Shannon Entropy: A Possible Intrinsic Target Property

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The Journal of Parapsychology, Vol. 58, December 1994

SHANNON ENTROPY: A POSSIBLE
INTRINSIC TARGET PROPERTY

By EpwIn C, May, S. JAMEs P. SPOTTISWOODE,
AND CHRISTINE L. JAMES

ABSTRACT: We propose that the average total change of Shannon’s entropy is a candidate for
an intrinsic target property. An intrinsic target property is one that is completely independent of
psychological factors and can be associated solely with a physical property of the target. We
analyzed the results of two lengthy experiments that were conducted from 1992 through 1993
and found a significant correlation (r, = 0.337, df= 31, ¢= 1.99, p < 0.028) with an absolute
measure of the quality of the anomalous cognition (AC). In addition, we found that the quality
of the AC was significantly better for dynamic targets than for static targets (f= 1.71, df= 36, p
S$ 0.048). The 1993 correlation with the change of entropy replicated a similar finding from our
1992 study. Using monte carlo techniques, we demonstrate that the observed correlations were
not due to some unforeseen artifact with the entropy calculation, but perhaps the correlation
can be accounted for because of the difference in some other measure between static and
dynamic targets. The monte carlo results and the significant correlations with static targets in
the 1992 study, however, suggest otherwise. We describe the methodology, the calculations, and
correlations in detail and provide guidelines for those who may wish to conduct similar studies.

INTRODUCTION

The psychophysical properties of the five known senses are well known
(Reichert, 1992). At the “front end,” they share similar properties. For
example, each system possesses receptor cells that convert some form of
energy (e.g., photons for the visual system, sound waves for the audio sys-
tem) into electrochemical signals. The transfer functions are sigmoidal;
that is, there is a threshold for physical excitation, a linear region, and a
saturation level above which more input produces the same output. How
these psychophysical reactions translate to sensational experience is not
well understood, but all the systems do possess an awareness threshold
similar to the subliminal threshold for the visual system. ;

Since all the known senses appear to share these common properties, it
is reasonable to expect that if anomalous cognition (Ac)! is mediated

1The Cognitive Sciences Laboratory has adopted the term anomalous mental ph a
instead of the more widely known psi. Likewise, we use the terms anomalous cognition and
anomalous perturbation for ESP and PK, respectively. We have done so because we believe

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through some additional “sensory” system, then it, too, should share similar
properties at the preperceptual cellular front end. For example, a putative
AC sensory system should possess receptor cells that have a sigmoidal trans-
fer function and exhibit threshold and saturation phenomena. As far as we
know, there are no candidate neurons in the peripheral systems whose
functions are currently not understood. So, if receptor cells exist, it is likely
that they will be found in the central nervous system. Since 1989, our labo-
ratory has been conducting a search for such receptor sites (May, Luke,
Trask, & Frivold, 1990); that activity continues.

There is a second way in which receptorlike behavior might be seen in
lieu of a neurophysiology study. If either an energy carrier for AC or some-
thing that correlated with it were known, then it might be possible to infer
sigmoidal functioning at the behavioral level as opposed to the cellular
level. Suppose we could identify an intrinsic target property that correlated
with AC behavior. Then, by manipulating this variable, we might expect to
see a threshold at low magnitudes and saturation at high magnitudes.

To construct such an experiment, it is mandatory that we eliminate, as
much as possible, all extraneous sources of variance and adopt an absolute
measure for the AC behavior (Lantz, Luke, & May, 1994). We can reduce
one source of variance by considering what constitutes a good target in an
ACG experiment. Delanoy (1988) reported a survey of the literature for
successful AC experiments and categorized the target material according to
perceptual, psychological, and physical characteristics. Except for trends
related to dynamic, multisensory targets, she was unable to observe system-
atic correlations of AC quality with her target categories.

Watt (1988) examined the target question from a theoretical perspective.
She concluded that the “best” AC targets are those that are meaningful,
have emotional impact, and contain human interest. Those targets that
have physical features that stand out from their backgrounds or contain
movement, novelty, and incongruity are also good targets.

