catastrophe theory in social psychology: some applications ...people.oregonstate.edu/~flayb/my...

CATASTROPHE THEORY IN SOCIAL PSYCHOLOGY: SOME APPLICATIONS TO ATTITUDES AND SOCIAL BEHAVIOR'

by Brian R. Flay Northwestern Universiiy

Much past mathematical modeling of psychological processes has assumed (a) smooth and continuous change in behavior or cognitions or, if not, @) simple step bctions or thresholds. Many psychological phenomena which are not smooth and continuous, or do not meet the assumptions of simple step functions, seem to demonstrate the properties of the cusp or butterfly catastrophes. Catastrophe models have already been proposed for many psychological phenomena In this paper catastrophe models are proposed for social behavior, attitude change, and some other related processes. These models synthesize many diverse and sometimes seemingly contradictory findings and suggest some unique hypotheses. The difRculties of testing catastrophe models are discussed and some meana for improving empirical tests are suggested. It is concluded that catastrophe models hold promise for theoretical development in social psychology wherever high quality measurement and scaling techniques are available or can be developed.

KEY WORDS: catastrophe theory, psychological phenomena, attitude theory, empirical testing, abrupt change.

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INTRODUCTION

om TYPES of phenomena observed by S psychologists have not always been amenable to mathematical modeling. In particular, until recently mathematical models have been limited to those phenomena where change is (a) smooth and continuous or (b) can be represented by simple step functions or thresholds. However, many psychological phenomena are not continuous or do not meet the assumptions of simple step functions such as stationarity and independence of path. Catastrophe theory promises to be useful for describing a restricted set of such phenomena.

In the remaining sections of this introduction the properties of psychological phenomena that have not always been amenable to mathematical modeling are introduced. Thew properties seem to be the same as those of the cusp catastrophe. Ad- ditional properties require consideration of the butterfly catastrophe which is also in-

' Parts of this mated have been presented previ- ously at the American Psychological Association An- nual Convention, Toronto, 1978. The author was par- tially funded by a Fulbright/Hays Fellowship during the preparation of this paper. He would like to thank Thomas D. Cook for continuing critical interest in the

troduced briefly (Zeeman, 1976a; Wood- cock & Davis, 1978; Flay, 1977).

Properties not easily modeled Many phenomena in psychology appear

to be discontinuous. Dogs and other animals suddenly change from attacking an adversary to fleeing. People suddenly change from complying with a request, or agreeing with an attitudinal position, to reacting negatively against it (reactance). Anorectics with bulimia suddenly change from purging and fasting to gorging. Manic depressives suddenly change from feelings of great joy to feelings of great depression. There are many other examples.

In each case, the sudden change in the behavioral variable is not accompanied by abrupt or large changes in the presumed causes. Instead, the presumed c a w s have been changing and continue to change slowly and smoothly. For example, if rage and fear are presumed causes of fight or flight (Lorenz, 1967; Zeeman, 1976a), then

author's work, and Loren Cobb, Peter Frey, k e n Henuigan, Mark Lewis, Tim Poston, Don Saari, and Ian Stewart for helpful comments on an earlier draft.

Now at the Health Studies Department, Univer- sity of Waterloo.

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Q 1978 James G. MBer. M.D.. Ph.D, Editor

336 BRIAN R. FLAY

there is no abrupt or large change necessary in either one of these causal variables or control factors, or independent variables, for the abrupt change in the behavioral variable, the dependent variable, to occur. It is as if some threshold has to be reached before behavior abruptly changes from one extreme to the other.

Some discontinuous phenomena are bimodal as well as extreme. For subjects ex- periencing the extremes of bimodal behavior, the possible neutral range of behaviors sometimes are not possible-from mild sub- mission through neutral to mild growling behaviors for dogs feeling both high fear and high rage, from satiety to eating in response to degree of hunger for anorectics. That is, some responses are inaccessible.

The degree of fear and rage present when animals abruptly change from fight to flight appears to be different from that present when they suddenly change from flight to fight. For anorectics, the degree of hunger present when they change from gorging to fasting is different from that present when they change from fasting to gorging (Zee- man, 1976a). In each case, the type of behavior that is being displayed at one point in time will be maintained until the control factor(& have changed a long way, albeit very slowly, before changing to the opposite type of behavior. The property of the changes in the behavioral variable taking place at different levels of the control factors, depending on the direction of change, is known as hysteresis. The requirement that there be little change in the behavioral variable until there has been a large, but smooth, change in the control factor(s) is an instance of a delay rule.

In the case of animals' aggressive behavior, if fear and rage are both at low levels, behavior should be neutral. If only fear is increased, then it is easy to predict that flight wil l result. If only rage is increased, attack will result. What happens if both fear and rage are increased together? The property of divergence probably will be displayed. A minor initial variation in the control factors will lead to a subsequent large divergence in behavior to one extreme or the other, fight or flight. In the case of anorectics, degree of abnormality in reac-

tions to hunger signals leads to divergence to fasting or gorging (Zeeman, 1976a).

