Precognitive Dreams: Bifurcations Due to Tolerance of Ambiguity and Dream Frequency

What Precognitive Dreams are Made of: The Nonlinear Dynamics of

Tolerance of Ambiguity, Dream Recall, and Paranormal Belief [1]

Rense Lange

Illinois State Board of Education

Michael Schredl

Central Institute of Mental Health

James Houran

Southern Illinois University School of Medicine

Abstract

Using a framework derived from nonlinear dynamics, two studies investigated a cusp model of precognitive dreaming using the GEMCAT II software for catastrophe estimation. As predicted, Study I (N = 50) and Study II (N = 59) both indicated that tolerance of ambiguity and dream recall loaded significantly on the cusp’s latent bifurcation variable (Y), whereas belief in the paranormal functioned as an asymmetry variable (X) (all p < .05). The validity of the proposed model is supported by the findings that a competing linear model, as well as an alternative cusp formulation in which the roles of the X and Y indicators were reversed provided a significantly poorer fit to the data in both experiments (p < .001) than the hypothesized cusp formulation. The differential fit of the three models is reflected both in the models’ (Pseudo-) R² values and via standard non-parametric tests over the models’ squared residuals. The findings support the hypothesis that some experiences of precognitive dreams (represented by the statement “There have been events that I dreamed about before the event occurred”) are illusions, i.e., coincidences between the contents of dreaming and waking experience that are noticed due to frequency of dream recall and given credence due to the combined effects of belief in the paranormal and a tolerance of ambiguity.

Keywords: precognitive dreams, cusp catastrophe, tolerance of ambiguity, belief in the paranormal, GEMCAT II

A precognitive dream is defined as a dream that seemingly includes knowledge about the future which cannot be inferred from actually available information (Stowell, 1995). Precognitive dreams have been reported throughout history; famous examples are the pharaoh’s dream of seven fat and seven meager cows (Hendricks, 1989) and Bishop Lanyi’s dream of the assassination of Archduke Franz Ferdinand at the beginning of World War I (Clericus, 1918). Although the phenomenon is often considered “paranormal” it occurs quite frequently:17.8 % to 38 % persons of large samples of individuals reported that they experienced at least one precognitive dream (Palmer, 1979; Haraldsson, 1985; Ross & Joshi, 1992; Thalbourne, 1994). Most studies indicate that women report more precognitive dreams than men, while the frequency of precognitive dreaming declines with age. The percentage of persons who believe that precognitive dreaming is possible is even larger, and estimates range from 63 % to 98 % (Thalbourne, 1984; Haraldsson, 1985).

Table 1: Criteria for precognitive dreams (Bender, 1966)

Criteria

1.Dream must be told or recorded before its fulfillment

2.Dream must include enough details to render chance fulfillment unlikely

3.The possibility of interference from actual knowledge must be excluded

4.Self-fulfilling prophecies must be excluded

5.Telepathic influences can not explain the occurrence of the precognitive dream

Despite the frequent occurrence of precognitive dreaming, scientific exploration of the phenomenon faces a number of issues. First, most spontaneously occurring precognitive dreams do not fulfill the criteria depicted in Table 1 which are necessary to differentiate precognitive dreaming from other phenomena such as deja vu experiences, telepathic dreams, memory distortions and merely chance occurrences. Besterman (1933) found that only two out of 45 reported precognitive dreams (4.4 %) met these criteria, i.e., the high prevalence rates which were obtained by questionnaires using items such as “Do you have dreams that later come true?“ may be an overestimation in comparison to the occurrence of precognitive dreams fulfilling the criteria of Table 1. While arguably all experiences of precognitive dreams are subsumed by this question, so are instances of non-precognitive type dreams. For instance, an individual could be dreaming about a dentist appointment to occur the next morning due to the ensuing anxiety. However the use of this general question can help in constructing theoretical models that can be tested later when more precise measures of precognitive dreaming are developed. Secondly, given the experimental evidence for psi (e.g., Bem & Honorton, 1994; Radin, 1997)[2], researchers face the problem that strong support for the precognition hypothesis challenges many people’s fundamental world views regarding causality. Thus, Mischo’s (1985) observation that support for this hypothesis can elicit anxieties seems very plausible. Third, the verification of a theory is according to the critical rationalism not possible (Popper, 1978), i.e., even a large amount of supportive evidence (confirmed examples of precognitive dreaming) cannot prove the theory. Of course, this problem concerns theory testing in general (Meehl, 1978).

