# Dynamic Brain Networks with Prescribed Functional Connectivity

U. Casti, G. Baggio, D. Benozzo, S. Zampieri, A. Bertoldo, A. Chiuso The
authors are with the Department of Information Engineering, University of
Padova, Italy. This research was supported by the DEI Proactive grant
“Personalized whole brain models for neuroscience: inference and validation”
from the Department of Information Engineering, University of Padova.

###### Abstract

In this paper, we consider stable stochastic linear systems modeling whole-
brain resting-state dynamics. We parametrize the state matrix of the system
(effective connectivity) in terms of its steady-state covariance matrix
(functional connectivity) and a skew-symmetric matrix $S$. We examine how the
matrix $S$ influences some relevant dynamic properties of the system.
Specifically, we show that a large $S$ enhances the degree of stability and
excitability of the system, and makes the latter more responsive to high-
frequency inputs.

##  I Introduction

Various network models have been proposed in neuroscience for describing the
brain organization and activity [1, 2]. A commonly used one is the Structural
Connectivity (SC) model, which is based on brain anatomy. This model builds a
graph whose nodes represent brain regions (i.e., populations of strongly
interconnected neurons) and edges anatomical connections between them.
Although the SC model proved to be an extremely useful description of the
brain, its interpretative power is limited by its static nature.

Modern noninvasive imaging techniques, such as functional magnetic resonance
imaging (fMRI), electroencephalography (EEG), and magnetoencephalography (MEG)
generate data, in the form of time series, describing the neural activity of
brain regions [3]. It can be seen that the activity of some of them appear to
be correlated and hence we can build a graph linking regions that exhibit such
a correlation. The network obtained in this way is called Functional
Connectivity (FC), since this coupling can be interpreted as a functional
cooperation between regions. Notice that the correlation between regions
changes over time according to the task the brain is accomplishing. For this
reason, much research on this field has focused on the resting-state
Functional Connectivity (rsFC), that is obtained from measurements recorded in
the absence of stimuli or tasks [4]. The degree of correlation between regions
can be mathematically specified in different ways. One simple and natural way
is by means of the statistical correlation between the brain activity signals.
In whatever way it is obtained, the FC matrix is always symmetric and hence
unable to capture the causal relationship between brain regions.

A model that attempts to unveil these causal interactions is the so-called
Effective Connectivity (EC) model. It is a generative model in the sense that
it provides a mathematical description by which, in principle, it is possible
to replicate the observed signals. Different dynamical models can be employed
to this aim, the simplest one being a linear stochastic model known as linear
Dynamic Causal Model (linear DCM) [5]. In this case, finding the EC consists
in finding the linear system that best fits the observed data (which is the
standard goal of system identification), e.g., see [6, 7, 8]. In this setup,
the EC model is specified by the state interaction matrix of the estimated
linear system and the covariance matrix of its driving noise.

Notice that, since an EC model can in principle simulate the brain activity,
it can also predict the FC in the form of the (steady-state) covariance matrix
associated with the linear stochastic system. However, the EC model contains
more information than its associated FC. How to extract this additional
information and how to translate it in terms of macroscopic properties of the
brain is currently an open problem in neuroscience. Motivated by this problem,
in this paper we consider linear stochastic systems with fixed FC (i.e. with
fixed steady-state covariance) and analyze how the dynamic properties of the
system are affected by the remaining degrees of freedom of the model, which
are shown to be encoded in the entries of a skew-symmetric matrix.

Contribution. The starting point of our analysis is a parametrization of the
state matrix $A$ of a stable linear stochastic system in terms of a positive
definite matrix $\Sigma$, representing its steady-state covariance, and a
skew-symmetric matrix $S$. We then examine how the “size” of $S$ affects some
dynamic properties of the system. Specifically, we show through analytical and
numerical results that, as $S$ increases, (i) the stability margin of the
system, as measured by the modulus of the largest real part of the eigenvalues
of $A$, grows, (ii) the transient amplification (or excitability) of the
system, as quantified by the numerical abscissa of $A$, increases, (iii) the
spectral energy of the system shifts from low to high frequencies. Altogether,
our findings suggest that $S$ may play a crucial role in shaping brain
dynamics.

Related work. A few works in neuroscience tried to extract, interpret, and
analyze information from EC that goes beyond functional relationships between
brain regions. In particular, in [9] the authors use the skew-symmetric part
of the EC matrix to build a hierarchy between brain regions. More closely
related to our work are [10, 11], which propose the Differential Covariance
(DC) as a tool to infer directionality of interactions between brain regions.
The DC is a matrix whose entries describe the correlation between the activity
of a brain area and the variation of the activity occurring in another one. As
it will be clear later, the DC matrix is strongly related to the matrix $S$
investigated in this paper.

Notation. Given $x\in\mathbb{C}$, $\mathrm{Re}[x]$ and $\mathrm{Im}[x]$ denote
the real and imaginary part of $x$, respectively. The symbol $\imath$ stands
for the imaginary unit. Given a matrix $A\in\mathbb{C}^{n\times m}$ we denote
with $A^{\top}$ the transpose of $A$ and with $A^{*}$ the conjugate transpose
of $A$. For an Hermitian matrix $A=A^{*}$, $\lambda_{\max}(A)$ and
$\lambda_{\min}(A)$ denote the largest and smallest eigenvalue of $A$,
respectively. A positive definite (semidefinite) matrix $A$ is denoted by
$A\succ 0$ ($A\succeq 0$, respectively). We let $\|A\|$ denote the 2-norm of a
matrix, $\mathrm{tr}(A)$ the trace of $A$, $I_{n}$ the $n$-dimensional
identity matrix (the subscript $n$ will be dropped when clear from the
context), and $\mathrm{diag}(d_{1},\dots,d_{n})$ the diagonal matrix with
entries $d_{1},\dots,d_{n}$ on the diagonal. We say that a matrix
$A\in\mathbb{R}^{n\times n}$ is Hurwitz stable if all the eigenvalues of $A$
have strictly negative real part.

##  II Preliminaries

We consider the continuous-time linear time-invariant stochastic system

$\displaystyle\dot{x}(t)$ $\displaystyle=Ax(t)+w(t),$ (1)

where $x\in\mathbb{R}^{n}$ is the vector containing the states of the network
nodes, $A\in\mathbb{R}^{n\times n}$ is the state interaction matrix, and $w$
is a zero-mean white noise vector with positive definite covariance matrix
$\mathbb{E}[w(t)w(t)^{\top}]=:\Sigma_{w}\succ 0$.111Rigorosuly speaking, the
equation (1) should be intended in the differential form
$\mathrm{d}x=Ax\mathrm{d}t+\mathrm{d}w$, where $\mathrm{d}w$ is a Wiener
process, i.e., a process with stationary orthogonal increments, e.g. see [12,
Chap. 3, Sec. 4].

When modeling resting-state brain activity, (1) is referred to as linear DCM
[5], in which $x$ contains the states of brain regions, $A$ is the EC matrix
encoding causal relationships between brain regions, and
$\Sigma_{w}=\sigma^{2}I$, $\sigma\in\mathbb{R}$. The DCM model also contains
an output equation of the form $y(t)=[h*x](t)+v(t),$ where
$y\in\mathbb{R}^{n}$ is the Blood-Oxygen-Level-Dependent (BOLD) signal,
$h\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ is the hemodynamic function mapping
brain activity to the BOLD signal, the symbol $*$ stands for convolution, and
$v$ is white measurement noise.

In this paper, $A$ is assumed to be Hurwitz stable. This ensures the existence
of a positive definite steady-state or stationary state covariance matrix:

$\displaystyle\Sigma$
$\displaystyle:=\lim_{t\to\infty}\mathbb{E}\left[x(t)x(t)^{\top}\right]\succ
0,$

which can be computed as the solution of the algebraic Lyapunov equation [12,
Thm. 6.1]:

$\displaystyle A\Sigma+\Sigma A^{\top}+\Sigma_{w}=0.$ (2)

In linear DCM, $\Sigma$ can be seen as a measure of functional strength
between brain regions since it is related to the steady-state covariance of
the output which (up to a normalization) coincides with the FC. Therefore,
$\Sigma$ can be thought of as a proxy of the FC.

##  III Parametrizing systems with prescribed $\Sigma$

Consider the system (1) and define the sets

$\displaystyle\begin{aligned} \mathcal{A}&:=\\{A\in^{n\times n}\,:\,A\text{
Hurwitz stable}\\},\\\ \mathcal{P}&:=\\{P\in^{n\times n}\,:\,P=P^{\top}\succ
0\\},\\\ \mathcal{S}&:=\\{S\in^{n\times n}\,:\,S=-S^{\top}\\}.\end{aligned}$
(3)

Our first result provides a parametrization of Hurwitz stable state matrices
$A$ in terms of a pair of matrices, namely a positive definite matrix and a
skew-symmetric one.

###### Theorem 1

(Parametrization of Hurwitz stable $A$’s) Consider the system (1) and the sets
defined in (3). For any $\Sigma\in\mathcal{P}$ and $S\in\mathcal{S}$,

$\displaystyle
A=\left(-\frac{1}{2}\Sigma_{w}+S\right)\Sigma^{-1}\in\mathcal{A},$ (4)

and $\Sigma$ solves (2). Further, the map

$\displaystyle\begin{aligned}
f\colon\mathcal{P}\times\mathcal{S}&\to\mathcal{A},\\\ \ (\Sigma,S)&\mapsto
A=\left(-\frac{1}{2}\Sigma_{w}+S\right)\Sigma^{-1},\end{aligned}$ (5)

is a bijection.