"he cusp catastrophe The processes described above all display

the following five properties (see Fig. 1): (1) The behavior is bimodal for some

values of the control factors; (2) Abrupt, catastrophic, changes are ob-

served between one mode of behavior and another;

(3) There is hysteresis, that is, the abrupt change from one mode of behavior to another takes place at different values of the control factors depending on the direction of change;

(4) There is an inaccessible zone of behavior for some values of the control factors;

(5) The possibility of divergent behavior is implied.

When all these properties are found together with the mathematical properties of genericity and structural stability (Poston, 1978b; Poston & Stewart, 1978a), a condi- tion known mathematically as a singularity is present and a cusp catastrophe is implied.

Fig. 1 shows the behavior or response surface of the cusp catastrophe, together with its projection onto the control space or bifurcation set. The control variables can be represented as the normal (a) and splitting (b) factors shown in Fig. 1 in which case a splitting model is indicated. In this model, the control factor, a, is called the normal factor because at the back of the surface, where the splitting factor, b, is negative, the behavioral variable, x, is dependent only on the magnitude of a and increases continuously with it. The control factor, b, is called the splitting factor because increases in its magnitude leads to a progressively larger divergence between the top and bottom surfaces of the response surface.

If the control factors are rotated through 45", which can be done without changing any of the properties of the model, then they are represented as the conflicting factors a! and /3 of Fig. 1, and a conflicting model is indicated. A conflicting model is so-called because the two control factors each lead to conflicting responses. An ex-

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CATASTROPHE THEORY APPLICATIONS IN PSYCHOLOGY 337

BEHAVIOR OR RESPONSE SURFACE UPPER

SURFACE

b (spliiiine tactor)

FIG. 1. The properties of mme psychological phenomena are the same as those of the cusp catastrophe. Here the behavior or response surface and its projection onto the control space and bifurcation set of the cusp catastrophe are shown. The two control factors can be represented as the normal and splitting factors, a and b, respectively, or as two conflicting factors, a and 8. Such a rotation of the control factors does not change the properties of the behavior surface becaws the ‘local results from catastrophe theory hold true for a wide class of coordinate transformations” (Saari 1977, p. 158).

ample of such a model is Zeeman’s (1976a) catastrophe model may be necessary. The model of aggremion in which fear and rage butterfly catastrophe models processes are the conflicting control factors. with four control factors. It has many of the

same properties as the cusp, and has the The butterfly catastrophe richest spectrum of applications after the

If there are more than two control factors cusp (Zeeman, 1976a). When the fourth determining a response, a more complex control factor of the butterfly catastrophe,

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338 BRIAN R. FLAY

d, is positive, the butterfly response space looks much like the cusp response surface except that instead of being folded along curves it is folded along entire surfaces.

Variations in the third control factor, the bias factor, c, have the effect of moving the s-shaped curve to the right and down if c is negative and to the left and up if c is positive, assuming that the upper response surface is on the right as in Fig. 1. Fig. 2 shows this for two values of c. In practical applications, bias means that the abrupt changes or catastrophes in the behavioral variables occur at different biased values of the normal and splitting or conflicting factors, depending on the value of the bias factor. For example, if there were negative bias, the normal factor would need to in- crease much more before there were any catastrophic change in the response vari-

able than it would if there were positive bias.

It should be noted that bias can also be introduced into the cusp catastrophe without the need for a butterfly catastrophe. However, this is bias due to the scale properties of the control factors or to the introduction of dummy control factors that do not change the properties of the model. If the properties of the model are altered to include a trimodality of behavior rather than a bimodality, then a butterfly catastrophe is indicated.

Trimodality in the behavioral variable introduces a third stable surface that inter- penetrates into the original fold of the cusp catastrophe. This happens when the butterfly factor, d, is negative. The resulting surface is shown in Fig. 3. Variations in the bias factor, c, have the effect of moving this

\ \

'/ V i POSITIVE BIAS \

AT POSITIVE BIAS

AT NEGATIVE BIAS

FIG. 2. The butterfly Catastrophe d a c e when the butterfly factor, d, is negative, showing the effects of bias, c.

