\chapter{Experiment I }\label{kapitel:ExperimentI}

\section{Introduction: Aims and Hypotheses}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Methods}

\subsection{Participants}

Altogether 26 right-handed participants (13 female, 13 male)
took part in this study. 24 of them were college students from
the University of Dortmund, while two worked at the Institute for
Occupational Physiology at the University of Dortmund. Their mean age 
was 23.6 yrs (SD 2.76 yrs, range: 19-29 yrs).
Participants received ten euro or class credits for taking part.
Data from three female participants from each group had to be
excluded from further analysis because of incorrect task handling (see chapter \ref{ExcludedEWAT1}).
13 male and seven female subjects (mean age 24.2 yrs, SD 2.74 yrs, age
range: 20-29 yrs) were able to complete the task correctly. Group
''Explicit Knowledge`` consisted of seven male and three female persons
(mean age 23.9 yrs, SD 2.69 yrs, age range: 20-29 yrs), while
group ``No Explicit Knowledge'' was formed by six male and four female
participants (mean age 24.4 yrs, SD 2.91 yrs, age range: 20-29
yrs). There was no significant difference between the two groups
concerning age $(\emph{t}_{(18)} = -.40, \emph{p} > .10)$ and gender
(Pearson $\chiČ(1) = .64, p > .10$).

\subsection{Apparatus}
\label {AufbauEWAT1}
All experimental sessions took place at the Institute for
Occupational Physiology at the University of Dortmund.
Participants were seated in front of a 17-inch flat screen (EIZO
FlexScan) with a resolution of 1280 x 1024 pixels an a refreshing
rate of 60 Hz. One cm on the screen contained 37.9 pixels. The
distance between subjects and the screen was 60 cm. Between the
screen and the participant a writing pad (SummaSketch III
Professional) with a size of 18 x 12 inches and a resolution of
500 lines per inch was positioned. Participants used a pen to draw
on the writing pad with their right hand. An attached cover
prevented participants from seeing their hand while drawing. To
prevent participants from resting their right arm on the writing
pad during drawing, the height of the table with the writing pad
on it was only ??? cm. One lamp illuminating the wall behind the
monitor and another light source on the ceiling were the only
light sources to prevent reflections on the monitor. OHM???

%\begin{figure}[htbp]
%\begin{center}
%\includegraphics[width=8cm]{kapitel/graphiken/LaborEWAT1}
%\caption{.} \label{LaborEWAT1}
%\end{center}
%\end{figure}


\subsection{Task and procedure}
\label{ProcedureEWAT1}
The experiment started with an introduction phase in which both
groups were informed about the course of the task. Group
``Explicit Knowledge`` was informed about the occurrence of a
change in visuomotor coupling by 20 percent which was indicated by
a colour change of the cursor from white to red during the trials.
Changes in visuomotor coupling were explained in words an with the
help of drawings (see figure \ref{InstruktionEWAT1}).

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=8cm]{kapitel/graphiken/InstruktionEWAT1.pdf}
\caption{Drawing used in the instruction phase to explain change
in transformation for the group ``Explicit knowledge''.}
\label{InstruktionEWAT1}
\end{center}
\end{figure}

Participants of this group were also told to adapt their
movements, that means to draw circles with a smaller radius, when
visuomotor coupling was changed.\\
Group ''No explicit Knowledge`` did not get any information about
changes in transformation. Also the colour of the cursor did not
change during the experiment for this group. The original
introductions for both groups can be found
in appendix \ref{IntroductionEWAT1} . \\
After the instruction phase participants completed a training
phase to get used to working with the writing pad. A white circle
with an eccentricity of 9° of visual angle and therefor a radius
of 4.72 cm became visible in the center of the monitor.
Participants were asked to move the cursor around this circle in
continuous circles. The cursor was visualized by a white dot with
a radius of 0.21 cm. Participants were told to draw circles of
about the size of the white circle in a comfortable way (see
figure \ref{uebungEWAT1}). If drawing was too slow (less than 1.1
seconds per circle) or too fast (more than 2.5 seconds per
circle) training phase was repeated until a proper velocity was
achieved. After completing 5 circles, the white circle disappeared
and participants continued drawing for another 10 circles. Drawing
velocity from this training phase was used to compute an
appropriate velocity for the test phase. For this purpose the mean
of durance for drawing the sixth to fifteenth circle was
calculated and saved.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=8cm]{kapitel/graphiken/uebungEWAT1.pdf}
\caption{Screenshot from training session. Participants moved the
white dot around the white circle.} \label{uebungEWAT1}
\end{center}
\end{figure}

