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RESEARCH ARTICLE
Nailing Jelly:
The Replication Problem Seems to Be Unsurmountable—
Two Failed Replications of the Matrix Experiment
H W
Poznan Medical University, Department of Pediatric Gastroenterology, Poznan, Poland;
University Witten/Herdecke, Department of Psychology, Witten, Germany;
hange Health Science Institute, Berlin, Germany
hwalac@gmail.com, https://harald-walach.de +49-30-46906958
K A. K
Technical University Chemnitz, Department of Psychology, Chemnitz, Germany
P S
Technical University Chemnitz, Department of Psychology, Chemnitz, Germany
H V
Change Health Science Institute, Berlin, Germany
T H
Medical University Regensburg, Department of Psychosomatic Medicine, Regensburg, Germany
W L
Parapsychology Counselling, Freiburg, Germany
Submitted December 17, 2020; Accepted September 12, 2021; Published December 30, 2021
https://doi.org/10.31275/20212031
Creative Commons License CC-BY-NC
Abstract-—We have reported previously on positive eects found in the
matrix experiment (Walach et al., 2020). This is a setup where a random
event generator (REG) drives a display, which participants are instructed
to “inuence” at will, i.e., in a psychokinesis (PK) setup. The dierence of
this matrix experiment from standard micro-PK REG experiments was
that instead of the deviation from randomness, a large array of 2025 cor-
Journal of Scientific Exploration, Vol. 35, No. 4, pp. 788–828, 2021 0892-3310/21
Two Failed Replications of the Matrix Experiment 789
relations between the behavior of the participant and the behavior of
the REG was tested. This previous experiment was signicant, and we
devised a consensus protocol, which was deposited before commence-
ment, according to which we conducted two independent replications
with the same experimental setup and equipment. In the rst experiment
64 participants conducted the experiment in one location under the ex-
perimental guidance of KK (power = 0.88), in the second experiment 40
participants conducted the experiment in another location under the
experimental guidance of HV (power = 0.69). The analysis used a non-
parametric randomization test with 10,000 iterations. Neither of the two
experiments was signicant. While in the rst experiment a very small,
but non-signicant eect was found, in the second experiment no eect
was detectable. We discuss the ndings in the context of the larger de-
bate around replicability of parapsychological (PSI) research results and
our theoretical model. The replication problem and this failed replica-
tion is likely part of the systematic nature of such eects. This makes it
unlikely that experimental research alone will be successful in the long
run in demonstrating PSI eects. Our conclusion is that the matrix ex-
periment in and of itself is not a replicable paradigm in PSI research.
INTRODUCTION
Parapsychologists were probably among the rst social science
researchers to understand the necessity to publish all negative ndings
and to insist on replications. Replications come in various forms:
Identical replications are rather rare and refer to the replication of the
very same experiment, including all materials, methods, procedures,
and statistical analyses except that either the same research group
or other researchers run a set of dierent subjects. Mostly, however,
replications also vary some kind of procedure, thereby making
replications conceptual (Schmidt, 2016). We established previously an
experimental paradigm which we hoped would lend itself to replication
(Walach et al., 2020). We report here on two replication experiments
that failed to replicate the previous result.
Parapsychology and Replication
The idea of replicating experiments is, of course, to exclude
accidental ndings, false positives, and small statistical uctuations
being identied as systematic eects. Physicists have long been taken
790 Harald Walach et al.
as examples of doing natural science rigorously, and they demand at
least 5 sigma or standard deviations of the standard normal curve and
associated probabilities as conrmations from various experiments until
they accept an eect as veridical (Abbot & LIGO Scientic Collaboration
and Virgo Collaboration, 2016; Grote, 2018; Horton, 2015). They usually
achieve this by accumulating data through multiple observations or
through multi-lab experiments, as in the LIGO experiments that
detected gravitation waves. Thereby, the very same experimental
setup, including all analysis pipelines and data-collection procedures
are standardized and logged. The idea behind this procedure is that
experiments are detectors for stable and local signals that may be weak
but can be eventually separated from noise. Local signals travel at, or
slower than, the speed of light and appear regular, i.e., exhibit lawful
behavior. The laws might either already be known and predict certain
signals, as in the case of gravitation waves that were long predicted by
the standard model of cosmology, but the signals might be very weak
or very rare, like gravitation waves, or researchers might surmise that
unknown laws underly hitherto undetected local signals. This is what
some researchers in the parapsychology research community assume
(Carr, 2015a; May et al., 1996; May et al., 2018; Radin, 2018).
In parapsychology we are looking for eects whose nature we do
not understand, because there is no accepted theory in the rst place.
Some theories with testable consequences have been proposed, such
as “decision augmentation theory” (DAT), which supposes that all PSI
eects are precognitive eects, where anomalous cognition of future
events is used to augment decision making (May et al., 1995, 2000,
1996). Apart from the fact that this model has problems explaining
makro-PK, spuk, and poltergeist phenomena, it also has some
empirical evidence against it (Dobyns & Nelson, 1998). Observational
theories use some form of argument from a von-Neumann–
Wigner-type of interpretation of quantum physics, in which human
consciousness is central in collapsing the wave function (Houtkooper,
2002; Walker, 1975, 1979, 2011 [1974]). In this family of theories, it is
the joint observational eect of those looking at the data that actually
produce the eect. While this might be a theoretical option, it depends
on the interpretation of the measurement process and the acceptance
of a dualistic model, none of which are currently universally accepted
Two Failed Replications of the Matrix Experiment 7 91
(Bierman, 2010). Finally, a model very similar to ours is the Conscious
Induced Restoration of Time Symmetry (CIRTS) model (Bierman, 2010).
This starts from the assumption that most physical theories, with the
exception of thermodynamics and special relativity, are time symmetric,
and that the brain sustaining consciousness might be a system that
restores time symmetry through providing time-negative eects as in
precognition. The CIRTS model assumes that there is a kind of signal
or informational element in PSI-eects that are, however, bounded by
physical theory. The model which we favor (see below) is also derived
from physical theory and hence a potential scientic candidate for PSI-
modeling, but more strongly than all other models assumes that signal
coding is strictly prohibited.
Are PSI eects due to causal and local signals, i.e., obeying the
special theory of relativity? Are the signals of a known type, i.e., belonging
to the four types of exchange particles of the four basic known forces
in the universe, for instance are we looking for photons as exchange
particles of the electromagnetic force, or for other particles (Penrose,
2004), or are we looking for completely dierent, yet nevertheless
local causes and signals of a physical nature? Or are we even looking
for completely dierent types of signals that cannot be encompassed
within the standard model of physics and hence would, if discovered
and proven as stable and replicable, entail a widening of our worldview
similar to that produced by the advent of quantum mechanics? Some
physical concepts that use higher dimensional models of space and
time than relativity theory and quantum theory would suggest this
(Carr, 2015a, 2015b; Heim, 1989).
