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) R^{2} values
and via standard nonparametric 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.Selffulfilling 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 nonprecognitive 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, indepth 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 rereading of the dreams in order to match them to
subsequent events to a period without rereading 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 rereading 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, 198990). 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 twogroup design. Group A keeps a dream diary over a oneweek
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 psimissing effect, i.e. target pictures were less often chosen
than nontarget pictures. Sixtyeight 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 multisensory environment
around this word, using items which were directed to visual, auditory,
gustatory, olfactory and tactilekinesthetic inputs to which the subject was
exposed. Three independent judges rated each of the eight dream protocols
against the eight words/descriptions of the multisensory 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. Ringtailed doves, ordinary doves, Canadian geese...” (Krippner
et al., 1972, p. 278, third dream report, postsleep 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 egoweakness (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, pseudohallucinations, 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 insightoriented 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, symbolicmetaphorical references, demand characteristics of
the situation, and prior beliefs and expectations (Lange, Houran, Harte, & Havens, 1996; Houran & Lange,
1997; Lange & Houran, 1996; Houran, Lange, & CristHouran, 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,
199293; 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 nonchance 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 (FrenkelBrunswick, 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 (R^{2}
= 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
threedimensional 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 Z^{3} = 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 crosssection of
the pleat created when two opposite ends of a piece of fabric are pushed
together. That is, the shape of a crosssection 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 viceversa. 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 Xaxis. 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 threedimensional 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 nonstandard 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 viceversa (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
nonbelievers 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 (nonlinear) 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 PseudoR^{2} 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 
x_{1} 
Belief in the paranormal 
Paranormal Belief subscale of AEI (Kumar et al., 1994) 
y_{1} 
Tolerance of ambiguity 
Rasch version of MacDonald’s AT20 (Lange & Houran, 1999a) 
y_{2} 
Dream Recall 
Schredl et al (1997) 
z_{1} 
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 = a_{0} + a_{1}x_{1}, (3)_{}
Y = b_{0} + b_{1}y_{1} + b_{2}y_{2},
Z = g_{0} + g_{1}z_{1}
where the symbols x_{1}, y_{1}, y_{2},
and z_{1} denote the indicator variables listed Table 2. It is
difficult to determine beforehand whether the offsets a_{0}, b_{0}, and g_{0} 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, g_{1} is set to 1.0 throughout
this paper.
Each GEMCAT run provides a PseudoR^{2}
index of fit and an associated test of statistical significance (see, e.g., Bates & Watts, 1988). Such R^{2} values are approximate
only because they are derived by treating catastrophes as if they were linear
models. It should be noted, however, that the PseudoR^{2} index
assumes continuous measures of X, Y, and Z. Since our measure of Z is
twovalued (see Method) we expect that the PseudoR^{2} 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 M_{1}, M_{2}, … (or two populations C_{1}, C_{2},
… ) can be tested by standard nonparametric 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 M_{1}, M_{2}, … 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 Cusp_{1}.
Although the finding of a high PseudoR^{2} index is not
inconsistent with Cusp_{1}, 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, Cusp_{2}, in which the roles of the X and Y
indicator are reversed. That is, in Cusp_{2} tolerance of ambiguity and
dream recall are X indicators, whereas paranormal belief is a Y indicator. Our
hypotheses entail that Cusp_{1} should provide a better fit to the data
than the alternative model Cusp_{2} (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
Cusp_{1}, Cusp_{2}, 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:
z_{1} = w_{0}
+ w_{1}x_{1} + w_{2}y_{1} + w_{3}y_{2} (4)
Study I
Method
Fifty undergraduate
psychology students (27 women and 23 men, M _{age} = 25.8, SD_{age}
= 8.5, range = 1855 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
(x_{1}) was derived from the Anomalous Experience Inventory (AEI, Kumar, Pekala, & Gallagher,
1994). The reported reliability (KR20 index) of this
subscale is 0.77 (Gallagher, Kumar, & Pekala, 1994). Our measure of Precognitive
Dreaming (z_{1}) 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 (y_{1}) was measured using Lange and Houran’s (1999a)
18item Rasch version of MacDonald’s
(1970) AT20 scale. The reported reliability of this
intervallevel 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, 19961997) to measure Dream Recall (y_{2}). 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 b_{0} and g_{0} in Equation 3 were superfluous
and they were therefore omitted (Note 2). Consequently, our main hypothesis is
represented by Cusp_{1} as is shown below:
Cusp_{1}:
X = a_{0} + a_{1}x_{1}, (5)_{}
Y = b_{1}y_{1} + b_{2}y_{2},
Z = g_{1}z_{1}
To test whether the assignment of the indicator
variables to either X or Y is correct (see Table 2), we define a model Cusp_{2}
in which the X and Y indicator variables are swapped relative to Equation 5
while keeping the X offset. In other words:
Cusp_{2}:
X = b_{0} + b_{1}' y_{1} + b_{2}' y_{2}, (6)
Y = a_{1}' x_{1},
Z = g_{1}z_{1}
While g_{1} 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 x_{1},
y_{1}, y_{2}, and z_{1} 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) 


(Pseudo) 