After finishing the training, the test phase of the experiment
started. Participants started each trial by moving the cursor
(visualized by a white arrow) to a yellow rectangle. This
rectangle was positioned above the twelve o'clock position of the
circle participants had to draw. When they had reached this
starting position and had pushed the tip of the pen onto the
drawing pad, the trial started. The cursor symbol changed from
arrow to a white dot with a radius of 0.21 cm. Furthermore, a
turquoise coloured dot became visible and began to move on a
circular path with a radius of 4.72 cm with the velocity
calculated from the training session of the participant.
Participants were told to follow the turquoise dot with the white
dot by drawing circles on the writing pad. It was emphasized that
the turquoise dot was only there to help participants drawing
circles of the same size with the same velocity. This part of each
trial will be referred to as synchronisation phase. See figure
\ref {synchroEWAT1} for an example of this phase.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=14.6cm]{kapitel/graphiken/AblaufEWAT1.pdf}
\caption{Four screenshots from synchronisation phase. Participants
followed the turquoise dot with the white dot. Pictures show
scenes from the completion of one circle (from left to right
picture).} \label{synchroEWAT1}
\end{center}
\end{figure}


After five full circles, the turquoise dot disappeared when it had
reached the 12 o'clock position. Participants had to continue
drawing circles in the same size and with the same velocity
without any additional orientation on the screen. In each trial
seven full standard circles had to be accomplished after the
synchronisation phase. Drawing one standard circles means that one
circle with the radius of 4.72 cm and the velocity computed in the
training session is completed. Due to changes in velocity and size
individuals varied
in the number of completed circles in the continuation phase.\\
This part of each trial is called continuation phase. Only in this
part of the trial, transformation changes were applied. Figure
\ref {AblaufTrialEWAT1} summarizes the procedure of each trial.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=14.6cm]{kapitel/graphiken/AblaufTrialEWAT1.pdf}
\caption{Procedure of each trial. After starting the trial by
moving the cursor to the starting position, synchronisation phase
is started. After following the turquoise dot for five circles,
continuation phase starts. In this phase changes in transformation
are applied in trials with transformation changes (blue). Trials without change in
transformation are presented in red.}
\label{AblaufTrialEWAT1}
\end{center}
\end{figure}


\subsection{Application of change in transformation}

\subsubsection{Trials with change in transformation}
\label{ChangeTrialsEWAT1}
The position of the dot on the screen ($P_{S}$) representing the
position of the pen on the writing pad ($P_{H}$) was computed by
multiplying the current position on the writing pad with a gain
factor g at each point of time (t) when the cursor position on the
screen was refreshed (refreshing rate 60 Hz).
\begin{equation}
P_{S}(x,y)=g(t)\cdot P_{H}(x,y)
\end{equation}
This gain factor was exactly one at each point of time t in trials
without changes in transformation and also before onset of change
in transformation in trials with change in transformation. With
the onset of change in transformation, the gain factor was
computed using the parameters

\begin{itemize}
    \item \emph{t}, representing the current point of time,
    \item \emph{$g_0$}, marking the amplitude of the gain factor when half of the gain change is reached ($g_0=1.1$),
    \item a, representing half of the amplitude of gain change ($a=0.1$);
    \item $\tau$, standing for the time span from the beginning of gain change and the point of time when $g_0$ is reached ($\tau=0.5$ standard circles),
    \item $t_{up}$, marking the point of time, when $g_0$ is reached while gain is ascending and
    \item $t_{down}$, marking the point of time, when $g_0$ is reached while gain is descending.
\end{itemize}

To prevent the change in transformation from being consciously
detected a sigmoid function was used to compute the gain factor g
during starting point of change in transformation and the point of
time when maximum gain change was reached. Also the descent of
gain was computed with the help of this function. Table
\ref{GainComputationEWAT1} shows how the gain factor g was
computed at each point of time.

% Table generated by Excel2LaTeX from sheet 'Tabelle1'

\begin{table}[htbp]
\caption{Computation of the gain factor g depending on the point
of time t during trials with change in transformation.}
\begin{small}
\begin{center}
\begin{tabular} {|c||c|}
 \hline
  {\bf Point of time} & {\bf Computation of gain} \\
\hline \hline
$t<(t_{up}-\tau)$ & $g(t)=g_0-a$ \\
\hline
$t>(t_{down}+\tau)$ & $g(t)=g_0-a$ \\
\hline
$t>(t_{up}+\tau)$ and  &  \\
$t<(t_{down}-\tau)$ & \raisebox{1.5ex}[-1.5ex]{$g(t)=g_0+a$} \\
\hline
 $t\geq(t_{up}-\tau)$  and & \\
$t\leq(t_{up}+\tau)$ &
\raisebox{1.5ex}[-1.5ex]{$g(t)=g_0+a\cdot\sin(-(0.5\cdot\pi)+
0.5\cdot\pi\cdot(t-t_{up}+\tau)/\tau)$}\\
\hline $t\geq(t_{down}-\tau)$ and &\\
$t\leq(t_{down}+\tau)$ &
\raisebox{1.5ex}[-1.5ex]{$g(t)=g_0-a\cdot\sin(-(0.5\cdot\pi)+
0.5\cdot\pi\cdot(t-t_{up}+\tau)/\tau)$} \\
\hline