In trying to understand these findings and develop a methodology for
target selection for process-oriented research, we have constructed a meta-
phor. Figure 1 shows three conceptual domains that contribute to the vari-
ability in AC experiments. The engineering metaphor of source,
transmission, and detector allows us to assign known contributors to the
variance in specific domains. Without controlling or understanding these
sources, interpreting the results from process-oriented research is problem-,
atical, if not impossible.

that these terms are more naturally descriptive of the observables and are neutral in
that they do not imply mechanisms, These new terms will be used throughout this paper.

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Shannon Entropy 3

Figure 1. Information-transfer metaphor.

For example, suppose that the quality of an AC response actually de-
pended upon the physical size of a target, and that affectivity was also a
contributing factor. That is, a large target that was emotionally appealing
was reported more often more correctly. Obviously, both factors are impor-
tant in optimizing the outcome; however, suppose we were studying the
effect of target size alone. Then an “attractive” small target might register as
well as a less attractive large target, and the size dependency would be
confounded in unknown ways.

Our metaphor allows us to assign variables, such as these, to specific
elements. Clearly, an individual’s psychological response to a target is not an
intrinsic property of a target; rather, it is a property of the receiver.” Likewise,
size is a physical property of the target and is unrelated to the receiver.
Generally, this metaphor allows us to lump together the psychology, person-
ality, and physiology of the receiver and consider these important factors as
contributors to a detector “efficiency.” If it is true that an emotionally ap-
pealing target is easier to sense by some individuals, we can think of them as
more efficient at those tasks. In the same way, all physical properties of a
target are intrinsic to the target and do not depend on the detector effi-
ciency. Perhaps, temporal and spatial distance between target and receiver
are intrinsic to neither the target nor the receiver, but rather to the trans-
mission mechanism, whatever that may be. .

2A person’s psychological reaction to a target (i.e., “detector” efficiency) is an impor-
tant contributing factor to the total response, as indicated in the references sited above;
however, it is possible to reduce this contribution by careful selection of the target pool
material.

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More than just nomenclature, our metaphor can guide us in designing
experiments to decrease certain variabilities in order to conduct meaning-
ful process-oriented research. Some of the methodological improvements
seem obvious. If the research objective is to understand the properties of
AC rather than how an AC ability may be distributed in the population, then
combining results across receivers should be done with great caution. To
understand how to increase high jumping ability, for example, it makes no
sense to use a random sample from the general population as high jumpers,
rather, find a good high jumper and conduct vertical studies (no pun in-
tended). So, too, is it true in the study of AC. We can easily reduce the
variance by asking given receivers to participate in a large number of trials
and not combining their results.

May, Spottiswoode, and James (1994) suggest that by limiting the num-
ber of cognitively differentiable elements within a target, the variance can
also be decreased. A further reduction of potential variance can be realized
if the target pool is such that a receiver’s emotional/ psychological response
is likely to be more uniform across targets (i.e., reducing the detector vari-
ance as shown in Figure 1).

Having selected experienced receivers and attended to these methodo-
logical considerations, we could then focus our attention on examining
intrinsic target properties. If we are successful at identifying one such prop-
erty, then all process-oriented AC research would be significantly improved,
because we would be able to control a source of variance that is target
specific. The remainder of this paper describes the analysis of two lengthy
studies that provide the experimental evidence to suggest that the average
of the total change of Shannon’s entropy may be one such intrinsic target

property.
APPROACH

The methodological details for the two experiments can be found in
Lantz, Luke, and May (1994). In this section, we focus on the target calcula-
tions and the analysis techniques.

Shannon Entropy: A Short Description

Building upon the pioneering work of Leo Szilard (1925/1972,
1929/1972), Shannon and Weaver (1949) developed what is now called
information theory. This theory formalizes the intuitive idea of information
that there is more “information” in rare events, such as winning the lottery,
than in common ones, such as taking a breath. Shannon defined the en-

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Shannon Entropy 5

tropy for a given system as the weighted average of the probability of occur-
rence of all possible events in the system. Entropy, used in this sense, is
defined as a measure of our uncertainty, or lack of information, about a
system. Suppose, for example, we had an octagonal fair die (i.e., each of the
eight sides is equally likely to come up). Applying Equation 1, below, to this
system gives an entropy of three bits, which is in fact the maximum possible
for this system. If, on the other hand, the die were completely biased so that
the same side always came up, the entropy would be zero. In other words, if
each outcome is equally likely then each event has the maximum surprise.
Conversely, there is no surprise if the same side always lands facing up.