In view of the methodological issues raised above, three approaches have been applied to the investigation of precognitive dreams: investigating spontaneously occurring precognitive dreams, diary studies and experimental studies.

Spontaneously Occurring Precognitive Dreams

Large surveys of spontaneous cases (Rogers, 1923; Saltmarsh, 1934; Rhine, 1954; Hearne, 1984; Ryback & Sweitzer, 1990) have shown that the themes of precognitive dreams are mostly negative, e.g. death of a relative, one’s own death, natural disaster, war and accidents. Stevenson (1961) and Stowell (1997a, b) reported that these dreams are vivid, intense and have a personal significance for the dreamer. Schriever (1987), however, analyzed 115 dreams of a single subject and found no differences between well confirmed precognitive dreams (see Table 1) and insufficiently confirmed dreams regarding emotional intensity, vividness and personal significance. Therefore, the preponderance of negatively toned precognitive dreams may be explained by the bias that persons reported solely impressive cases to the researcher. This is comparable to the emotions of ordinary dreams since surveys eliciting dreams retrospectively found a preponderance of negative dreams whereas diary studies or laboratory awakenings yielded a balanced ratio between positive and negative dream emotions (Schredl & Doll, 1998). Since the researcher cannot investigate the spontaneous precognitive dream of an event after its occurrence, in-depth inquiries (see Table 1) are often not possible. Barker (1967), for example, placed a newspaper announcement looking for precognitions about the Aberfan disaster (where 144 persons were killed) and found that only 3 out of the 76 responses met the criteria in Table 1. A similar study was conducted by Murray and Wheeler (1937). After the kidnapping of the Lindbergh baby, they published a newspaper request throughout the country which yielded over 1300 dreams prior to the discovery of the baby’s body. Only seven dreams contained several general details (e.g. location in a wood, buried) matching with the actual circumstances of the baby’s death. These authors questioned the possibility of precognition.

A major problem of this kind of research is the lack of control samples to estimate the base rate of such dream themes. Yet, despite the methodological problems inherent to this approach, it seems valuable to investigate spontaneous cases in the future; not as evidence but as material which contributes to the theorizing about the phenomenon.

Diary Studies

During a dream diary study the participants record their dreams every morning upon awakening (thereby fulfilling criterion 1 of Table 1) and they are instructed to compare subsequent waking events to the dream content (Dunne, 1927; Besterman, 1933; Bender, 1966; Schriever, 1987; Sondow, 1988). This approach, which elicits a larger number of precognitive dreams per participant than spontaneous cases, indicates that precognitive dreams are not dominated by negative emotions (Schriever, 1987). Sondow (1988) analyzed 96 precognitive dreams (out of a series of her own 943 dreams) and found that there is an exponential relationship between the occurrence of a precognitive link and the time interval between dream and the actual waking event (cf. Green, 1960; Orme, 1974; de Pablos, 1998). About 41 % of the precognitive dreams were linked to an event of the following day (Sondow, 1988). To rule out the possibility of memory artifacts, Sondow (1988) compared a period with intense re-reading of the dreams in order to match them to subsequent events to a period without re-reading the dreams. In both cases she found an exponential decrease as a function of the time interval, but significantly more matches were found in the re-reading period. This supports the hypothesis that memory factors play a crucial role, i.e. the participants have to keep the dream in mind in order to find links to waking events.

Interestingly, dream research has revealed a similar pattern (exponential decline) for the temporal references of dream elements (e.g. Botman & Crovitz, 1989-90). The hypothesis of Dunne (1927) and Jackson (1967) that references to future events occur as often as references to past events is not supported by empirical data since about 10 % of dreams were found to be precognitive (Besterman, 1933; Bender, 1966; Sondow, 1988), whereas for 25 % to 75 % of the dream elements references to past events were detected (Strauch & Meier, 1996). Although research has shown that dream content is affected by the study’s design (cf. Schredl, 1999a), no systematic investigation has been carried out to test whether intense looking for matches between waking life and dreams changes dream content. It seems plausible, however, that the extensive reading of newspapers affects dream content such that people dream more about global events, thereby heightening the chances for future matches.