The parapsychological database is jagged so far. While we do
have many extremely intriguing phenomena on a phenomenal level
(Braude, 1986, 2017; Grosso, 2016), strong and well-documented
cases, and highly signicant meta-analyses summarizing research
elds or experimental paradigms across researchers, variations, and
time (Cardeña, 2018; May & Marwaha, 2018, 2019a, 2019b), critics are
also correct in pointing out that it is not possible to name one single
parapsychological experiment as foolproof and resistant to experimental
replication (Alcock, 2003; Reber & Alcock, 2020). It is also true that we
have a replication crisis in psychology in general, i.e., the inability to
externally replicate experiments that were thought to be proven (Open
792 Harald Walach et al.
Science Collaboration, 2015; Schooler, 2011). This replication crisis
aects all sciences, according to a survey where 90% of polled scientists
say that there is a problem with replicability (Munafò et al., 2017). It
is notorious in medicine despite the fact that medical interventions
are widely used and believed in (Horton, 2015; Ioannidis, 2005). So
why bother about the lack of replicability in parapsychology? Perhaps
parapsychology is even more replicable than standard science, but only
more controversial and hence less accepted (Radin, 2018; Schwartz et
al., 2018)?
Assumptions Behind Replicability, Synchronicity, Generalized
Quantum Theory, and Generalized Entanglement
Replicability, we observed above, makes the implicit assumption
that we are dealing a) with local, causal signals that are b) regular,
following some lawful rule and c) are therefore always available for
experimental control and manipulation. In nal consequence they
would be amenable to human engineering, once the rules and the
lawful behavior are fully discovered. We started our research from the
assumption that the lack of replicability is part of the systematic nature
of parapsychological eects. In other words, we assumed that the eects
of parapsychology might be lawful, but not of a local-causal nature. This
sounds like a contradiction, but it is not. The focus of science has so far
been mainly on local-causal signals, because once they are discovered,
they can be put to use: We have used electricity, once we discovered the
nature of the electromagnetic force. We can use the gravitational force,
e.g., by sending satellites into orbit. We even made use of the strong
atomic force that keeps atoms together, when we started to engineer
atomic ssure. We make use of the knowledge of the weak force in
isotope calculations, Geiger counters, and the like. So, what would be a
lawful, yet not causal-local event?
In the realm of psychology, Carl-Gustav Jung and Wolfgang
Pauli, the physicist, discussed exactly such phenomena, i.e., lawful
yet not causal relationships, under the umbrella term “synchronicity”
(Atmanspacher & Primas, 2006; Atmanspacher et al., 1995a; Jung, 1952;
Manseld, 1995; Peat, 1992). These are material events in the material
world that in their occurrence appear to be without cause, i.e., they
Two Failed Replications of the Matrix Experiment 7 93
happen “accidentally” or “randomly”, yet they have a correlation
with the psychological state of a person who relates to those events.
Someone calling on the phone, while a person is desperately in need
of this contact would be a typical example. This phenomenon has also
given rise to a series of experiments (Sheldrake & Smart, 2003) which,
in our view, demonstrate that this phenomenon actually exists but is
not of a causal nature (Schmidt et al., 2004a), exactly as Jung and Pauli
would have postulated.
Jung and Pauli wanted synchronicity to be seen as a type of
lawful relationship that is complementary to causal relationships. They
codied this in their famous quaternity (Jung in a letter to Pauli on
November 30, 1950, in Meier, 1992, p. 64; Meier, 2001), depicted in
Figure 1. This also implies that synchronistic relationships that are due
to psychic meaning-making or constellation of an archetype, as Jung
called it, and are in a way part of a “deeper” structure of reality than
local causes. For synchronistic, correlational relationships are part of
this primordial level of “indestructible” energy. Similar to this ontic
level of “indestructible energy”, that some physicists call the endo-
physical level of unbroken unity (Atmanspacher et al., 1995b; Primas,
1994a, 1994b), there are also relationships that pertain to this level and
might be put to use (Lucadou, 2019). They might be lawful, but they are
not causal in nature. The causality principle only operates on the level of
the space–time continuum, or on the level of exo-physics, where clear
delineations and determinations can be made, because the original
unity is broken into measured and measurable parts.
Figure 1. The quaternity Jung suggested to Pauli: Local causes are complementary to
correlational relationships or synchronicity in a similar way as the space-
time continuum is complementary to indestructible energy.
794 Harald Walach et al.
In that sense, attempts to uncover a purported causality in this
realm is futile, simply because there are no local causes operative here,
but only formal and nal causes, to speak in Aristotelian terminology.
Another way of putting this is that there are likely only correlations
of a lawful but not causal nature. Indeed, there are examples in the
physical world for such a type of relationship as well, namely quantum
entanglement correlations (Atmanspacher et al., 2002; d’Espagnat,
1997; Schrödinger, 1935; Shimony, 1989; Stillfried, 2010). These are
quite lawful, but not causal in the sense that the lawfulness of these
correlations is not mediated by any exchange particles of force or
energy. This is something that Einstein had already observed and this is
the reason why he called them “spooky actions at a distance” (Einstein
et al., 1935). They remained purely hypothetical for a long time, derivable
from the formalism, but no one knew whether they are “real”. This
dispute was settled aer John Bell derived his famous inequalities as a
boundary condition for joint probabilities that are mutually exclusive,
obeying locality conditions (Bell, 1987). This inequality gave rise to an
operationalization and an experimental test which nally claried the
issue (Aspect et al., 1982a; Aspect et al, 1982b). There are indeed non-
local correlations, i.e., lawful, yet not causally mediated regularities in
nature.
Physical quantum correlations have been empirically documented
as factual beyond any reasonable doubt (Handsteiner et al., 2017; Ma
et al., 2012; Salart et al., 2008; Stefanov et al., 2002). Meanwhile, they
are the basis for various new elds of research and application, from
quantum computing to quantum encryption. And some think they
are the basis for our neural operations as well (Hamero & Penrose,
2014). Therefore, it might be reasonable to assume that such lawful,
yet a-causal and non-local relationships could also play a role in the
wider area of human aairs or in macroscopic nature. However, this
would necessitate that either physical entanglement correlations that
are normally only detectable under highly controlled and articial
conditions can also be preserved to some degree in the macroscopic
environment; or that there is an equivalent to physical entanglement
correlations that are exactly those meaningful correlations Jung spoke
of, but not necessarily of a physical nature. Such correlations might
be, for instance, systemic, i.e., pertaining to the general setup of a
Two Failed Replications of the Matrix Experiment 7 95
system of dierent physical constituents, and not only strictly physical
in nature (Atmanspacher et al., 2002; Lucadou, 1995, 2015b). This is the
path some of us have chosen, in assuming that there is a generalized
form of entanglement that is operative in various types of systems,
provided they have a certain structure (Filk & Römer, 2011; Walach &
von Stillfried, 2011a, 2011b). We assume that parapsychological eects
are due to such correlations, lawful, yet not causal, regular, yet not local
(Lucadou, 2015b; Walach et al., 2014).
The No-Signal-Transfer (NT) Axiom and the Development of the
Matrix Experiment
A corollary of this assumption is that if such correlations are
mistaken for causal-local regularities and could be potentially used
as such they will either change channel, i.e., show up in the control
condition, or they will reverse signs, i.e., become signicantly weaker,
or are seen in dierent parameters. The reason for these observations
is given by the fact that physical entanglement correlations must not
be used as causal signals, and this can be formally proven (Lucadou et
al., 2007). We therefore assume that this no-signal-transfer axiom (NT-
axiom) also holds in the generalized case, although here it cannot be
proven to be true, but is assumed to hold.