\end{tabular}
\label{GainComputationEWAT1}
\end{center}
\end{small}
\end{table}


Figure \ref{gainFormelEWAT1} shows the described parameters in the
course of gain change over time. The red curve represents gain
values at each point of time. Each trial with change in
transformation was divided in five phases depending on the status
of the gain factor g (before change in transformation, ascent of
gain, during change in transformation, descent of gain, after
change in transformation). In figure \ref{gainFormelEWAT1} these
phases are also shown.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=14.6cm]{kapitel/graphiken/gainFormelEWAT1.pdf}
\caption{Gain values for each point of time are represented by the
red curve. The parameters used to compute gain at each point of
time are displayed (see text for details). Trials were divided into five phases. }
\label{gainFormelEWAT1}
\end{center}
\end{figure}

During maximum change of transformation between positions on the writing pad and positions on the screen,
the gain factor was 1.2, meaning that in this period of time distances made on the writing pad were 20 percent
longer on the screen. During this phase of each trial with change in transformation, participants had to reduce
the size of the drawn circles by 20 percent. Because the radius of the circle in the synchronisation phase was r=4.72 cm,
participants had to reduce the size of drawn circles by 0.94 cm to a radius of r= 3.78 cm.\\
Figure \ref{onsetEWAT1} shows the variation in onset of change in transformation during continuation phase and the
number of trials before, during and after change in transformation. Onset varied in 0.1 steps between 1 to 1.9
completed circles in the continuation phase. It took one standard circle to ascend the gain factor from 1 to 1.2.
The maximal gain factor 1.2 was applied for two standard circles. It took one more standard circle to descend the gain factor back to 1.
 After change in transformation, another 2.1 to 3 standard circles had to be accomplished (depending on the onset of change),
 before the trial was finished.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=14.6cm]{kapitel/graphiken/OnsetEWAT1.pdf}
\caption{Variation in onset of change in transformation and duration of each phase of the trial (in standard circles).}
\label{onsetEWAT1}
\end{center}
\end{figure}

\subsubsection{Trials without change in transformation}

To be able to compare trials with (change trials) and without
changes in transformation (no change trials), all trials without
change in transformation were matched to trials with change in
transformation according to the onset of change in transformation.
Although it was only necessary for change trials all trials were
marked by a point of onset of change. So there existed a pair of
change and no change trials with the same point of onset in each
block. To compute dependent variables over the course of time for
no-change trials the values of gain change at each point of time
for the paired change trial were used to divide no change trials
into the five phases.

\subsection{Design}

Altogether, three blocks with 20 trials each had to be
accomplished. In ten of these trials a change in visuomotor
coupling occurred, while there was no change in the other ten
trials. The occurrence of change trials and no change trials was
permuted for every participant in each block. Onset of change was
varied in 0.1 steps between 1 and 1.9 standard circles after the
beginning of the continuation phase, so that ten different onsets
of changes resulted (see figure \ref{onsetEWAT1}). The onset of
change was also permuted for each block and every participant.\\
Each day an experiment took place, participants of only one group were 
tested. Group was changed every day using the ABBA-Design. 


\subsection{Analysis of data}

\subsubsection{Determination of the dependent variable ``Radius''}
\label{RadiusBeschreibenEWAT1}
As the dependent variable, the radius of the circles drawn onto
the writing pad was analyzed. For a continous computation of the radius during each trial, it was determined for each three subsequent points of the movement path. With three subsequent points $P_{1}(x_{1}, y_{1}), P_{2}(x_{2}, y_{2})$ and $P_{3}(x_{3}, y_{3})$ a circle can be described with a fixed center point $P_0(x_0, y_0)$ and a radius r.
With the help of equation 
\begin{equation}
r^{2}= (x-x_0)^{2} + (y-y_0)^{2}
\end{equation}
radius could be computed for each point, if the position of the center point is known. With the help of three subsequent points $P_{1}, P_{2}$ and $P_{3}$ coordinates of the center point and therefor also the radius were computable.\\
For data accumulation, each drawn circle was divided into four similarly sized segments of circular arcs (see figure \ref{Kreissegmente}). Mean values were computed for all radiuses belonging to one segment. That way, every completed circle could be described by four mean values of radius.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=8cm]{kapitel/graphiken/kreissegmente.pdf}
\caption{Division of each drawn circle into four segments. For each segment mean values of radiuses $r_{n}$ are calculated.}
\label{Kreissegmente}
\end{center}
\end{figure}