Jackson (1967) observed that the procedure of connecting dream elements with subsequent waking events by the participants does not allow an estimation of chance coincidences. Therefore, he suggested a two-group design. Group A keeps a dream diary over a one-week period and is then exposed to a special waking experience (unusual for all participants) and records another week their dreams. Group B also keeps dream diaries over two weeks but they are not exposed to the special waking experience. The advantage of this controlled design is the possibility to estimate the effect of the unusual experience on subsequent dreams, using the contents of the dreams of group B as a base rate. Laboratory studies (see next section) and a few diary studies have partly applied Jackson’s suggestions. In Hearne’s (1986) study, two independent judges rated possible links between 52 dreams of a single subject to newspaper reports of the following 28 days. To control for chance coincidences newspaper reports of the previous and following year were included. Statistically significant correspondences were found and some dream reports showed striking matches (Hearne, 1986). Harley (1989) utilized 20 sets of four pictures from which one picture was chosen by random selection procedures. The participants were asked to choose – according to their dreams – the picture that would be selected. Harley (1989) obtained a significant psi-missing effect, i.e. target pictures were less often chosen than non-target pictures. Sixty-eight dream reports of 23 participants were gathered by Emery (1991) who asked her participants to dream about the cover of the magazine “Newsweek” which would be published two weeks ahead. Although Emery (1991) obtained 11 good matches, her results have to be interpreted with caution since no control condition was introduced.

Experimental Laboratory Studies

The research group at the Maimonides hospital in New York carried out studies on telepathy and dreaming (Ullman, Krippner & Vaughan, 1977) and two studies on precognitive dreams (Krippner et al., 1971; 1972). Participant in both studies was Malcolm Bessent, an English “sensitive” with a history of reported spontaneous precognitive dreams. After eight sleep laboratory nights with REM awakenings for dream collection, a word from the dream element list of Hall and Van de Castle’s (1966) book was chosen at random. An experimenter created a multi-sensory environment around this word, using items which were directed to visual, auditory, gustatory, olfactory and tactile-kinesthetic inputs to which the subject was exposed. Three independent judges rated each of the eight dream protocols against the eight words/descriptions of the multi-sensory experience. Five out of eight trials were rated correctly (p = .00018; Krippner et al., 1971). The second study with a slightly different design also yielded five hits (p = .0012; Krippner et al., 1972). Among the most impressive hit was the following dream account which was rated with an average agreement of 98.3 % by the three judges:

“Bob Morris does research on animal behavior and more specifically birds.... He’s been doing various research and studies with birds and he’s taken me out to see his sanctuary place where all the birds are kept.... I remember seeing various different kinds of doves. Ring-tailed doves, ordinary doves, Canadian geese...” (Krippner et al., 1972, p. 278, third dream report, post-sleep interview)

The subsequent waking experience consisted of a slide show with various pictures of birds selected at random.

Despite the fact that the above phenomena were studied in an artificial rather than a natural context, the two studies showed positive results. However, both studies used only a single subject and future research is needed therefore to expand these findings.

Influencing Factors

The occurrence of precognitive dreams correlates with dream recall frequency in general (Haraldsson, 1975; Palmer, 1979; Kohr, 1980; Thalbourne, 1984; 1994; Houran & Lange, 1998; Schredl, 1999b). This is expected, since a person who hardly recalls any dream will not be able to experience a precognitive dream. Equally plausible is the strong relationship between the reporting of precognitive dreams and a positive attitude towards parapsychological phenomena (e.g. Houran & Lange, 1998). The causality in this relationship, however, remains unclear. That is, persons interested in parapsychology may pay more attention to their dreams in order to detect links to subsequent events. Conversely, however, it may also be possible that a person who experienced an intense precognitive dream (maybe negatively toned) would develop an interest in parapsychology trying to understand the experienced phenomenon better.

Bender (1966) proposed that persons with ego-weakness (a psychoanalytic notion) are more likely to experience precognitive dreams and Schredl’s (1999b) finding that persons with thin boundaries reported precognitive dreams more often than persons with thick boundaries - even when dream recall frequency is statistically controlled - support and expand this notion. The boundary construct (Hartmann, 1991) is a broad concept; persons with thin boundaries are creative, empathic, vulnerable, have intense but stressful relationships, etc. This view is supported by Ross and Joshi’s (1992) finding of a positive correlation between the experience of parapsychological phenomena (including precognitive dreams) and the occurrence of dissociative experiences (e.g. depersonalisation, pseudo-hallucinations, psychogenic fugue). In addition, the reporting of paranormal experiences was related to sexual abuse during childhood (Ross & Joshi, 1992). Since severe traumata can cause dissociative disorders such as multiple personality disorder which is characterized by a confusion of mental boundaries, it may be hypothesized that traumata can make boundaries more permeable. Consistent with this hypothesis, Hearne (1984) found elevated scores of neuroticism in persons who responded to a newspaper announcement asking for precognitive dreams. Since Hearne’s sample was biased, i.e., persons responding to a newspaper announcement, this finding cannot be generalized. In a similar way, the above cited studies should be interpreted with caution since these studies utilized questionnaires to measure the occurrence or frequency of precognitive dreaming. However, as stated earlier, such questionnaires do not satisfy the criteria in Table 1. Therefore, future research applying more sophisticated measurement and inference techniques are needed to evaluate the previous research on personality and precognitive dreaming.