Constant 


Belief 


Ambiguity 


Recall 


Dreaming 

R^{2} 
df_{1} 
df_{2} 
F 

Linear (Eq. 4) 
W^{a} 
0.778 
*^{b} 
W 
0.059 
* 
W 
0.027 
* 
W 
0.047 

Z 
1.000 
^{c} 
0.320 
3 
46 
7.221 
*** 





















Cusp_{2} (Eq. 6) 
X 
0.556 

Y 
0.097 

X 
0.033 

X 
0.055 

Z 
1.000 

0.334 
3 
46 
7.672 
***^{d} 





















Cusp_{1} (Eq. 5) 
X 
0.041 
* 
X 
0.011 
* 
Y 
0.024 
*** 
Y 
0.067 
*** 
Z 
1.000 

0.949 
3 
46 
282.793 
***^{d} 
SE_{1} 

0.052 


0.052 


0.004 


0.020 


 






* p < .05 ** p < .01 *** p < .001 (applies also to ASL levels)
a)
Each indicator weight is prefixed
by the (latent) variable to which it contributes (see Equations 4, 5, and 6)
b) The significance levels for the linear model weights derives from standard ttests, while those for Cusp_{1} and Cusp_{2} reflect ASL values derived from 500 bootstrap replications
c) This weight was fixed at the value 1 for reasons described in the text.
d) Pseudo values (see text)
Table 4: Pairwise comparisons of the fit of Cusp_{1}, Cusp_{2}, and the linear model in Study I (N = 50).
Cusp_{1} 
5.333* 


Cusp_{2} 
0.246 
5.633* 


Linear 
Cusp_{1} 

* p < 0.001
Note: Shown are the normal deviates (zvalues) produced by Wilcoxon’s Signed Rank test. A positive entry indicates that the model identified by the row label is better than that identified by the column label.
Model Identification. Table 3 (left side of
table) shows the parameters of the Linear model (Eq. 4), Cusp_{1} (Eq.
5) and Cusp_{2} (Eq. 6) as well as their R^{2} type
indices of fit (right side of table). As expected, Cusp_{1} (PseudoR^{2}
= 0.949) shows a better fit to the data than
either the Linear model (R^{2} = 0.320)
or the alternative cusp formulation, Cusp_{2} (PseudoR^{2}
= 0.334). As pointed out earlier, the PseudoR^{2} values may be
inflated due to the binary nature of Z, and the three models were therefore
also compared via nonparametric tests. Consistent with the obtained order of
the (Pseudo) R^{2} values, the results of Kendall’s W
test over these three models’ squared residuals indicates that their fit
differs significantly (W = 0.395, c^{2}_{2} = 39.250, p <
0.001). As is shown in Table 4, subsequent pairwise comparisons between the
three models indicate that Cusp_{1} performs significantly better than
other two models (p < .001), while Cusp_{2} and the linear
model show a similar performance (p > .20). Thus, Cusp_{2}
and the linear model can both be rejected, while the proposed assignment of the
indicator variables to the latent variables X and Y in Cusp_{1} is
justified. For these reasons, the remainder of this section deals exclusively
with the best fitting model, Cusp_{1}.
Figure 4: Actual vs. Predicted in Study I (Z by Y)
As predicted, Table 3 shows
that Tolerance of Ambiguity (b_{1} = 0.024, ASL
< 0.001) and Dream Recall (b_{2} = 0.067, ASL < 0.001) both contribute
positively to the latent bifurcation variable Y in Cusp_{1}. That is,
as is shown in Y by Z crosssection in Figure 4, greater Tolerance of Ambiguity
and greater Dream Recall increase the magnitude of the bifurcation in
Precognitive Dreaming. Analogous to the theoretical situation depicted in
Figure 1, the observed values of Z in Figure 4 diverge with increasing Y,
thereby producing the expected bimodal distribution of Z. Note that our data
only occupy the rightmost part of Figure 2 as the Z values do not meet for the
lowest values of Y. Table 3 further shows that Paranormal Belief contributes
significantly negative to the latent variable X (a_{1} = 0.011, ASL <
0.05) while the value of the X offset is statistically significant (a_{0} = 0.041, ASL <
0.05).
Study II
The first study involved
several decisions and refinements of Cusp_{1} that were contingent on
properties of the particular sample being used. Therefore, we replicated Study
I using a new sample of respondents consisting of 59 undergraduate psychology
students (28 women and 31 men, M _{age} = 24.5, SD_{age}
= 8.6, range = 1859 yrs.). The same questionnaires were administered and
identical procedures were followed as in Study I.
Table 5: The indicator weights’ plugin values and model fit of the linear model and the cusp catastrophe model in Study II (N = 59).





Paranormal 

Tolerance of 

Dream 


Precognitive 
(Pseudo) 


(Pseudo) 




Constant 


Belief 


Ambiguity 


Recall 


Dreaming ^{c} 

R^{2} 
df_{1} 
df_{2} 
F 


Linear (Eq. 4) 
W^{a} 
0.046 
*^{b} 
W 
0.029 
* 
W 
0.002 
* 
W 
0.078 

Z 
1.000 
^{} 
0.102 
3 
55 
2.077 
























Cusp_{2} (Eq. 6) 
X 
0.650 

Y 
0.094 

X 
0.020 

X 
0.002 

Z 
1.000 

0.780 
3 
52 
64.950 
***^{d} 























Cusp_{1} (Eq. 5) 
X 
0.157 
* 
X 
0.018 
* 
Y 
0.029 
*** 
Y 
0.053 
*** 
Z 
1.000 

0.959 
3 
55 
426.813 
***^{d} 

SE_{1} 

0.037 


0.004 


0.004 


0.021 


 







t(107) ^{e} 

1.811 
^{f} 

0.137 


0.917 


0.476 


 