To gain characteristic values for each phase of the trial, mean values of all segment radiuses belonging to each phase were determined. Due to high possibility of error, first and last segment of each trial were excluded from the analysis. The first phase of each trial, ``Before change in transformation'', consisted of at least one and at most 1.9 standard circles. Because the first segment was always excluded, at least three and at most seven segments contributed to the computation of the mean value for this phase. \\
Phase ``During change in transformation'' always consisted of one circle. Thus four segments were used to compute mean values.\\
2.1 to three circles were performed during phase ``After change in transformation''. Depending on the onset of change and considering, that the last segment of each trial was always excluded from analysis, from eight to eleven segments could be used to calculate mean value for the last phase of each trial.\\
Phases ``Ascent of gain'' and ``Descent of gain'' consisted of four segments each. For these both phases mean values could not be computed, because segments were not comparable due to the increasing or decreasing gain factor. This was the reason for only analyzing phases ``Before change in transformation'', ``During change in transformation'' and ``After change in transformation'' regarding the dependent variable radius. 


\subsubsection{Determination of invalid segments, trials and participants}
\label{InvalidEWAT1}
A number of different criteria were used to categorize valid and invalid data. In the following section these criteria are described on the levels of segments, trials and participants.\\
As mentioned above, every first and last segment of each trial was excluded from analysis. Furthermore, very short segments (less than 15 values), segments with a mean radius of more than 30 cm or a tangetial velocity of less than five seconds were classified as invalid segments and not included in further analyses.\\
A trial was categorized as invalid, when it was not possible to calculate mean values for all three phases. This was the case if there were less than three valid segments in phases ``Before change in tranformation'' or ``During change in transformation'' or if there were less than seven valid segments in phase ``After change in transformation''. If only one of the three mean values was not computable in a trial, this trial was excluded.\\
Participants data was completely excluded from analyses, if there were more invalid than valid trials in at least one type of trial of at least one experimental block.

\subsubsection{Analysis of radius}

For each participant, a mean value of radius for each phase in both conditions was calculated.
To analyze this radius of drawn circles for both groups, a 2 x 3 x
2 x 3 repeated measures ANOVA was computed with the dependent
variable ``Radius'', the between-factor ``Group'' (Explicit knowledge, No explicit knowledge) and
the within-factors ``Block'' (1-3), ``Type of trial`` (Change trial, No change trial) and ``Phase'' (Before, during and after change in transformation). See figure \ref{DesignEWAT1} for the ANOVA design.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=14.6cm]{kapitel/graphiken/DesignEWAT1.pdf}
\caption{ANOVA design for the analysis of radius.}
\label{DesignEWAT1}
\end{center}
\end{figure}


\subsubsection{Interview}

Participants in both groups received an interrogation after having
finished all three experimental blocks. Group ``Explicit
knowledge'' was primarily asked about formal aspects of the
instruction and whether they were able to use the given
information during the test phase.\\
Participants in group ``No explicit knowledge'' were asked eight
questions concerning the possible detection of changes in
transformation. While the first question was unspecific (``Did
anything unusual occur during the experiment?''), participants
attention was drawn to the different aspects of awareness of
changes in transformation in the following questions. The last
question explained the change in transformation and its
consequences for the participants behaviour and asked whether this
principle was consciously detected. See appendix for a
presentation of both questionnaires.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}

\subsection{Data Analysis and excluded data}
\label{ExcludedEWAT1}

Data was filtered using a Butterworth filter ($4^{th}$ degree) and a Nyquist frequency of 2.5 Hz. \\
Data from six female participants had to be excluded from analyses due to a large number of invalid trials (more invalid than valid trials). Three of them had been assigned to the ``Explicit knowledge'' group, while the other three belonged to group ``No explicit knowledge''. Number of invalid trials of the remaining 20 participants was analysed using a repeated measures ANOVA with the factors Group (``Explicit knowledge'', ``No explicit knowledge''), Type of trial (Change trials, No change trials) and Block (1-3). Results showed no significant differences between blocks, types of trials or group regarding number of invalid trials. 