A largely unexplored field was recently investigated by Schredl (1999b). In two studies, several aspects of dreaming were related to the occurrence of precognitive dreams. The results indicated that persons with intense dreams and persons who reported that dreams affect their waking life (e.g. dreams which are helpful in solving problems, dreams affecting daytime mood, dreams stimulating creativity), also report more often precognitive dreams. Therefore, the experience of precognitive dreams seems to be part of a more general aspect of dream life which can be seen as a thin boundary between dreaming and subsequent waking life.

Personal Significance and Clinical Relevance

Stowell (1995) pointed out that while research in precognitive dreaming is focused on finding evidence for the existence of the phenomenon, studies investigating the effect of such dreams on the inner life of persons herself/himself are scarce. Case reports (Hastings, 1977; Sondow, 1988; Schredl, 1996; Stowell, 1997a, b) indicated that precognitive dreams can serve as preparation to the following event, e.g. the death of a close relative. In other words, the expression of emotions in a dream may facilitate the mourning process.

Halliday (1987) stressed the clinical relevance of precognitive dreaming since he observed that some persons suffer from precognitive dreams (his term is prophecy nightmares). Schredl (1996), for example, reported a case where the female dreamer experiences several dreams of muddy water and in the following few days a relative or close friend has died. This woman developed anxieties regarding her dreams because of their capacity to predict such negative events. Halliday (1987) suggested the use of insight-oriented therapeutic approaches as used also for ordinary dreams such as Gestalt techniques or the imagery rehearsal approach developed for the treatment of nightmares (Krakow & Neidhardt, 1992). Halliday (1987) further suggested directly testing the possibility of chance coincidence and addressing the possible superstition that dreams are seen as causes for subsequent events (sometimes found in young children).

The Present Research

The above review emphasizes the need to implement research designs and develop models for precognitive dreaming that have strong ecological validity for purposes of generalization, while maintaining a strict quantitative focus similar to experimental approaches. Moreover, any proposed model should account for the patterns of findings derived from studies of spontaneous cases and dream diaries. For the reasons outlined below, the approach adopted in the present research does not assume that experiences of precognitive dreaming are necessarily parapsychological in origin.

Contextual Variables. Despite the strict criteria given in Table 1 for identifying ostensibly genuine precognitive dreams, the fact remains that the contents of dreams are inherently ambiguous (Hobson, 1997) and that additional information is typically required to achieve a coherent interpretation of those contents. Research indicates that such additional information is frequently derived from the dreamer’s broadly defined “context,” which includes both state and trait factors (for an overview, see e.g., Houran, 1998). In this sense, the labeling of dream contents as “precognitive” is simply an interpretation of ambiguous dream stimuli. We have found that the interpretation of ambiguous stimuli is predictably guided by contextual variables such as the perceiver’s psychophysical state, embedded environmental cues, symbolic-metaphorical references, demand characteristics of the situation, and prior beliefs and expectations (Lange, Houran, Harte, & Havens, 1996; Houran & Lange, 1997; Lange & Houran, 1996; Houran, Lange, & Crist-Houran, 1997). Consistent with the findings by Krippner et al. (1971, 1972), it seems likely therefore that dreams are subject to the same influences as well (Houran, 1998).

Labeling. Belief in the paranormal strongly affects the recall and interpretation of ambiguous experience (Smith, 1992-93; Wiseman & Morris, 1995; Wiseman, Jeffreys, Smith, & Nyman, 1999). For example, laboratory experiments have confirmed that believers in the paranormal are less inclined than nonbelievers to consider coincidences as being the result of mere chance (Blackmore & Troscianko, 1985; Brugger, Landis & Regard, 1990; Blackmore, 1992), while believers are also more inclined to claim a relationship between coincidences and their own thoughts and actions (Brugger, Regard & Landis, 1991; Brugger, Regard, Landis, Cook, Krebs, & Niederberger, 1993). Moreover, the labeling of events and information can have a strong effect on people’s subsequent attitudes (Eiser, 1990) by making them resistant to change (Lange & Fishbein, 1983, Fishbein & Lange, 1990). Accordingly, we expect that once a non-chance interpretation of ambiguous events has been established, such interpretations continue to play an active role in believers’ cognitions.

