# DECISION AUGMENTATION THEORY: TOWARD A MODEL OF ANOMALOUS MENTAL PHENOMENA

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Decision
Decision Augmentation Theory:
Toward a Model
of
Anomalous Mental Phenomena
by
Edwin C. May, Ph.D
Science Applications International Corporation
Menlo Park, CA
Jessica M. Utts, Ph.D.
University of California, Davis
Department of Statistics
Davis, CA
and
S. James P. Spottiswoode
Science Applications International Corporation (Consultant)
Menlo Park, CA
Abstract
Decision Augmentation Theory (DAT) holds that humans integrate information obtained by anoma-
lous cognition into the usual decision process. The result is that, to a statistical degree, such decisions
are biased toward volitional outcomes. We introduce DAT and define the domain for which the model
is applicable. In anomalous mental phenomena research, DAT is applicable to the understanding of
effects that are within a few sigma of chance. We contrast the experimental consequences of DATwith
those of models that treat anomalous perturbation as a causal force. We derive mathematical expres-
sions for DAT and causal models for two distributions, normal and binomial. DAT is testable both retro-
spectively and prospectively, and we provide statistical power curves to assist in the experimental design
of such tests. We show that the experimental consequences of DAT are different from those of causal
models except for one degenerate case.
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Decision Augmentation Theory: Toward a ode of
Introduction
We do not have positive definitions of the effects that generally fall under the heading of anomalous
mental phenomena (AMP).* In the crassest of terms, AMP is what happens when nothing else should, at
least as nature is currently understood. In the domain of information acquisition, or anomalous cogni-
tion (AC), it is relatively straightforward to design an experimental protocol (Honorton et al., 1990,
Hyman and Honorton, 1986) to assure that no known sensory leakage of information can occur. In the
domain of causation, or anomalous perturbation (AP), however, it is very difficult, if not impossible
(May, Humphrey, and Hubbard, 1980 and Hubbard, Bentley, Pasturel, and Issacs, 1987); thus, making
the interpretation of results equally difficult.
We can divideAP into two categories based on the magnitude of the putative effect. Macro-AP include
phenomena that generally do not require sophisticated statistical analysis to tease out weak effects
from the data. Examples include inelastic deformations in strain gauge experiments, the obvious bend-
ing of metal samples, and a host of possible "field phenomena" such as telekinesis, poltergeist, tele-
portation, and materialization. Conversely, micro-AP covers experimental data from noisy diodes, ra-
dioactive decay and other random sources. These data show small differences from chance expectation
and require statistical analysis.
One of the consequences of the negative definitions of AMP is that experimenters must assure that the
observables are not due to "known" effects. Traditionally, two techniques have been employed to guard
against such interactions:
(1) Complete physical isolation of the AP-target system.
(2) Counterbalanced control and effort periods.
Isolating physical systems from potential "environmental" effects is difficult, even for engineering spe-
cialists. It becomes increasingly problematical the more sensitive the Macro-AP device. For example
Hubbard, Bentley, Pasturel, and Issacs (1987) monitored a large number of sensors of environmental
variables that could mimic AP effects in an extremely isolated piezoelectric strain gauge. Among these
were three-axis accelerometers, calibrated microphones, and electromagnetic and nuclear radiation
monitors. In addition, the sensors were mounted in a government-approved enclosure to assure no
leakage (in or out) of electromagnetic radiation above a given frequency, and the enclosure itself was
levitated on an air suspension table. Finally, the entire setup was locked in a controlled access room
which was monitored by motion detectors. The system was so sensitive, for example, that it was possible
to identify the source of a perturbation of the strain gauge that was due to innocent, gentle knocking on
the door of the closed room. The financial and engineering resources to isolate such systems rapidly
become prohibitive.
The second method, which is commonly in use, is to isolate the target system within the constraints of
the available resources, and then construct protocols that include control and effort periods. Thus, we
trade complete isolation for a statistical analysis of the difference between control and effort periods.
The assumption implicit in this approach is that environmental influences of the device will be random
* The Cognitive Sciences Laboratory has adopted the term anomalous menial phenomena instead of the more widely known psi.