%Table \ref{EntfernteTrialsEWAT1} summarizes mean values and standard deviation of number of invalid trials for both types of trial in each block for each group.
%
%\begin{table}[htbp]
%\caption{Mean values and standard deviations (in brackets) of number of invalid trials for both types of trial in each block for each group.}
%\begin{small}
%\begin{center}
%\begin{tabular} {|c||c||c||d{2.2} d{1.2}|}
% \hline
%  &  &  & \multicolumn{2}{c|}{\bf Number of }\\
% \raisebox{1.5ex}[-1.5ex]{\bf{Type of trial}}&\raisebox{1.5ex}[-1.5ex]{\bf{Block}}&\raisebox{1.5ex}[-1.5ex]{\bf{Group}}&\multicolumn{2}{c|}{\bf{invalid trials}}\\
% \hline\hline
% & &Explicit knowledge&1.50 &( .97)\\
% &\raisebox{1.5ex}[-1.5ex]{1} &No explicit knowledge&1.10 &( .74)\\
%  \cline{2-5}&&Explicit knowledge&1.10 &( .99)\\
%  \raisebox{1.5ex}[-1.5ex]{Change trials}&\raisebox{1.5ex}[-1.5ex]{2}&No explicit knowledge& 1.20 &( .92)\\
%   \cline{2-5}&&Explicit knowledge& 1.20 &(1.22)\\
%  &\raisebox{1.5ex}[-1.5ex]{3}&No explicit knowledge& 1.10& (1.10)\\
%  \hline \hline
%  &&Explicit knowledge& 1.70 &(1.06)\\
%  &\raisebox{1.5ex}[-1.5ex]{1} &No explicit knowledge& 1.20& (1.14)\\
%   \cline{2-5}&&Explicit knowledge& 1.50 &( .85)\\
%   \raisebox{1.5ex}[-1.5ex]{No change trials}&\raisebox{1.5ex}[-1.5ex]{2} &No explicit knowledge& 1.00& (1.05)\\
%   \cline{2-5}&&Explicit knowledge& 1.40& ( .70)\\
%  &\raisebox{1.5ex}[-1.5ex]{3} &No explicit knowledge& 1.40 &(1.08)\\
%  \hline
%\end{tabular}
%\label{EntfernteTrialsEWAT1}
%\end{center}
%\end{small}
%\end{table}

\subsection{Movement time}

Participants needed between 1.40 and 2.28 seconds to draw one circle in the training section (mean time 1.83 sec, SD .26).  Mean duration of continuation phase of each trial was 12.83 seconds (SD 1.79 secs, range: $9.79-15.95$ sec). There were no significant differences in time to complete a circle in the trainings section ($t_{(18)}= .57$, $p>.10$) as well as the duration of continuation phase for participants in group ``Explicit knowledge'' and ``No explicit knowledge'' ($t_{(18)}= .57$, $p>.10$). See table \ref{ZeitenEWAT1} for mean time and standard deviation in each group.

\begin{table}[htbp]
\caption{Mean values and standard deviations (in brackets) for time to draw one circle during the training session and duration of continuation phase in seconds for both groups.}
\begin{small}
\begin{center}
\begin{tabular} {|c||d{2.2} d{2.2}||d{2.2} d{1.2}|}
 \hline
 & \multicolumn{2}{c||}{\bf Explicit } & \multicolumn{2}{c|}{\bf No explicit }\\
 \raisebox{1.5ex}[-1.5ex]{\bf{Time component}}& \multicolumn{2}{c||}{\bf knowledge} &\multicolumn{2}{c|}{\bf knowledge}\\
\hline \hline
Time to draw one circle during training (sec)&  1.87& ( .27)&  1.80& ( .25)\\
\hline\hline
Duration of continuation phase (sec)& 13.06& (1.87)& 12.60& (1.77)\\
\hline

\end{tabular}
\label{ZeitenEWAT1}
\end{center}
\end{small}
\end{table}
 

\subsection{Awareness of change in transformation in group ``No explicit knowlege''}

To check for a possible detection of changes in transformation with participants in group ``No explicit knowledge'', these participants were interrogated after having finished the last test block. None of the participants declared that they had experienced anything unusual during the test phase in first place. Asked in more detail about the course of trials, five of them answered the last question with ``yes'', meaning that they had realized that there was a change in transformation and that they had to adapt their movements in some trials. Four of these five participants also claimed that they had not drawn circles of the same size all the time. This result shows that five participants became at least partially aware of the change in transformation, although none was able to report about it without external cues. Two separat analyses of radius will be made, one including all participants of group ``No explicit knowledge''(n=20), the other one only with the five participant who had not become aware of the change in transformation (n=15).