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FIG. 3. The butterfly catastrophe surface when the butterbly factor, d, is positive, illustrating the third stable heet. iew surface to the left, positive bias, or to he right, negative bias, and reducing its rea until it disappears in a swallowtail atastrophe at extreme bias. The inter- Benetrating surface is also reduced in area s the magnitude of d is reduced. At d = 0 ; disappears. It is apparent from illustra- ions of the butterfly catastrophe (Wood- ock & Poston, 1974, Fig. 8B, p. 21) that the hird stable area is only of sigdicance at xtreme negative values of d and near neu- ral values of the bias ( c ) and splitting ( b ) tctors. Thus, it is not likely to be very nportant in psychological applications un- 1 high quality measurement and scaling re available and detailed predictions are msidered. The catastrophe theory models pre-

?nted and reviewed in this journal and lsewhere (Flay, 1977; Isnard & Zeeman, 374; Poston & Stewart, 1978a; Zeeman,

1977) demonstrate that the theory has many possible applications and promises to be a rich source of model building ideas. Many diverse, and sometimes seemingly contradictory, phenomena can be parsimo- niously described in one coherent model. Such parsimonious models appear to be of great synthetical, descriptive, and pedagogical value, providing a rich source of predictions and new hypotheses to be tested, and sometimes even leading to the development of new theories. These wide ranging claims for catastrophe modeling have already been met in the physical sciences but remain to be met in the social sciences (Poston & Stewart, 1978a).

MODELS IN SOCIAL PSYCHOLOGY

Any psychological phenomenon that dis- plays the properties of divergence, bimo-

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340 BRIAN R. FLAY

dalit y, discontinuity, inaccessibility, or hysteresis probably can be modeled by catastrophe theory. If one of these properties is observed, the others should be anticipated. If several are present, then a cusp catastrophe model can be developed, but all are likely to be found if catastrophe theory is entirely appropriate. In addition, if there are conditions which suggest the presence of a third stable mode of behavior, then a butterfly catastrophe is indicated.

The models proposed and discussed in this paper are not meant to be exhaustive of the possible applications of catastrophe modeling in social psychology. Indeed, ex- haustiveness would be impossible-other applications have already been proposed (Erwin & Dai, 1977,1978; Isnard & Zeeman, 1974; Zeeman, 197613; Zeeman, Hall, Harri- son, Marriage, & Shapland, 1976), and others are no doubt possible. Our intention here is to demonstrate possible applications of catastrophe theory to some important area9 of social psychology, including social behavior, attitude formation and change, and related areas.

Other psychological phenomena for which catastrophe theory models have already been proposed include Pavlovion conditioning (Frey & Sears, 1978), cognitive development (Saari, 1977), the perception of ambiguous figures (Poston & Stewart, 1978b; Shafer, 1976), psychoanalytic trans- actions (Galatzer-Levy, in press), the reso- lution of psychological crisis (Lewis, 1977, 1978), intrapersonal conflict (Cowan, 1977, 1978), changes of state such as sleeping- waking (McFarland & Ashton, 1978), and aggression, anorexia nervosa, and others by Zeeman (1976a, 1976b, 1977; Isnard & Zee- man, 1974). Many of these models have been reviewed elsewhere (Flay, 1977).

As with most of the cited presentations, this presentation is highly speculative and does not provide new empirical evidence to support the models proposed. While pre- vious research results appear to support the models and the predictions derived from them, the models will ultimately stand or fall on the basis of research designed spe- cifically to test them.

Within the vast social psychological literature many diverse findings regarding the relationship between the cognitive compo-

nent of attitudes and behavior, and regarding attitude change, have been reported. There seems to be little consistency in the degree of relationship reported between attitudes and behavior (Calder & Ross, 1973; Fishbein & Ajzen, 1975; Wicker, 1969). Few of the findings on attitude change have been consistently replicated while many more have received little replication and sometimes appear contradictory (Cook & Flay, 1978; McGuire, 1969; Ronis, Baum- gardner, Leippe, Cacioppo, & Greenwald, 1977). The butterfly catastrophe models for predicting behavior from attitudes and for attitude change proposed here promise parsimonious incorporation of many of the diverse, and sometimes apparently conflicting, findings and theories.

Social behavior The most well known attempt to deter-

mine the relationship between attitudes and behavior has been by Fishbein and Ajzen (1974, 1975). They have proposed that behavioral intentions, and through them behavior, are determined by attitudes toward the behavior or object and perceived social norms, i.e., social pressure. While their linear regression model has received moderate support, an average across about 15 studies of almost 50 percent of the variance in behavioral intentions remains to be explained (Flay, 1976).

If attitudes toward behavior are highlj positive and social pressure to perform tht behavior is high, Fishbein’s model predictr that the behavior will be performed. On tht other hand, if attitudes toward the behavioi are highly negative, social pressure to per form the behavior will probably arouse psy chological reactance (Brehm, 1966, 1972 and the behavior will not be performed Indeed, sometimes the opposite appearin! behavior wil l result. If attitudes toward thc behavior are rather neutral, and social pres sure to perform it is high, it seems reason able to expect that the eventual behaviora response could go either way, suggestin; bimodality. Divergence, hysteresis, and bi modality in behavior when social pressur. is high are also suggested in the literature on self-perception theory (Bem, 1972), re actance (Brehm, 1966, 1972), commitmen (Brickman, in preparation; Kiesler, 1971:



dissonance (Festinger, 1957; Wicklund & Brehm, 19761, and conformity (Asch, 1956; Crutchfield, 1955; Deutwh & Gerard, 1955; Gerard, 1965; Kiesler & Kiesler, 1969). All this suggests social preasure as a splitting factor in a cusp catastrophe model. The relationship between attitudes toward the behavior and the overt behavioral response, i.e., the response or dependent variable, may safely be assumed to be monotonic, suggesting attitudes toward the behavior as a normal factor.