Likewise, we use the terms anomalous cognition and anomalous perturbation for r3P and PK, respectively. We have done so
because we believe that these terms are more naturally descriptive of the ohservahlcs and arc neutral with regard to mecha-
nisms. These new terms will he used throughout this paper.
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Decision Augmentation Theory: Toward a Model of AMP V9. 22 April 1994
and uniformly distributed in both the control and effort conditions, while AP will tend to occur in the
effort periods. Our arguments in favor of an anomaly, then, are based on statistical inference and we
must consider, in detail, the consequences of such analyses, one of which implies a generalized model
for AMP.
Background
As the evidence forAMP becomes more widely accepted (Bem and Honorton, 1994, Utts, 1991, Radin
and Nelson, 1989) it is imperative to determine the underlying mechanisms of the phenomena. Clearly,
we are not the first to begin thinking of potential models. In the process of amassing incontrovertible
evidence of an anomaly, many theoretical approaches have been examined; in this section we outline a
few of them. It is beyond the scope of this paper, however, to provide an exhaustive review of the
theoretical models of AMP; a good reference to an up-to-date and detailed presentation is Stokes
(1987).
Brief Review of Models
Two fundamentally different types of models have been developed: those that attempt to order and
structure the raw observations inAMP experiments (i.e., phenomenological), and those that attempt to
explainAMP in terms of modifications to existing physical theories (i.e., fundamental). In the history of
the physical sciences, phenomenological models, such as the Snell's law of refraction or Ampere's law
for the magnetic field due to a current, have nearly always preceded fundamental models of the phe-
nomena, such as quantum electrodynamics and Maxwell's theory. In producing useful models ofAMP it
may well be advantageous to start with phenomenological models, of which DAT is an example.
Psychologists have contributed interesting phenomenological approaches. Stanford (1974a and 1974b)
proposed PSI-mediated Instrumental Response (PMIR) as a descriptive model. PMIR states that an
organism usesAMP to optimize its environment. For example, in one of Stanford's classic experiments
(Stanford, Zenhausern, Taylor, and Dwyer 1975) subjects were offered a covert opportunity to stop a
boring task prematurely if they exhibited unconscious AP by perturbing a hidden random number gen-
erator. Overall, the experiment was significant in the unconscious tasks; it was as if the participants
were unconsciously scanning the extended environment for any way to provide a more optimal situation
than participating in a boring psychological task!
As an example of a fundamental model, Walker (1984) proposed a literal interpretation of quantum
mechanics in that since superposition of eigenstates holds, even for macrosystems, AMP might be due
to macroscopic examples of quantum phenomena. These concepts spawned a class of theories, the so-
called observation theories, that were based either upon quantum formalism conceptually or directly
(Stokes, 1987). Jahn and Dunne (1986) have offered a "quantum metaphor" which illustrates many
parallels between AMP and known quantum effects. Unfortunately, these models either have free pa-
rameterswith unknown values, or are merely hand waving metaphors and therefore have not led to test-
able predictions. Some of these models propose questionable extensions to existing theories. For ex-
ample, even though Walker's interpretation of quantum mechanical formalism might suggest wave-like
properties of macrosystems, the physics data to date not only show no indication of such phenomena at
room temperature but provide considerable evidence to suggest that macrosystems lose their quantum
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coherence above 0.5 Kelvins (Washburn and Webb, 1986) and no longer exhibit quantum wave-like be-
havior.
This is not to say that a comprehensive model of AMP will not eventually require quantum mechanics as
part of its explanation, but it is currently premature to consider such models as more than interesting
speculation. The burden of proof is on the theorist to show why systems, which are normally considered
classical (e.g., a human brain), are, indeed, quantum mechanical. That is, what are the experimental
consequences of a quantum mechanical system over a classical one?
Our Decision Augmentation Theory is phenomenological and is a logical and formal extension of Stan-
ford's elegant PMIR model. In the same manner as early models of the behavior of gases, acoustics, or
optics, it tries to subsume a large range of experimental measurements into a coherent lawful scheme.