* p < .05 ** p < .01 *** p < .001 (applies also to ASL values)
a)
Each indicator weight is
prefixed by the (latent) variable to which it contributes (see Equations 4, 5,
and 6)
b) The significance levels for the linear model weights derives from standard ttests, while those for Cusp_{1} and Cusp_{2} reflect ASL values derived from 500 bootstrap replications
c) This weight was fixed at the value 1 for reasons described in the text.
d) Pseudo values (see text)
e) The t value was computed based on the standard error _{}, where the subscripts I and I refer to quantities pertaining to Studies I and II, respectively.
f) p
< .10 (2sided)
Table 6: Pairwise comparisons of the fit of Cusp_{1}, Cusp_{2}, and the linear model in Study II (N = 59).
Cusp_{1} 
6.514* 

Cusp_{2} 
5.510* 
4.544* 

Linear 
Cusp_{1} 
* p < 0.001
Note: Shown are the
normal deviates (zvalues) produced by Wilcoxon’s Signed Rank test. A
positive entry indicates that the model identified by the row label is better than
that identified by the column label.
Figure 5: Actual vs Predicted Study II (Z by Y)
Results
The results of Study II are
highly similar to those obtained earlier.
First, as is shown in Table 5,
Cusp_{1} again provided the best fit to the data (PseudoR^{2}
= 0.959), followed by Cusp_{2} (PseudoR^{2}
= 0.780) and the linear model (R^{2}
= 0.102). Consistent with the large
differences among the (Pseudo) R^{2} values, the
difference in fit among the three models is highly significant (W =
0.546, c^{2}_{2}
= 64.373, p < 0.001), while Table 5 shows that Cusp_{1} again
fits significantly better than both Cusp_{2} and the linear model (p
< 0.001). However, in contrast to Study I, Cusp_{2} now performs
significantly better than the linear model (p < 0.001). Third,
inspection of Tables 3 and 5 reveals that the indicator weights for Cusp_{1}
are highly similar to those in Studies I. To obtain a formal test of this
observation we performed ttests for independent samples over the four
free indicator weights for Cusp_{1} in the two studies. As is described
in Footnote e of Table 5, this test is based on the standard errors of
the indicator weights (SE_{1}) as produced by GEMCAT II’s
bootstrap procedure. The tvalues in the bottom row of Table 5 indicate
that all four tests fail to reach statistical significance (all p >
0.05, 2sided). Finally, note that the Y by Z crosssection shown
in Figure 5 strongly resembles its counterpart in Study I.
Ancillary Analyses
General. Analysis of the combined
data of Studies I and II (N = 109) yielded results that generally agree with
past findings related to the frequency of precognitive dreaming. However, while
a greater proportion of women (0.66) reported precognitive dreams than men
(0.59), this difference fails to reach statistical significance (p =
.301 by Fishers’ exact test). Likewise, the mean age of those reporting
precognitive dreams was less (M = 24.29 yrs.) than that of those not
reporting such dreams (M = 26.49 yrs.). Again, however, the difference
was not statistically significant (t(107) = 1.30, p > .19).
Finally, as might be expected, dream frequency was significantly greater (t(107)
= 3.06, p < .01) for
selfreported precognitive dreamers (M = 4.60) than for nonprecognitive
dreamers (M = 3.76).
Gender. To further explore the
effect of gender, we created one data set containing only men (N = 51)
and another containing only women (N = 58) by combining all available
cases of Studies I and II. Next, Cusp_{1} was fitted to each data set.
Table 7 reveals that men show a slightly better fit (R^{2} =
0.967) than women (R^{2} = 0.941) and a MannWhitney U
test over the residuals indicated that this difference in fit is statistically
significant (z = 2.181, p < 0.05). Also, the box plots in
Figure 6 show that men’s residuals have more outliers and a slightly smaller
standard deviation (SD = 0.090) than those for the women (SD =
0.116). However, men’s and women’s indicator weights all have the same signs
and they are of comparable magnitudes. Further, all pairwise comparisons of
these weights by gender failed to reach statistical significance as indicated
by the values of the t statistics shown in the bottom row of Table 6
(all p > 0.10). Finally, the YZ crosssections shown in Figures 7 and
8 show highly similar patterns of actual vs. predicted values for men and women.
We conclude therefore that Cusp_{1} applies about equally well to both
sexes.
Table 7: The indicator weights’ plugin values and model fit of Cusp_{1} for men (N = 51) and women (N = 58) as estimated based on the combined data sets of Studies I and II.






Indicator Variables 




Fit Statistics 





Latent X 


Latent Y 

Z 












Tolerance of 
Dream 

Precognitive 
Pseudo 


Pseudo 


Gender 
Model 
Constant 

Belief 

Ambiguity 

Recall 

Dreaming ^{b} 
R^{2} 
df_{1} 
df_{2} 
F 

Men 
Cusp 
0.099 
*** ^{a} 
0.012 
** 
0.026 
*** 
0.066 
* 
1.000 
0.967 
3 
47 
460.678 
*** 

SE_{Cusp} 
0.034 

0.004 

0.003 

0.017 







Women 
Cusp 
0.169 
*** 
0.018 
** 
0.030 
*** 
0.045 
* 
1.000 
0.941 
3 
54 
287.954 
*** 