Tolerance of Ambiguity. A series of studies (Lange & Houran, 1998, 1999b) indicated that an emotional and perceptual personality variable called “tolerance of ambiguity” plays an important role in the processing of ambiguous stimuli. Intolerance of ambiguity reflects the tendency to resort to black and white solutions characterized by premature closure, often at the neglect of consensual reality. In essence, an intolerance of ambiguity results in rapid and overconfident judgment of equivocal stimuli or events. As such, it is associated with perceiving ambiguous situations as threatening, whereas a tolerance of ambiguity is associated with perceiving ambiguous situations as desirable (Frenkel-Brunswick, 1949; Budner, 1962).

Consistent with this characterization, we (Lange & Houran, 1998, 1999b) have found that those with a high tolerance of ambiguity express little or no anxiety when presented with ambiguous stimuli. By contrast, individuals with a low tolerance of ambiguity often react with anxiety or fear. This aforementioned research also consistently identified tolerance of ambiguity as a correlate of belief in the paranormal. Houran and Williams (1998) subsequently discovered that tolerance of ambiguity was related only to specific types of paranormal beliefs and experiences, namely those that involve a reinterpretation of internal and physiological experience [e.g., beliefs suggesting the mind can expand beyond its usual boundaries, memories of reincarnation, visual apparitions, and vestibular alterations]. Among those paranormal beliefs and experiences associated with tolerance of ambiguity was precognitive dreaming (as measured by the question “There have been events that I dreamed about before the event occurred”). Path analyses reported by Houran and Lange (1998) extended this initial finding and established that precognitive dreaming was a byproduct of both belief in the paranormal and frequency of dream recall.

Nonlinearity. The various relations involving tolerance of ambiguity referred to above involve linear effects that were expressed exclusively in terms of standard bivariate correlation coefficients or regression weights. However, recent research (Lange & Houran, 2000) showed that tolerance of ambiguity has nonlinear effects in certain cases and that prediction is improved substantially by adopting a nonlinear approach as based on the mathematical theories by Thom (1975) pertaining to catastrophe models. Specifically, it was found that a cusp catastrophe significantly outperformed a standard linear regression model in predicting the formation of paranormal beliefs (R² = 0.954 vs. 0.305). Other studies (Houran & Lange, 1998; Lange & Houran, 2000) indicate that experiences of precognitive dreaming and the formation of paranormal beliefs share several important variables, and we propose therefore to study precognitive dreaming from a catastrophe perspective as well. In essence, catastrophe models describe how small changes in a systems independent variables can have large and discontinuous effects on a response (dependent) variable. Analogous to the sudden buckling of a beam under an increasing load, Lange and Houran’s (2000) results suggest that the labeling of a dream as precognitive occurs rather abruptly as more otherwise inexplicable explanations accumulate and are seized upon. As we discuss in greater detail in the following sections, the cusp model also explains why such changes are largely irreversible.

Starting with the work by Zeeman (1974), catastrophe models have a long history in the social sciences (Guastello, 1995). However, statistical problems have restricted their spread in the life sciences (Alexander, Herbert, DeShon, & Hanges, 1992; Oliva, Desarbo, Day, & Jedidi, 1987), and catastrophe models have not previously been used in research on precognitive dreams. For this reason, the following sections provide an introduction to these models as relevant to the present purposes as well as an overview of the statistical techniques needed for fitting catastrophe models. An elementary introduction to applied catastrophe theory can be found in Brown (1995), while Guastello (1995) provides a comprehensive review of the available psychological literature.

The Cusp Catastrophe

The most widely studied catastrophe model is the “cusp,” and it is this model that was used successfully in the study by Lange and Houran (2000). The cusp catastrophe is a three-dimensional curve whose first derivative is given by the equation:

(1)

Note that if, as in the present paper (see Method section), Z takes on only two possible values, then the variable Z can be rescaled such that Z³ = Z. In this case Equation 1 reduces to:

(2)

In Equations 1 and 2, Z represents the dependent variable, whereas X and Y represent two control variables. Because these two variables have quite different effects on Z, we first describe the pairwise relations between X and Z and between Y and Z.