Hopefully this process will lead the way to the uncovering of deeper mechanisms. In fact DAT leads to
the idea that there may be only one underlying mechanism of all AMP effects, namely a transfer of in-
formation between events separated by negative time intervals.
Historical Evolution of Decision Augmentation
May, Humphrey, and Hubbard (1980) conducted a careful random number generator (RNG) experi-
ment. What makes this experiment unique is the extreme engineering and methodological care that
was taken in order to isolate any potentially known physical interactions with the source of randomness.
It is beyond the scope of this paper to describe this experiment completely; however, those specific de-
tails which led to the idea of Decision Augmentation are important for the sake of historical complete-
ness.
May, Humphrey, and Hubbard were satisfied in that RNG study, that they had observed a genuine sta-
tistical anomaly. In addition, because of an accurate mathematical model of the random device and the
engineering details of the experiment, they were equally satisfied that the deviations were not due to
any known physical interactions. They concluded, in their report, that some form of AMP-mediated
data selection had occurred. They named it then Psychoenergetic Data Selection.
Following a suggestion by Dr. David R. Saunders of MARS Measurement and Associates, we noticed
in 1986 that the effect size in binary RNG studies varied on the average as the square root of the number
of bits in the sequence. This observation led to the development of the Intuitive Data Sorting model that
appeared to describe the RNG data to that date (May, Radin, Hubbard, Humphrey, and Utts, 1985).
The remainder of this paper describes the next step in the evolution process. We now call the model
Decision Augmentation Theory (DAT).
Decision Augmentation Theory-A General Description
Since the case forA C-mediated information transfer is now well established, it would be exceptional if
we did not integrate this form of information gathering into the decision process. For example, we rou-
tinely use real-time data gathering and historical information to assist in the decision process. Perhaps,
what is called intuition may play a important role. Why, then, should we not includeAC information?
DAT holds thatAC information is included along with the usual inputs that result in a final human deci-
sion that favours a "desired" outcome. In statistical parlance, DAT says that a slight, systematic bias is
introduced into the decision process by AC.
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Decision Augmentation Theory: Toward a Model of AMP p
This philosophical concept has the advantage of being quite general. We know of no experiment that is
devoid of at least one human decision; thus, DAT might be the underlying basis for AMP To illustrate
the point, we describe how the "cosmos" determines the outcome of a well-designed, hypothetical ex-
periment. To determine the sequencing of an RNG experiment, suppose that the entry point into a
table of random numbers will be chosen by the square root of the barometric pressure as stated in the
weather report that will be published seven days hence in the New York Times. Since humans are notori-
ously bad at predicting or controlling the weather, this entry point might seem independent of a human
decision; but why did we "chose" seven days in advance? Why not six or eight? Why the New York Times
and not the London Times? DATwould suggest that the selection of seven days, the New York Times, the
barometric pressure, and square root function were optimal choices, either individually or collectively,
and that other decisions would not lead to as significant an outcome.
Other non-technical decisions may also be biased by AC in accordance with DAT. When should we
schedule a Ganzfeld session; who should be the experimenter in a series; how should we determine a
specific order in a tri-polar protocol?
It is important to understand the domain in which a model is applicable. For example, Newton's laws
are sufficient to describe the dynamics of mechanical objects in the domain where the velocities are very
much smaller than the speed of light, and where the quantum wavelength of the object is very small
compared to the physical extent of the object. If these conditions are violated, then different models
must be invoked (e.g., relativity and quantum mechanics, respectively).
The domain in which DAT is applicable is when experimental outcomes are in a statistical regime (i.e., a
few standard deviations from chance). In other words, does the measured effect occur under the null
hypothesis? This is not a sharp-edged requirement and DAT becomes less apropos the more a single
measurement deviates from mean-chance-expectation (MCE). We would not invoke DAT, for exam-
ple, as an explanation of levitation if one found the authors hovering near the ceiling!
All this maybe interesting philosophy, but DAT can be formulated mathematically and subjected to rig-
orous examination.