SE_{Cusp} 
0.046 

0.006 

0.005 

0.025 







Men vs. Women 
t(105) ^{c} 
1.227 
^{d} 
0.850 

0.693 

0.697 







* p < .05 ** p < .01 *** p < .001 (including ASL values)
a) The significance levels of each indicator weight as estimated by the ASL over B = 500 bootstrap replications.
b) The weight of this indicator was fixed at the value 1 for reasons described in the text.
c) Computed using the error _{}for each pair of free indicator weights.
d) p > 0.10 (2sided)
Figure 6: Boxplot of Residuals by Gender (Study I + II).
Figure 7: Z by Y for Men (Study I + II, N = 51)
Figure 8: Z by Y for Women (Study I + II, N = 58).
Linear Models vs the Cusp. To highlight the
differences between a linear model of precognitive dreaming and a cusp
formulation we consider Equation 2 in more detail. In particular, since our measure
of Precognitive Dreaming is binary (i.e., Z = 0 or Z = 1), it is instructive to
consider the behavior of Cusp_{1} for these two possible values.
First, note that substitution of
Z = 0 (indicating the absence of Precognitive Dreaming) into Equation 2 yields
0  X  0.Y = 0, which simplifies to (Note 3):
X = 0. (7)
The most important feature of this equation is that
it does not contain the variable Y, i.e., the values of Tolerance of
Ambiguity and Dream Recall are irrelevant while Paranormal Belief is the only
variable responsible for keeping Z = 0. As such, Equation 7 strongly supports
the conclusions reached in previous research (Lange
& Houran, 1997; Lange & Houran, 1998; Lange & Houran, 1999b) that belief is the single most important factor in
the interpretation of ambiguous stimuli as “paranormal.”
Second, substitution of Z = 1 (indicating the occurrence of
Precognitive Dreaming) into Equation 2 yields 1  X  1.Y = 0, or:
Y = 1  X. (8)
In other words, the occurrence of Precognitive
Dreaming presupposes a negative correlation between the latent variables X and
Y and the data support this conclusion since r_{XY} = .488 (p
< .001) for Z = 1. Note that we saw earlier that Paranormal Belief contributed
significantly negative to X, and thus the sign of this variable is actually positive
in Equation 8. As a result, Tolerance of Ambiguity, Dream Recall, and
Paranormal Belief variables all contribute positively to Precognitive
Dreaming.
Figure 9: A comparison of the linear model and Cusp_{1} using all available data points (Study I + II, N = 109).
Taken together, Equations 7
and 8 imply that the functionality of the various indicator variables changes
depending on the value of the variable Z. Figure 9 contrasts this behavior with
that of the linear model by plotting the predicted values of the linear model
and those for Cusp_{1} against the latent bifurcation variable Y using
all available cases (N = 109). This figure illustrates that the changes
in the predictions produced by the linear model can only be proportional to
changes in the predictor variables, while for each value of the predictor
variable Y there is only one prediction Z. However, since Z is bimodal, many of
the predictions produced by the linear model simply do not correspond to the
data. As a result, the overall fit is poor. For instance, the R^{2}
fit of the linear model over all 109 cases is only 0.178.
By contrast, cusp
catastrophes divide their predictor variables into two types (X and Y).
Variations in X values allows an appropriate cusp to produce different
predictions for the same value of Y (hysteresis), thereby avoiding generating
nonoccurring data values. It is precisely for this reason that Cusp_{1}
yielded a superior fit relative to the linear model.
Summary and
Discussion
As anticipated, the
nonlinear models showed a significant improvement in fit over a competing
linear model in both experiments. In other words, the findings strongly support
the basic hypothesis that precognitive dreaming is better described by a cusp
catastrophe than by the linear models such as used in Houran and Lange (1998).
However, the cusp formulation proved successful at a more detailed level as
well. For instance, tolerance of ambiguity and dream recall loaded
significantly on the latent Y variable. Thus, as hypothesized, these variables
govern the cusp’s bifurcation behavior that corresponds to the binary measure
of precognitive dreaming. Next, belief in the paranormal functioned as an
asymmetry variable (X). Since variables of this type govern the overall level
of the cusp response surface, we conclude that paranormal beliefs mainly serve
to support the interpretation of otherwise ambiguous dream contents as
precognitive. Further, the preceding interpretation is supported by the finding
that an alternative cusp formulation in which the roles of the X and Y
indicators are reversed provided a significantly poorer fit to the data in both
experiments. Also, additional analyses showed that gender did not play a
substantial role in the model fit, thereby increasing the generality of the
cusp formulation. Finally, although the cusps’ PseudoR^{2}
values of fit are possibly inflated due to the binary nature of the dependent
variable Z, we note that the relative magnitudes of these statistics agree with
the order implied by the results of standard nonparametric tests over the
models’ squared residuals.
These results are consistent
with the hypothesis that some experiences of precognitive dreams 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.
Although this proposal is not new, we believe that ours is the first
empiricallyderived, process model that explains the specific psychological
mechanisms behind these experiences. Further, our nonlinear model accounts for