###### Proof:

By construction, the matrices $\Sigma$ and $A$ in (4) satisfy the Lyapunov
equation (2). Since $\Sigma\succ 0$ and $\Sigma_{w}\succ 0$, it follows that
$A\in\mathcal{A}$ [13, Thm. 8.2]. To prove that (5) is a bijection observe
that, for any $A\in\mathcal{A}$, there exists a unique $\Sigma\in\mathcal{P}$
satisfying (2) [13, Thm. 8.2]. Further, it holds

$\displaystyle A\Sigma$ $\displaystyle=\frac{1}{2}\left(A\Sigma+\Sigma
A^{\top}\right)+\frac{1}{2}\left(A\Sigma-\Sigma A^{\top}\right)$ (6)
$\displaystyle=-\frac{1}{2}\Sigma_{w}+S,$ (7)

where $S:=\frac{1}{2}\left(A\Sigma-\Sigma A^{\top}\right)\in\mathcal{S}$.
Thus, for any $A\in\mathcal{A}$ there exists
$(\Sigma,S)\in\mathcal{P}\times\mathcal{S}$ such that (4) holds, that is, (5)
is surjective. Finally, consider $A_{1},A_{2}\in\mathcal{A}$ and let
$\Sigma_{1},\Sigma_{2}\in\mathcal{P}$ be the corresponding solutions to (2)
and $S_{1}:=\frac{1}{2}\left(A_{1}\Sigma_{1}-\Sigma_{1}A_{1}^{\top}\right)$,
$S_{2}:=\frac{1}{2}\left(A_{2}\Sigma_{2}-\Sigma_{2}A_{2}^{\top}\right)$. If
$A_{1}=A_{2}$, then $\Sigma_{1}=\Sigma_{2}$ by the uniqueness of the solution
to (2) and, from (7), $A_{1}\Sigma_{1}=A_{2}\Sigma_{2}$ implies $S_{1}=S_{2}$.
This proves injectivity of (5) and concludes the proof. ∎

As a consequence of the previous result, all state matrices of (1) generating
a fixed steady-state covariance $\Sigma\succ 0$ are parametrized by the skew-
symmetric matrix $S$ of Equation (4). Further, for a fixed $A\in\mathcal{A}$
in (1) it is possible to obtain the corresponding parameters $\Sigma$ and $S$
as follows: $\Sigma$ corresponds to the (unique) solution to (2) and $S$ to
the skew-symmetric part of $A\Sigma$, i.e., $S=\frac{1}{2}\left(A\Sigma-\Sigma
A^{\top}\right)$.

###### Remark 1

(Statistical interpretation of $S$) The skew-symmetric matrix $S$ in (5) can
be also characterized from a statistical viewpoint. In fact, it can be shown
that

$\displaystyle\lim_{t\to\infty}\mathbb{E}[\dot{x}(t)x(t)^{\top}]$
$\displaystyle=A\Sigma=-\frac{1}{2}\Sigma_{w}+S,$

The latter equation yields

$\displaystyle
S=\frac{1}{2}\Sigma_{w}+\lim_{t\to\infty}\mathbb{E}[\dot{x}(t)x(t)^{\top}].$

In [10] the matrix $\lim_{t\to\infty}\mathbb{E}[\dot{x}(t)x(t)^{\top}]$ is
termed Differential Covariance (DC). Note in particular that when $\Sigma_{w}$
is diagonal (as in the DCM model) the off-diagonal entries of $S$ coincide
with those of the DC.

###### Example 1

(Matrices $S$ in EC mice data) We consider the experimental EC matrices
employed in [14] which are estimated from BOLD recordings in mice. In Fig. 1,
we plot the norm of the $S$ matrix constructed from the parametrization in Eq.
(4) for a sample of $N=20$ subjects, and compare it with an $S$ extracted from
randomly generated stable matrices (see the caption of Fig. 1 for details).
The mice ECs exhibit a norm of $S$ substantially larger than that of the
random case. In the next sections, we elucidate the role played by a “large”
$S$ on some relevant dynamic properties of the system. $\triangle$

Figure 1:  The figure shows the norm of $S$ in the parametrization (4) for
mice ECs matrices and randomly generated Hurwitz stable matrices. The entries
of the random matrices are drawn independently from a standard normal
distribution. To enable a fair comparison with the mice ECs, these matrices
are shifted and rescaled so that they possess the same maximum real part
(stablity margin) and imaginary part of the eigenvalues of mice ECs (averaged
over the $N$ subjects). The box plot shows the results obtained for $20$
realizations of the random ensemble and $20$ mice subjects with
$\Sigma_{w}=I$.

##  IV Role of $S$ in the stability of (1)

We consider the effect of large perturbations on the skew-symmetric term $S$
in (4) on the stability of $A$. We first note that, for $S=0$, the eigenvalues
of $A$ are real since $\Sigma_{w}\Sigma^{-1}$ is similar to a symmetric
matrix. Further, the average of the eigenvalues is independent of $S$, namely

$\frac{1}{n}\mathrm{tr}\left(A\right)=-\frac{1}{2n}\mathrm{tr}\left(\Sigma_{w}\Sigma^{-1}\right),$

since $S\Sigma^{-1}$ is similar to a skew-symmetric matrix and therefore has
trace equal to zero. When the entries of $S$ increase, the eigenvalues of $A$
become complex and their real part tend to get closer to the average
$-\frac{1}{2n}\mathrm{tr}\left(\Sigma_{w}\Sigma^{-1}\right)$, which means that
the system becomes more stable, i.e., the largest real part of the eigenvalues
of $A$ becomes more negative. This is clearly seen in the 2-dimensional case,
as discussed below.

###### Example 2

(2-dimensional system) Consider the parametrization of $A\in^{2\times 2}$
given in Theorem 1 and let

$S=\alpha\begin{bmatrix}0&1\\\ -1&0\end{bmatrix},\ \ \ \alpha\geq 0.$

We fix $\Sigma_{w}=\sigma^{2}I$ and we take
$\Sigma^{-1}=\mathrm{diag}(d_{1},d_{2})$ with $d_{1}\geq d_{2}>0$. Then

$A=\begin{bmatrix}-\frac{\sigma^{2}}{2}d_{1}&\alpha d_{2}\\\ -\alpha
d_{1}&-\frac{\sigma^{2}}{2}d_{2}\end{bmatrix}.$

The eigenvalues of $A$ are the roots of its characteristic polynomial

$p(\lambda)=\lambda^{2}+\frac{\sigma^{2}}{2}(d_{1}+d_{2})\lambda+\left(\frac{\sigma^{4}}{4}+\alpha^{2}\right)d_{1}d_{2}.$

It follows that the eigenvalues are real if
$0\leq\alpha\leq\left(d_{1}-d_{2}\right)/\sqrt{d_{1}d_{2}}$, and complex
conjugate with identical real part $-\frac{\sigma^{2}}{4}(d_{1}+d_{2})$,
otherwise. The behavior of the eigenvalues as a function of $\alpha\geq 0$ is
illustrated in Fig. 2.  $\triangle$

$\mathrm{Re}(\lambda)$$\mathrm{Im}(\lambda)$$\alpha\uparrow\uparrow$$-\frac{\sigma^{2}}{2}d_{1}$$-\frac{\sigma^{2}}{2}d_{2}$$-\frac{\sigma^{2}}{4}\left(d_{1}+d_{2}\right)$
Figure 2: Behavior of the eigenvalues of $A$ for the system in Example 2 and
increasing values of $\alpha\geq 0$.

The next result provides asymptotic formulas for the eigenvalues of $A$ when
the entries of $S$ grow, in the general $n$-dimensional case.