The proposed model predicts that if behavior is at one extreme or the other, at the front of the upper or lower response surface of Fig. 1, it will not change much without large changes in the control factors. That is, the model makes the almost banal prediction that the best predictor of future behavior is often past behavior (Owens, 1968, Russell, 1945). The prediction is no longer banal, however, because the catastrophe model leads to the specification of conditions under which it is true, and suggests why there is sometimes a low relationship between attitudes toward a behavior and overt behavior. When behavior is at the front of the response surface, i.e., under conditions of high social pressure or highly salient social norms, new circumstances and information may lead to changes in attitudes without accompanying changes in behavior. Under these conditions, past behavior should be a better predictor of future behavior than present attitudes.

The cusp catastrophe model of social behavior also predicts that overt behavior will be more accurately predicted from attitudes toward that behavior if one has already experienced the behavior than if one has not. This prediction has support both in tests of Fishbein’s model (Flay, 1976; Songer-Nocks, 1976) and in the other literature cited above.

While revising this paper it was discov- ered that two models similar to this have been proposed independently. Chidley (1976) proposed a model of consumer behavior in which purchasing behavior is predicted from attitudes toward the product and social pressure to use it. Tesser (1977, 1978) proposed a model of social behavior remarkably similar to the one proposed here except that it is tied much more closely

to conformity behavior. Tesser has begun experimental tests of the model using the Crutchfield (1955) conformity paradigm.

In earlier formulations of his model, Fish- bein (1967) included personal norms, that is, what one personally feels he or she should do, as a predictor. While this com- ponent was later dropped from the model, there is some evidence (Carlson, 1968; Flay, 1976; Schwartz & Tessler, 1972) that it is sometimes of importance. Personal norms may be included as a bias factor in a butterfly catastrophe model. The model would then account for the common sense prediction that “I am likely to do something that I personally feel should be done even if my attitudes about it are negative.” For example, if it is important to me to be a good researcher and I feel that I should learn statistics to do so, then I am likely to learn statistics even if I hate the subject.

This model seems to account for many diverse findings in the literature. To be anything more than descriptive, however, the processes that lead to attitudes and norms affecting behavior the way they do will need to be specified. That is, in the language of catastrophe theorists, we need to find the dynamics underlying the relationships in the model. Theories of consistency, commitment, reactance, and dissonance, among others, are likely to provide important leads here (Tesser, 1977). The model should be relatively easy to test using either correlational data for the analytical methods suggested by Cobb, (1978) and Lewis, (1977,1978) or experimental manip- ulations of the control factors (Frey & Sears, 1978; Tesser, 1977).

Attitude change It would be desirable to develop one

model for predicting both behavior and attitudes that would take account of the prob- able two-way cause and effect relationship between behavior and the cognitive com- ponent of attitudes. When greater theoretical progress has been made, an umbilic catastrophe model, i.e., one that involves two response or dependent variables and two or more causal, control, or independent variables, may be derived that wil l satisfy these conditions. In the meantime, however, a separate butterfly catastrophe

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342 BRIAN R. FLAY model of attitude formation and change may be proposed (Flay, 1978).

When a persuasive communication is presented to people, they are usually observed to change their attitudes in the direction of the communication. That is, attitude change is usually monotonic with the content, consequences, or power of a message. Although a precise operational defi- nition of these concepts is lacking, the content or power of a message is proposed as the normal factor in a catastrophe model in which attitude is the response variable. When people are highly committed to an attitudinal position, or highly involved in an issue, their attitudes are usually observed to be bimodal and extreme. Thus, involvement or commitment is suggested as a splitting factor in a catastrophe model.

This cusp catastrophe model accounts for findings that highly involved or committed people are more resistant to persuasive attempts (Sherif, Sherif & Nebergall, 1965; Sherif, et al., 1973) than less involved people. An anecdote1 example of this phenomenon is provided by the recent Carter- Lance Affair. If President Carter had been less committed to his budget director, he probably would have asked Lance to resign earlier in the proceedings. If information opposing a committed person’s position be- comes very strong and can no longer be denied, an abrupt change to an opposite extreme position will occur. An example of this is the loss of support for Richard Nixon. People who had made a commitment to Nixon, e.g., precinct captains, were slower to withdraw their support for him. The greater their commitment, the longer they took to withdraw their support and the more extremely their position shifted when it did change. Such predictions are further supported in the literature on experimental attitude change (Sherif et al., 1965, 1973) and commitment (Kiesler, 1971; Halverson & Pallak, 1978).