In the case of images, a similar analysis can be used to calculate the
entropy. For simplicity, consider a black and white image in which the
brightness, or luminance, of each picture element, or pixel, is measured on
a scale from zero to 255, that is, with an eight-bit binary number. Equation
1 can again be used to arrive at a measure of the picture’s entropy. As with
the other sensory systems where gradients are more easily detected, we shall
show that the gradient of Shannon’s entropy is correlated with AC perform-
ance far better than the entropy itself.

In other sensory systems, receptor cells are sensitive to incident energy
regardless of “meaning”, which is ascribed as a later cognitive function.
Shannon entropy is also devoid of meaning. The pixel analysis ignores
anything to do with cognitive features. From this point of view, a photo-
graph of a nuclear blast is, perhaps, no more Shannon-entropic than a
photograph of a kitten; it all depends on the intensities, which were used to
create the photographs. Thus, it is not possible to give a prescription on
how to chose a high change-in-entropy photograph based on its pictorial
content. Perhaps, after much experience, it may be possible to recognize
good targets from their intensity patterns; at the moment we do not know
how to accomplish this.

Target Calculations

Because of the analogy with other sensory systems, we expected that the
change of entropy would be more sensitive than the entropy alone would
be. The target variable that we considered, therefore, was the average total
change of entropy. In the case of image data, the entropy is defined as:

Ns (1)
Se = - ¥ pi log 2 (pm),

m=0

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where #,., is the probability of finding image intensity m of color k. In a
standard, digitized, true color image, each pixel (ie., picture element)
contains eight binary bits of red, green, and blue intensity, respectively. That
is, N, is 255 (ie., 281) for each k, k= 7, g, 6. For color, & the total change of
the entropy in differential form is given by:

OSk
ot

(2)

dS, =|VS,- FI + dt.

The first term corresponds to the change of the entropy spatially across
a single photograph of video frame. Imagine a hilly plane in entropy space;
this term represents the steepness of the slope of the hills (i.e., the change
between adjacent macropixels, as defined below). The second term adds
time changes to the total change. Not only does the entropy change across
a scene, but a given patch of the photograph changes from one scene to the
next. Of course, this term is zero for all static photographs.

We must specify the spatial and temporal resolution before we can com-
pute the total change of entropy for a real image. Henceforth, we drop the
color index, k, and assume that all quantities are computed for each color
and then summed.

To compute the entropy from Equation 1, we must specify empirically
the intensity probabilities, pm. In Lantz, Luke, and May’s 1993 experiment,
the targets were all video clips that met the following criteria:

1. Topic homogeneity. The photographs contained outdoor scenes of
settlements (e.g., villages, towns, cities, etc.), water (e.g., coasts, rivers and
streams, waterfalls, etc.), and topography (e.g., mountains, hills, deserts,
etc.).

2. Size homogeneity. Target elements are all roughly the same size. That
is, there are no size surprises such as an ant in one photograph and the
moon in another.

3. Affectivity homogeneity. As much as possible, the targets included
materials thst invoke neutral affectivity.

For static targets, a single characteristic frame from a video segment was
digitized (i.e., 640 x 480 pixels) for eight bits of information of red, green,
and blue intensity. The video image conformed to the NTSC standard as-
pect ratio of 4 x 3, and so we arbitrarily assumed an area (i.e., macropixel)
of 16 x 12 = 192 pixels from which we calculated the pm. Since during the’
feedback phase of a trial the images were displayed on a Sun Microsystems
standard 19-inch color monitor, and since they occupied an area approxi-
mately 20 x 15 cm square, the physical size of the macropixels was approxi-
mately 0.5 cm square. Since major cognitive elements were usually not

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Shannon Entropy 7

smaller than this, this choice was reasonable—192 pixels were sufficient to
provide a smooth estimate of the pm.