###### Theorem 2

(Behavior of eigenvalues of $A$ for large $S$) Consider the parametrization of
$A$ given in Theorem 1 and let $S=\alpha\bar{S}$, where $\alpha\in$ and
$\bar{S}\in\mathcal{S}$, $\bar{S}\neq 0$. Let $\\{\lambda_{i}\\}_{i=1}^{n}$ be
the eigenvalues of $A$ and $\mu_{i}$, $u_{i}$ be the (imaginary) eigenvalues
and the (normalized) eigenvectors of $\Sigma^{-1/2}\bar{S}\Sigma^{-1/2}$.
Assume that the eigenvalues of $\Sigma^{-1/2}\bar{S}\Sigma^{-1/2}$ are simple.
Then, as $\alpha\to\infty$,

$\displaystyle\mathrm{Re}[\lambda_{i}]\to-\frac{1}{2}u_{i}^{*}\Sigma^{-1/2}\Sigma_{w}\Sigma^{-1/2}u_{i},$
(8) $\displaystyle\alpha^{-1}\mathrm{Im}[\lambda_{i}]\to\mathrm{Im}[\mu_{i}].$

###### Proof:

Observe that $A$ is similar to the matrix

$\displaystyle A^{\prime}$
$\displaystyle=-\frac{1}{2}\Sigma^{-1/2}\Sigma_{w}\Sigma^{-1/2}+\alpha\Sigma^{-1/2}\bar{S}\Sigma^{-1/2}$
$\displaystyle=\frac{1}{\varepsilon}\left(-\frac{\varepsilon}{2}\Sigma^{-1/2}\Sigma_{w}\Sigma^{-1/2}+\Sigma^{-1/2}\bar{S}\Sigma^{-1/2}\right),$

with $\varepsilon:=\alpha^{-1}$. Consider the matrix
$-\frac{\varepsilon}{2}\Sigma^{-1/2}\Sigma_{w}\Sigma^{-1/2}+\Sigma^{-1/2}\bar{S}\Sigma^{-1/2}$
and let $\\{\mu_{i}(\varepsilon)\\}$ and $\\{u_{i}(\varepsilon)\\}$ be its
eigenvalues and corresponding eigenvectors. The latter eigenvalues and
eigenvectors are analytic functions of $\varepsilon$ in a neighbourhood of
$\varepsilon=0$, since the eigenvalues of $\Sigma^{-1/2}\bar{S}\Sigma^{-1/2}$
are simple by assumption [15], and we can consider their Taylor expansion
around $\varepsilon=0$:

$\mu_{i}(\varepsilon)=\mu_{i,0}+\mu_{i,1}\varepsilon+\cdots,\quad
u_{i}(\varepsilon)=u_{i,0}+u_{i,1}\varepsilon+\cdots,$

where $\mu_{i,0}=\mu_{i}$ is imaginary. Then,

$\displaystyle\left(-\frac{\varepsilon}{2}\Sigma^{-1/2}\Sigma_{w}\Sigma^{-1/2}+\Sigma^{-1/2}\bar{S}\Sigma^{-1/2}\right)$
$\displaystyle(u_{i,0}+u_{i,1}\varepsilon+\cdots)$
$\displaystyle=(\mu_{i,0}+\mu_{i,1}\varepsilon+\cdots)$
$\displaystyle(u_{i,0}+u_{i,1}\varepsilon+\cdots).$

By equating the terms of the same order in the previous expression, we obtain

$\displaystyle\Sigma^{-1/2}\bar{S}\Sigma^{-1/2}u_{i,0}=\mu_{i,0}u_{i,0},$
$\displaystyle\Sigma^{-1\\!/2}\bar{S}\Sigma^{-1\\!/2}u_{i,1}\\!\\!-\\!\\!\frac{1}{2}\Sigma^{-1\\!/2}\Sigma_{w}\Sigma^{-1\\!/2}u_{i,0}\\!=\\!\mu_{i,1}u_{i,0}\\!+\\!\mu_{i,0}u_{i,1}.$

The first equation is equivalent to
$u_{i,0}^{*}\Sigma^{-1/2}\bar{S}\Sigma^{-1/2}=\mu_{i,0}u^{*}_{i,0}$, which
substituted in the second one (premultiplied by $u_{i,0}^{*}$) yields
$-\frac{1}{2}u_{i,0}^{*}\Sigma^{-1/2}\Sigma_{w}\Sigma^{-1/2}u_{i,0}=\mu_{i,1}u_{i,0}^{*}u_{i,0}.$
The latter identity in turn implies

$\displaystyle\mu_{i,1}$
$\displaystyle=-\frac{u_{i,0}^{*}\Sigma^{-1/2}\Sigma_{w}\Sigma^{-1/2}u_{i,0}}{2\|u_{i,0}\|^{2}}$
$\displaystyle=-\frac{1}{2}u_{i,0}^{*}\Sigma^{-1/2}\Sigma_{w}\Sigma^{-1/2}u_{i,0}\in.$
(9)

Consider now the matrix
$\frac{1}{\varepsilon}(-\frac{\varepsilon}{2}\Sigma^{-1/2}\Sigma_{w}\Sigma^{-1/2}+\Sigma^{-1/2}\bar{S}\Sigma^{-1/2})$.
As $\varepsilon\to 0$ the eigenvalues of this matrix satisfy
$\frac{1}{\varepsilon}\mu_{i}(\varepsilon)\to\frac{1}{\varepsilon}(\mu_{i,0}+\varepsilon\mu_{i,1})=\mu_{i,1}+\frac{1}{\varepsilon}\mu_{i,0}$.
Then,

$\alpha^{-1}\mathrm{Im}[\lambda_{i}]\to\mathrm{Im}[\mu_{i,0}]=\mathrm{Im}[\mu_{i}].$

Moreover, since $\mu_{i,0}$ is purely imaginary, as $\varepsilon\to 0$, from
(9) we have

$\displaystyle\mathrm{Re}\left[\frac{1}{\varepsilon}\mu_{i}(\varepsilon)\right]$
$\displaystyle\to\mathrm{Re}\left[\frac{1}{\varepsilon}(\mu_{i,0}+\varepsilon\mu_{i,1})\right]=\mu_{i,1}$
$\displaystyle=-\frac{1}{2}u_{i,0}^{*}\Sigma^{-1/2}\Sigma_{w}\Sigma^{-1/2}u_{i,0}.$

The thesis follows by noting that $\varepsilon\to 0$ is equivalent to
$\alpha\to\infty$ and $\\{u_{i,0}\\}$ coincide with the (normalized)
eigenvectors of $\Sigma^{-1/2}\bar{S}\Sigma^{-1/2}$. ∎

Theorem 4 states that while the real parts of the eigenvalues of $A$ tends to
a constant as the size of $S$ grows, their imaginary part diverge to infinity
with rate given by the eigenvalues of $\Sigma^{-1/2}\bar{S}\Sigma^{-1/2}$. It
is also possible to prove that the limit values of the real parts fall within
the range of the minimum and maximum eigenvalues of the matrix
$-\frac{1}{2}\Sigma^{-1/2}\Sigma_{w}\Sigma^{-1/2}$. This implies that the real
parts of the eigenvalues do not exceed the bounds given by the eigenvalues of
the matrix $A$ associated with $\alpha=0$.

Moreover, numerical simulations suggest that the real parts of the eigenvalues
of $A$ tend to concentrate around the mean as $\alpha$ increases (see also
Fig. 3 for a 4-dimensional example).

Figure 3: Behavior of the eigenvalues of $A$ for increasing values of
$\alpha\geq 0$ for a $4$-dimensional system the asymptotic branches
corresponds to the values
$-\frac{1}{2}u_{i}^{*}\Sigma^{-1/2}\Sigma_{w}\Sigma^{-1/2}u_{i}$ of Equation
(8).

##  V Role of $S$ in the excitability of (1)

While the previous analysis is strictly linked to the asymptotic behavior of
the system (1), in this section we will focus on a different metric that
characterizes the transient evolution of the system. This metric is the
numerical abscissa of $A$, defined as:

$\omega(A):=\lambda_{\max}\left(\frac{A+A^{\top}}{2}\right).$

The numerical abscissa quantifies the initial growth rate of the trajectories
generated by (1). Precisely, it holds [16]:

$\frac{\mathrm{d}\|e^{At}\|}{\mathrm{d}t}\bigg{|}_{t=0}=\omega(A).$ (10)

In view of (10), we say that a system is excitable when $\omega(A)>0$, and
non-excitable otherwise. Excitability has been shown to be beneficial for
information transmission and controllability of linear dynamical systems [17,
18].

The following example provides insights into the relationship between the
numerical abscissa of $A$ and the size of $S$.

###### Example 3

(2-dimensional system, cont’d) Consider the system of Example 2. The numerical
abscissa $\omega(A)$ can be expressed in closed form as:

$\omega(A)=-\frac{\sigma^{2}}{4}(d_{1}+d_{2})+\frac{\sqrt{\sigma^{4}+2\alpha^{2}}}{4}\left(d_{1}-d_{2}\right).$

When $d_{1}=d_{2}$ the system is non-excitable for all $\alpha$. When
$d_{1}\neq d_{2}$ it is excitable for
$\alpha>\sigma^{2}\frac{\sqrt{d_{1}d_{2}}}{d_{1}-d_{2}}$. Further, as $\alpha$
increases, the numerical abscissa grows unbounded as a linear function of
$\alpha$, namely $\omega(A)\approx\frac{d_{1}-d_{2}}{2\sqrt{2}}\alpha$.
$\triangle$

$\alpha$$\omega\left(A\right)$$-\frac{\sigma^{2}}{2}d_{2}$$\tan(\frac{d_{1}-d_{2}}{2\sqrt{2}})$
Figure 4: Behavior of $\omega(A)$ for increasing values of $\alpha\geq 0$ for
the system in Example 3. The dashed lines are the lower and upper bound in
Theorem 3.

The following theorem generalizes the results of the previous example to the
$n$-dimensional case.