Development of a Formal Model
While DAT may have implications for AMP in general, we develop the model in the framework of un-
derstanding experimental results. In particular, we consider AP vs AC in the form of DAT in those ex-
periments whose outcomes are in the few-sigma, statistical regime.
We define four possible mechanisms for the results in such experiments:
(1) Mean Chance Expectation. The results are at chance. That is, the deviation of the dependent vari-
able meets accepted criteria for MCE. In statistical parlance, we have measurements from an un-
perturbed parent distribution with unbiased sampling.
(2) Anomalous Perturbation. Nature is modified by some anomalous interaction. That is, we expect a
causal interaction of a "force" type. In statistical parlance, we have measurements from a perturbed
parent distribution with unbiased sampling.
(3) Decision Augmentation. Nature is unchanged but the measurements are biased. That is, AC in-
formation has "distorted" the sampling. In statistical parlance, we have measurements from an
unperturbed parent distribution with biased sampling.
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(4) Combination. Nature is modified and the measurements are biased. That is, both AP and AC are
present. In statistical parlance, we have conducted biased sampling from a perturbed parent dis-
tribution.
General Considerations
Since the formal discussion of DATis statistical, we will describe the overall context for the development
of the model from that perspective. Consider a random variable, X, that can take on continuous values
(e.g., the normal distribution) or discrete values (e.g., the binomial distribution). Examples ofX might
be the hit rate in an RNG experiment, the swimming velocity of cells, or the mutation rate of bacteria.
Let Ybe the average computed over n values of X, where n is the number of items that are collectively
subjected to an AMP influence as the result of a single decision-one trial. Often this maybe equivalent
to a single effort period, but it also may include repeated efforts. The key point is that, regardless of the
effort style, the average value of the dependent variable is computed over the n values resulting from
one decision point. In the examples above, n is the sequence length of a single run in an RNG experi-
ment, the number of swimming cells measured during the trial, or the number of bacteria-containing
test tubes present during the trial.
Assumptions for DAT
We assume that the parent distribution of a physical system remains unperturbed; however, the mea-
surements of the physical system are systematically biased by some AC-mediated informational pro-
cess.
Since the deviations seen in experiments in the statistical regime tend to he small in magnitude, it is safe
to assume that the measurement biases might also he small; therefore, we assume small shifts of the
mean and variance of the sampling distribution. Figure 1 shows the distributions for biased and un-
biased measurements.
Figure 1. Sampling Distribution Under DAT.
The biased sampling distribution shown in Figure 1 is assumed to be normally distributed as:
u
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Decision Augmentation heor : o
where the notation means that Z is distributed as a normal distribution with am e an ofItz and a standard
deviation of aZ.
Assumptions for an AP Model
For comparison sake, we develop a model for AP interactions. With a few exceptions reported in the
poltergeist literature,AP appears to be a relatively "small" effect in laboratory experiments. That is, we
do not readily observe anomalous and obvious mental interactions with the environment. Thus, we be-
gin with the assumption that a putative AP force would give rise to a perturbational interaction. What
we mean is that given an ensemble of entities (e.g., binary bits, cells), a force acts, on the average, equal-
lyon each member of the ensemble. We call this type of interaction perturbational AP (PAP).
Figure 2 shows a schematic representation of probability density functions for a parent distribution un-
der the PAP assumption and an unperturbed parent distribution. In the PAP model, the perturbation
induces a change in the mean of the parent distribution but does not effects its variance. We parameter-
ize the mean shift in terms of a multiplier of the initial standard deviation. Thus, we define anAP-effect
size as:
(fi l - fr o)
EAP - ao
where ?t and ?o are the means of the perturbed and unperturbed distributions, respectively, and where
oo is the standard deviation of the unperturbed distribution.
Figure 2. Parent Distribution for Perturbational AP
For the moment, we consider eAp as a parameter which, in principle, could he a function of a variety of
variables (e.g., psychological, physical, environmental, methodological). As we develop DAT for specif-
ic distributions and experiments, we will discuss this functionality of &,4p.