\subsection{Radius}

\subsubsection{Analyses of radius including all participants of group ``No explicit knowledge''}
\label{Analyse1EWAT1}
Predictions for ANOVAs are normal distributions, homogeneity of variances, and
specifically for repeated measures designs sphericity of the dependent variable.\\


Figure \ref{Ergebnisse1EWAT1} depicts mean radiuses for both groups before, during and after change in transformation for both types of trial.
Analysis revealed a significant main effects of Block $(F_{(2,36)}=5.43, p<.01)$, Type of trial $(F_{(1,18)}=54.91, p<.001)$ and Phase $(F_{(2,36)}=56.32, p<.001)$, as well as significant interaction effects of Group x Type of trial $(F_{(1,18)}=13.85, p<.01)$, Group x Phase $(F_{(2,36)}=6.10, p<.01)$ and Type of trial x Phase $(F_{(2,36)}=71.66, p<.001)$. The interaction effect of Group x Type of trial x Phase $(F_{(2,36)}=7.65, p<.01)$ also was significant and is also visible in figure \ref{Ergebnisse1EWAT1}).


\begin{figure}[htbp]
\begin{center}
\includegraphics[width=14.6cm]{kapitel/graphiken/Ergebnisse1EWAT1.pdf}
\caption{The significant interaction effect Group x Type of trial x Phase is shown. Mean radius of drawn circles for group ``Explicit knowledge''(left) and group ``No explicit knowledge''(right) before, during and after change in transformation are displayed (n=20).}
\label{Ergebnisse1EWAT1}
\end{center}
\end{figure}

The main effect of Block is caused by a linear increase of radius of drawin circles with time. Because factor Block does not interact with any other factors, it can be assumed that found results can be generalized over all blocks.\\
Main effect of Type of trial shows that radius in all no change trials tends to decrease linearly in course of the trial, while change trials display a decrease of radius during change in transformation and an increase of radius after it.\\
The significant interaction effect of Group x Type of trial x Phase reveals that participants in group ``Explicit knowledge'' decrease radius of drawn circles during change in transformation more than participants without explicit knowledge. Contrasts computed between change trials and no change trials comparing both groups in the course of the three phases revealed significant differences $(F_{(1,18)}=8.97, p<.01)$. Figure \ref{Ergebnisse1EWAT1} makes it clear what this effect means. It can be interpreted that adaptation of radius during change in transformation in change trials in group ``Explicit knowledge'' happens in larger extend than adaptation in group ``No explicit knowledge'', although an adaptation of radius during change in transformation can also be found. Contrasts comparing radius before and after change in transformation with radius during change in transformation in change trials reveal significant effects for group ``Explicit knowledge'' ($(F_{(1,18)}=96.50, p<.001)$) as well as for group ``No explicit knowledge'' ($(F_{(1,18)}=25.69, p<.001)$). While radius before change in transformation in change trials is significantly larger than radius after change in transformation for group ``Explicit knowledge''($(F_{(1,18)}=13.82, p<.01)$), there was no such difference for group ''No explicit knowledge'' ($(F_{(1,18)}=1.68, p>.10)$). Radius in no change trials decreased linearly over the course of each trial for both groups. There was no significant contrast regarding course of no change trials in both groups ($(F_{(1,18)}=0.07, p>.10)$). See table \ref{MWErgebnisse1EWAT1} for mean values of radius in each phase, type of trial and group.\\


\begin{table}[htbp]
\caption{Mean values of radius (in cm) for both types of trial in each phase for each group (n=20).}
\begin{small}
\begin{center}
\begin{tabular} {|c||c||c||c|}
 \hline
 \bf{Type of trial}& {\bf Phase} & {\bf Group } & {\bf Radius}\\
 \hline\hline
 & &Exlicit knowledge& 4.67 \\
  & \raisebox{1.5ex}[-1.5ex]{Before change }&No explicit knowledge& 4.76 \\
   & &Exlicit knowledge& 3.88 \\
 \raisebox{1.5ex}[-1.5ex]{{Change trials}} & \raisebox{1.5ex}[-1.5ex]{During change}&No explicit knowledge&4.37 \\
   & &Exlicit knowledge& 4.45 \\
  & \raisebox{1.5ex}[-1.5ex]{After change}&No explicit knowledge& 4.68 \\
   \hline \hline
 & &Exlicit knowledge&4.71 \\
  & \raisebox{1.5ex}[-1.5ex]{Before change}&No explicit knowledge&4.75 \\
   & &Exlicit knowledge&4.64 \\
 \raisebox{1.5ex}[-1.5ex]{{No change trials}} & \raisebox{1.5ex}[-1.5ex]{During change}&No explicit knowledge& 4.70 \\
   & &Exlicit knowledge&4.61 \\
  & \raisebox{1.5ex}[-1.5ex]{After change}&No explicit knowledge& 4.67 \\
   \hline 
   