the general relationship between dream recall and the experience of
extrasensory perception (Haraldsson, 1975; Palmer, 1979; Kohr, 1980;
Thalbourne, 1984; 1994; Houran & Lange, 1998; Schredl, 1999b), and it
explains the exponential decline effect associated with precognitive dreaming
(Green, 1960; Orme, 1974; Sondow, 1988; de Pablos, 1998) as an artifact of
memory, i.e., less “hits” are reported over time because ambiguous dream
contents are gradually forgotten and this prevents the percipient from finding
correspondences with waking experience. Moreover, aside from reporting or
researchers’ bias, it is understandable why negative themes and strong
emotional responses may be predominant in experiences of precognitive dreams.
Vivid and intense dream content often induces strong emotional reactions in the
dreamer, and consequently these types of dreams are more easily remembered than
dreams with mundane content. Our model does not directly address the findings
of the Maimonides parapsychological dream studies, which arguably had more to
do with apparent telepathy (cf. Bem & Honorton, 1994) than precognitive
dreaming. Still, our results have implications for enhancing the methodologies
of laboratory studies of precognitive dreaming. Our model suggests that
experiences of precognitive dreaming occur because the contents of both waking
and dream experience are largely ambiguous and therefore susceptible to
numerous interpretations or selective reconstruction. Further, it appears that
the types of information most likely to be experienced as “precognitive”
involve situations that correspond to normal waking experience (we stress that
crises and other negative or fluke events are also part of a continuum of
normal waking experience). Therefore, we suggest that the second criterion in
Table 1 (i.e., a “dream must include enough details to render chance
fulfillment unlikely”) should be more stringent. Specifically, targets for
laboratory studies should be as unambiguous as possible while simultaneously
being of a variety that does not easily correspond to normal waking experience.
We anticipate that applying these precautions in laboratory studies will
safeguard against chance hits due to the variance in the randomization and
judging procedures.
While the proposed cusp
model explains some forms of precognitive dreaming (or perhaps the general
tendency to notice chance correspondences between dreaming and waking
experience), it is not clear whether it applies to cases that satisfy all the
criteria listed earlier in Table 1. For instance, we did not differentiate between the exact contents of
the dreams, nor did we take into account the effect these dreams had on the
participants. Such factors are likely to be relevant since Stowell’s (1997a, b)
phenomenological study indicated that at least four different types of
precognitive dreams can be distinguished. A first type provides information
about nontraumatic situations; a second type brings guidance to the dreamer;
another relates negative situations about which it is impossible to take
action; a final type provides information about situations where intervention
is possible. Thus, we propose that future research might profitably investigate
whether our model applies differentially to the four types of precognitive
dreams as distinguished by Stowell (1997a, b). In addition, some methodological
improvements are needed. We suggest that our dichotomous measure of
precognitive dreaming be replaced with a measure of the frequency of
precognitive dreaming that better conforms to the criteria in Table 1. For
example, we are planning a replication study in which a frequency of dream
recall measure is administered to a sample of selfdescribed “psychics” who
claim to have recurrent precognitive
dreams that meet the explicit criteria in Table 1. Further, because only one of
our indicator variables (i.e., the Rasch version of the AT20, Lange and
Houran, 1999a) is guaranteed to yield data at an interval level of measurement,
the present results may depend partly on the particular operationalizations of
the indicator variables. Further, in light of previous findings indicating age
and gender effects in precognitive dreaming, we propose that any biasing
effects in all indicator variables should be eliminated using the
“purification” methods outlined in Lange, Irwin and Houran (2000; cf. Lange,
Thalbourne, Houran, & Storm, in press).
On a more general level, we
have again found that some types of perceived paranormal experiences are best
modeled as a process of misattribution. This finding agrees with those who
suggest that paranormal beliefs and experiences are misinterpretations of
normal events (see Irwin, 1999, p 280). Yet, paranormal beliefs and experiences
should not be regarded as a pathological or abnormal behavioral phenomenon.
Rather, such beliefs and experiences can serve adaptive functions. For instance,
Irwin (1999) noted that “By incorporating a system of paranormal beliefs, the
individual has a cognitive framework for effectively structuring many events
and experiences in life so that they appear comprehensible and thereby able to
be mastered, at least intellectually” (p. 291). Accordingly, we hypothesize
that some experiences of precognitive dreaming are not paranormal glimpses of
the future but rather are the normal byproducts of people’s attempts to
understand ambiguous stimuli in the present. The fact that the type of
“precognitive dream” described by item # 29 on the Anomalous Experiences