Calculation of E(Z2)
We compute the expected value and variance of Z2 under MCC, PAP, and DAT for the normal and bino-
mial distributions. The details of the calculations can he found in the Appendix; however, we summa-
rize the results in this section. Table I shows the results assuming that the parent distribution is normal.
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Decision Augmentation Theory: oward a A p
Table 1.
Normal Parent Distribution
Quantity
E(Z2)
1 + E,;pn
?4+aZ
Var(7.2)
2(1 + 2F2 Ap n)
2(aT + 2?4a=)
Table 2 shows the results assuming that the parent distribution is binomial. In this calculation, PO is the
binomial event probability and a0 = ,/p0(1-p0).
Table 2.
Binomial Parent Distribution
Quantity
E(Z2)
Var(Z2)
2+-1--(1 -6(yo)
0
I +E,,lp(n- 1) + L0(1 -2p0)
?4+(4
2(1 + 2F;,1 n)
2((1+ 2?Z(4)
* The variance shown assumes p0 = 0.5 and n >> 1. See the Appendix for other cases.
We wish to emphasize at this point that in the development of the mathematical model, the parameter
eAp for PAP, and the parameters ?Z, and az in DAT may all possibly depend upon n; however, for the
moment, we assume that they are all n-independent. We shall discuss the consequences of this assump-
tion below.
Figure 3 displays these theoretical calculations for the three mechanisms graphically.
Figure 3. Predictions of MCE, PAP, and DAT.
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Within the constraints mentioned above, this formulation predicts grossly different outcomes for these
models and, therefore, is ultimately capable of separating them, even for very small perturbations.
Retrospective Tests
It is possible to apply DAT retrospectively to any body of data that meet certain constraints. It is critical
to keep in mind the meaning of n-the number of measures of the dependent variable over which to
compute an average during a single trial following a single decision. In terms of their predictions for
experimental results, the crucial distinction between DAT and the PAP model is the dependence of the
results upon n; therefore, experiments which are used to test these theories must be those in which ex-
periment participants are blind to n. In a follow-on to this theory-definition paper, we will retrospec-
tively apply DAT to as many data sets as possible, and examine the consequences of any violations of
these criteria.
Aside from these considerations, the application of DAT is straight forward. Having identified the unit
of analysis and n, simply create a scatter diagram of points (Z2 n) and compute a least square fit to a
straight line. Tables I and 2 show that for the PAP model, the square of theAP-effect size is the slope of
the resulting fit. A student's t-test may be used to test the hypothesis that theAP-effect size is zero, and
thus test for the validity of the PAP model. If the slope is zero, these same tables show that the intercept
maybe interpreted as anAC strength parameter for DAT The follow-on paper will describe these tech-
niques in detail.
Prospective Tests
A prospective test of DAT will not only test the AMP hypothesis against mean chance expectation, but
will also test for a PAP contribution. In such tests, n should certainly he a double-blind parameter and
take on at least two values. If you wanted to check the prediction of a linear functional relationship
between n and the E(Z2) that is suggested by PAP model, the more values of n the better. It is not pos-
sible to separate the PAP model from DAT at a single value of n.
In any prospective test, it is helpful to know the number of runs, N, that are necessary to determine with
95% confidence, which of the two models best fits the data. Figure 4 displays the problem graphically.
L. - 1.645 n
Figure 4. Model Predictions for the Power Calculation.
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Decision Augmentation Theory: Toward a Mo el o
Under PAP, 95% of the values of Z2 will be greater than the point indicated in Figure 4. Even if the
measured value of Z2is at this point, we would like the lower limit of the 95% confidence interval for this
value to be greater than the predicted value under the DAT model. Or:
ZAP 1.645 - 1.960 ? EAC (ZZ).
FN FN
Solving for N in the equality, we find:
z
r 3.605 a,,P (1)
N = LEAP (Z') - EAC (ZZ)
Since aAp ? 'Ac, this value of N will always be the larger estimate than that derived from beginning with
DAT and calculating the confidence intervals in the other direction.