This NT-axiom states:
If a system is governed by non-local correlations but is treated as if the
correlations were local causes, and if a signal is extracted from it, or
could be extracted in principle, then those purported signals will break
down in a second experiment, or when so used.
This means that experiments on systems constituted by such
non-local correlations that are repeated, constitute a violation of the
NT-axiom and are likely to demonstrate a breakdown of such eects,
either by a dwindling of the eect size or by demonstrating paradoxical
eects, such as having the eect show up in the control group, or
changing its sign. It also means: This is only applicable to replications of
experiments. We will return to this problem in the Discussion.
Indeed, there is a series of empirical hints that testify to the
ubiquity of this phenomenon in parapsychology. A recent example is a
796 Harald Walach et al.
commissioned identical replication of a previously reported experiment
in which mental eort of trained meditators was supposed to aect an
interference pattern in a standard double-slit optical setup (Radin et al.,
2012). The strictly preregistered and controlled study came out negative
(Walleczek & von Stillfried, 2019), although an eect can be seen in
a completely dierent channel, in the variance. The same was found
in the multisite replication of the PEAR Lab’s micro PK-experiment
(Jahn & Dunne, 1987). Although a case can be made that the PEAR
Lab’s database was largely due to the eect of some gied subjects,
the consortium replication between Princeton, Freiburg, and Giessen
was predened as a large replication study of the PEAR Lab procedure
in a protocol, in which Walter von Lucadou also predicted the negative
result.
The replication was negative (Jahn et al., 2000), but secondary
parameters that were not logged in the protocol, variance and non-
linearity parameters, were clearly signicant (Pallikari, 2001). Maier and
colleagues had the same experience in a series of PK- and priming
experiments (Dechamps & Maier, 2020; Maier et al., 2014; Maier &
Dechamps, 2018; Maier et al., 2018). This has also been observed in
other datasets from parapsychology (Bierman, 2000).
We started from the assumption that, if this lack of causal stability
is to be expected there might be a workaround by testing for some kind
of indirect parameter that would prohibit the coding of a causal signal.
Standard experiments that use a control group are, by default, cause
detectors and thereby allow the coding of a signal. For instance, they
yield a result which in a strict replication experiment could be used to
code a signal: Scores above or below the mean of the rst experiment
would be the signal to code a 1 or a 0. This, however, would violate the
NT-axiom and hence would cause the eect to vanish or change track,
if our model is correct. Therefore, we sought an experimental setup
that would be as immune as possible to such potential violation.
One such setup was developed by Walter von Lucadou in what
he called the “Matrix Experiment”, meanwhile referred to as the
“Correlation Matrix Method – CMM” (Lucadou, 1974, 1986, 1987a,
1987b, 1991, 2006, 2015a; Lucadou et al., 1987). The idea here is not to
dene a clear outcome parameter which would be prone to violating
the NT-axiom, as it would allow for signal coding, but to use an array
Two Failed Replications of the Matrix Experiment 7 97
of variables in a correlation matrix. The correlation matrix reects the
correlation of the interaction of human intentions or human behavior
with a physical system that is otherwise locally decoupled. In our case
the physical system was a micro-psychokinesis (micro-PK) experiment.
A computer displayed a fractal, a Julia-set, whose change—growth or
shrinkage—was driven by a random event generator (REG). That means
the behavior of the fractal could not have been inuenced by ordinary
means of interaction. However, human participants were instructed
to do so by their intentionality and were asked to move the sampling
process of the random event generator forward by pressing either
one of two keys on the keyboard of the laptop computer that ran the
experiment. These keystrokes represent the psychological or behavioral
variables, while the behavior of the physical system represents
the physical variables. These variables can be correlated across all
experiments and all participants and yield a correlation matrix. If the
correlation matrix contains a signature of the intentional eect or the
entanglement eect of participants with the experiment or physical
system, then we would expect more signicant correlations than by
statistical chance expectation or in a control experiment that is run
without a participant present.
Indeed, von Lucadou’s previous experiments were supportive of
this idea and produced more signicant correlations than expected
by chance and more than seen in a control matrix. Thus, we set out
to replicate this experimental setup with a larger, well-controlled
experiment. We rebuilt the hardware and soware—the random
event generator and the control soware—from scratch and enlarged
the matrix into a matrix of 45 psychological and 45 physical variables
(because there were 5 such variables per run and 9 runs made up an
experiment), yielding a matrix of 2025 cells. We created a robust non-
parametric system of statistical evaluation by simulating 10,000 such
experiments and deriving the statistical signicance from it. This rst
large replication in two labs yielded a signicant but fragile eect, as
signicance broke down in reasonably improved methods of analysis
(Walach et al., 2020).
We then convened an international consortium of experts to arrive
at a consensus protocol. This protocol followed our original one quite
closely with a few exceptions (see Methods section below) and formed
798 Harald Walach et al.
the basis for future replications. One such replication was conducted
by Karolina Kirmse as part of her master’s thesis under the supervision
of Peter Sedlmeier. Another replication was conducted by Hans Vogt
and Harald Walach. Both replications came out negative. We report on
these replications in this paper and will end with a few ideas about
potential ways forward and why we think it will be a dicult challenge
to experimentally prove anomalistic eects using experimental models
(Rabeyron, 2020).
METHODS
We used a predened protocol that was the result of a consensus
meeting of experts. The studies reported here are in fact replications
of the parent study (Walach et al., 2020). The protocol was dened
and published beforehand on the Open Science Framework platform
(https://osf.io/cx2tf/). Since it is described there in detail, we will
only summarize the most important elements here. The experiment
is a comparatively strict replication of the parent study, as the same
equipment, the same material, and the same procedures were used
with only a few exceptions that are described below. The criterion for
a successful replication was a signicant result as determined by a
statistical randomization test (see below). Since we do not assume a
stable, causal eect, a standard power analysis is not part of the protocol
but can only be provided as a post-hoc analysis.
Material and Participants
We used the same equipment as in the parent experiment. KK was
lent one of the four REGs that were used for the rst experiment and
received a copy of the soware program that operated the experiment.
This soware program was custom written in C following the rst code
which was programmed in Basic. This program operated the experiment
automatically, prompted the experimenter and the participants for
inputs, and wrote the data into a le.
The rst replication was conducted by KK in Dresden with a broad
group of participants recruited mainly in public spaces, the second by
HV in Witten with a group of students of psychology, gaining course
credits through their participation. The experiment was advertised as an
Two Failed Replications of the Matrix Experiment 7 9 9
experiment in extraordinary facilities and was conducted face to face,
one aer the other. Before the experiment started, the experimenter
switched on the computer and the equipment with a lead time of half
an hour to allow for dri and warming up.
The experimenter greeted the participant and briey explained
the experiment and handed out a consent form, as well as a very
short questionnaire. The questionnaire data were deliberately not
used, as in the previous experiment, but had the function of involving
the participant with the experiment. In Experiment 1, however, this
questionnaire was extended and evaluated for exploratory purposes
(see below). When the participants were ready, the experimenter started
the program and le them alone. The participant could take as long as
necessary. They had the instruction to “inuence the movement of the
fractal on the screen” in the indicated direction and knew that they had
to press either of two shi keys on the computer keyboard to move the
sampling process forward. Each time either one of these shi keys was
pressed, the REG was sampled and the result was used to generate a
movement of the fractal displayed on the screen. The sampling process
was ltered by a Markov chain instead of the frequently used XOR-lter.