For this macropixel size, the target frame was divided into a 40 x 40 array.
The entropy for the (4,7)th macropixel was computed as:

N-1

Sij = - »y Pm loge (pn);
m=0

where p,, is computed empirically only from the pixels in the (3% J)
macropixel and m is the pixel intensity. For example, consider the white
square in the upper left portion of the target photograph shown in Figure
2.

Figure 2. City with a mosque.

The green probability distribution for this macropixel (3, 3) is shown in
Figure 3. The probability density and the photograph itself indicate that
most of the intensity in this macropixel is near zero (i.e., no intensity of
green in this case). In a similar fashion, the S¥ are calculated for the entire’
scene. Since iand j range from 0 to 40, each frame contains a total of 1,600
macropixels.

We used a standard image processing algorithm to compute the two-di-
mensional spatial gradient for each of the 1,600 macropixels. The first term

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Figure 3. Green intensity distribution for the city target (macropixel 3, 3).

in Equation 2 was approximated by its average value over the image and was
computed by the relations shown as Equations 3.

IVS, l=) +

= (Sis, j+1— Sij+ 1) + 2085 j41 — Si f+) + (Si-1,j+1- Si-1,j-0

dSij
dx

“i (Si+1,j+1— Si-1, j+1) + 2(Si41,; — Si-1,)) + (Si+1,j-1 - Si-1,j-1)

The total change of entropy for the dynamic targets was calculated in
much the same way. The video segment was digitized at one frame per
second. The spatial term of Equation 2 was computed exactly as it was for
the static frames. The second term, however, was computed from differ-
ences between adjacent, 1-second frames for each macropixel. Or,

OSiz A Sij(t) _ | Sig 4+ At) - Sj (4) (4)
ot At At ,

where Atis one over the digitizing frame rate. We can see immediately that
the dynamic targets will have a larger AS than do the static ones because
Equation 4 is identically zero for all static targets.

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Shannon Entropy 9

In Lantz, Luke, and May’s 1992 experiment, the static targets were digit-
ized from scanned photographs. This difference and its consequence will
be discussed below.

AC Data Analysis

Rank-order analysis in Lantz, Luke, and May’s (1994) experiment dem-
onstrated significant evidence for AC; however, this procedure does not
usually indicate the absolute quality of the AC. For example, a response that
is a near-perfect description of the target receives a rank of 1. But a response
that is barely matchable to the target may also receive a rank of 1. Table 1
shows the rating scale that we used to assess the quality of the AC responses,
regardless of their rank.

To apply this subjective scale to an AC trial, an analyst begins with a score
of 7 and determines if the description for that score is correct. If not, then
the analyst tries a score of 6, and so on. In this way the scale is traversed from
7 to 0 until the score-description seems reasonable for the trial.

TABLE 1
o—7 PoINT ASSESSMENT SCALE

Score Description

7 Excellent correspondence, including good analytical detail, with
essentially no incorrect information

6 Good correspondence with good analytical information and relatively
little incorrect information

5 Good correspondence with unambiguous unique matchable
elements, but some incorrect information

4 Good correspondence with several matchable elements intermixed
with incorrect information

3. Mixture of correct and incorrect elements, but enough of the former
to indicate receiver has made contact with the site

2 Some correct elements, but not sufficient to suggest results beyond
chance expectation

1 Little correspondence
No correspondence

For all analyses in the 1992 and 1993 studies, we decided a priori to use°
only the upper half of the rating scale. As the strength of the AC functioning
increases, by definition there is less incorrect information (i.e., noise). In
other words, the noise contribution to each score level decreases in some

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unknown way as the the AC increases. Thus, we limited the noise contribu-
tion by using only the upper half of the scale in the analysis.

ANOMALOUS COGNITION EXPERIMENT— 1992

In Lantz, Luke, and May’s 1992 experiment there were no significant
interactions between target condition (i.e., static vs. dynamic) and sender
condition (i.e., sender vs. no sender); therefore, they combined the data for
static targets regardless of the sender condition (i.e., 100 trials). The sum-
of-ranks was 265 (i.e., exact sum-of-rank probability of p < 0.007, effect size
= 0.248). The total sum-ofranks for the dynamic targets was 300 (ie., pS
0.50, effect size = 0.000).