Inventory can largely be reduced to the dynamic interactions of three main
variables (i.e., belief in the paranormal, dream recall, and tolerance of
ambiguity) bolsters this conclusion. Of course, only time will tell if our
hypothesis is correct.
Reference
Notes
1)
It
is generally advisable that a simpler catastrophe model, the Fold, should be
tested first (Lange & Houran,
2000). However, this consideration does not apply in the
present case because the dependent variable Z (Precognitive Dreaming) is
operationalized by a binary measure (see Method section). In this case, the fit
of a Fold catastrophe is identical to that of a linear model with the same
predictor variables.
2)
The version of GEMCAT II used in the present research (V 1.3) is the
result of a progressive series of enhancements of this program (Lange, 1998; Lange et., 1999a, b) and it is anticipated that a newer version will
become available later. A copy of the latest GEMCAT II, together with
instructions for its use, can be downloaded from the internet web site: <http://www.sbm.temple.edu/~oliva/cattheory.htm>.
3)
Although Equation 7 is true when Equation 2 fits
perfectly, the actual value of X depends on the extent to which GEMCAT II is
able to minimize the residual error (e) in the
probabilistic version of Equation 1, i.e., (Z^{3}  X  ZY)^{2}
= X^{2} = e^{2}. In support of our
contentions, X^{2} and e^{2} are identical for Z
= 0 (M = .002, SD = .0019, r = 1.000). By
contrast, the median values of X^{2} and e^{2} differ almost by a
factor 10 (Md_{X2} =8.279E04, Md_{e2} = 7.452E03, z = 5.286, p < .001) for Z = 1 as indicated by a Wilcoxon’s Signed
Rank Test while their correlation (r = .161,
p > .18) fails to reach statistical significance.
References
Alexander, R., Herbert, G.,
DeShon, R., & Hanges, P. (1992). An examination of leastsquares regression
modeling of catastrophe theory. Psychological
Bulletin, 111, 366374.
Barker, J. C. (1967).
Premonitions of the Aberfan disaster. Journal
of the Society for Psychical Research, 44,
169181.
Bates, D. M., & Watts,
D. G. (1988). Nonlinear regression
analysis and its applications. New York: Wiley & Sons.
Bem, D. J., & Honorton,
C. (1994). Does psi exist? Replicable evidence for an anomalous process of
information transfer. Psychological
Bulletin, 115, 418.
Bender, H. (1966). The
Gotenhafen case of correspondence between dreams and future events: A study of
motivation. International Journal of
Neuropsychiatry, 2, 398407.
Besterman, T. (1933). Report
of inquiry into precognitive dreams. Proceedings
of the Society for Psychical Research, 41,
186204.
Blackmore, S. (1992).
Psychic experiences: Psychic illusions. Skeptical
Inquirer, 16, 367376.
Blackmore,
S., & Troscianko, T. (1985). Belief in the paranormal: Probability
judgments, illusory control, and the “chance baseline shift.” British Journal of Psychology, 76, 459468.
Bollen, K. A. (1989). Structural equations with latent variables.
New York: Wiley.
Botman, H. I., &
Crovitz, H. F. (198990). Dream reports and autobiographical memory. Imagination, Cognition and Personality, 9, 213224.
Brown, C. (1995). Chaos and catastrophe theories. Thousand
Oaks, CA: Sage.
Brugger, P., Landis, T.,
& Regard, M. A. (1990). A ‘sheepgoat effect’ in repetition avoidance:
Extrasensory perception as an effect of subjective probability? British Journal of Psychology, 76, 459468.
Brugger, P., Regard, M. A.,
& Landis, T. (1991). Belief in the paranormal and illusory control: A
replication. Journal of Psychology, 125, 501502.
Brugger, P., Regard, M. A.,
Landis, T., Cook, N., Krebs, D., & Niederberger, J. (1993). ‘Meaningful’
patterns in visual noise: Effects of lateral stimulation and the observer’s belief
in ESP. Psychopathology, 26, 261265.
Budner, S. (1962).
Intolerance of ambiguity as a personality variable. Journal of Personality, 30,
2950.Clericus, J. (1918). Zum Wahrtraum des Bischofs Lanyi von Großwardein. Psychische Studien, 45, 465468.
Cobb, L. (1981). Parameter
estimation of the cusp catastrophe model. Behavioral
Science, 23, 7578.
De Pablos, F. (1998).
Spontaneous precognition during dreams: Analysis of a oneyear naturalistic
study. Journal of the Society for
Psychical Research, 62, 423433.
Dunne, J. W. (1927). An experiment with time. New York:
Macmillan.
Efron, B., & Tibshirani,
R. J. (1993). An introduction to the
bootstrap. New York: Chapman & Hall.
Eiser, J. R. (1990). Social judgment. Pacific Grove, CA:
Brooks/Cole.
Emery, M. (1991).
Programming the precognitive dream. Association
for the Study of Dreams Newletter, 8,
715.
Fishbein, M., & Lange,
R. (1990). The effects of crossing the midpoint on belief change: A replication
and extension. Personality and Social
Psychology Bulletin, 16,189199.
FrenkelBrunswik, E. (1949).
Intolerance of ambiguity as an emotional and perceptual personality variable. Journal of Personality, 18, 108143.
Gallagher, C., Kumar, V. K.,
& Pekala, R. J. (1994). The anomalous experiences inventory: Reliability
and validity. Journal of Parapsychology,
58, 402428.
Green, C. (1960). Analysis
of spontaneous cases. Proceedings of the
Society for Psychical Research, 53,
97161.
Guastello, S. (1982).
Moderator regression and the cusp catastrophe: Application to twostage
personnel selection, training, therapy, and policy evaluation. Behavioral Science, 27, 259272.
Guastello, S. J. (1995). Chaos, catastrophe, and human affairs.
Mahwah, New Jersey: Lawrence Erlbaum.
Hall, C. S., & Van de
Castle, R. L. (1966). The content
analysis of dreams. New York: AppletonCenturyCrofts.
Halliday, G. (1987). Therapy
for prophecy nightmares. Journal of
