levinson, R. W. (1986). Assessing the role of emotion in marriage. Behavioral Assessment, 8, 31-48.

Gottman, J. M., Murray, J. D., Swanson, C. C., Tyson, R., & Swanson, K. R. (2002). The mathematics of marriage: Dynamic nonlinear models. Cambridge, MA: MIT Press.


Guastello, S. J. (1995). Chaos, catastrophe, and human affairs: Applications of nonlinear dynamics to work, organizations, and social evolution. Mahwah, NJ: Erlbaum.

Guastello, S. J. (2000). Symbolic dynamic patterns of written exchange: Hierarchical structures in an electronic problem solving group. Nonlinear Dynamics, Psychology, and Life Sciences, 4, 169-188.

Guastello, S. J. (2002). Managing emergent phenomena: Nonlinear dynamics in work organizations. Mahwah, NJ: Lawrence Erlbaum Associates.

Guastello, S. J., & Bock, B. R. (2001). Attractor reconstruction with principal components analysis: Application to work flows in hierarchical organizations. Nonlinear Dynamics, Psychology, and Life Sciences, 5, 175-192.

Guastello, S. J., Bock, B. R., Caldwell, P., & Bond, R. W. Jr. (2005). Origins of group coordination: Nonlinear dynamics and the role of verbalization and personnel replacement. Nonlinear Dynamics, Psychology, and Life Sciences, 9, 175-207.

Guastello, S. J., & Guastello, D. D. (1998). Origins of coordination and team effectiveness: A perspective from game theory and nonlinear dynamics. Journal of Applied Psychology, 83, 423-437.

Guastello, S. J., Hyde, T., & Odak, M. (1998). Symbolic dynamic patterns of verbal exchange in a creative problem solving group. Nonlinear Dynamics, Psychology, and Life Sciences, 2, 35-58.

Haken, H. (1984). The science of structure: Synergetics. New York: Van Nostrand Reinhold.

Haken, H. (1988). Information and self-organization: A macroscopic approach to complex systems. New York: Springer-Verlag.

Hartkamp, N., & Schmitz, N. (1999). Structures of introject and therapist-patient interaction in a single case study of inpatient psychotherapy. Psychotherapy Research, 9, 199-215.

Jacobson, N. S., Gottman, J. M., Waltz, J., Rushe, R., Babcock, J., & Holtzworth-Munroe, A. (1994). Affect, verbal content, and psychophysiology in the arguments of couples with a violent husband. Journal of Consulting and Clinical Psychology, 62, 982-988.

Kaplan, K. B., Burch, N. R., & Bloom, S. W. (1964). Physiological covariation and sociometric relationships in small peer groups. In P. H. Leiderman & D. Shapiro (Eds.), Psychobiological Approaches to Social Behavior (pp. 92-109). Stanford, California: Stanford University Press.

Levinson, R. W., & Gottman, J. M. (1983). Marital interaction: Physiological linkage and affective exchange. Journal of Personality and Social Psychology, 45, 587-597.


Levinson, R. W., & Reuf, A. M. (1992). Empathy: A physiological substrate. Journal of Personality and Social Psychology, 63, 234-246.

Malmo, R. B., Boag, T. J., & Smith, A. A. (1957). Physiological study of personal interaction. Psychosomatic Medicine, 19, 105-119.

Margolin, G., John, R. S., & Gleberman, L. (1988). Affective responses to conflictual discussions in violent and nonviolent couples. Journal of Consulting and Clinical Psychology, 56, 24-33.

Mendes, W. B., Blascovich, J., Lickel, B., & Hunter, S. (2002). Challenge and threat during social interactions with white and black men. Personality and Social Psychology Bulletin, 28, 939-952.

Nowak, A., & Vallacher, R. R. (1998). Dynamical social psychology. New York: Guilford Press.

Ott, E., Sauer, T., & Yorke, J. A. (Eds.). (1994). Coping with chaos. New York: Wiley.

Pincus, D. (2001). A framework and methodology for the study of nonlinear, self-organizing family dynamics. Nonlinear Dynamics, Psychology, and Life Sciences, 5, 139-176.

Pincus, D., & Guastello, S. J. (2005). Nonlinear dynamics and interpersonal correlates of verbal turn-taking patterns in a group therapy session. Small Group Research, 36, 635-677.

Rieke, M. L., Guastello, S. J., & Conn, S. R. (1994). Empathy and interpersonal skills. In S. R. Conn & M. L. Rieke (Eds.), The 16PF Fifth Edition Technical Manual (pp. 163-182). Champaign, IL: Institute for Personality and Ability Testing.

Riggio, R. E. (1989). Social Skills Inventory: Manual. Palo Alto, CA: Consulting Psychologists Press.

Riggio, R. E., Tucker, J., & Coffaro, D. (1989). Social skills and empathy. Personality and Individual differences, 10, 93-99.

Ruelle, D. (1991). Chance and chaos. Princeton, NJ: Princeton University Press. Shockley, K. (2003, October). Application: Cross-RQA and interpersonal

synchrony. In M. Riley & G. Van Orden (Co-Chairs), Workshop: Nonlinear methods for psychology. Fairfax VA: George Mason University and National Science Foundation.

Strogatz, S. (2003). Sync: The emerging science of spontaneous order. New York: Hyperion.

Thieler, J., & Eubank, S. (1993). Don’t bleach chaotic data. Chaos, 3, 771-782. Wiggins, S. (1988). Global bifurcations and chaos. New York: Springer-Verlag.

© 2006 society for chaos theory in psychology & life...

Documents