Entropy Analysis

To examine the relationship of entropy to AC, two analysts inde-
pendently rated all 100 trials (i.e., 20 each from five receivers) from_the
static-target sessions using the rating scale shown in Table 1, post hoc.? All
differences of assignments were verbally resolved; thus, the resulting scores
represented a reasonable estimate of the visual quality of the AC for each
trial.

We had specified, in advance, that for the correlation with the change of
target entropy, we would only use the section of the post hoc rating scale
that represented definitive, albeit subjective, AC contact with the target
(i.e., scores 4 through 7). Figure 4 shows a scatter diagram for the post hoc
rating and the associated AS for the 28 trials with static targets that met this
requirement. Shown also is a linear least-squares fit to the data and a Spear-
man rank-order correlation coefficient (7% = 0.452, df= 26, t= 2.58, p< 7.0
x 10°).

This strong correlation suggests that AS is an intrinsic property of a static
target and that the quality of an AC response will be enhanced for targets
with large AS. It is possible, however, that this correlation might be a result
of AS and the post hoc rating independently correlating with the targets’
visual complexity. For example, an analyst is able to find more matching
elements (i-e., a higher post hoc rating) in a visually complex target than in
a visually simple one. Similarly, AS may be larger for more complex targets. |
If these hypotheses were true, the correlation shown in Figure 4 would not

3This was conduced post hoc because we did not realize until after the judges com-
pleted their blind analysis and had been given feedback on the study outcome that a rating
scale is more sensitive than ranking. We used this result to form a hypothesis that we tested
in the second study.

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Shannon Entropy 11

support the hypothesis that AS is an important intrinsic target property for
successful AC.

Figure 4. Correlation of post hoc score with static target AS.

To check the validity of the correlation, we used a definition of visual
complexity, which was derived from a fuzzy set representation of the target
pool. We had previously coded by consensus 131 different potential target
elements for their visual impact on each of the targets in the pool. We
assumed that the sigma-count (i.e., the sum of the membership values over
all 131 visual elements) for each target is proportional to its visual complex-
ity. A description of the fuzzy set technique and a list of the target elements
may be found in May, Utts et al. (1990).

The Spearman rank correlation between target complexity and post hoc
rating was small (75 = 0.041, ¢= 0.407, df= 98, p < 0.342). On closer inspec-
tion this small correlation was not surprising. Although it is true that an
analyst will find more matchable elements in a complex target, so also are
there many elements that do not match. Since the rating scale (i.e., Table 1)
is sensitive to correct and incorrect elements, the analyst is not biased by-
visual complexity.

Since the change of Shannon entropy is derived from the intensities of
the three primary colors (i.e., Equation 1) and is unrelated to meaning,
which is inherent in the definition of visual complexity, we would not expect

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a correlation between AS and visual complexity. We confirmed this expecta-
tion when we found a small correlation (7 = 0.028, ¢=-0.277, df= 98, ps
0.609).

Visual complexity, therefore, cannot account for the correlation shown
in Figure 4; thus, we are able to suggest that the quality of an AC response
depends upon the spatial information (i.e., change of Shannon entropy) in
a Static target.

A single analyst scored, post hoc, the 100 responses from the dynamic
targets using the scale in Table 1. Figure 5 shows the scatter diagram for the
post hoc scores and the associated AS for the 24 trials with a score greater
than 3 for the dynamic targets. We found a Spearman correlation of 7% =
0.055 (¢= 0.258, df= 22, ps 0.399).

Figure 5. Correlation of post hoc score with dynamic target AS.

This small correlation is not consistent with the result derived from the
static targets; therefore, we examined this case carefully. The total sum of
ranks for the dynamic-target case was exactly mean chance expectation,
which indicates that no AC was observed (Lantz, Luke, & May, 1994). May, ”
Spottiswoode, and James (1994) propose that the lack of AC might be
because an imbalance of, what they call, the target-pool bandwidth. That is,
the number of different cognitive elements in the dynamic pool far ex-
ceeded that in the static pool. This imbalance was corrected in the 1993

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Shannon Entropy 13

study and is analyzed below. Regardless, we would not expect to see a corre-
lation if there is no evidence of AC.