Contemporary Psychotherapy, 17,
217224.
Haraldsson, E. (1975). Reported
dream recall, precognitive dreams, and ESP. In Research in parapsychology 1974. Metuchen: Scarecrow Press.
Haraldsson, E. (1985).
Representative national surveys of psychic phenomena. Journal of the Society for Psychical Research, 53, 145158.
Harley, T. A. (1989). PSI
missing in a dream clairvoyant experiment. Journal
of the Society for Psychical Research, 56,
17.
Hartmann, E. (1991). Boundaries in the mind. New York: Basic
Books.
Hastings, A. C. (1977).
Dreams of future events: Precognitions and perspectives. Journal of the American Society of Psychosomatic Dentistry and Medicine,
24, 5160.
Hearne, K. M. T. (1984). A
survey of reported premonitions and of those who have them. Journal of the Society for Psychical
Research, 52, 261270.
Hearne, K. M. T. (1986). An
analysis of premonitions deposited over one year from an apparently gifted
subject. Journal of the Society for
Psychical Research, 53, 376382.
Hendricks, L. L. (1989). Discovering my biblical dream heritage.
San Jose: Resource.
Hobson, J. A. (1997).
Dreaming as delirium: A mental status analysis of our nightly madness. Seminars in Neurology, 17, 121128.
Houran, J. (1998).
Complement to “Ratio of male and female characters in a dream series.” Perceptual and Motor Skills, 86, 14691470.
Houran, J., & Lange, R.
(1997). Hallucinations that comfort: Contextual mediation of deathbed visions. Perceptual and Motor Skills, 84, 14911504.
Houran, J., & Lange, R.
(1998). Modeling precognitive dreams as meaningful coincidences. Perceptual and Motor Skills, 83,
14111414.
Houran, J., Lange, R., &
CristHouran, M. (1997). An assessment of contextual mediation in trance states
of shamanic journeys. Perceptual and
Motor Skills, 85, 5965.
Houran, J., & Williams,
C. (1998). Relation of tolerance of ambiguity to global and specific paranormal
experience. Psychological Reports, 83, 807818.
Irwin, H. J. (1999). An introduction to parapsychology (3^{rd}
ed.). Jefferson, NC: McFarland.
Jackson, M. P. (1967).
Suggestions for a controlled experiment to test precognition in dreams. Journal of the American Society for
Psychical Research, 61, 346353.
Kohr, R. L. (1980). A survey
of PSI experiences among members of a special population. Journal of the Amercian Society for Psychical Research, 74, 395411.
Krakow, B., & Neidhardt,
J. (1992). Conquering bad dreams and
nightmares: A guide to understanding, interpretation and cure. New York:
Berkeley.
Krippner, S., Honorton, C.,
& Ullman, M. (1972). A second precognitive dream study with Malcolm
Bessent. Journal of the American Society
for Psychical Research, 66,
269279.
Krippner, S., Ullman, M.,
& Honorton, C. (1971). A precognitive dream study with a single subject. Journal of the American Society for
Psychical Research, 65, 192203.
Kumar, V. K., Pekala, R. J.,
& Gallagher, C. (1994). The Anomalous
Experiences Inventory. Unpublished
Psychological Test. West Chester, PA: West Chester University.
Lange, R. (1998, July 28 
August 1). GEMCAT II : A
distributionfree method for testing the statistical properties of Thom's catastrophe
equations as defined by sets of arbitrary latent variables. Paper presented
at the Eighth Annual Conference of the Society for Chaos Theory in Psychology
and the Life Sciences, Boston, MA.
Lange, R. (1999). A cusp
catastrophe approach to the prediction of temporal patterns in the kill dates
of individual serial murderers. Nonlinear
Dynamics, Psychology, & Life Sciences, 3, 143159.
Lange, R., & Fishbein,
M. (1983). The effects of category differences on belief change and agreement
with the source of persuasive communication. Journal of Personality and Social Psychology, 44, 933941.
Lange, R., & Houran, J.
(1996). Role of contextual mediation in direct versus reconstructed angelic
encounters. Perceptual and Motor Skills,
83, 12591270.
Lange, R., & Houran, J.
(1997). Death anxiety and the paranormal: The primacy of belief over
experience. Journal of Nervous and Mental
Disease, 185, 584586.
Lange, R., & Houran, J.
(1998). Delusions of the paranormal: A haunting question of perception. Journal of Nervous and Mental Disease, 186,
637645.
Lange, R., & Houran, J.
(1999a). Scaling MacDonald's AT20 using itemresponse theory. Personality and Individual Differences, 26,
467475.
Lange, R., & Houran, J.
(1999b). The role of fear in delusions of the paranormal. Journal of Nervous and Mental Disease, 187, 159166.
Lange, R., & Houran, J.
(2000). Modeling Maher’s attribution theory of delusions as a cusp catastrophe.
Nonlinear Dynamics, Psychology, and Life
Sciences, 4,
235254..
Lange, R., Houran, J., Harte,
T. M., & Havens, R. A. (1996). Contextual mediation of perceptions in
hauntings and poltergeistlike experiences. Perceptual
and Motor Skills, 82, 755762.
Lange, R., Irwin, H. J.,
& Houran, J. (2000). Topdown purification of Tobacyk's Revised Paranormal
Belief Scale. Personality and Individual
Differences, 29,
131156.
Lange, R., Oliva, T., &
McDade, S. (1999a, June 1316). Multivariate
catastrophe modeling with GEMCAT II as applied to organizational product
adoption. Paper presented at the AMA
ART Forum, Sante Fe, NM.
Lange. R., McDade, S., and Oliva, T.A.(2000). A catastrophe model of
organizational product adoption. Manuscript submitted for publication.
Lange, R., Oliva, T. A.,
& McDade, S. R. (1999b). An algorithm
for estimating multivariate catastrophe models: GEMCAT II. Manuscript
submitted for publication.
Lange, R., Thalbourne, M.
A., Houran, J., & Storm, L. (in press). The Revised Transliminality Scale: Reliability and validity data from a