Figure 1: The relation between the dependent variable (Z) and the asymmetry variable (X) while holding Y constant.

Figure 2: The relation between the dependent variable (Z) and the bifurcation variable (Y) while holding X constant.

As is shown in Figure 1, varying a cusp’s asymmetry (or “normal”) variable X for a fixed (but relatively high) level of Y produces a curve that resembles a cross-section of the pleat created when two opposite ends of a piece of fabric are pushed together. That is, the shape of a cross-section changes from approximately linear to an overlapping fold. In the folded portion, variations in X can cause dramatic effects since the value of Z may jump from the bottom plane to the top plane or vice-versa. Because the locations on the curve between these markers are inaccessible such jumps (up or down) occur at the markers only. Consequently, the Z values corresponding to the inaccessible region should not occur empirically (or, at least, they should occur infrequently), resulting in a bimodal distribution of Z. Although Z may show abrupt changes, its behavior is dampened by the fact that the system continues to adhere to the Z plane it currently occupies, unless it is “pushed too far” along the X-axis. As a result, the system exhibits hysteresis, i.e., a lag in behavior that acts as a form of memory.

Figure 2 shows the effects of variations in the bifurcation (or “splitting”) variable Y when X is held constant. Note that the change in Z has two characteristic features. First, as the value of Y moves away from the origin, a point is reached where the function bifurcates (i.e., Z takes one of two values). Second, once bifurcation has occurred, the magnitude of the differences between these two possible values increases with increasing Y. The preceding discussion indicated that the system does not arbitrarily jump between the two values of Y. Instead, all changes are completely determined by the preceding values of the X and Y variables, and the resulting dynamics can be studied in longitudinal research designs (Guastello, 1982; Lange, 1999; Lange, McDade and Oliva, 2000). However, longitudinal designs are not always feasible nor desirable and several applications have been reported in which the system of interest is observed only once (Guastello, 1995). In such instances, Z observations should form two layers, and -- due to the effects of hysteresis – one may find that two cases with identical X and Y control values exhibit different values for the dependent variable Z.

Figure 3: The Complete Three Dimensional Cusp Shape.

Figure 3 depicts the complete three-dimensional cusp that results when the effects of X and Y on Z are considered simultaneously. It can be seen that X and Y bear a nearly linear relation to Z for low values of the asymmetry variable X. In addition, the figure shows the existence of two different nearly linear (or, at least, continuous) planes for extreme values of X, regardless of the value of the bifurcation variable Y. However, these two variables produce highly discontinuous changes in Z for high values of Y combined with intermediate values of X. This particular combination of X and Y values constitutes the “bifurcation set” as indicated by the darkened area on the XY plane. As pointed out above, absent prior information concerning Z, the (X, Y) pairs in this set are associated with seemingly unpredictable and erratic changes in the dependent variable. We note that the cusp catastrophe derives its name from the characteristically peaked projection of the folded Z surface onto the XY plane. It is precisely this part of the cusp that is of greatest interest for the present purposes.

Hypotheses

It should be obvious that precognitive dreaming represents the variable Z in our cusp catastrophe formulation. The question arises however how belief in the paranormal, tolerance of ambiguity, and dream recall as identified by Houran and Lange (1998) should be mapped onto a cusp’s X and Y control variables. We advance the following hypotheses.

First, based on the findings by Lange and Houran (2000) we hypothesize that tolerance of ambiguity constitutes a bifurcation variable (Y) rather than an asymmetry variable. In particular, we expect that the interpretation of dreams as precognitive is likely to be avoided by those shunning non-standard explanations (which would be facilitated by intolerance of ambiguity). Second, in order to notice that one’s dreams might be related to subsequent waking experience, it is clearly necessary that such dreams be recalled in the first place. Thus, greater dream recall should increase the likelihood of noticing any correspondences, if only by chance. We hypothesize therefore that frequency of dream recall plays a role similar to tolerance of ambiguity, and it should therefore act as a bifurcation variable (Y). Finally, previous research strongly suggests that prior belief shapes the interpretation of ambiguous experiences, and not vice-versa (Lange & Houran, 1997; Lange & Houran, 1998, 1999b). Accordingly, we assume that the very interpretation of a dream as precognitive presupposes a belief in the paranormal, while the absence of such beliefs is a major factor in the rejection of dreams as precognitive. Thus, belief in the paranormal should contribute to the asymmetry variable (X) rather than the bifurcation variable. As a result, believers are expected to occupy the “high” plane of the cusp in Figure 3 whereas non-believers should be more likely occupy the “low” plane.