###### Theorem 3

(Relation between $\omega(A)$ and size of $S$) Consider the parametrization of
$A$ given in Theorem 1 and let $S=\alpha\bar{S}$, where $\alpha\in$,
$\bar{S}\in\mathcal{S}$, $\bar{S}\neq 0$. It holds

$\lambda_{\min}(P)+\alpha\lambda_{\max}(M)\leq\omega(A)\leq\lambda_{\max}(P)+\alpha\lambda_{\max}(M),$
(11)

where

$\displaystyle
P\\!:=\\!-\frac{1}{4}\left(\Sigma_{w}\Sigma^{-1}+\Sigma^{-1}\Sigma_{w}\right),\
M\\!:=\\!\frac{1}{2}\left(\bar{S}\Sigma^{-1}-\Sigma^{-1}\bar{S}\right).$

###### Proof:

First, note that $P$ and $M$ are symmetric, and $(A+A^{\top})/2=P+\alpha M.$
The bounds in (11) now follows by Weyl’s inequality [19, Thm 4.3.1], which
states that for symmetric matrices $X$, $Y$,
$\lambda_{\max}(X+Y)\geq\lambda_{\min}(X)+\lambda_{\max}(Y)$,
$\lambda_{\max}(X+Y)\leq\lambda_{\max}(X)+\lambda_{\max}(Y)$. ∎

From the bounds in Theorem 3, it follows that $\omega(A)$ grows linearly with
the size of $S$, as quantified by the parameter $\alpha$, if
$\lambda_{\max}(M)>0$. The latter condition is satisfied if and only if $M\neq
0$. In fact, the skew-symmetry of $\bar{S}$ yields $\mathrm{tr}(M)=0$, which
implies that $M$ is either the zero matrix or has at least a strictly positive
eigenvalue. Fig. 4 shows the behavior of $\omega(A)$ together with the bounds
in Theorem 3 as a function of $\alpha$ for the 2-dimensional system of Example
3.

To conclude, we note that the lower bound in Theorem 3 yields a sufficient
condition on $S$ to guarantee excitability. Namely, if $M\neq 0$ and
$\alpha>-\lambda_{\min}(P)/\lambda_{\max}(M)$, then $\omega(A)>0$ and the
system is excitable.

##  VI $S$ and frequency-domain behavior of (1)

In this section, we examine the effect of $S$ on the frequency-domain
properties of (1). Specifically, we analyze how the size of $S$ affects the
energy distribution of the system across different frequencies, as determined
by its power spectral density [20, Chap. 10]:

$\Phi\left(\imath\omega\right)=\left(\imath\omega
I-A\right)^{-1}\Sigma_{w}\left(\imath\omega I-A\right)^{-*},\ \
\omega\in\mathbb{R}.$

We observe that $S$ has no effect on the overall energy of the system since it
holds [12, Chap. 5, Sec. 2]

$\Sigma=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\Phi\left(\imath\omega\right)\mathrm{d}\omega,$
(12)

and the steady-state covariance of the system $\Sigma$ is unaffected by $S$.
However, $S$ can in principle redistribute the energy across different
frequency bands. This is indeed the case for two-dimensional systems, where
$S$ shifts the energy from low to high frequencies, as we illustrate in the
next example.

###### Example 4

(2-dimensional system, cont’d) Consider the system of Example 2. The trace of
the power spectral density of the system is given by

$\mathrm{tr}\left[\Phi\left(\imath\omega\right)\right]=\sigma^{2}\frac{2\omega^{2}+\zeta(\alpha^{2})}{\left[\omega^{2}-d_{1}d_{2}\zeta(\alpha^{2})\right]^{2}+\omega^{2}\frac{\sigma^{4}}{4}(d_{1}+d_{2})^{2}},$
(13)

with $\zeta(\alpha^{2}):=\alpha^{2}+{\sigma^{4}}/{4}$. From (13), it follows
that for any fixed $\omega\in\mathbb{R}$,
$\mathrm{tr}\left[\Phi\left(\imath\omega\right)\right]\to 0$ as
$\alpha\to\infty$, which means that as $\alpha$ grows, the energy of the
system moves from low to high frequencies. To better understand how $\alpha$
modifies the shape of $\mathrm{tr}\left[\Phi\left(\imath\omega\right)\right]$,
we note that (13) can be viewed as the modulus of the frequency response of a
fictitious second-order scalar system featuring a zero at
$-\sqrt{\zeta(\alpha^{2})/2}$. As standard practice, we can investigate the
emergence of resonance phenomena in the system by neglecting the effect of the
zero. That is, we can approximate (13) as

$\mathrm{tr}\big{[}\tilde{\Phi}(\imath\omega)\big{]}=\frac{\sigma^{2}}{\left[\omega^{2}-d_{1}d_{2}\zeta(\alpha^{2})\right]^{2}+\omega^{2}\frac{\sigma^{4}}{4}(d_{1}+d_{2})^{2}}.$
(14)

Fig. 5 shows a comparison between (13) and (14). Using the approximation in
(14) and restricting the analysis to nonnegative frequencies, it can be shown,
after some calculations, that for
$\alpha>\alpha_{\mathrm{th}}:=\sigma^{2}\sqrt{d_{1}^{2}+d_{2}^{2}}/\left(2\sqrt{2d_{1}d_{2}}\right)$,
$\mathrm{tr}\left[\Phi(\imath\omega)\right]$ exhibits a resonance peak at
frequency
$\omega_{r}=\sqrt{8d_{1}d_{2}\alpha^{2}-\sigma^{4}\left(d_{1}^{2}+d_{2}^{2}\right)}/\left(2\sqrt{2}\right)$.
We conclude that as $\alpha$ (that is, the size of $S$) increases the system
tends to amplify input signals at one particular frequency, and this frequency
grows linearly with $\alpha$.  $\triangle$

Figure 5: Plot in a logarithmic scale of the trace of the frequency response
for the $\mathrm{tr}[\Phi(\imath\omega)]$ and its approximate version
$\mathrm{tr}[\tilde{\Phi}(\imath\omega)]$ for different values of $\alpha$, in
particular $\alpha=\alpha_{\mathrm{th}}-0.1,10,100$, and $d_{1}>d_{2}>0$ have
been generated random between $0$ and $10$. $\alpha_{\mathrm{th}}$ denotes the
threshold at which the system exhibits resonance.

The observations made in the previous example can be generalized as follows.

###### Theorem 4

(High-pass filtering effect of $S$) Consider the parametrization of $A$ given
in Theorem 1, with $S=\alpha\bar{S}$, $\alpha\in$, $\bar{S}\in\mathcal{S}$,
$\bar{S}\neq 0$. Let

$\Phi\left(\imath\omega,\alpha\right)=\left(\imath\omega
I-A\right)^{-1}\Sigma_{w}\left(\imath\omega I-A\right)^{-*}.$

The following hold:

  1. 1.

$\exists\,\bar{\omega}\\!>\\!0$ s.t.
$\mathrm{tr}\left[\Phi\left(\imath\omega,0\right)\right]\\!>\\!\mathrm{tr}\left[\Phi\left(\imath\omega,\infty\right)\right]$,
$\forall\omega$, $|\omega|\\!\leq\\!\bar{\omega}$;

  2. 2.

If $\bar{S}$ is full rank,
$\mathrm{tr}\left[\Phi\left(\imath\omega,\alpha\right)\right]\to 0$ as
$\alpha\to\infty$, $\forall\omega\\!\in\mathbb{R}$.

###### Proof:

Let us define $F(\imath\omega,\alpha):=\left(\imath\omega
I-A_{\alpha}\right)^{*}\Sigma_{w}^{-1}\left(\imath\omega I-A_{\alpha}\right)$
so that $\Phi\left(\imath\omega,\alpha\right)=F(\imath\omega,\alpha)^{-1}$.
Then, it holds

$F(\imath\omega,\alpha)=\frac{\Sigma^{-1}\Sigma_{w}\Sigma^{-1}}{4}+\widetilde{F}\left(\imath\omega,\alpha\right),$
(15)

where

$\displaystyle\widetilde{F}\left(\imath\omega,\alpha\right):=$
$\displaystyle\omega^{2}\Sigma_{w}^{-1}+\omega\alpha\imath\left(\Sigma_{w}^{-1}\bar{S}\Sigma^{-1}+\Sigma^{-1}\bar{S}\Sigma_{w}^{-1}\right)$
(16)
$\displaystyle+\alpha^{2}\Sigma^{-1}\bar{S}^{\top}\Sigma_{w}^{-1}\bar{S}\Sigma^{-1}.$

By setting $\omega=0$, (15) becomes

$\displaystyle F\left(0,\alpha\right)$
$\displaystyle=\frac{\Sigma^{-1}\Sigma_{w}\Sigma^{-1}}{4}+\alpha^{2}\Sigma^{-1}\bar{S}^{\top}\Sigma_{w}^{-1}\bar{S}\Sigma^{-1}.$
(17)

Note that $\Sigma^{-1}\bar{S}^{\top}\Sigma_{w}^{-1}\bar{S}\Sigma^{-1}\succeq
0$ since $\Sigma_{w},\Sigma\succ 0$. This implies that, for all $\alpha>0$,
$F\left(0,\alpha\right)\succeq F\left(0,0\right)\succ 0$, or, equivalently,
$F\left(0,0\right)^{-1}\succeq F\left(0,\alpha\right)^{-1}\succ 0$. In
addition, if $\alpha>0$, there is at least one eigenvalue of
$F\left(0,0\right)^{-1}$ that is not an eigenvalue of
$F\left(0,\alpha\right)^{-1}$.222Indeed, assume by contradiction that
$F\left(0,0\right)^{-1}$ and $F\left(0,\alpha\right)^{-1}$ have the same
eigenvalues. This implies that $F\left(0,0\right)$ and
$F\left(0,\alpha\right)$ have the same eigenvalues, and therefore the same
trace. However, from (17), if $\alpha>0$,
$\mathrm{tr}[F\left(0,\alpha\right)]>\mathrm{tr}[F\left(0,0\right)]$ since
$\bar{S}\neq 0$. Therefore, for all $\alpha>0$,
$\mathrm{tr}[F\left(0,0\right)^{-1}]>\mathrm{tr}[F\left(0,\alpha\right)^{-1}].$
Point 1) now follows from the latter inequality and the continuity of
$\Phi\left(\imath\omega,\alpha\right)$ with respect to $\omega$.