ANOMALOUS COGNITION EXPERIMENT — 1993

The details of the 1993 study may also be found in Lantz, Luke, and May
(1994). In that study, they included a static versus dynamic target condition,
and all trials were conducted without a sender. They changed the target
pools so that their bandwidths were similar. They also included a variety of
other methodological improvements, which are not apropos to this discus-
sion.

Lantz, Luke, and May selected a single frame from each dynamic target
video clip, which was characteristic of the entire clip, to act as its static
equivalent. The static and dynamic targets, therefore, were digitized with
the same resolution and could be combined for the correlations.

For each response, a single analyst conducted a blind ranking of five
targets—the intended one and four decoys—in the usual way. Lantz, Luke,
and May computed effect sizes in the same way as in the 1992 study.

Three receivers individually participated in 10 trials for each target type,
and a fourth participated in 15 trials per target type. Lantz, Luke, and May
reported a total average rank for the static targets of 2.22 for 90 trials for an
effect size of 0.566 (pS 7.5 x 10°); the exact same effect size was reported
for the dynamic targets.

Entropy Analysis

Unlike the 1992 experiment, in the 1993 experiment an analyst who was
blind to the correct target choice used the scale shown in Table 1 to assess
each response to the same target pack that was used in the rank-order
analysis. The average total change of Shannon’s entropy (i.e., Equation 2)
was calculated for each target as described above. Figure 6 shows the corre-
lation of the blind rating score with this gradient. The squares and dia-
monds indicate the data for static and dynamic targets, respectively.

The key indicates the Spearman correlation for the static and dynamic
targets combined. In addition, since the hypothesis was that AC would
correlate with the total change of the Shannon entropy, Figure 6 shows only |
the scores in the upper half of the scale in Table 1 for the 70 trials of the
three independently significant receivers. The static target correlation was
negative (7 = —0.284, t=-1.07, df= 13, p< 0.847,) and the correlation from
the dynamic targets was positive (75 = 0.320, t= 1.35, df= 16, p< 0.098). The
strong correlation for the combined data arises primarily from the entropic

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difference between the static and dynamic targets. The rating scores were
signficantly stronger for the dynamic targets than for the static ones (f=
1.71, df= 36, p< 0.048).

Figure 6. Correlations for significant receivers.

As a control check for possible unforeseen artifacts, we conducted a
monte carlo analysis as follows. The actual blind rating scores greater than
3 were correlated with the gradient of Shannon’s entropy from a target
chosen randomly from the pool of the appropriate target type (i.e., only
from the static pool or dynamic pool if ratings were originally from a static
or dynamic target, respectively). After 100 such monte carlo trials, we found
the Spearman rank correlation was 7 = —0.0501 + 0.311. If we assume that
the standard deviation for the actual data is the same as that in the monte
carlo calculation (i.e., % = 0.337 + 0.311), we find a significant difference
between these cross-match controls and the actual data (tdi = 4.98, df= 62,
pS 2.68 x 10°), This analysis assumes that the visual correspondence be-
tween a response and its intended target remains the same, but the gradient
of the entropy is random. Thus, it appears that the data correlation does not
arise from an artifact. ,

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Shannon Entropy 15

GENERAL CONCLUSIONS

To understand the differences between the results in the two experi-
ments, we redigitized the static set of targets from the 1992 experiment with
the same hardware and software that was used in the 1993 study. With this
new entropy data, the correlation dropped from a significant 0.452 to 0.298,
which is not significant (¢ = 1.58, df= 26, p < 0.063). When we combined
these data with the static results from the 1993 experiment (i.e., significant
receivers), the static correlation was 7% = 0.161 (t= 1.04, df= 41, p< 0.152).
The correlation for the static targets from the 1992 experiment combined
with the significant static and dynamic data from the 1993 experiment was
significant (7 = 0.320, df= 59, t = 2.60, p< 0.006). These post hoc results are
shown in Figure 7. The combined data from the two experiments, including
all receivers and all scores greater than four, give a significant correlation (7
= 0.258, df= 64, t= 2.13, ps 0.018).

Figure 7. Correlations for combined experiments.