This was done for two reasons. First, Markov-chain ltering makes a
process smoother and look more natural. Most natural processes, like
the weather, are Markov processes, i.e., they contain one or two lags of
memory. Second. the Markov process preserves some of the physical
properties of the REG. Perceptually, this resulted in the appearance
of a very smooth movement of the fractal. What the participant did
not know was that when both shi keys were pressed, the sampling
process would go on until one of them or both keys were released. The
sampling process was repeated 80 times, since each sub-run consisted
of 80 such trials, and 3 sub-runs with three dierent instructions
made up one experiment. Each run was associated with a specic
instruction to either grow or shrink the movement of the fractal or
keep it constant. These instructions were conveyed by red arrows on
the screen, and each instruction was repeated 3 times at random. Thus,
a full experiment consisted of 3*3 sub-runs with 80 trials or 720 data-
points. In contrast to the parent experiment, each participant conducted
only one experiment.
800 Harald Walach et al.
Outcome Variables
For creating the 45 x 45 correlation matrix, ve behavioral psycho-
logical and ve physical variables were generated.
The ve behavioral psychological variables were generated by the
behavior of the participants and dened as follows:
T1: Number of le key presses
T2: Number of right key presses
T3: Number of double key presses
DR: Mean time between key presses, i.e., speed
DV: Mean variance between key presses, i.e., constancy of
behavior
The ve physical variables were associated with the behavior of the
random event generator (REG) and derived from the following values:
TR: Number of times the output of the Markov-chain parsing of
the REG yielded “1” during one run, i.e., the physical behavior
of the REG ltered by the Markov chai
DT: The number of steps the fractal display deviated from the
experimental instruction in either direction or from the
central position, i.e., this is the summarized number of steps
the fractal deviated from the goal
KR:
deviation of the actual physical output of the Markov chain
from an ideal Markov chain, measured as the deviation of the
theoretical autocorrelation function from the experimental
autocorrelation function of the sub-run calculated over 10 steps
ZT: mean voltage output of the REG at channel 4 out of eight;
this channel was dened a priori as the one where the voltage
would be recorded, because it was the middle channel and
hence least likely to be aected by currency changes due
to physical switching processes; the other channels were
measured but the data not checked and analyzed
ZV: the standard deviation of this voltage output at channel 4, i.e.,
of the variable ZT.
There was continuous voltage applied to the Zener diode which
triggered a current. This randomly changing current was converted
by analogue–digital converter. Each time a key press was enacted the
converter was sampled. If the number of bits was smaller than the
previous one the outcome was 0, if it was larger the outcome was 1,
Two Failed Replications of the Matrix Experiment 801
and if it was equal a new sampling was initiated.
The 5 variables were calculated for each run per participant. As each
participant had nine runs, this yielded 45 behavioral-psychological and 45
physical variables. These variables were correlated across all participants,
which together produced the 45*45 matrix with 2025 cells. These cells
were lled by the Spearman rank correlations coecients between the
respective variables across participants. We counted the number of
correlations signicant at the predetermined level of p < .1 (one-sided,
or .05 two-sided). This is arbitrary and followed previous practice and our
protocol. We also report sensitivity analyses for correlations signicant
at a lower p-value than that. The idea behind the testing procedure is
as follows: In each correlation matrix there is a number of correlations
signicant at a certain level by chance. For instance, in a matrix of 100
cells there would be 5 correlations expected to be signicant at the level
p = .05 or 10 at the level p = .1. Similarly, in a matrix of 2025 cells we
would expect 202 to 203 correlations to be signicant at a level p = .1.
Therefore, we counted the number of correlations signicant at the
level p ≤ .1 and tested (see below), whether this number of signicant
correlations found empirically was signicantly dierent from a chance
nding, using a randomization test, or dierent from the number of
correlations found in a control experiment.
Control Experiments
Aer each participant had nished his or her experiment, the
experimenter started a control experiment and then le the room.
The control experiment consisted of the physical equipment running
empty. This resulted in the generation and recording of the physical
variables (TR, DT, KR, ZT, ZV as described above) without interference
or interaction from a participant, sampling as many data points as
during a real experiment. The generated array of physical variables was
automatically written into a database, and the psychological variables
of the previous experiment copied into the control database as
corresponding psychological variables. Thus, each real experiment was
matched by a control experiment with the same set of psychological
variables, whereby all potential causal and non-causal eects were
transferred into the control database and correlated with a new set of
independent physical variables.
802 Harald Walach et al.
Special Features of the Two Experiments
Experiment 1, conducted by KK in Dresden, had, in addition, the
following features: Instead of performing just one control experiment
at the end of each session, a second control experiment was carried
out at the beginning. In this way, the scope of the comparison was
expanded. Furthermore, the questionnaire used in previous matrix
experiments was modied by adding state variables identied as
particularly psi-promoting (see Braud, 2002) and discussed by the matrix
experiment consortium against the background of the Organizational
Closure. These variables served as foundation of data to exchange the
psychological variables (keystrokes) with the questionnaire data, using
the “Phenomenology of Consciousness Inventory” (Pekala, 1995) in an
additional analysis and to perform explorative analyses to examine which
variables, determined as favoring psi eects, inuence the number of
signicant correlations. The questionnaire was implemented in an
online format on the computer where the experiment took place. In
addition, this questionnaire was continued aer the experiment had
been performed in order to allow a comparison of the participants’
states before and aer the experiment. The experimenter was blinded
to the responses during the experiment; only aer the experiment was
completed were the answers inspected.
Experiment 2, conducted by HV in Witten used two additional
features: There was a switch implemented that allowed the system to
choose between two types of REGs. One was the custom-made REG
that was also used by KK, identical to the ones from the rst experiment.
The second was an o-the-shelf REG called TrueRNG which can be
easily purchased and implemented via a USB-stick. The idea was to
see whether our elaborate sampling process would really be better or
whether we might be able to oer a simpler system for wider usage.
A coin toss at the beginning of the experiment decided which REG
would operate the experiment. The second feature referred to the
implementation of an assessment of absorption (Glicksohn, 2001;
Glicksohn et al., 1992; Watt & Tierney, 2013). It consists of measuring
objective time the experiment takes and then asking participants to
estimate the time they took to conduct the experiment. The dierence
can serve as a measure of absorption, as more deeply absorbed
Two Failed Replications of the Matrix Experiment 803
participants tend to underestimate the time (Sedlmeier et al., 2020).
Ethical clearances were given by the respective ethical boards.
Statistical Analysis and Data Preparation
Data analysis followed the predened protocol and consisted in a
randomization test as specied. Briey, an analysis script was written
in Matlab to reshue the data 10,000 times and to recalculate the
correlation coecients each time (see Appendix). For every permutation
step, the number of signicant correlations in the matrix was counted.
The number of times out of those 10,000 permutations where an
equal or larger number of correlations was found than in the empirical
matrix, divided by 10,000, yields an estimate of the true probability that
the empirical result or a more extreme one could have been found by
chance.