To prove point 2), we introduce the following notation:

$\displaystyle\tilde{a}:=\lambda_{\min}\left(\Sigma^{-1}\Sigma_{w}\Sigma^{-1}\right),\
\ \tilde{b}:=\lambda_{\min}\left(\Sigma_{w}^{-1}\right),$
$\displaystyle\tilde{c}:=\lambda_{\min}\left(\Sigma^{-1}\bar{S}^{\top}\Sigma_{w}^{-1}\bar{S}\Sigma^{-1}\right),\
\ \tilde{d}:=\lVert\Sigma^{-1}\bar{S}\Sigma_{w}^{-1}\rVert,$

where $\tilde{a},\tilde{b},\tilde{c},\tilde{d}>0$ from the the positive
definiteness of $\Sigma$, $\Sigma_{w}$, and the full rank condition of
$\bar{S}$. The following hold:

$\displaystyle\Sigma^{-1}\Sigma_{w}\Sigma^{-1}$
$\displaystyle\succeq\tilde{a}I,$ (18) $\displaystyle\Sigma_{w}^{-1}$
$\displaystyle\succeq\tilde{b}I,$ (19)
$\displaystyle\Sigma^{-1}\bar{S}^{\top}\Sigma_{w}^{-1}\bar{S}\Sigma^{-1}$
$\displaystyle\succeq\tilde{c}I.$ (20)

It remains to bound the term
$\omega\alpha\imath\left(\Sigma_{w}^{-1}\bar{S}\Sigma^{-1}+\Sigma^{-1}\bar{S}\Sigma_{w}^{-1}\right)$
in (16). First, we see that
$\Sigma_{w}^{-1}\bar{S}\Sigma^{-1}+\Sigma^{-1}\bar{S}\Sigma_{w}^{-1}$ is skew-
symmetric so that
$\imath\left(\Sigma_{w}^{-1}\bar{S}\Sigma^{-1}+\Sigma^{-1}\bar{S}\Sigma_{w}^{-1}\right)$
is an Hermitian matrix. So it holds

$\begin{split}&\imath\left(\Sigma_{w}^{-1}\bar{S}\Sigma^{-1}+\Sigma^{-1}\bar{S}\Sigma_{w}^{-1}\right)\preceq
2\tilde{d}I.\end{split}$ (21)

We also know that the eigenvalues of
$\imath\left(\Sigma_{w}^{-1}\bar{S}\Sigma^{-1}+\Sigma^{-1}\bar{S}\Sigma_{w}^{-1}\right)$
are symmetric with respect to zero. Thus, (21) yields the bounds
$-2\tilde{d}I\preceq\imath\left(\Sigma_{w}^{-1}\bar{S}\Sigma^{-1}+\Sigma^{-1}\bar{S}\Sigma_{w}^{-1}\right)\preceq
2\tilde{d}I$, which in turn implies

$\omega\imath\left(\Sigma_{w}^{-1}\bar{S}\Sigma^{-1}+\Sigma^{-1}\bar{S}\Sigma_{w}^{-1}\right)\succeq-2|\omega|\tilde{d}I.$
(22)

By applying (19), (20) and (22) to
$\widetilde{F}\left(\imath\omega,\alpha\right)$ in (15), we obtain

$\displaystyle\widetilde{F}\left(\imath\omega,S\right)$
$\displaystyle\succeq\left[\omega^{2}\tilde{b}-2|\omega|\alpha\tilde{d}+\alpha^{2}\tilde{c}\right]I$
$\displaystyle=\tilde{b}\underbrace{\left[\omega^{2}-2|\omega|\alpha{\frac{\tilde{d}}{\tilde{b}}}+\alpha^{2}{\frac{\tilde{c}}{\tilde{b}}}\right]}_{=:f\left(\imath\omega,\alpha\right)}I.$
(23)

Let $c:={\tilde{c}}/{\tilde{b}}$, $d:={\tilde{d}}/{\tilde{b}}$ and fix any
$\bar{\omega}>0$. Then, for all $|\omega|\leq\bar{\omega}$ and
$\alpha\geq\bar{\alpha}:={4d\bar{\omega}}/{c}$ we have that:

$\displaystyle f\left(\imath\omega,\alpha\right)=\omega^{2}-2|\omega|\alpha
d+\alpha^{2}c$ $\displaystyle\geq-2|\omega|\alpha
d+\alpha^{2}c=\alpha^{2}\left[c-2\frac{|\omega|}{\alpha}d\right]\geq\alpha^{2}\left[c-\frac{c}{2}\right]=\alpha^{2}\frac{c}{2},$

where the last inequality follows from
${|\omega|}/{\alpha}\leq{\bar{\omega}}/{\bar{\alpha}}=c/(4d)$. From the latter
bound and (VI),

$\widetilde{F}\left(\imath\omega,\alpha\right)\succeq\tilde{b}\,\alpha^{2}\frac{c}{2}I.$
(24)

By substituting (24) and (18) in (15) we get

$F\left(\imath\omega,\alpha\right)\succeq\left[\frac{\tilde{a}}{4}+\tilde{b}\alpha^{2}\frac{c}{2}\right]I\quad\text{
if }|\omega|\leq\bar{\omega}.$

Point 2) now follows from the latter inequality and the fact that
$\bar{\omega}$ can be chosen arbitrarily. ∎

Loosely speaking, Theorem 4 says that when $S$ is large the system attenuates
inputs at low frequencies. In particular, if $S$ has full rank and
sufficiently large size, low-frequency inputs are completely blocked by the
system. Notably, since the overall spectral energy is preserved as $S$ varies
(cf. Eq. (12)), Theorem 4 implies that as $S$ grows the spectral energy of the
system moves from low to high frequencies.

##  VII Concluding remarks

In this paper, we analyze the properties of stochastic linear systems
featuring a prescribed steady-state covariance. These systems are used to
model dynamic brain networks generating a fixed resting-state functional
connectivity. We show that the state matrix of these systems can be
parametrized by a skew-symmetric matrix $S$, and we study the role of $S$ in
the stability, excitability, and frequency-domain behavior of the system. By
means of theoretical and numerical results, we find that as the entries of $S$
grow the system becomes more stable, excitable, and similar to an high-pass
filter.

There are several intriguing directions of future research. In particular, it
would be interesting to examine in depth the statistical interpretation of $S$
of Remark 1 and analyze how $S$ affects other system properties, e.g.,
controllability.

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# How Participants Respond to Computer Delays

Désirée Scholz  Friedrich Schiller University Jena, Jena, Germany  Linda
Graefe<EMAIL_ADDRESS>Department of Methodology and Evaluation
Research, Friedrich Schiller University Jena, Jena, Germany  Thomas M. Prinz
<EMAIL_ADDRESS>Course Evaluation Service, Friedrich Schiller
University Jena, Jena, Germany

###### Abstract

Reaction time studies with computers investigate how and how quickly
participants respond to changing sensory input. They promise simple and
precise measurement of time and inputs and offer interesting insights into
human behavior. However, several previous studies have discovered imprecisions
in timing appearing as delays, depending on the browser, software and
programming used for conducting such studies. Since the accuaracy of the
collected data is widely discussed, we aim to provide new results on the
effect of unintended delays on participants’ behavior. For this purpose, a new
reaction time study was conducted. Computer delays were added to the
experiment to investigate their effects on participants’ performance and
repulsion. Minimal changes in participants’ behavior did occur and should be
furtherly investigated, as the power of this study was rather low and might
not have uncovered all underlying effects. The following report details our
study design and results and offers several suggestions for improvements in
further studies.

##  1 Introduction

Reaction time studies investigate how and how quickly participants react on
changing sensory inputs (Woods et al., 2015). Especially in human-computer
interaction (HCI) and psychology, researchers use computers for their
experiments since they promise cheap, accurate, and precise measurements of
time and inputs (Clark and Molnar, 1964; Ollesch et al., 2006). However, the
greatest possible accuracy would require instanteneous hardware and software
responses, which is almost never possible, as proven in numerous recent
studies with differing hardware/software (Anwyl-Irvine et al., 2020; Bridges
et al., 2020; Holden et al., 2020; Neath et al., 2011). Of course, no
researcher wants to conduct an unintended non-objective experiment, i. e.,
that unnoticed variations in the experiment’s computer system (such as delays)
influence the participants’ behavior and, therefore, the experimental results.

Although non-objective experiments are a topic of research, delays in a
computer’s response are usually considered random and, therefore, negligible.
This assumption is supported by the replication of general results, including
effect sizes, when comparing web- and lab-based research (Komarov et al.,
2013; Hilbig, 2016; Crump et al., 2013; de Leeuw and Motz, 2016). In contrast
to these reports, Woods et al. (2015) and Holden et al. (2020) discovered
discrepancies in the results of previous reaction time studies and a tendency
towards increasing reaction times over the course of several decades. The
impact of mechanical inaccuracy is usually investigated by comparing results
of different devices and / or approaches. One can also investigate response
patterns after artificially introduced delays in order to say whether they
directly influence participants’ behavior, as done by Bergmann et al. (1981),
Butler (1983), Schleifer and Amick (1989) or Szameitat et al. (2009). However,
the results of these studies vary: They show either large or no effects. The
reasons for these differences currently are a matter of speculation.