Suppose, from an earlier experiment, one can estimate a single-trial effect size for a specific value of n,
say n1. To determine whether the PAP model or DAT is the proper description of the mechanism, we
must conduct another study at an additional value of n, say n2. We use Equation 1 to compute how many
runs we must conduct at n2 to assure a separation of mechanism with 95% confidence, and we use the
variances shown in Tables I and 2 to compute ap. Figure 5 shows the number of runs for an RNG-like
experiment as a function of effect size for three values of n2.
We chose n. = 100 hits because it is typical of the numbers found in the RNG database and the values of
n2 shown are within easy reach of today's computer-based RNG devices. For example, assuming az =
1.0 and assuming an effect size of 0.004, one we derived from a publication of PEAR data (Jahn, 1982),
then at n1 = 100,?r = 0.004 x /100 =0.04andEAc(Z2) = 1.0016. Suppose n2 = 104. ThenEEp(Z2) =
1.160 and alp = 1.625. Using Equation 1, we find N = 1368 runs, which can be approximately obtained
from Figure 5. That is in this example, 1.368 runs are needed to resolve the PAP model from DAT at n2 =
104 at the 95% confidence level. Since these runs are easily obtained in most RNG experiments, an
ideal prospective test of DAT, which is based on these calculations, would be to conduct 1500 runs ran-
domly counterbalanced between n = 102 and n = 104 hits/trial. If the effect size at n = 102 is near 0.004,
than we would resolve the AP vsAC question with 95% confidence.
AC Effect Size at nj = 100 hits
Figure 5. Runs Required for RNG Effect Sizes
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Figure 6 shows similar relationships for effect sizes that are more typical of biologicalAP as reported in
the Former Soviet Union (May and Vilenskaya, 1994).
Similarly, for biological oriented AP experiments, we chose n j = 2 because use two simultaneous AP
targets is easily accomplished. If we assume an effect size of 0.3 and aZ = 1.0, at n2 = 10 we compute
EAC(Z2) = 1.180, E4p(Z2) = 1.900, aaAp = 2.366 and N = 140, which can be approximately obtained
from Figure 6.
We have included n2 = 100 in Figure 6, because this is within reach in cellular experiments although it is
probably not practical for most biological AP experiments.
Figure 6. Runs Required for Biological AP Effect Sizes
We chose nr = 2 units for convenience. For example in a plant study, the physiological responses can
easily be averaged over two plants and n2 = 10 is within reason for a second data point. A unit could be a
test tube containing cells or bacteria; the collection of all ten test tubes would simultaneously have to be
the target of the AP effort to meet the constraints of a valid test.
The prospective tests we have described so far are conditional; that is, given an effect size, we provide a
protocol to test if the mechanism forAMP is PAP or DAT An unconditional test does not assume any
effect size; all that is necessary is to collect data at a large number of different values of n, and fit a
straight line through the resulting Z2s. The mechanism is PA P if the slope is non-zero and may be DAT if
the slope is zero.
Discussion
We now address the possible n-dependence of the model parameters. A degenerate case arises if Eqp is
proportional to Vn; if that were the case, we could not distinguish between the PAP model and DAT by
means of tests on then dependence of results. If it turns out that in the analysis of the data from a vari-
ety of experiments, participants, and laboratories, the slope of a Z2vs n linear least-squares fit is zero,
then either e, p = 0.0 or EA p is exactly proportional to Vn depending upon the precision of the fit (i.e.,
errors on the zero slope). An attempt might be made to rescue the PAP hypothesis by explaining the Vn
dependence of Z2 in the degenerate case as a fatigue or other time dependence effects. That is it might
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be hypothesized that human participants might becomeAP-tired as a function of n; however, it seems
improbable that a human-based phenomena would be so widely distributed and constant and give ex-
actly the \/n dependency in differing protocols needed to imitate DAT. We prefer to resolve the degen-
eracy by wielding Occam's razor: if the only type of AP which fits the data is indistinguishable from AC,
and given that we have ample demonstrations of A C by independent means in the laboratory, then we
do not need to invent an additional phenomenon called AP. Except for this degeneracy, a zero slope for
the fit allows us to reject all PAP models, regardless of their n-dependencies.