McGuire’s (1964) theory of resistance to persuasion is also relevant here. Using an analogy of biological innoculation, McGuire assumes that resistance to persuasion is determined first by what one learns before a persuasive message is heard, and second by how well one learns the contents of the message. The learning variable most

strongly related to resistance to a persuasive message appears to be knowledge that defense of a persuasive attack on one’s beliefs is possible (Cook & Flay, 1978). Such knowledge is more likely to be available if one is committed to an attitudinal position or highly involved in an issue, so that the resistance to persuasion found in studies by McGuire (1961,1963; McGuire & Papageor- gis, 1961) and others (Rogers & Thistle- thwaite, 1969) would be predicted by the delay rule and hysteresis in the cusp catastrophe model. These properties also predict that, once a certain threshold of persuasive appeal has been reached, there will be an abrupt catastrophic change in attitude. This prediction appears to be derived uniquely from the cusp catastrophe model and remains to be tested.

These examples can also be used to illus- trate the relationship between the model of social behavior described earlier and the model of attitude change. High commitment is predicted in the model of attitude change to prevent or delay attitude change when new information is presented. Per- formance of behavior generally is assumed to lead to greater commitment or involvement. Whether or not a behavior is performed is somewhat dependent in the model of social behavior on attitudes as well as social pressure and prior behavior. An umbilic catastrophe model might even- tually be developed in which both attitudes and behavior are predicted from each other as well as the power of a persuasive message, social pressure, and commitment or involvement.

One area of inconsistent findings in the attitude change literature concerns primacy and recency effects. If two opposing persuasive messages are presented, one immediately after the other, final attitude is usually more closely linked to the second message, i.e., a recency effect is observed. However, final attitude sometimes is more closely linked to the position advocated by the first persuasive message; a primacy effect is observed (Miller & Campbell, 1959; Wilson & Miller, 1968). The cusp catastrophe model of attitude change predicts that the conditions necessary for obtaining a primacy effect would include a high level of involvement or commitment to the position advo-



cated in the first message. Then the attitude due to the first message would be at the front of the response surface and subject to the delay rule when a second, opposing message is presented. One way of inducing commitment to the position advocated by the first message is to simply ask for some attitudinal response after the first message is heard but before the second one is heard. The findings reported by Miller and Campbell (1959) and Wilson and Miller (1968) are consistent with this interpretation.

Given that the effects of a persuasive message usually decay over time (Cook & Flay, 1978), this model of attitude change predicts that attitude change due to a persuasive message will not persist over time unless involvement or commitment is high. If involvement is low, the attitude change caused by a message will be at the back of the response surface of Fig. 1. If the power of the message decays, as it generally does, attitude should decay isomorphically with it. If, on the other hand, involvement or commitment is high, the attitude change caused by a message wil l be to the front of the upper response surface of the catastrophe surface of Fig. 1. If the power of the message then decays, but commitment or involvement remains high, attitude will be subject to the delay rule, and will decay only slightly but remain on the upper surface. This prediction finds some support in the literature reviewed by Cook and Flay (1978). They suggest that involvement and behavioral commitment might be necessary for the temporal persistence of attitude change.

Other well known findings in the attitude change literature concern the effects of the credibility of the source of a persuasive message. A source of high credibility usually causes greater attitude change, but such change rapidly decays. A source of low credibility usually causes less attitude change, but such change is more likely to persist over time. An extremely low credibility or derogating source often will arouse reactance (Brehm, 1966, 1972). Attitude will change in the direction opposite to the message content. A bias factor to represent any message acceptance or rejection cue in the catastrophe model of attitude change

could account for these findings. The resulting butterfly model (Fig. 2)

also predicts a sleeper effect. This is a change in attitude that occurs some time after a persuasive communication rather than immediately after it. This effect will occur only if a rejection or discounting cue, e.g., low credibility source, inhibits initial attitude change because of the lowering and moving to the left of the s-shaped response surface and is then dissociated from the message over time while the message re- tains its persuasive power.

These conditions have been derived independently from the discounting cue hypothesis (Cook, Gruder, Hennigan, & Flay, 1978) and recently led to successfully set- ting up the conditions for obtaining a sleeper effect (Gruder, Cook, Hennigan, Flay, Alessis, & Halamaj, in press). If commitment or involvement is low, this model predicts that the sleeper effect will occur gradually over time, isomorphically with the dissociation of the discounting cue from the persuasive message. If commitment or involvement is high, however, this model predicts that the sleeper effect will occur suddenly at that point in time when dissociation is complete, and that it will be to a higher level than at lower levels of commitment or involvement. These predictions remain to be tested.