Their research emphasizes the need for further investigations on this topic,
as ignoring unintended delays could systematically affect participants’
experimental results and, therefore, the validity of the study.

Systematic differences in results can have strong impacts. For example, there
are reaction time tests in the medical context, where deviations from reaction
times of healthy patients provide diagnostic information. This diagnostic
information may provide access to help, interventions, and treatments.

False negative results can, therefore, potentially lead to long-term
deficiencies. In HCI research, reaction times are important to measure how and
how quickly users interact with computers. Hence, unintentional computer
delays in experiments could lead to misinterpretations of HCI phenomena. For
example, suppose that a prompt is displayed to a user in a software after a
predefined time when the user does not have interacted with the computer. If
delayed system responses prevent users from interacting with the software, the
prompt will be displayed incorrectly. In the worst case, reactions of users to
the software are interpreted as responses to the prompt and, therefore, lead
to unintended effects on the software. Such unintended reactions can have
strong effects, as Schmid et al. (2020) show.

This paper presents the results of a new reaction time study that investigates
the effects of computer delays on participants’ performance and repulsion. The
study used a simple experiment in which participants had to decide which of
two numbers was greater. Participants were divided into four groups: a control
group without additional delays and three experimental groups with different
variability of delays. We hypothesized that higher variability in computer
delays would lead to higher reaction times and repulsion among participants.
We also hypothesized that error rates would increase with additional delays,
especially when variability is high. A total of $100$ persons participated in
the study.

We were unable to confirm our main hypotheses. Additional explorative analyses
show minimal changes in participants’ overall response speed as well as an
apparent impact of age and gender in the behavior following the delays.

The paper is structured as follows: Section 2 provides an introduction to the
human-computer interaction process. Subsequently, it describes our newly
conducted experiment with our hypotheses in Section 3. Section 4 details the
analyses used and the results of the study. These results are further
discussed in Section 5. Finally, a conclusion is drawn in Section 6.

##  2 Theoretical Background

Previous research describe the general phenomenon of interruptions in
information technology. Addas and Pinsonneault (2015) distinguish
interruptions between _intrusions_ and _interventions_. Both are caused by
external factors (e. g., pop-ups of incoming messages). Intrusions do not
belong to the current primary task. Therefore, they do not lead to a long-term
change in behavior in relation to the task at hand. Interventions, on the
other hand, show deviations between the expected and the actual performance of
the primary task. As a result, they lead to long-term changes in behavior to
avoid further interventions. Addas and Pinsonneault (2015) further make
distinctions between different types of intrusions: Some are _informational_
but irrelevant in the current situation (e. g., at work), others require
_actions_ that have nothing to do with the primary task but are relevant to
the general situation, and others are caused by the computer (_system_
intrusions). In a controlled laboratory environment, as it often used in
psychological or HCI-related experiments, informational and action-related
intrusions are usually an intended part of the experiment. System intrusions
are usually unintended. In the present work, delayed responses (system
intrusions) are intentionally added in a reaction time study to investigate
their effects on participants.

Reaction time studies, as the name suggests, examine how and how quickly
participants respond to changing input signals. The time needed to react to
the changing input signal is generally referred to as _reaction time_.
Bergmann et al. (1981) propose a distinction in user response time that is
necessary to interpret it correctly in both natural and experimental settings.
They argue furthermore that the _human-computer interaction process_ (HCIP)
consists of two primary time-spans: The _user response time_ and the _system
response time_.

Figure 1: Response times in human-computer interaction processes

Figure 1 illustrates both time spans. The user response time comprises the
time from the beginning of a measurement $t_{s}$ ($s$ for start) to the point
in time $t_{a}$ ($a$ for action) when the desired action (e. g., clicking a
button) is completed. It can be divided into the _cognitive_ user response
time, which starts at $t_{s}$ and lasts until the point in time $t_{d}$ ($d$
for decision) when the user has made a decision about action to take, and the
immediately following _system access time_ , which comprises the performed
user action to be finished. The second time span of the HCIP, the system
response time, includes the time after the user action is completed $t_{a}$
until the point in time $t_{f}$ ($f$ for finish) when the result of the action
is presented (e. g., displaying a new text on a screen). While describing
their precautions to measure all possibly occuring delays Woods et al. (2015)
further divide the system response time into two time spans. We have included
them in the model as follows: First _system detection time_ , which describes
the time it takes for the computer to recognize the user’s action. The
remaining period between the point in time $t_{n}$ ($n$ for noticed), when the
computer recognized the action, and the time $t_{f}$, when the computer has
processed effects of the action, is the _system processing time_.

##  3 Methods

Due to the differences in the results of previous research efforts, we
conducted a study to gain new insights into delayed system responses in
reaction time studies. A main requirement of the experiment was to make the
task comparable to simple reaction time experiments from psychology and HCI
using a one-step cognitive decision and a two-choice response. As the
experiment was used in an online study due to Covid-19, it was ensured that
the experiment offered little opportunity for confusion or cheating by using
the guidelines suggested by Reips (2002) (e. g., randomization of data, one-
way passwords, etc.). The experiment was tested for conciseness on a group of
five volunteers and their comments were integrated into the final experiment.
After a second pretest with further eight participants, a high error rate was
noticed caused by a simple, remaining misunderstanding in the experimental
instructions. This was solved in the final experiment.

###  3.1 Experimental Setup

After all pretests, the final experiment had the following setting: Each
participant got a set of $320$ number pairs. Each number has four digits
($1000$ to $9999$) and differs in exactly one digit from the other number in
the pair (e. g., $1\mathbf{2}34$ and $1\mathbf{4}34$). The choice of numbers
of the same size with only one difference in digit is justified to avoid
longer reaction times caused by a higher complexity of the comparison (Nuerk
et al., 2004). Both numbers were displayed side by side in the internet
browser (cf. Figure 2). The participants had to decide which of the two
numbers was the greatest. If the left number is greater than the right one,
the participants had to use the _D_ key, otherwise _L_.

The entire experiment was divided into $7$ blocks; two blocks for practice
followed by five blocks for the experiment. The two practice blocks comprised
$10$ number pairs each, which were the same for all participants. The five
experimental blocks comprised $60$ pairs of numbers each. However, each
participant got all $300$ pairs in the experimental blocks in a randomized
order to avoid position effects (Troyer, 2011).

Three different experimental groups were designed to examine different kinds
of system delays. Due to the lack of data on average delay durations, we based
the mean duration on 1650ms, as suggested by Szameitat et al. (2009).
Participants of Group $1$ got a constant delay of $1650$ ms for each pair of
numbers (including the practice blocks); Group $2$ got a delay with low
variability ($\sigma=0.5s$) and a mean of $1650$ ms; finally, participants of
Group $3$ got a delay with high variability ($\sigma=1.5s$) and the same mean
of $1650$ ms. In addition to the three treatment groups, a fourth control
group (referred to as Group $0$) showed no intended delays.

The delays for Group $2$ and $3$ were calculated using $\gamma$-distributions
with the mean values and standard deviations given, in line with Bergmann et
al. (1981). The choice was made based on the unaffectedness by the lack of
negative values as well as the right-skew, since shorter delays are more
common than long delays (Butler, 1983). The distributions result in delays
from $562.1$ to $3206.9$ ms for Group $2$ and from $17.3$ to $7697.0$ ms for
Group $3$. The delays were calculated once and were the same $320$ delays for
all pairs of numbers (but in a random order) within each experimental group.

Figure 2 demonstrates the sequence of events for one trial.

Figure 2: Example Trial.

###  3.2 Measured Times and Data

The experimental set-up measured three different time intervals:

System time $\Delta t_{sys}$

describes the time between taking a new pair of numbers, waiting for the
intended delay, and displaying the pair in the browser for the participants.
In the HCIP model previously introduced, this can be interpreted as the system
processing time.

User response time $\Delta t_{user}$

comprises the time interval between displaying the current pair of numbers and
pressing either key _D_ or _L_. This includes the entire user response time
and the system detection time, as explained in the HCIP model.

Total experiment time $\Delta t_{total}$

starts with the system time measure and ends with the user response time
measure.

The system time $\Delta t_{sys}$ was measured to detect unintentional delays
beside the intended delays. That is, if $\Delta t_{delay}$ describes the
current delay for the number pair, then $\Delta t_{sys}-\Delta t_{delay}$ is
the unintentional delay. To avoid unintentional delays, the complete
experiment was loaded at once as a JavaScript/ECMAScript file such that no
reloading of sources over the web was required. The total experiment time
$\Delta t_{total}$ was measured to check for further unintentional delays in
the time measurements. A more accurate measurement of user response time
without the system detection time was not possible due to the necessity to
have a user input (key press _D_ or _L_) to perform a system action.