DAT is not limited to experiments that capture data from a dynamic system. DAT may also he the mech-
anism in protocols which utilize quasi-static target systems. In a quasi-static target system, a random
process occurs only when a run is initiated; a mechanical dice thrower is an example. Yet, in a series of
unattended runs of such a device there is always a statistical variation in the mean of the dependent
variable that may be due to a variety of factors, such as Brownian motion, temperature, humidity, and
possibly the quantum mechanical uncertainty principle (Walker, 1974). Thus, the results obtained will
ultimately depend upon when the run is initiated. It is also possible that a second-order DAT mecha-
nism arises because of protocol selection; how and who determines the order in tri-polar protocols. In
second order DAT there may be individuals, other than the formal subject, whose decisions effect the
experimental outcome and are modified by AC.
Finally, we would like to close with a clear statement of what is meant by DAT: the decisions on which
experimental outcomes depend are augmented by AC to capitalize upon the unperturbed statistical
fluctuations of the target system. In our follow-on paper, we will examine retrospective applications to a
variety of data sets.
Acknowledgements
Since 1979, there have been many individuals who have contributed to the development of DAT. We
would first like to thank David Saunders without whose remark this work would not have been. Beverly
Humphrey kept the philosophical integrity intact at times under extreme duress. We are greatly appre-
ciative of Zoltan Vassy, to whom we owe the Z-score formalism, to George Hansen, Donald McCarthy,
and Scott Hubbard for their constructive criticisms and support.
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References
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Decision Augmentation Theory: Toward a
Appendix
Mathematical Derivations for the Decision Augmentation Theory
In this appendix we develop the formalism for the Decision Augmentation Theory (DAT). We consider
cases for the mean-chance-expectation (MCE), anomalous perturbation (AP), and anomalous cogni-
tion (AC) under two assumptions-normality and Bernoulli sampling. For each of these three models,
we compute the expected values of Z and Z2, and the variance of Z2*
Mean Chance Expectation (MCE)
Normal Distribution
We begin by considering a random variable, X, whose probability density function is normal, (i.e., N(/co,
ap2)t). After many unbiased measures from this distribution, it is possible to obtain reasonable ap-
proximations to yo and ao2 in the usual way. Suppose n unbiased measures are used to compute a new
variable, Y, given by:
Yk =
Then Y is distributed as N(yo, a, 2), where ant = ao2/n. If Z is defined as
Z=Yk - fro
an ,
then Z is distributed as N(0, 1) and E(Z) is given by:
E = f ze 0.5z2dZ
Since Var(Z) = I = E(Z2) - E2(Z), then
EMCE(Z2) = 1 1z2e0572dz
TI~
The Var(Z2) = E(Z4) - E2(Z2) = E(Z4) - I. But
* We wish to thank Zoltan Vassy for originally suggesting the Z2 formalism.
t Throughout this appendix, this notation means: 2
N(? (72) = 1 e O.S~x a? l .
a2-
(1)
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ENfcr(Z") = 1 J z4e "o -2dz = 3.
2z
Var.ascr(Z2) = 2.
(3)
Bernoulli Sampling
Let the probability of observing a one under Bernoulli sampling be given hypo. After n samples, the
discrete Z-score is given by:
7=k-npo
(Y0 = I/Po(1 - PO),
and k is the number of observed ones (0 < k C n). The expected value of Z is given by:
Eu MCE(Z) _
(n)k(, _
pk(n,po) = k Popo)? k?
The first term in Equation 4 is the E(k) which, for the binomial distribution, is npo. Thus
EH MCE(Z)
(y0 Fn k=0
- nPo)Bk(n,Po), (4)
(k - nPo)13k(n,Po) = 0. (5)
The expected value of Z2 is given by:
Et1' (Z2) = Var(Z) + E2(Z),
MCF
Var(k - npo)
_ + 0,
na'o
z
EMCr(Z2) = ncr2 = 1.
0
As in the normal case, the Var(Z2) = E(Z4) - E2(Z2) = E(Z4) - 1. Buts
* Johnson, N. L., and S. Kotz, Discrete Distributions, John Wiley & Sons, New York, p. 51, (1969).