GEMCAT II

Analogous to the situation in structural modeling, our hypotheses imply that Y is a composite, or “latent,” variable that combines tolerance of ambiguity and dream recall as its indicator variables. Although latent variables pose no particular problem in linear modeling (see, e.g., Bollen, 1989), they greatly complicate the testing of specific hypotheses when dealing with (non-linear) catastrophe models (Brown, 1995; Cobb, 1981; Guastello, 1982), at least not without making strong additional assumptions (Alexander et al., 1992). However, Lange (1998) recently developed the GEMCAT II software that generalizes the model fitting approach pioneered by Oliva et al. (1987) while adding statistical tests of the model parameters. Extensive computer simulations reported in Lange, Oliva, and McDade (1999b) demonstrated the robustness of the GEMCAT II algorithm, even when very noisy indicator variables are used. This paper also introduced Pseudo-R² indices of fit, together with tests of statistical significance and competitive model testing.

Table 2: Definitions of the indicator variables and their operationalizations

Indicator variable	Description	Operationalization
x₁	Belief in the paranormal	Paranormal Belief subscale of AEI (Kumar et al., 1994)
y₁	Tolerance of ambiguity	Rasch version of MacDonald’s AT-20 (Lange & Houran, 1999a)
y₂	Dream Recall	Schredl et al (1997)
z₁	Precognitive dreaming	item # 29 on the AEI (Kumar et al., 1994)

In GEMCAT II, the variables X, Y, and Z may consist of arbitrary linear combinations of indicators variables, including fixed or variable offset values. Thus, in their most general form our hypotheses translate into the equations:

X = a₀ + a₁x₁, (3)

Y = b₀ + b₁y₁ + b₂y₂,

Z = g₀ + g₁z₁

where the symbols x₁, y₁, y₂, and z₁ denote the indicator variables listed Table 2. It is difficult to determine beforehand whether the offsets a₀, b₀, and g₀ are actually needed and empirical information is often needed to settle such questions (Brown, 1995). We further note that the algorithm requires that at least one of the Z indicator weights be fixed (Lange et al., 1999b; Oliva et al., 1987). Therefore, g₁ is set to 1.0 throughout this paper.

Each GEMCAT run provides a Pseudo-R² index of fit and an associated test of statistical significance (see, e.g., Bates & Watts, 1988). Such R² values are approximate only because they are derived by treating catastrophes as if they were linear models. It should be noted, however, that the Pseudo-R² index assumes continuous measures of X, Y, and Z. Since our measure of Z is two-valued (see Method) we expect that the Pseudo-R² obtained in the present research are inflated. To determine the statistical significance of the indicator weights GEMCAT II implements a bootstrap resampling approach (see, e.g., Efron & Tibshirani, 1993) which determines the statistical significance of the indicator weights by computing an Achieved Significance Level (ASL). Computing the ASL does not make any assumptions regarding the indicator weights’ sampling distributions and the resulting significance tests are distribution free. In addition, Lange et al. (1999c) describe how differences in fit for two or more models M₁, M₂, … (or two populations C₁, C₂, … ) can be tested by standard non-parametric tests (see, e.g., Marascuilo & McSweeney, 1977). Although this approach does not require that the models all be catastrophes, it does assume that they contain equal numbers of free parameters.

We point out that a different approach is required when M₁, M₂, … have unequal numbers of parameters and we refer the interested reader to the presentation in Lange et al. (1999c) for details.

An Overview

Since catastrophe theory has not previously been applied in precognitive dream research, we used Houran and Lange’s (1998) data set to refine the hypotheses summarized by Equation 3. The resulting formulation was then tested on a second, independent data set. Also, to establish more conclusively the validity of our basic hypotheses, we will consider two alternative formulations as well.

First, our hypotheses describe a cusp model in which belief in the paranormal functions as an asymmetry (X) variable, while tolerance of ambiguity and dream recall together play the role of the bifurcation variable (Y). In the remainder, we will refer to this model as Cusp₁. Although the finding of a high Pseudo-R² index is not inconsistent with Cusp₁, this alone would not exclude the possibility that paranormal belief is in fact a Y indicator or that tolerance of ambiguity and dream recall are really X indicators. Therefore, we will fit a second cusp model, Cusp₂, in which the roles of the X and Y indicator are reversed. That is, in Cusp₂ tolerance of ambiguity and dream recall are X indicators, whereas paranormal belief is a Y indicator. Our hypotheses entail that Cusp₁ should provide a better fit to the data than the alternative model Cusp₂ (Note 1).