The experiment might be challenged to be open to systematic
causal coding, for instance if someone used a certain strategy such
as hammering on the keyboard, or always alternating shi keys, there
might be causal correlations between physical and psychological
variables. Therefore, we dened a sensitivity analysis: We analyzed
only those correlations that are found in the time-forward or upper
part of the matrix. As the matrix unfolds 9 * 5 psychological variables
in rows and 9 * 5 variables in columns the correlation of the rst set
of physical variables with the second set of psychological variables is
a time-forward correlation of physical variables in the rst run with
psychological variables in the second run, which should preclude all
causality, as causality normally does not run backwards in time.
In the second experiment, the data for the TrueRNG in Experiment
2 were found to not conform to expected behavior (Appendix Figure 1
and Appendix Figure 2). Closer inspection revealed that this was due to
a newly acquired programming glitch when programming the switch
between the REGs that led to a buer overow for the data coming
from the TrueRNG. We normalized the data and aer normalization
they conformed well to chance expectation (Appendix Figure 3 and
Appendix Figure 4). The programming mistake was corrected for
subsequent usage.
804 Harald Walach et al.
RESULTS
Experiment 1—Dresden Experiment by KK
Sixty-four participants were recruited, 43 females (67%) and 21
males (33%). Due to the layout of the questionnaire, which used the
original one by Walter von Lucadou, age was only available in categories.
The category of 41 to 50 years was the modal one with 17 participants.
Two participants were below 20 years, six were below 30, and 14 were
between 31 and 40. Sixteen participants were between 51 and 60, seven
were between 61 and 70 and one person was older between 71 and 80.
The results of the statistical analysis of Experiment 1 can be seen
in Table 1. (Appendix Table 1 presents the data together with the results
of the control matrices.)
TABLE 1
Result of Statistical Analysis (Permutation Test with 10,000 Iterations)
of Experimental Matrix, Full 45*45 Matrix, Experiment 1.
Yellow: Signicant Results; Red: Missing Signicance
sig_th: theoretical signicance level at which the number of signicant correlations is
counted
z0: number of signicant correlations empirically found at respective level
n_sim: number of simulated matrices out of 10,000 with signicant correlations at or
above the number found empirically
p_sim: actual signicance level of observed number of correlations (n_sim/10,000)
z0_part: number of correlations in time-forward (upper) part of the matrix
n_part_sim: number of signicant correlations found in 10,000 simulations at
respective level in upper part of the matrix
p_part_sim: actual signicance level of observed number of correlations (n_part_
sim/10,000) in upper part of the matrix
The rst line of Table 1 presents the signicance level at which
the numbers of signicant correlations are counted, the second line
gives the empirically found number of signicant correlations at that
Two Failed Replications of the Matrix Experiment 805
level. The number of simulated matrices with signicant correlations
at or above the number found empirically out of 10,000 simulations
follows in the next line, and the p-level is given by this number divided
by 10,000. The red color indicates which one of those statistical tests
did not reach formal signicance, while the yellow color indicates
signicance. The lower part of Table 1 reports the same for the upper
diagonal of the correlation matrix, which contains only time-forward
correlations, i.e., the correlation of the physical variables in the rst
run with the psychological variables of the second run (abbreviated as
“part” in Table 1). This contains the causally independent parts of the
correlation matrix because they are time-forward.
Only for some of the levels of signicance were there more
signicant correlations than expected by chance (remember that p ≤
.1 was the predened level), namely for correlations at the level of p ≤
.02, p ≤ .0005, and p ≤ .0001. The number of signicant correlations
at p ≤ .05 and p ≤ .01 miss formal signicance by a small margin. The
number of signicant correlations at the predened level of p ≤ .1 is
not signicant.
While in the original experiment (Walach et al., 2020) we found
signicant correlations beyond chance even in the time-forward upper
part of the matrix, overall none could be determined in this case.
We also analyzed smaller matrices (27*45, 18*27) which correspond
to the setup of previous experiments by Walter von Lucadou and can
be considered as replications of the earlier experiments. None of them
showed any consistent and clear-cut results (Appendix Table 2 and
Appendix Table 3).
Experiment 2—Witten Experiment by HV
The experiment conducted by HV in Witten recruited 40 parti-
cipants, all of them students at the university and most of them
psychology students who received course credits. Thus, all of them
were between 18 and 30 years old. In that experiment we also measured
time and had participants estimate the time of the experiment. On
average, participants estimated the experiment as 0.4 minutes shorter
than it actually was, which is a sign of modest absorption or closure.
Twelve participants reckoned that the experiment took longer than it
806 Harald Walach et al.
actually took, and thus were likely not very involved. The result of the
statistical analysis is given in Table 2. Graphical representations of the
experimental matrices of Experiments 1 (KK) and 2 (HV), as well as one
of the control experiments (by KK) are presented in Figures 2, 3, and 4.
As can be seen, at none of the evaluated levels of signicance do
we nd more signicant correlations than expected by chance, neither
in the full matrix, nor in the partial one. The same is true for the smaller
matrices (27*45, 18*27 matrix; data not shown). Because there was no
systematic eect in the rst place, further analyses as to the ecacy of
the two dierent REGs or the importance of organizational closure,
measured as absorption were no longer useful. The control matrices
did not show a signicant eect either.
Taken together, none of the two experiments corroborates our
original ndings and the replication must be considered failed.
TABLE 2
Result of Statistical Analysis (Permutation Test with 10,000 Iterations)
of Experimental Matrix, Full 45*45 Matrix, Experiment 2.
Yellow: Signicant Results; Red: Missing Signicance
sig_th: theoretical signicance level at which the number of signicant correlations is
counted
z0: number of signicant correlations empirically found at respective level
n_sim: number of simulated matrices out of 10,000 with signicant correlations at or
above the number found empirically
p_sim: actual signicance level of observed number of correlations (n_sim/10,000)
z0_part: number of correlations in time-forward (upper) part of the matrix
n_part_sim: number of signicant correlations found in 10,000 simulations at
respective level
Two Failed Replications of the Matrix Experiment 807
Figure 2. Experimental Matrix of Experiment 1 (KK).
Figure 3. Experimental Matrix of Experiment 2 (HV).
Figure 4. Control Matrix of Experiment 1 (KK).
808 Harald Walach et al.
DISCUSSION
Our hope that we might be able to replicate our earlier positive
nding and those reported by Walter von Lucadou (Lucadou, 1986,
1987b, 1991, 2000; Lucadou et al., 1987; Walach et al., 2020) did not bear
out. The two experiments, reported here, were part of a concerted eort
to nd a replicable experimental model that would circumvent the NT
axiom. This prohibits signal coding for anomalous experiments which
are supposed to operate on the basis of generalized entanglement cor-
relations. Circumventing the NT axiom was not possible. By the same
token, our negative results also preclude an anomalous signal. For had
such an anomalous signal been there, we would have been able to see
it, as it would have driven one of our variables (TR) that measures the
deviation of the REG from randomness, and thus produced a series
of signicant correlations. This result has to be seen against poten-
tial weaknesses and against other results, partially positive and partially
negative.
A major weakness of our experiments is that they are compara-
tively small. So, one could argue that they did not have the necessary
power. While our predecessor experiment had 503 participants, these
new experiments only had 104 participants together. Using the eect
size of our predecessor experiment, approximately r = .38, our smaller
experiment had a power of 69% and the larger one a power of 88% to
detect the eect. Precisely because we assume that the eects are of
a non-classical, non-signal–like nature a classical power discussion is
beside the point, we contend. A classical power analysis assumes that
there is a stable eect that can be detected, given enough resources.