(6)
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Decision Augmentation Theory: Toward a Model o
EMCE(Za) _
(k - npo)4Bk(n,po)
= 3 + - (1 - 600).
n C12
0
Varl,~c1s(Z2) = 2 + nL (1 - 6a2) = 2 - 2 , (Po = 0.5).
0
(7)
Anomalous Perturbation (AP)
Normal Distribution
Under the perturbation assumption described in the text, we let the mean of the perturbed distribution
be given by,uo+ Fapap, where Fap is an AP strength parameter, and in the general case may he a function
of n and time. The parent distribution for the random variable, X, becomes N(1.to+ Fapap, 002). As in the
MCE case, the average of n independent values off, is Y- N(tto+ Fapco, an2). Let
Y =/10 +Fapap+AY,
dY = y - (ro + FapO ).
For a mean of n samples, the Z-score is given by
z 0 = Fapap + dY = Fap
a? a?
where i; is distributed as N(0, 1) and is given by J), / an. Then the expected value of Z is given by
Enir(Z) = EAp(Fap Jn + ~) = Fap + E(C) = Fap I (8)
and the expected value of Z2 is given by r
S)
EN (Z2) = EAp([Eap Jr` + C)2) = nfap + E(C2) + 2E, Fn E(p
= 1 + Fapn, (9)
since E( = 0 and E(1;2) = 1.
In general, Z2 is distributed as anon-central X2 with '1 degree of freedom and non-centrality parameter
nEp2, X2(1, n ,p2). Thus, the variance of Z2is given by`
Varvt,(Z2) = 2(1 + 2nFap). (10)
Bernoulli Sampling
As before, let the probability of observing a one under MCE be given hypo, and the discrete Z-score be
given by:
* Johnson, N. L., and S. Kotz, Continuous Univariate Distributions-2, John Wiley & Sons, New York, p. 134, (1970).
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Decision Augmentation Theory: oward a o e o
k - npo
aov
where k is the number of observed ones (0 < k < n). Under the perturbation assumption, we let the
mean of the distribution of the single-bit probability be given by pi = po + eapoo, where rap is an AP
strength parameter. The expected value of Z is given by:
Erir(Z) _
- npo)Bk(n,pr),
Bk(n,pl) = ()p(1 -Pi)? k.
The expected value of Z becomes
E",,(Z) = 1 ZkBk(n,Pl) - nPo
ao~n k=0
_ (Pi PO /n
_ Fap?n.
Go
Since eap = E(Z)/V, so rap is also the binomial effect size. The expected value of Z2 is given by:
EB,,(Z2) = Var(Z) + E2(Z),
Var(k - npo) z
o + eQPn,
n 0
1
_ p'( - pi + rapn.
- (2
Expanding in terms of pi = p0 + eapo0,
E;,,(Z2) = 1 + sap(n - 1) + ao (1 - 2Po)-
(12)
If Po = 0.5 (i.e., a binary case) and n > 1, then Equation 12 reduces to the E(Z2) in the normal case,
Equation 9.
We begin the calculation of Var(Z2) by using the equation for the jth moment of a binomial distribution
mt = ~Lj t,I(q + Pc`)nl I r=o.
Since Var(Z2) = E(Z4) - E2(Z2), we must evaluate E(Z4). Or,
n
EA~,(Z4) = na4 7, (k - npo)4pk(n,pi).
Ok=0
Expanding n -200 -4(k - npo)4, using the appropriate moments, and subtracting E2(Z2), yields
Var(Z2) = Co + C, n + C_, n y'.
(13)
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Decision u men a ion r
f, 2
Co =2-36Lap+10EQp+8E~?.o(1 2po)(1 -2Eap)+6i
0
3
Cl = 4EaP(1 - Ea 4) + 4 ao (i - 2p ), and
C_1 = 48 - 6[Eap - 3]2 + 12 cso (1 - 2po) + (1 027Lap) + ? (1 - 2P0)(12P0 - 12po + 1).
0 0
Under the condition that Lap 1, e