\end{tabular}
\label{MWErgebnisse1EWAT1}
\end{center}
\end{small}
\end{table}

To be able to draw conclusions about the amount of adaptation of radius during change in transformation relative to radiuses of circles before and after change in transformation, difference values of radius were computed. To account for the linear decrease , that was visible in no change trials and that seemed to occur in each trial, the value of the negative trend was subtraced from each difference value between change and no change condition in each phase. Difference values and amount of adaptation of radius for each group for the change between during and before, after and during as well as after and before change in transformation are displayed in table \ref {Differenzen1EWAT1}. Group ``Explicit knowledge'' decreased the size of radius by .72 cm during change in transformation compared to radiuses before change in transformation was applied, which represents a reduction of radius of 15 percent. Meanwhile, group ``No explicit knowledge'' diminished the radius of their drawn circles by .34 cm or seven percent during change in transformation. When looking at the difference values between phases ``During change'' and ``After change'', it becomes obvious that group ``No explicit knowledge'' was able to adjust the radius to the level it was before change in transformation was applied, considering the linear decrease. However, group ``Explicit knowledge'' was not able to fully regain the radius of circles that they had produced before the onset of change. Radius of circles after change in transformation was reduced by -.12 cm or three percent regarding radiuses before the change.

\begin{table}[htbp]
\caption{Difference values and amount of adaptation of radius (in cm) between all phases of trial for both groups in cm.}
\begin{small}
\begin{center}
\begin{tabular} {|c||c||d{5.2}||d{5.2}||d{5.2}|}
 \hline
 &&\multicolumn{1}{c||}{\bf Change}&\multicolumn{1}{c||}{\bf No}&\multicolumn{1}{c|}{\bf Adaptation}\\
 \raisebox{1.5ex}[-1.5ex]{\bf{Group}}& \raisebox{1.5ex}[-1.5ex]{{\bf Phases}} & \multicolumn{1}{c||}{\bf  trials} & \multicolumn{1}{c||}{\bf change trials} & \multicolumn{1}{c|}{\bf of radius}\\
 \hline\hline
 &During-Before&$-.79$&$-.07$&$-.72$\\
 \cline{2-5}\raisebox{1.5ex}[-1.5ex]{Explicit} &After-During&$ .57$&$-.03$&$ .60$\\
  \cline{2-5}\raisebox{1.5ex}[-1.5ex]{knowledge}&After-Before&$-.22$&$-.10$&$-.12$\\
   \hline \hline
 &During-Before&$-.39$&$-.05$&$-.34$\\
  \cline{2-5}\raisebox{1.5ex}[-1.5ex]{No} &After-During&$ .31$&$-.03$&$ .34$\\
  \cline{2-5}\raisebox{1.5ex}[-1.5ex]{explicit knowledge}&After-Before&$-.08$&$-.08$&$0$\\
 \hline
\end{tabular}
\label{Differenzen1EWAT1}
\end{center}
\end{small}
\end{table}

\subsubsection{Analyses of radius including only participants of group ``No explicit knowledge'' without awareness}


To examine whether the small amount of awareness of change in transformation gained by five participants of group ``No explicit knowledge'' had an effect on the adaptation of radius, a second repeated measures ANOVA with the same factors Group, Block, Type of trial and Phase was computed for only those five participants without awareness and all participants of group ``Explicit knowledge''. Predictions???


Analyses showed that again a significant main effect of Type of trial ($(F_{(1,13)}=32.53, p<.001)$ and of Phase $(F_{(1,13)}=30.19, p<.001)$ could be found. The main effect of Block was not significant ($(F_{(2,26)}=1.22, p>.10)$, although there was a positive linear trend. Again, significant interaction effects of Group x Type of trial $(F_{(1,13)}=6.76, p<.05)$, Group x Phase $(F_{(2,26)}=3.89, p<.05)$ and Type of trial x Phase $(F_{(2,26)}=38.93, p<.001)$. The interaction effect of Group x Type of trial x Phase $(F_{(2,36)}=7.65, p<.01)$ also was significant as in the previous analysis (see figure \ref{Ergebnisse2EWAT1}).