We do not think that this is the case. This is the reason why in our con-
sortium protocol power analysis is not part of the protocol, but only
denition of recruitment procedures and a preclusion of optional stop-
ping. As we argue below, power is not the decisive issue, as there are
various instances of strongly powered and well-prepared replications
that were unsuccessful.
This lack of success in replication is not a problem of personal
factors, as these experiments were conducted by two independent
groups following the same protocol and using the same equipment.
Rather, it feeds into a stream of similar results: Walleczek and
Two Failed Replications of the Matrix Experiment 809
von Stillfried (2019) were unable to replicate the Radin double-slit
experiment, a careful replication in which Radin himself was involved,
conducted the experiment according to a predened protocol, and
analyzed the data according to previous standards. Rabeyron was unable
to replicate Bem’s retro-priming results (Rabeyron, 2020). Maier and
colleagues were unable to replicate earlier results and found higher-
level regularities, i.e., an eect that moves in a kind of sinusoidal wave
from positivity to negativity and potentially back again (Dechamps &
Maier, 2020; Maier & Dechamps, 2018; Maier et al., 2018). The matrix
experiment was repeated in a dierent form by Grote, who could
not nd clear-cut eects either (Grote, 2015, 2017). A newly designed
experiment by Grote which replicated the general setup of the matrix
experiment with new equipment and 200 participants was unsuccessful.
This demonstrates that power does not seem to be the issue. However,
an analysis of correlations of the same physical data of this experiment
with dierent psychological variables, in that case questionnaire data
obtained from each participant before the experiment, was marginally
signicant (p = 0.064) (Grote, 2021). Jolij and Bierman conducted two
replications of Bem’s retropriming paradigm, but found no eect (Jolij
& Bierman, 2019). However, when they analyzed the questionnaire data
that were also taken together with the psi data in a matrix analytical
approach, they found a signicant result (p < 0.03) in one experiment,
the smaller one with 61 participants, and a borderline signicant eect
in the second study (p = 0.06) with 222 participants. This is again a clear
hint that the decisive question is not about power.
These results have to be seen together with experiments by Ana
Borges in Edinburgh who has conducted three experiments herself with
clearly positive results and one commissioned by another experimenter
with negative results (Ana Borges, personal communication and
unpublished Ph.D. thesis, The University of Edinburgh, Department of
Psychology).
The results of the experiments of Ana Borges can only be really
discussed once they are fully published. Meanwhile one might suppose
that in those experiments we are dealing with an experimenter eect, as
the study conducted by a second experimenter who was indierent to
the results was clearly negative, while the studies conducted by Borges
herself, who is enthusiastic about this work, were positive. We had
810 Harald Walach et al.
such a setup implicitly in our experiments: KK tended toward hoping
to nd positive results, while HV was pretty indierent toward the
results of the experiment. Was the negative eect of HVs experiment
a negative experimenter eect? As the CIRTS theory would suggest, all
experiments might be in principle tests of experimenter PK (or the lack
of it) (Bierman, 2008).
A strength and weakness of our experiment at the same time was
the statistical analysis. The Monte-Carlo simulation of potential dierent
matrices produces an empirical distribution against which statistical
inferences can be made without any parametric assumptions and is thus
a straightforward, non-parametric analysis. It is comparatively stable:
The p-values change maximally by 10-3 and the values of signicant
simulated matrices by around 30, if the 10,000 iterations are repeated
30 times, i.e., instead of 1,069 signicant matrices which translates into
p = 0.106 we would have 1,099 signicant matrices which translates
into p = 0.109. It also corrects for potential causal biases, as these are
destroyed in the permutations.
But such an analysis also destroys the intricate network between
potential causal and non-local correlations, making the analysis
conservative. The type of analysis chosen and dened in the protocol
actually uses only the experimental matrix. One could also use dierence
scores between the experimental and the control matrix and other
metrics for the statistical analysis. We have done that for exploratory
purposes. But this does not change the result.
An optimal analysis might be able to use some dierence metric
between the control and the experimental matrix. One might argue
that the eect is embedded within the whole experiment and not
only within the experimental matrix. Thus, some dierence measure
between the two matrices might be better able to capture the eect.
This is for a subsequent analysis of the data to decide.
In our view, the results seem to suggest a decline eect as observed
by Maier and colleagues: Our own rst experimental results were the
stimulus for further work. They were very positive. The experiments
of Ana Borges were immediate successor experiments timewise and
were also positive. KK’s experiment was next and had a small, nearly
signicant eect. HVs experiment was the last in this series and had a
zero eect. This supports a decline eect and contradicts our expectation
Two Failed Replications of the Matrix Experiment 811
that the matrix method might help to mitigate such a decline. A decline
eect is a prediction of our model (Lucadou, 2015b; Lucadou et al.,
2007; Walach et al., 2014): The NT axiom states that whenever eects
due to generalized entanglement correlations are mistaken as causal
eects and could be used for signal transmission. the eects go away
(decline), or change channel, i.e., become visible in another parameter
not tested, or change sign, i.e., become obvious in the control group.
Obviously, the NT axiom (Lucadou et al., 2007) cannot be circum-
vented as we had hoped. It may take longer before a decline comes
into eect. But eventually there is no experimental system that gener-
ates its own comparison standard through a control group that can
elude it. For no matter how complex the system or how many degrees
of freedom, eventually there will always be an option to code a signal.
In our case it would have been the number of signicant correlations.
We had similar experiences with other experimental models. A
careful pilot study of a DMILS replication, in which we tried to replicate
the originally successful DMILS studies of Schlitz and Braud (Braud
& Schlitz, 1983; Braud & Schlitz, 1991; Schlitz & Braud, 1997), yielded
a strong positive eect of r = .35, which was, however, not tested
statistically as per protocol (Schmidt et al., 2001). A large replication with
sucient participants for detecting a much smaller eect failed utterly
(Schmidt, 2002; Schmidt et al., 2002). We replicated the Grinberg-
Zylberbaum study in which he had claimed that a visual stimulation of
one subject had introduced transferred evoked potentials in the EEG of
a spatially distant, but connected participant (Grinberg-Zylberbaum et
al., 1994). In our study we could not nd transferred potentials as such,
but signicant deviations from chance expectations (Wackermann et
al., 2003). Harald Walach commissioned two large-scale replications in
the same lab, which were clearly positive, but never published (Claudio
Naranjo, personal communication; he had conducted the studies but
was prohibited from publishing the data by Wackermann, the former
head of the lab). We thought we had a replicable, if complicated
paradigm and conducted another replication which was meant to be
completely foolproof against fraud and artifacts, as it was between
subjects separated by about 800 kms. But we could not nd the eect
in its original signature. We found an eect in the alpha frequency band
which was signicant in three studies. However, the relevance of this
812 Harald Walach et al.
eect remains unclear as it only showed up aer averaging thousands
of trials. Instead, we saw an unexpected anticipatory or precognition
eect (Hinterberger et al., 2008, 2007). The reverse priming study by
Daryl Bem (Bem, 2011) did not prove to be as replicable as hoped either
(Jolij & Bierman, 2019; Rabeyron, 2014; Ritchie et al., 2012).