Social support has been discussed often in the attitude change literature (Cook & Flay, 1978; McGuire, 1969) as being a very important variable determining the degree of attitude change and whether or not the change is enduring. It seems that social support is no more than an acceptance, if present, or rejection, if absent or negative, cue and thus is a biasing factor. High social support should lead to greater attitude change with a greater probability. Low or negative social support should lead to less or negative attitude change, with a lower probability of positive attitude change. These predictions have been well supported in the attitude change literature as well as by work derived from the attribution frame- work proposed by Kelley (1967).

A fourth control factor, d, to introduce the third stable surface of the butterfly catastrophe (Fig. 3) seems necessary to ex- plain some other attitudinal phenomena

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344 BRIAN R. FLAY

such as dissonance (Collins & Hoyt, 1972) and the foot-in-the-door technique (Freed- man & Fraser, 1966; Seligman, Bush, & Kirsch, 1976).

One possibility is that commitment to an attitudinal position and ego involvement in an issue ought to be differentiated from each other with the splitting factor b being limited to commitment and the butterfly factor d being concerned with involvement. The resulting model could account for the dissonance and foot-in-the-door phenomenon (Flay, 1977). However, the predictions become rather involved and it is not alto- gether clear that commitment and involvement are independent from each other (Halverson & Pallak, 1978).

Without some direct research support for the model as so far presented and the predictions derived from it, it seems pointless to develop the model further. The model, as already presented, promises to be very parsimonious, subsuming into one model many diverse theories and findings in the attitude change literature. The possibility of a more complex model being even more parsimonious is promising but one must be wary not to continue fitting additional variables to a model without a clear conceptual understanding of the processes involved. It would be easy to accomodate more research findings with a more complex model without achieving any greater understanding or statistically significant improvement (using tests of the goodness of fit). Where one competing model is simpler than another, involving singularities of lower order, the simpler should probably be preferred by the principle of Occam’s Razor, at least until experimental tests demonstrate the neces- sity of greater complexity (Poston, 1978a).

Other attitude-related phenomena Many attitude-related phenomena

should be amenable to description by catastrophe theory models that are closely related to the model of attitude change. Some of these phenomena are suggested briefly below.

Interpersonal attraction. Interper- sonal attraction can be modeled by a catastrophe model in much the same way as attitude change. What A thinks of B should depend on what information A has about B

and on how involved A is with B. If A is highly involved with €3, for example, in a job situation, but knows very little about B, then it will be difficult to predict whether A’s attitude toward B will be positive or negative, the bimodality. Small changes in A’s information could lead to a catastrophic change from positive to negative feelings or vice versa.

In the model of interpersonal attraction the most likely biasing factor is one we typically think of when we think of interpersonal attraction: similarity. It can be viewed as an analogue of acceptance or rejection cues. If the object person is very similar to me, I am more likely to like him or her, regardless of how little information I have, i.e., there is a positive bias.

This model is directly analogous to the model of attitude change and similar implications and predictions follow.

Student evaluation of courses. An- other phenomenon that can easily be modeled by the cusp catastrophe, and for which there might be suitable empirical data with which to test the model, concerns student evaluation of courses. Frey and Flay (1978) have suggested that the overall evaluation of a course, x, is a function of the pedagogical skill of the instructor, a, i.e., the instructor’s ability to get the ma- terial across to the students, and the degree of the students’ involvement in or commitment to the course, b. Such a model would predict the phenomenon of either very high or very low overall ratings given to courses where workload, i.e., commitment, is rated as high. That is, bimodality will be observed and, according to Frey’s (in press) data, this is a reasonable expectation.

In the model of student evaluations, a biasing factor might be something like the perceived personal warmth or rapport of the instructor. This leads to the prediction that instructors with a high degree of personal warmth, i.e., positive bias, are more likely to have their courses evaluated highly. This is verifiable with existing data. Moreover, instructors with a high degree of personal warmth can afford to impose a greater workload without jeopardizing course evaluations.

A model similar to this might be applied to the evaluation of research proposals and

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ournal manuscripts, and other peer review 3yStems.

Worker satisfaction. It is easy to mod- ify the above model to describe worker satisfaction. Replace “pedagogical skill of the instructor” with “competence or skill of the supervisor or management.” Again the splitting factor will be the workers’ involvement in, or commitment to, their jobs. The biasing factor is the rapport between the worker and the supervisor or, alternatively, the self-esteem of the worker, how well he thinks he can meet the demands of his supervisor.

Subject artifacts. In work on subject behavior in psychological experiments (Flay & Hamid, 1977; Rosenthal & Rosnow, 1969; Weber & Cook, 1972), it has been suggested that subjects sometimes react according to three roles. The good subject is one who attempts to figure out the researcher’s hypothesis and then provide data to support it and so please the researcher. The negativistic subject is one who attempts to figure out the researcher’s hypothesis and then provide data to dis- prove it. The faithful subject is usually not concerned with the researcher’s hypothesis, and simply provides true and unadulterated responses. There is an obvious bimodality between the good and negativistic subject roles. The faithful subject role suggests a third stable role somewhere between, yet independent of, both extremes.