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=14.6cm]{kapitel/graphiken/Ergebnisse2EWAT1.pdf}
\caption{The significant interaction effect Group x Type of trial x Phase is shown. Mean radius (in cm) of drawn circles for group ``Explicit knowledge''(left) and group ``No explicit knowledge''(right) before, during and after change in transformation are displayed (n=15).}
\label{Ergebnisse2EWAT1}
\end{center}
\end{figure}

Again, a number of contrasts were computed. At first, contrasts between change trials and no change trials comparing both groups in the course of the three phases again revealed significant differences $(F_{(1,13)}=4.67, p<.05)$. 
contrasts comparing radius before and after change in transformation with radius during change in transformation in change trials revealed significant effects for group ``Explicit knowledge'' ($(F_{(1,13}=71.72, p<.001)$) as well as for group ``No explicit knowledge'' ($(F_{(1,13)}=9.45, p<.01)$). While radius before change in transformation in change trials was significantly larger than radius after change in transformation for group ``Explicit knowledge''($(F_{(1,13)}=11.52, p<.01)$), there was again no such difference for group ''No explicit knowledge'' ($(F_{(1,13)}=0.12, p>.10)$). In no change trials radius decreased linearly over the course of trial for both groups. Once more, no significant contrast regarding course of no change trials in both groups ($(F_{(1,13)}=0.07, p>.10)$) could be found. Table \ref{MWErgebnisse2EWAT1} summarizes mean values of radius in each phase, type of trial only for group ``No explicit knowledge''. For information on group ``Explicit knowledge'', consider table \ref{MWErgebnisse1EWAT1}.

\begin{table}[htbp]
\caption{Mean values of radius (in cm) for both types of trial in each phase for group ``No exlicit knowledge'' (n=20).}
\begin{small}
\begin{center}
\begin{tabular} {|c||c||c|}
 \hline
 \bf{Type of trial}& {\bf Phase} &  {\bf Radius}\\
 \hline\hline
   & Before change in transformation& 4.69\\
 Change trials & During change in transformation& 4.32\\
  & After change in transformation&4.66 \\
   \hline \hline
  & Before change in transformation& 4.71\\
 No change trials   &During change in transformation & 4.68\\
& After change in transformation& 4.64\\
   \hline 
   
\end{tabular}
\label{MWErgebnisse2EWAT1}
\end{center}
\end{small}
\end{table}

Also for these data, difference values between phases were computed and compared for both groups. Hence information about group ``Explicit knowledge'' was already given in section \ref {Analyse1EWAT1}, only data from group ``No explicit knowledge'' is described here (see table \ref{Differenzen2EWAT1}). Adaptation to change in transformation was similar to data of all ten participants forming group ``No explicit knowledge''. During change in transformation radius was diminished by .34 cm (seven percent). Between radiuses after change in transformation and before it only a marginal difference of -.04 cm (about one percent of the radius before change was applied) was detected.


\begin{table}[htbp]
\caption{Difference values and amount of adaptation of radius (in cm) between all phases of trial for group ``No explicit knowledge'' (n=5) in cm.}
\begin{small}
\begin{center}
\begin{tabular} {|c||d{5.2}||d{5.2}||d{5.2}|}
 \hline
 &\multicolumn{1}{c||}{\bf Change}&\multicolumn{1}{c||}{\bf No}&\multicolumn{1}{c|}{\bf Adaptation}\\
 \raisebox{1.5ex}[-1.5ex]{{\bf Phases}} & \multicolumn{1}{c||}{\bf  trials} & \multicolumn{1}{c||}{\bf change trials} & \multicolumn{1}{c|}{\bf of radius}\\
 \hline\hline
During-Before&$-.37$&$-.03$&$-.34$\\
\hline
After-During&$ .34$&$-.04$&$ .38$\\
\hline
After-Before&$-.03$&$-.07$&$.04$\\
   \hline
\end{tabular}
\label{Differenzen2EWAT1}
\end{center}
\end{small}
\end{table}


To directly compare both subgroups of participants of group ``No explicit knowledge'', a further repeated measures ANOVA was computed only for this group (n=10) with the between factor Awareness (Awareness, No awareness) and the between factors Block, Type of trial and Phase. Results showed no significant main effect of Awareness and no interaction effects where Awareness was involved as a factor. Figure \ref{AwarenessEWAT1} contrasts mean radiuses for participants without awareness (green) and with awareness (orange) for change trials (left) and no change trials (right). People who had become somewhat aware of changes generally tend to draw larger circles, although differences are in the range of one cm.\\
From this analyses the conclusion can be drawn, that the amount of awareness gained by some of the participants from group ``Explicit knowledge'' does not lead to a different behaviour from participants without this awareness during the task. 

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=14.6cm]{kapitel/graphiken/AwarenessEWAT1.pdf}
\caption{Comparison of participants with (orange) and without (green) awareness in group ``No explicit knowledge'' for change trials (left) and no change trials (right).}
\label{AwarenessEWAT1}
\end{center}
\end{figure}



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\section{Discussion}

\subsection{Summary of Results}

\subsection{Interpretation of Results}

\subsection{Suggestions for future studies}

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