We conclude that the quality of AC appears to.correlate linearly with the
average total change of the Shannon entropy, which is an intrinsic target
property. ‘

These two experiments may raise more questions than they answer. If our
conservative approach—which assumes that AC functions similarly to the ©
other sensory systems—is correct, we would predict that the AC correlation
with the frame entropy should be smaller than that for the average total

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16 The Journal of Parapsychology

change of the entropy. We computed the total frame entropy from the pj
which we computed from all of the 640 x 480 pixels. The resulting correla-
tion for the significant receivers in the 1993 experiment was % = 0.234 (t=
1.34, df= 31, p< 0.095). This correlation is considerably smaller than that
from the gradient approach, but not significantly so. We computed the
average of the Sj for the 1,600 macropixels as a second way of measuring the
spatial entropic variations. We found a significant Spearman’s correlation of
1% = 0.423 (t= 2.60, df= 31, p< 0.007) for the significant receivers in the 1993
experiment. The difference between the correlation of the quality of the AC
with the frame entropy and with either measure of the spatial gradient is not
significant; however, these large differences are suggestive of the behavior
of other sensory systems (i.e., an increased sensitivity with change of the
input).

We have quoted a number of different correlations under varying cir-
cumstances and have labeled these as post hoc. For example, hardware
limitations in 1992 prevented us from combining those data with the data
from 1993. Thus, we recalculated the entropies with the upgraded hardware
in 1993 and recomputed the correlations. Our primary conclusions, how-
ever, are drawn only from the static results from the 1992 experiment and
the confirmation from the combined static and dynamic 1993 results.

Since we observed a significant AC-score difference in the 1993 study,
perhaps our correlations with AS arise because of some other nonentropy
related property such as motion. Yet, the monte carlo results demonstrated
that the correlations vanished when the AC scores were kept constant and
random crosstarget values were used for AS. Because the randomizations
were exclusively within target type, the correlation between the AC scores
and AS should have remained significant had some nonentropy factor dis-
tinguishing static and dynamic targets been operative. In addition, the
within-static-target AS in the 1992 study significantly correlated with the the
AC quality.

We conclude that we may have identified an intrinsic target property that
correlates with the quality of anomalous cognition. Our results suggest a
host of new experiments and analyses before we can come to this conclu-
sion with certainty. For example, suppose we construct a new target pool
that is maximized for the gradient of Shannon’s entropy yet meets reason-
able criteria for the target-pool bandwidth. If the Shannon information is
important, than we should see exceptionally strong AC. We also must im-_
prove the absolute measure of AC. Although dividing our 0-to-7 rating scale
in two makes qualitative sense, it was an arbitrary decision. Rank order
statistics are not as sensitive to correlations as are absolute measures (Lantz,
Luke, & May, 1994); but, perhaps, if the AC effect size is significantly in-
creased with a proper target pool, the rank order correlations will be strong

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Shannon Entropy 17

enough. A more sensitive and well-defined rating scale should also improve
the analysis. It may be time consuming, but it is also important to under-
stand the dependency of the correlation on the digitizing resolution. In the
first experiment, we digitized the hard-copy photographs using a flatbed
scanner with an internal resolution of 100 dots per inch and used 640 x 480
pixels for the static and dynamic targets in the second experiment. Why did
the correlation drop for the static targets by nearly 35% when the digitizing
resolution decreased by 20%?

We noticed, post hoc, that the correlations exhibit large oscillations
around 0 below the cutoff score of 4. If we assume there is a linear relation-
ship between AC scores and the total change of Shannon entropy, we would
expect unpredictable behavior for the correlation at low scores because
they imply chance matches with the target and do not correlate with the
entropy.

To determine if we are observing behavioral evidence for receptor-like
functioning for the detection of AC, we must identify threshold and satura-
tion limits. This can be accomplished in future experiments.

It is absolutely critical to confirm our overall results and to provide an-
swers to some of the enigmas from our experiment. If we have identified an
intrinsic target property, then all of our research can precede more effi-
ciently. Consider the possibilities if we were able to construct a target pool
and eliminate a known source of variance. Psychological and physiological
factors would be much easier to detect. Because of the availability of inex-
pensive video digitizing boards for personal computers, replication at-
tempts are easily within the grasp of research groups with modest operating
budgets.

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