Besides the measured times, the participants also had to state their age and
gender, as both could be a cause of prolonged reaction times independent of
the experimental condition (Gale et al., 2016; Hultsch et al., 2002; Jorm et
al., 2004; Teleb and Awamleh, 2012). In addition, after each block (practice
as well as experimental), a question was asked about the participants’
repulsion, which was rated on a 6-point Likert scale. At the end, there was
again a question on repulsion in relation to the whole experiment.

Sometimes, participants tend to find the experimental hypothesis during the
study that influences their behavior (Rosenthal and Fode, 1963). Therefore, a
question regarding the suspected hypothesis/question was added to the study. A
control analysis was conducted that excluded all participants who correctly
assumed the experimental manipulation.

###  3.3 Hypotheses

Taking into account the experimental conditions, the following assumptions
were made:

Hypothesis 1

An higher variability in system response time is linked to an higher user
response time.

Hypothesis 2

Error rates are larger in conditions with delays in comparison to the control
group.

Hypothesis 3

The higher the variability in system response time, the higher the repulsion.

##  4 Analyses/Results

Participants were invited to take part in the experiment described in the
previous section. To avoid any effects of previous knowledge about the goal of
the study, participants were invited with a cover story to _investigate the
effects of age and gender on number perception_. The actual aim of the study
was not presented until the individual participants completed their
experiments. They had a chance to withdraw their consent to participation at
any point during the experiment. All participants were recruited via various
university mailing lists in Germany, the website of the German psychology
magazine _Psychologie Heute_ , and personal requests to friends and colleagues
unfamiliar with the project.

The results are divided into the following parts: First, the composition of
the groups and some general results are presented. Afterwards, the influences
of the covariates on the measures are described. Subsequently, the results
test our hypotheses and provide some alternative exploratory analyses. At the
end, the effect on the results of participants’ assumptions about the study
hypotheses are considered.

###  4.1 Composition of Groups and General Results

The general results and composition of the groups differ in results about
participants, distribution of system response times, performance, and
repulsion of the participants.

####  4.1.1 Participants

At the end of recruitment, $141$ people took part in the study. One of them
had to be excluded because of the age111According to law in the country where
the study was conducted, the data of minors may only be used for studies if
their custodians have given their explicit agreement.. Of the remaining $140$
participants, only $100$ actually completed the entire study. They were
randomly assigned to the four groups, but the final group sizes were
unbalanced ($N_{X=0}=29$, $N_{X=1}=27$, $N_{X=2}=20$, $N_{X=3}=24$).
Implications of the unbalanced group sizes and the dropout of $40$
participants are discussed later in Section 5.

Most participants were female ($N=66$), only a small minority chose the gender
option _diverse_ ($N=5$). Since there were no diversified participants in
Group $3$ and the overall data situation is insufficient (Matsuno and Budge,
2017), gender-specific differences can only be interpreted between male and
female participants. The groups did not significantly differ in the
distribution of these two genders ($\chi^{2}(3,N=95)=1.09$, $p=0.78$).

The age of the participants ranged from $19$ to $64$ years ($M=30.32$ years,
$SD=10.45$ years), with no significant differences in mean between the groups
($F(3,96)=0.361$, $p=0.781$).

####  4.1.2 Distribution of System Response Times

We calculated unintended system response times by reducing the measured system
time $\Delta t_{sys}$ by the intentionally added delays. This resulted in 320
measurements of unintended system response time per person, which allowed to
identify its distribution for each participant. Over all participants, the
mean system response time varied from $0.23$ ms up to $11.63$ ms ($M=5.5$ ms,
$SD=4.07$ ms), while the standard deviation ranged from $0.05$ ms to $9.26$ ms
($M=3.05$ ms, $SD=2.36$ ms). There is a significant difference in mean system
response time across groups ($F(3,38)=108.6$, $p<0.001$) and standard
deviation of system response time across groups ($F(3,39)=80.343$, $p<0.001$).
Although participants probably did not notice these short intervals, possible
side effects are nevertheless discussed in Section 5.

####  4.1.3 Overall Performance of Participants

The average user response time per trial (pair of numbers) of a single
participant was between $1224$ ms and $1509$ ms ($M=1368$ ms, $SD=54.34$ ms).
The values were calculated using the median to avoid being influenced by
individual outliers. Although a comparison with other studies is difficult due
to differences in the task specification, the research assumes that these are
realistic distributions for a mathematical task (Whelan, 2008; Restle, 1970).

The errors made by participants were minimal, with $50$% of the participants
making less than $9$ mistakes ($M=13.71$, $SD=18.28$). The lowest number of
errors was $1$, the highest was recorded with $125$ errors.

####  4.1.4 Repulsion Scores

Aggregated repulsion across all blocks ranged from $0$ to $4.8$, where $0$
means _no repulsion_ and $5$ means _very high repulsion_ ($M=1.3$, $SD=1.28$).
In other words, some participants experienced no repulsion at all, while no
participant experienced very high repulsion constantly.

###  4.2 Influence of Covariates on Measures

The study measured two covariates, _age_ and _gender_ of participants. Their
influence on measurements is described below.

####  4.2.1 Age

Using a correlative approach, age was found to significantly influence both
reaction time and error rate. With increasing age, participants appeared to
become slower, indicated by $r(98)=.27$, $p=0.006$. This is consistent with
previous findings (Gale et al., 2016; Fozard et al., 1994; Hultsch et al.,
2002; Woods et al., 2015). However, the correlation between age and error rate
was found to be negative with $r(98)=-0.23$, $p=0.022$, indicating either the
absence of cognitive decline with increasing age or an increase in diligence.
The latter is further supported by a significant negative correlation between
error rates and reaction times with $r(98)=-.40$, $p<0.001$. A linear
regression model using both age and error rate as additive factors explains
19.6% of overall variance in user response time ($NER$ is the number of
errors):

$E(\Delta t_{user}|Age,NER)=1390.212-6.313\cdot NER+5.805\cdot Age$ (1)

The effects of age on performance were limited, as the correlation between age
and repulsion was not significant ($p=0.339$).

####  4.2.2 Gender

As already mentioned, the following results only compare male and female
participants.

No significant effects on performance in reaction time ($t(61)=-0.988$,
$p=0.327$) or error rates ($t(38)=1.056$, $p=0.298$) were discovered. As this
task is a more cognitive one, these results are consistent with the absence of
gender differences in intelligence (Colom et al., 2000).

A t-Test was used to examine the effects of gender on repulsion and yielded
significant results ($t(61)=-2.5176$, $p=0.015$). Females apparently rated
their repulsion higher than male participants, with a mean difference of
$0.65$. These findings also complement previous research on differences in the
perception and expression of emotions (Deng et al., 2016; Thomsen et al.,
2005).

###  4.3 Effects of Delay Manipulation

Several tests were carried to investigate the hypotheses introduced in Section
3.3.

####  4.3.1 Hypothesis 1

_A higher variability in system response time is linked to an higher user
response time._ An one-way t-test was used to examine whether the mean user
response time of Group $3$ was significantly higher than the mean user
response time of Group $2$. In addition, since the homogeneity of variance was
verified, the t-test is considered robust to non-normality and unequal group
sizes according to the guidelines for empirical sciences (Bortz and Schuster,
2010). The analysis yielded a non-significant result $t(40)=-0.468$,
$p=0.321$. _Therefore, the variability of the system response time cannot be
considered as an influencing factor on the user response time in this study._

####  4.3.2 Hypothesis 2

_Error rates are larger in conditions with delays in comparison to the control
group._ Since this hypothesis suggests a difference in means of all
experimental groups in comparison to the control group, a one-factorial Anova
was used to investigate the differences in error rates. Homogeneity of
variance was verified so that unbalanced group sizes were of little importance
(Grace-Martin, 2020). The result was not significant ($F(3,96)=0.258$,
$p=0.855$). _The assumption of an unequal mean error rate between the groups
cannot be confirmed._

####  4.3.3 Hypothesis 3

_The higher the variability in system response time, the higher the
repulsion._ In this case, all experimental groups were compared pairwise with
another one-factorial Anova, assuming that unbalanced group sizes have no
influence due to the homogeneity of variance (Grace-Martin, 2020). The
resulting $F(2,68)=0.450$ and $p=0.639$ imply that _no group mean in repulsion
differs significantly from another group mean, leading to the rejection of
this hypothesis_.

###  4.4 Alternative Analyses

The previous analyses treated the effects of covariates as less important (1)
due to the lack of research on the influence of participants’ age and gender
on perceptions of system response times and (2) due to non-significant
differences in the distribution of these variables between groups. Therefore,
we exploratively re-examined the impact of system response times per se on
trial response time (instead of looking at them group-wise). In a second
additional analysis, we investigated the suitability of several complex linear
regression models to explain the total time for all experimental trials.

####  4.4.1 Trial User and System Response Time

All $30{,}000$ experimental measurements of system response times (including
manually added delays, i. e., $\Updelta t_{sys}$) and user response times
($\Updelta t_{user}$) were used for correlative analysis and linear
regression. The correlation appeared to be small ($r(29998)=-.07$) but
significant ($p<0.001$). To avoid bias due to outliers, a second correlative
approach was taken, excluding all user response times lower than $291.633$ ms
and higher than $2500.613$ ms. Such values could be caused either by
accidental, reflexive typing or by (environmental) distractions (Whelan,
2008). $93.3$% of the data was used for the second analysis, revealing a still
significant correlation of $r(28095)=-.13$, $p<0.001$. Accordingly, the
following linear regression model explains about $1.5$% of the total variance
in user response times:

$E(\Updelta t_{user}|\Updelta t_{sys})=1454-0.044\cdot\Updelta t_{sys}$ (2)

However, the results of this analysis have to be interpreted with caution,
since the number of trials is quite large, which could influence the reports
of significance, but derived from repeated measurements on the 100
participants.