Rasch topdown purification procedure. Consciousness and Cognition.
MacDonald, J., A.P. (1970).
Revised scale for ambiguity tolerance: Reliability and validity. Psychological Reports, 26, 791798.
Marascuilo, L. A., &
McSweeney, M. (1977). Nonparametric and
distributionfree methods for the social sciences. Monterey, CA:
Brooks/Cole.
Meehl, P. E. (1978).
Theoretical risks and tabular asteriks: Sir Karl, Sir Ronald, and the slow
progress of soft psychology. Journal of
Consulting and Clinical Psychology, 16,
806834.
Milton, J., & Wiseman,
R. (1999). Does psi exist? Lack of replication of an anomalous process of
information transfer. Psychological
Bulletin, 125, 387391.
Mischo, J. (1985). Paranormale
erfahrungen im traum. Zeitschrift für
Parapsychologie und Grenzgebiete der Psychologie, 27, 116141.
Murray, H. A., &
Wheeler, D. R. (1937). A note on the possible clairvoyance of dreams. Journal of Psychology, 3, 309313.
Oliva, T. A., Desarbo, W. S.,
Day, D. L., & Jedidi, K. (1987). GEMCAT: A general multivariate methodology
for estimating catastrophe models. Behavioral
Science, 32, 121137.
Orme, J. E. (1974).
Precognition and time. Journal of the
Society for Psychical Research, 47,
351365.
Palmer, J. (1979). A
community mail survey of psychic experiences. Journal of the American Society for Psychical Research, 73, 221251.
Popper, K. R. (1978). Logik der forschung. Tübingen: Mohr.
Radin, D. I. (1997). The conscious universe: The scientific truth
of psychic phenomena. San Francisco, CA: HarperCollins.
Rogers, L. W. (1923). Dreams and premonitions. Chicago: Theo
Book Co.
Rhine, L. E. (1954).
Frequency and types of experience in spontaneous precognition. Journal of Parapsychology, 18, 93123.
Ross, C. A., & Joshi, S.
(1992). Paranormal experiences in the general population. Journal of Nervous and Mental Disease, 180, 357361.
Ryback, D., & Sweitzer,
L. (1990). Wahrträume. München:
Knaur.
Saltmarsh, H. F. (1934).
Report on cases of apparent precognition.
Proceedings of the Society for Psychical Research, 42, 49103.
Schredl, M. (1996). Hör auf deine träume. Küttingen:
MidenaVerlag.
Schredl, M. (1999a). Die nächtliche traumwelt: Eine einführung in
die psychologische traumforschung. Stuttgart: Kohlhammer.
Schredl, M. (1999b). Präkognitive träume: Überblick über die
forschung und zusammenhang zum traumerleben. Manuscript submitted for
publication.
Schredl, M., & Doll, E.
(1997). Autogenic training and dream recall. Perceptual and Motor Skills, 84, 13051306.
Schredl, M., & Doll, E.
(1998). Emotions in diary dreams. Consciousness
and Cognition, 7, 634646.
Schredl, M., Jochum, S.,
& Sougoenet, S. (1997). Dream recall, visual memory, and absorption in
imaginings. Personality and Individual
Differences, 22, 291292.
Schredl, M., &
Montasser, A. (199697). Dream recall: State or trait variable? Part II: State
factors, investigations and final conclusions. Imagination, Cognition and Personality, 16, 239261.
Schriever, F. (1987). A
30year "experiment with time:" Evaluation of an individual case
study of precognitive dreams. European
Journal of Parapsychology, 7,
4972.
Smith, M. D. (199293). The
effect of belief in the paranormal and prior set upon the observation of a
“psychic” demonstration. European Journal
of Parapsychology, 9, 2434.
Sondow, N. (1988). The decline
of precognized events with the passage of time: Evidence from spontanous
dreams. Journal of the American Society
for Psychical Research, 82, 3351.
Stevenson, I. (1961). An
example illustrating the criteria and characteristics of precognitive dreams. Journal of the American Society for
Psychical Research, 55, 98103.
Stowell, M. S. (1995).
Researching precognitive dreams: A review of past methods, emerging scientific
paradigms, and future approaches. Journal
of the American Society for Psychical Research, 89, 117151.
Stowell, M. S. (1997a).
Precognitive dreams: A phenomenological study  Part I: Methodology and sample
cases. Journal of the American Society
for Psychical Research, 91,
163220.
Stowell, M. S. (1997b).
Precognitive dreams: A phenomenological study  Part II: Discussion. Journal of the American Society for
Psychical Research, 91, 255304.
Strauch, I., & Meier, B.
(1996). In search of dreams  Results of
experimental dream research. Albany: State University of New York Press.
Thalbourne,
M. A. (1984). Some correlates of belief in psychical phenomena: A partial
replication of the Haraldsson findings. Parapsychological
Review, 15, 1315.
Thalbourne,
M. A. (1994). The SPR centenary census: II. The survey of beliefs and
experiences. Journal of the Society for
Psychical Research, 59, 420431.
Thom, R. (1975). Structural stability and morphogenesis.
New York: BenjaminAddisonWesley.
Ullman, M., Krippner, S.,
& Vaughan, A. (1977). Traumtelepathie
(Dream telepathy 1973). Freiburg: Aurum.
Wiseman, R., Jeffreys, C.,
Smith, M., & Nyman, A. (1999). The psychology of the seance, from
experiment to drama. Skeptical Inquirer,
23, 3032.
Wiseman, R., & Morris,
R. L. (1995). Recalling pseudopsychic demonstrations. British Journal of Psychology, 86,
113125.
Zeeman, E. (1974). On the
unstable behavior of stock exchanges. Journal
of Mathematical Economics, 1, 3949.
[1] We thank Josh Flanders for his assistance in the data collection. Address correspondence to J. Houran at 301 University Court West, Springfield, IL., 62703, USA. Electronic mail may be sent to <rlange3726@aol.com>
[2] We note that recent research by Milton and Wiseman (1999) challenges the robustness of particular types of reported “psi” effects obtained under experimental conditions.