Second, we pointed out earlier that Equations 1 and 2 are equivalent because Z represents a binary variable in the present research. Thus, if Y in Equation 1 takes on small values only, a (nearly) linear equation results (i.e., Z » X) that can be fitted by standard regression techniques. Although the finding of a superior fit of such a linear model does not really invalidate our hypotheses, it also provides very little reason to prefer the more complex cusp formulation over a simpler linear model. To address this issue, competitive model test will be performed between Cusp₁, Cusp₂, and a standard regression model which predicts precognitive dreaming as the weighted linear sum of tolerance of ambiguity, belief in the paranormal, and frequency of dream recall. That is:

z₁ = w₀ + w₁x₁ + w₂y₁ + w₃y₂ (4)

Study I

Method

Fifty undergraduate psychology students (27 women and 23 men, M _age = 25.8, SD_age = 8.5, range = 18-55 yrs.) completed the following questionnaires in random order. The symbols between parentheses refer to the row labels in Table 2.

Belief in the Anomalous/ Paranormal (x₁) was derived from the Anomalous Experience Inventory (AEI, Kumar, Pekala, & Gallagher, 1994). The reported reliability (KR-20 index) of this subscale is 0.77 (Gallagher, Kumar, & Pekala, 1994). Our measure of Precognitive Dreaming (z₁) is represented by the response (true/ false) to the statement “There have been events that I dreamed about before the event occurred,” which is contained in the Anomalous/ Paranormal Experience subscale of the AEI (derived from factor and cluster analyses). Responses of “true” were scored as 1, whereas responses of “false” were scored as 0. To maintain the “paranormal” context of the statement, it was left embedded in the AEI. Tolerance of Ambiguity (y₁) was measured using Lange and Houran’s (1999a) 18-item Rasch version of MacDonald’s (1970) AT-20 scale. The reported reliability of this interval-level scale is 0.68. Finally, we used an English language translation of Schredl’s frequency of dream recall questionnaire (see e.g., Schredl & Doll, 1997; Schredl, Jochum, & Sougoenet, 1997; Schredl & Montasser, 1996-1997) to measure Dream Recall (y₂). This measure reflects respondents’ answers to the question “How often have you recalled your dreams during the last few months?” with the following response options: “almost every morning,” “several times a week,” “about once a week,” “two or three times a month,” “about once a month,” “less than once a month,” or “never.” These responses were scaled as 6, 5, 4, 3, 2, 1, and 0, respectively.

Results

Preliminaries. A series of preliminary GEMCAT II runs indicated that offsets b₀ and g₀ in Equation 3 were superfluous and they were therefore omitted (Note 2). Consequently, our main hypothesis is represented by Cusp₁ as is shown below:

Cusp₁:

X = a₀ + a₁x₁, (5)

Y = b₁y₁ + b₂y₂,

Z = g₁z₁

To test whether the assignment of the indicator variables to either X or Y is correct (see Table 2), we define a model Cusp₂ in which the X and Y indicator variables are swapped relative to Equation 5 while keeping the X offset. In other words:

Cusp₂:

X = b₀ + b₁' y₁ + b₂' y₂, (6)

Y = a₁' x₁,

Z = g₁z₁

While g₁ has the value 1 in Equation 5 as well as 6, the values of the a’ and b’ in Equation 6 almost certainly differ from the corresponding a and b weights in Equation 5. However, the definitions of the indicators x₁, y₁, y₂, and z₁ remain unchanged, i.e., they continue to correspond to the definitions given in Table 1.

Table 3: The indicator weights’ plugin values and model fit of the linear model and the cusp catastrophe model in Study I (N = 50).

Paranormal

Tolerance of

Dream

Precognitive

(Pseudo-)

Constant

Belief

Ambiguity

Recall

Dreaming

R²

df₁

df₂

Linear (Eq. 4)

W^a

-0.778

*^b

0.059

0.027

0.047

1.000

0.320

7.221

***

Cusp₂ (Eq. 6)

-0.556

0.097

0.033

0.055

1.000

0.334

7.672

***^d

Cusp₁ (Eq. 5)

0.041

-0.011

0.024