It seems we have enough controversial data and failed replications.
It is important to note at this point: Failed replications and positive
results in meta-analyses do not contradict each other. It is possible that
in a long series of experiments some very careful negative replications,
although they might be important, either do not (Schmidt et al., 2004b),
or only partially (Bösch et al., 2006) inuence the summary result of the
meta-analysis, because many other positive results are published or
because eects that have been negative in the hands of one research
group recover in other labs (Bierman, 2001). This is to be expected
under the NT axiom, since it only applies to strict replications. As soon
as parameters are changed, and they usually are when other groups
replicate an experiment, it is, technically speaking, a new experiment,
even though it might use the same experimental model and will be
analyzed under the same umbrella by meta-analysts. Thus, one way
out of the conundrum would be to conduct replications as conceptual
replications, changing important elements in an experimental paradigm
so as to prevent it from being a direct replication which could be used
for signal coding.
Another thought might be worth considering: If our hypothesis
is correct and generalized entanglement correlations exist and are the
basis for most, if not all PSI phenomena, then we need to consider the
fact that in real life they are normally always embedded in a series of
local-causal correlations which also support and frame them, like water
is supported in a sponge (Lucadou, 2019). In the experimental situation
we are trying to separate the two out, squeezing the sponge, as it were,
and then are surprised to nd the structure and the water gone.
Thus, the current situation is an impasse: The directly replicable
paradigm that critics demand seems to be impossible. The fact that so
many studies have been conducted by dierent groups and in slightly
varying designs allows meta-analysts to draw positive conclusions.
Hence, both skeptics and proponents of PSI are right and wrong at
the same time. The “Dodo bird verdict” which has beset psychotherapy
Two Failed Replications of the Matrix Experiment 813
research is valid here as well: All have won and all must have prizes
(Luborsky et al., 2002; Rosenzweig, 1936). It has been pointed out that
this constitutes a paradox: If PSI is real, as a lot of the data suggest,
then by the same token it cannot be proven experimentally, because
the experimental paradigm presupposes the possibility of partitioning
reality into independent segments, which is exactly what PSI negates
(Rabeyron, 2020).
What this series of replications together with other evidence
shows, is in our view that a causal, signal-theoretical interpretation of
PSI is unlikely. It rather strengthens, even though indirectly, an analysis
and theoretical model that assumes these eects to be instances
of generalized entanglement correlations, or similar processes.
If so, critics will remark: Why is it that entanglement correlations
could be empirically proven in the physical case, but not in such a
generalized case as in parapsychology? The answer to this question is
straightforward: In the physical case we have a very strong formalism
that allows the derivation of expectation values or empirical bounds
that are theoretically dened, such as Bell’s inequalities. This dened
frame is not given in the generalized case because the model is not
strong enough and does not contain enough quantitative terms that
would allow such a derivation. In the physical case, only combinations
of for example polarization angles are measured, and whether they
are correlated or not is not determined by an experimental control
group of dierent or incompatible angles, but by the violation of
Bell’s inequalities, i.e., by the theoretical distribution of two joint
probabilities. This is structurally completely dierent from determining
the control standard by a control experiment. As long as we do not have
an equally strong theoretical framework, we will not be able to provide
a straightforward proof of the facticity of generalized entanglement
correlations.
Proponents of remote viewing experiments oen lament about the
inadequacy of experimenting with people who have no special gi for
PSI, as is the rule in experiments like ours (May et al., 2018). They liken it
to trying to judge musical prowess in an average group of people, some
of whom might be musically gied while the majority won’t be, diluting
the end result. Experimenting with gied people might help avoid this
pitfall. However, it was estimated that this will be maximally one or
814 Harald Walach et al.
two in one hundred (May et al., 2018). While this argument is certainly
convincing in part, it conates two distinct points: Working with gied
people is certainly a good idea. But this does not preclude failures,
as the failed replication by Walleczek and von Stillfried (2019) showed.
The remote viewing experiment is not an experiment in the sense the
term is used here. And this might be the reason why remote viewing
experiments cannot violate the NT axiom and hence can produce quite
stable results (Targ, 2019; Targ & Katra, 2000).
In remote viewing there is no control standard that is produced by
the experiment. The control is the expectation of no special information
transferred, which is a generic null-expectation. Therefore, it can be
replicated at will. The NT axiom would only come into eect in the
counterfactual situation, which by denition never exists, if the same
person were to target the very same target twice. But the same remote
viewer will not normally do this, and once a target is described there is
no point in having this repeated. Also, in experimental setups that are
similar, targets and participants are normally changed, thus implicitly
avoiding the NT axiom. Therefore, some free-response remote viewing
or telepathy studies might be able to eschew the NT axiom, but all
studies that produce their own control standard in a control group and
are replicated as an exact replication will have the same problems as
we experienced. Unfortunately, remote viewing and Ganzfeld telepathy
studies belong to a category where a lot of expert knowledge, material,
and facilities are necessary and hence do not lend themselves to
the type of classroom experiment that is set up quickly and easily to
demonstrate telepathy.
Thus, we might have to live with the fact that a denitive
experimental paradigm is very dicult, if not impossible, to have. As
long as a paradigm incorporates enough changes, for instance by way
of conceptual replication, or changing variables, or outcome measures
each time it is conducted, it may eschew the NT axiom. But by the
same token it will also be less convincing to skeptics, who will keep
demanding a strict replication. Thus, skeptics will likely have an easy
life: They won’t be bullied into acceptance by a foolproof experimental
paradigm of PSI, because it simply may not exist. So, is experimenting,
then, unnecessary and a waste of time and resources? Probably not,
because it might teach us about higher order parameters, such as the
Two Failed Replications of the Matrix Experiment 815
recovery time it takes until an eect bounces back, or about the amount
of change necessary to make an experiment conceptually a new one
(Dechamps & Maier, 2020; Maier et al., 2018). Or it might help decide
between theoretical options (Bierman, 2010). Or it might yield a higher
class of models that not only predict when an eect might appear, but
also when it will go away. Experimenting might thus also produce the
parameters necessary to build a fuller model that contains enough
richness to derive a formally more stringent theory.
But we should probably give up the hope that the intellectual ght
about whether anomalous cognition eects or PSI is real, can be won
with the brute force of rational argument and experimental evidence
alone. This is very rarely the nal arbiter anyway, even for very mundane
questions, where social movements, intellectual fashions, generic
worldviews, political considerations are oen much more important
(Latour, 1999). Perhaps a mixed approach will be best: devising clever
experiments, avoiding the pitfalls of the NT axiom by changing
procedures in replications, not forgetting qualitative real-world studies,
observations of natural occurrence of PSI and analytical arguments
combating the prevailing naturalistic stance that is more of a dogma
than an intellectual necessity (van Fraassen, 2016; Williams & Robinson,
2016). All this together might help opening up the community for the
possibility of PSI. Producing a nal proof is likely a vain expectation, as
our results show.
Our conclusion is: The matrix experiment is likely not a replicable
experiment. The NT axiom that prohibits signal transfer in systems that
are built on correlations might be operative even in this sophisticated
experimental design. This makes likely that such eects are not of a
local-causal nature. In addition, artefacts might be operative in this
highly complex study. There might be other regularities involved
which we do not understand as yet, but we can preclude signals with
a high likelihood, else we would have seen their eect. Future studies
should determine if conceptual replications of the matrix experiment
changing important elements and parameters can avoid the NT axiom.