In short, we have trimodal behavior which might be appropriately described by a catastrophe model. The control factors for such a model are likely to be factors that have been found to be involved in subject behavior. Some examples are demand characteristics, the normal factor, evaluation apprehension, the splitting factor, and suspicion of deception, the bias factor. The tendency to volunteer might be represented by the butterfly factor, d, and give rise to the third stable surface of Fig. 3. Consistent with the literature, the model predicts that response to demands charac- terizes the good subject role. Negativistic subject behavior often is accompanied by suspicion of deception or some other rejection cue. Volunteers usually are faithful subjects. Finally, evaluation apprehension is likely to bias responses, with high levels

leading to a greater probability of adopting the good subject role.

Note on attitude-related models. The models of attitude change, interpersonal attraction, student evaluations, worker satisfaction, and subject artifacts are highly similar. Further work in these areas and a greater understanding of the more complex catastrophe models might one day lead to development of one more general model which can be applied to all attitude-related phenomena. This would mean even greater parsimony.

DISCUSSION

Models of social behavior, attitude change, and other attitude-related phenomena have been proposed as examples of the types of catastrophe theory models that could be useful in social psychology. The models presented have been shown to be very parsimonious, synthesizing many diverse, and sometimes seemingly contradictory findings and theories from the vast social psychology literature. For example, many different viewpoints, theories, and findings from the attitude change literature were able to be subsumed into one model. It was even suggested that many diverse attitude-related phenomena might be en- compassed by a single parsimonious model.

In many instances, the models promise to describe processes as they are already known more comprehensively, and possibly more accurately, than any other single model. Such modeling makes it easier for the relationships between various patterns of findings to be understood, leading to new predictions that can be tested, that sometimes might even lead to the development of new theory (Poston & Stewart, 1978a).

The ability to synthesize is due not only to the generality of the model, but also to its precision. Precise predictions can be derived and experimentally tested. However, as with practically every other catastrophe theory model proposed in psychology, those proposed here are only speculative and not yet supported by independent research data. Alternative models can be proposed (Flay, 1977) and all remain to be further developed and tested, and either accepted, rejected, or modified on the basis of such tests, Because of the wide generality of

Behavioral Science, Volume 23, 1978

346 BRIAN R. FLAY catastrophe theory, it is imperative that new models be thoroughly tested. Until models are experimentally verified, catastrophe theory is no more than a suggestive mathematical metaphor which can provide neat and parsimonious accounts of diverse and seemingly contradictory empirical findings. Such accounts have descriptive and pedagogical value, but to be scientifically useful they need to be exposed to experimental verification.

It is not a weakness of catastrophe theory that most psychological models proposed to date have not been tested. Poston and Stewart (1978a) have reviewed models from the physical sciences which have withstood experimental tests acceptable even to the critics of catastrophe theory in the social sciences (Sussmann & Zahler, 1978).

The works of Frey and Sears (1978) and Lewis (1977, 1978) are notable exceptions within psychology because they have gone beyond speculation to demonstrate that psychological models can be tested and verified, albeit with greater difficulties than is typical in the hard sciences. Frey and Sears' work is based on the rabbit eyelid conditioning paradigm where there is a great deal of experimental control and it is easy to design the experiments necessary to provide strong tests of a catastrophe model. Lewis' work on psychological crisis is based on correlational, questionnaire type data. He has developed and demonstrated statistical analysis procedures based on cluster analysis and multiple regression that appear to be appropriate for locating folds and singularities.

Other weaknesses of catastrophe theory models in the social and biological sciences raised by Sussmann (1976; Sussmann & Zahler, 1977, 1978; Zahler & Sussmann, 1977) have been adequately dealt with by other authors in this journal (Poston & Stewart, 1978b) and elsewhere (Flay, 1977; Poston, 1978b; Poston & Stewart, 1978a, 1978~). Overall, it may be safely concluded that catastrophe theory has strengths which, if' properly understood and applied, should lead to advances in those fields where it is applied. The weaknesses of past models need not be the weaknesses of future models, particularly if the model de-

velopers are experts in the area of applica tion as well as having a good understandin, of the mathematics underlying catastroph, theory. In short, catastrophe theory shoulc not be prejudged as being of no value to thc social sciences until strong tests of the pro posed models have been conducted.

The social sciences, psychology among them, often have greater difficulty than thc physical sciences testing mathematicaj models. Some (Sussmann & Zahler, 1978) have suggested that mathematical models have no place in the social sciences. The major reason for this suggestion is the state of the art in measurement and scaling in these fields. Rather than arguing that mathematical models should not be used at all in the social sciences, they probably mean to argue that mathematical models should not be used in those areas where interval measurement and scaling procedures are not well developed. At the present time this might rule out some of psychology, but certainly not all of it.

Frey and Sears (1978) have been wise or fortunate to work in an area of psychology where precise measurement is possible. They are able to accurately scale and measure all parameters needed for their catastrophe theory model of conditioning and thereby fit data to response surfaces. In very few other areas of psychology will this level of precision be possible in the near future. That does not mean that we should not strive for it.

Social scientists should be encouraged to develop their measurement and scaling techniques so that their work can become more precise, rather than being discouraged from using catastrophe theory models because their measurement techniques are inadequate. Social scientists should be warned, however, that the experimental verification of catastrophe theory models might end up being a prolonged exercise in the development of refined measurement and scaling techniques. This should not discourage them from attempting such tests. Contributing improved measurement techniques, as well as testing a model, has more value than testing a model alone. Already too many promising models which have led to precise predictions, e.g., Mc-

Behavioral Science. Volume 23.1978

CATASTROPHE THEORY APPLICATIONS IN PSYCHOLOGY 347 Guire's (1968) two-factor theory of attitude change, are in danger of dying a quiet death because of the lack of necessary measurement and scaling techniques. The advance- ment of theoretical and experimental psychology to higher levels of sophistication requires this development.

Predictions specified only at the nominal or ordinal levels of measurement are, in some ways, probably less difficult to test than ones specified at the interval or ratio levels. For example, one can postulate the existence of two response sheets represent- ing two qualitatively different types of behavior for the cusp catastrophe and test for them using techniques derived from cluster analysis and multiple regression like those suggested by Lewis (1977, 1978). One can postulate a bimodal distribution in the response variable at high values of a splitting factor and test for this as Cobb (1978) as well as Lewis have suggested. Or one can predict, for example, a higher value of the response variable a t positive values of a bias factor and lower values of the response variable at negative bias and then manipu- late bias experimentally to test the predictions. Most areas of psychology have developed measurement to a level sufficient for tests of such hypotheses.

There is one difficulty with experimental tests of catastrophe theory models and predictions that is applicable to all types of modeling but nevertheless deserves special attention here. An inability to confirm the research hypothesis must often remain ambiguous in its interpretation: was it because the hypothesis was false or because of in- sensitive measurement and/or manipula- tions of control factors? Tests of catastrophe theory models should be set up to make the interpretation of no difference findings as unambiguous as possible. This can be iccomplished by paying careful attention to all the conditions that should be met for 3 strong test of any hypothesized effect. %t, the necessary theoretical conditions 'or the effect to occur should be identified md measures taken to ensure that they are net. Second, power tests should be con- lucted to ensure that the statistical tests ised are powerful enough to detect the iypothesized effect if it occurs. Third, all

forces that might countervail against the occurrence of the hypothesized effect should be identified and minimized in the experimental design. A good example of the applications of these principles is provided in the recent review and tests of the absolute sleeper effect predicted from the discounting cue hypothesis (Cook et al., 1978; Gruder et al., in press).

Catastrophe models of greater complexity than those reviewed here will be possible in the future by either: (a) introducing more control factors or (b) attempting to predict more than one response variable from the same or a larger set of control factors. For example, if the butterfly models of behavior and of attitude change are verified, re- searchers might derive one umbilic catastrophe model for predicting both behavior and attitude simultaneously. Such models are not easy to understand at the present state of the art, but as our knowledge advances they no doubt will become so. How useful is catastrophe theory for

modeling, hypothesis generation, and theory development in social psychology? Re- cent developments in the physical sciences described by Poston and Stewart (1978a) suggest that catasbophe theory has the promise to be useful in all these ways. This is already apparent in those areas of psychology where measurement is less of a problem, e.g., rabbit eyelid conditioning, and in some other areas with the development of new statistical techniques, e.g., Lewis' use of cluster analysis and multiple regression techniques.

The models proposed here have led to some new and unique hypotheses and hold promise of leading to more, which are amenable to experimental tests. The corrobo- ration of new hypotheses often will lead to the modification of existing theories and sometimes the development of new ones.

Criticisms directed at past models are being successfully overcome by: (a) increased awareness that catastrophe theory models must lead to new and unique predictions and be experimentally tested to be of ultimate scientific value; (b) greater understanding of both the substantive areas of application and the mathematics of catastrophe theory by those who wish to de-

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348 BRIAN R. FLAY

velop catastrophe models; (c) increased sophistication and precision of measurement and scaling techniques in psychology; and (d) the development of statistical methods appropriate for testing catastrophe theory models. As long as these developments continue, catastrophe theory seems to have the potential to be of value to social psychology.

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(Manuscript received January 30, 1978; revised June 9, 1978)


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