In addition, further research eorts could advance the experimental
setup of the matrix experiment/CMM, transferring it to other psi areas.
816 Harald Walach et al.
ACKNOWLEDGMENTS
This work was funded by Bial Grant 400/14. The study by KK was a
master’s thesis at the Technical University Chemnitz. We thank Nikolaus
von Stillfried for important conceptual and logistic help, especially with
the expert symposium that allowed the formulation of the consensus
protocol. We thank Torkel Falkenberg who helped secure co-funding
for the expert-symposium and the meeting venue. We are grateful to
all the experts who participated and who helped formulate this protocol
in various Delphi rounds.
Author Statements: HW developed the original protocol,
secured funding, organized Experiment 2, and wrote the rst dra
of the manuscript. He participated in discussing and nalizing the
manuscript. KK and HV conducted the experiments and collected
data. PS supervised Experiment 1. WvL was the senior advisor in this
project, provided his material, and helped with the understanding of
the experimental procedures and discussion of results. TH conducted
the statistical analyses. All authors participated in writing, reviewing,
and approving the nal manuscript.
Conict of Interests: None of the authors has a conict of interest.
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APPENDIX
Appendix Figure 1. Distribution of sampling of True RNG.
Appendix Figure 2. Distribution of sampling of our traditional RNG.
Appendix Figure 3. REG-output of all REGs before normalization.
Appendix Figure 4. REG-output of all REGs aer normalization.
826 Harald Walach et al.
APPENDIX TABLE 1
Number of Signicant Matrix Elements in the 45 x 45 Experimental Matrix
Compared to the Control Matrices C1 and C2 and to Chance Expectation Depending
on Signicance Level. Experiment 1 by KK, Original Analysis.
Because the data of this analysis were based on KK’s own analytic strategy which is slightly
dierent from that of TH who evaluated the data for this experiment statistically, some numbers
deviate from Table 1.
APPENDIX TABLE 2
Statistical Analysis of Experiment 1 – 27*45 Matrix;
Randomization Test with 10,000 Iterations
sig_th: theoretical signicance level at which the number of signicant correlations is counted
z0: number of signicant correlations empirically found at respective level
n_sim: number of simulated matrices out of 10,000 with signicant correlations at or above the
number found empirically
p_sim: actual signicance level of observed number of correlations (n_sim/10,000)
z0_part: number of correlations in time-forward (upper) part of the matrix
n_part_sim: number of signicant correlations found in 10,000 simulations at respective level in
upper part of the matrix
p_part_sim: actual signicance level of observed number of correlations (n_part_sim/10,000) in
upper part of the matrix
27x45
sig_th 0.1 0.05 0.02 0.01 0.005 0.002 0.0010.0005 0.0002 0.0001
full z0 163.00 96.00 45.00 21.00 8.00 7.00 3.00 2.00 0.00 0.00
full n_sim 697 439 511 1118 2482 585 990 676 1716 937
full p_sim 0.0697 0.0439 0.0511 0.1118 0.2482 0.0585 0.099 0.0676 0.1716 0.0937
part z0_part 83.00 36.00 13.00 8.00 1.00 1.00 0.00 0.00 0.00 0.00
part n_part_sim 405 1817 2831 1903 5948 2727 3172 1879 838 446
part p_part_sim 0.0405 0.1817 0.2831 0.1903 0.5948 0.2727 0.3172 0.1879 0.0838 0.0446
Two Failed Replications of the Matrix Experiment 82 7
APPENDIX TABLE 3
Statistical Analysis of Experiment 1 – 18*27 Matrix;
Randomization Test with 10,000 Iterations
sig_th: theoretical signicance level at which the number of signicant correlations is counted
z0: number of signicant correlations empirically found at respective level
n_sim: number of simulated matrices out of 10,000 with signicant correlations at or above the
number found empirically p_sim: actual signicance level of observed number of correlations
(n_sim/10,000)
0_part: number of correlations in time-forward (upper) part of the matrix
n_part_sim: number of signicant correlations found in 10,000 simulations at respective level in
upper part of the matrix
p_part_sim: actual signicance level of observed number of correlations (n_part_sim/10,000) in
upper part of the matrix
APPENDIX TABLE 4
Number of Signicant Matrix Elements in the 45 x 9 Varied Experimental Matrix
with Psychological Variables Obtained by Questionnaire Compared to the Control
Matrices C1 and C2 and to Chance Expectation Depending on Signicance Level;
Experiment 1 by KK, Original Analysis
Note. The number of correlations were calculated between 45 physical variables (TR, DT, KR, ZT,
ZV x 9 runs) and 9 psychological variables (joy, love, anger, grief, fear, arousal, inner dialogue,
direction of attention, absorption), reecting the states of consciousness of the participants
measured with the Phenomenology of Consciousness Inventory (PCI) (Pekala, 1995).
18x27
sig_th 0.1 0.05 0.02 0.01 0.005 0.002 0.001 0.0005 0.0002 0.0001
full z0 63.00 33.00 18.00 6.00 3.00 3.00 1.00 1.00 0.00 0.00
full n_sim 1667 1876 1015 2631 2381 722 1071 490 788 427
full p_sim 0.1667 0.1876 0.1015 0.2631 0.2381 0.0722 0.1071 0.049 0.0788 0.0427
part z0_part 30.00 11.00 4.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00
part n_part_sim 1606 3888 3778 4961 5211 2732 1600 870 376 215
part p_part_sim 0.1606 0.3888 0.3778 0.4961 0.5211 0.2732 0.16 0.087 0.0376 0.0215
828 Harald Walach et al.
Program Code for the Permutation Test in Matlab:
for n = 1:10000
% random permutations
EPh2 = EPh(:,randperm(size(EPh,2)));
CPh2 = CPh(:,randperm(size(CPh,2)));
EPs2 = EPs(:,randperm(size(EPs,2)));
CPs2 = CPs(:,randperm(size(CPs,2)));
% calculation of correlation matrix
[E_rho, E_p] = corr( EPh2’,EPs2’, ‘type’, ‘Spearman’, ‘rows’, ‘all’, ‘tail’,’both’);
[C_rho, C_p] = corr( CPh2’,EPs2’, ‘type’, ‘Spearman’, ‘rows’, ‘all’, ‘tail’,’both’);
nc= size(E_p,1)*size(E_p,2);
ti=0;
sig_th = [.1, .05,.02,.01, .005, .002, .001, .0005, .0002, .0001];
n_soll = sig_th.*nc;
for p_th = sig_th
ti=ti+1;
psig=(E_p<p_th);
n0_exp(ti) = sum(sum(psig));
psig=(C_p<p_th);
n0_cont(ti) = sum(sum(psig));
z0(ti)=(n0_exp(ti)-n0_cont(ti))/sqrt(2*n0_cont(ti)*(1-n0_cont(ti)/nc));
y0(ti)=(n0_exp(ti)-n0_cont(ti))/sqrt((n0_exp(ti)*(1-n0_exp(ti)/nc))+(n0_
cont(ti)*(1-n0_cont(ti)/nc)));
d0(ti)=n0_exp(ti)-n0_cont(ti);
e0(ti)=n0_exp(ti);
end
…
end