####  4.4.2 Complex Models with Predictors and Interactions

Five models seemed interesting to compare their ability to describe the data,
in particular the time to complete all experimental trials, the number of
errors, and repulsion:

Model 1

Group as predictor.

Model 2

Group and age as predictors (without interactions).

Model 3

Group, age, and gender as predictors (without interactions).

Model 4

Group, age, and gender as predictors (interaction between group and age).

Model 5

Group, age, and gender as predictors (interactions between all variables).

For the comparison of these (multiple) linear regression models, the maximum-
likelihood-ratio-test was used, and all models were specified as reference
group models, with _control_ as the referenced group. This gives rise to
multiple indicator variables $I_{group=x}$ that amount $1$ whenever $group=x$
and $0$ in all other cases.

##### Overall time to complete the experiment

Comparing all five models in terms of the user response time, Model 2 turned
out to be most suitable at describing the user’s time to complete the
experiment ($\chi^{2}(1,N=95)=8.239$, $p=0.004$). Three parameters differ
significantly from zero with an $\alpha$-level of $.05$. The model below
explains $13$% of the total variance with ∗ identifying significant
coefficients:

$\displaystyle E(\Updelta t_{user}|\Updelta t_{sys},Age)=$ $\displaystyle
402196.6^{*}-49387.5\cdot I_{group=1}\;-$ (3) $\displaystyle 53977.8\cdot
I_{group=2}-53729.6^{*}\cdot I_{group=3}\;+$ $\displaystyle 2617.6^{*}\cdot
Age$

##### Number of errors (NER)

Both Model 2 ($\chi^{2}(1,N=95)=4.932$, $p=0.026$) and Model 5
($\chi^{2}(7,N=95)=40.455$, $p<0.001$) are comparatively more suitable than
the baseline model. Comparing these two models individually, Model 5 fits even
better ($\chi^{2}(11,N=95)=43.443$, $p<0.001$). However, only four of the
model’s parameters are significant, as shown in Table 1. Overall, Model 5
explains $41$% of the variance in the number of errors (NER) between
participants.

Table 1: Significant regression coefficients for describing the number of
errors. Regression coefficients
---
Description |  | Coefficient | t-Value | p-Value
Parameter | $I_{group=1}$ | 186.6 | 5.093 |  $<$ 0.001
Interaction |  $I_{group=1}$ and Age | -5.3 | -4.622 |  $<$ 0.001
Interaction |  $I_{group=1}$ and $I_{gender=female}$ | -214.1 | -5.330 |  $<$ 0.001
Interaction |  $I_{group=1}$ and Age and $I_{gender=female}$ | 6.0 | 4.757 |  $<$ 0.001

##### Repulsion

For the relationship between repulsion and all measured variables, Model 3
seems to be most suitable ($\chi^{2}(1,N=95)=7.220$, $p=0.007$). The four
significant parameters of this model, explaining $13$% of the variance in
repulsion, are described in Table 2.

Table 2: Regression coefficients in a model describing repulsion. Regression
Coefficients
---
Description |  | Coefficient | t-Value | p-Value
Intercept | $group=control$ | 1.50 | 3.368 |  $<$ 0.001
Parameter | $I_{group=2}$ | -0.88 | -2.495 | 0.014
Parameter | $I_{gender=female}$ | 0.71 | 2.651 | 0.010

###  4.5 Checking Participants’ Assumptions of Study Hypotheses

$79$ participants made assumptions about the research question of the study.
The assumptions were finally divided into $7$ categories. The interested
reader can find all categories with some examples in Table 3. After repeating
the analyses described in the Sections 4.1 to 4.3, we replicated the general
results in terms of their significance or insignificance. In the case of
significant results, the difference was marginal.

Table 3: Categorized guesses of participants on possible research questions. The examples are translated into English. The category _Critical_ describes all hypotheses that require an understanding of the experimental manipulation. If participants made more than one suggestion, the most detailed (or critical) one was preferred. Category | N |  Examples
---|---|---
Concentration & Reaction | 14 |  (1) Difficulties concentrating on a task.
(2) Monotony and Perception.
Comparing numbers | 13 |  (1) Which digit causes most errors?
(2) Can digit switches influence the result?
Errors $\leftrightarrow$ Repulsion | 12 |  (1) Higher error rate causes higher repulsion.
(2) Relationship between error rate & aversion.
Critical | 11 |  (1) Higher repulsion due to longer delays.
(2) Patience & Attention.
Resistance to stress | 10 |  (1) Comparing numbers during time pressure.
(2) How stressful is flow disruption?
Duration $\leftrightarrow$ Performance | 6 |  (1) Higher errors with longer duration of the experiment.
(2) Higher errors after comparing many numbers.
Other | 13 |  (1) Repulsion and repetitive tasks.
(2) Younger participants have better performance for number comparison.

##  5 Discussion

In the following, the above-mentioned results of the study are discussed. The
discussion is divided into the following parts: First, the effects of
variations in system response times are detailed. This is followed by a
discussion about the reductions in user response times over the duration of
the experiment. Next, the discussion considers the impact of dropouts (break-
offs) and provide some analyses of key presses as an alternative indicator of
repulsion. Finally, the limitations of the present work are discussed.

###  5.1 Effects of Variation of System Response Times

Since none of our analyses led to significant results, one could assume that
manipulating the system response time has no impact on the participants.
However, when age and gender are included in a multiple regression model, a
significant influence of the experimental group (mainly Group $1$ and Group
$2$) is becoming apparent. Interestingly, this influence was _positive_ :
increasing the system response time led to a faster completion of the
experiment by the participants, almost one minute shorter than in the control
group without additional delays.

Group 1 with constant delays of $1650$ ms had several significant interaction
effects with age and female gender on participants’ individual error rate. As
previous studies do not detail the effects of combined covariates, this is a
new finding that should be further investigated. Another interesting finding
is the lack of significant repulsive responses for all groups except Group
$2$.

Although these findings were surprising and contradict our hypothesis, there
are plausible explanations for the results based on two different models of
human-computer interaction:

Processing Time:

In the interaction between humans and computers, both partners are limited in
their processing speed. As a result, a computer’s response should be fast but
slow enough for the human to actually process the information presented. Some
limitations lie in the memory capacity of the human (Shneiderman, 1984):
Although working memory can store information for up to $30$ seconds (Cowan,
2008), interferences with previously presented information are possible
(Jonides et al., 2008; Oberbauer, 2009).

With regard to the present study, the choice of the greater number might be
difficult due to the similar characteristics of the trials (two four-digit
numbers), as the old pair is still in mind. Therefore, adding delays grants
participants more time to lose focus on the old pair in memory. In part, this
approach is also supported by the significant negative correlation between
user response time and the number of errors. However, reflexive key presses or
general distractions from the environment cannot be excluded as a cause for
this correlation (Whelan, 2008).

Perceiving a natural interaction:

Miller (1968) argued that an interaction between humans and computers is
similar to an interaction between people, where only one partner has an active
goal. In our case, the participants’ main goal was to complete the experiment,
with sub-target goals being the completion of each trial (Dix et al., 2004).
Miller (1968) described that the perfect speed of the interaction process
between humans and computers depends on the nature of the goal/subtype of
interaction. Either way, the process should ideally be faster than four
seconds, preferably two. However, in some cases (e. g., when a new task is
requested), it can take up to fifteen seconds without this being perceived as
negative. Miller emphasizes that all times are based on his experience as
behavioral psychologist and there is no explanation of what would happen if
the computer react almost instantaneous. Galletta et al. (2004) have shown
that the optimal system reaction time is at $0.5$ seconds and, therefore,
_not_ instantaneous.

###  5.2 Decreased Response Time over the Experiment’s Duration

Previous studies report an increase in user response times with longer
experimental duration (Langner et al., 2009), which made us explore why our
study showed a different result.

The simplest explanation is that our experiment was simply too short to induce
fatigue effects (participants were able to complete the study in less than
half an hour if they decided not to take any of the breaks offered).
Therefore, the trend towards faster user response time could be due to the
training (Sanders and Hoogenboom, 1970). This explanation is further supported
by the loss of a significant linear trend when twenty practice trials are
included in the analyses, which familiarized participants with the task.

Another investigation concerned the effects of faster response times due to
practice, in particular, whether this effect is influenced by the system
response time and its variability. This was investigated by comparing the
correlations of the different groups. There was no significant trend in the
median user response time ($p_{1}=0.494$ und $p_{2}=0.062$) for either Group
$1$ or Group $2$. However, in Group $0$ and Group $3$, the trend towards
faster response times ($r(298)=-.12$, $p=0.033$ and $r(298)=-.17$, $p=0.003$,
respectively) was rediscovered. It seems that the trend of faster response
time correlates with the system response time when there is no such response
time or it has a high variability. However, this is a singular result. Further
research is needed to replicate these findings and investigate their causes.

###  5.3 Dropout (Break-offs)

Online studies are usually and unfortunately coupled with a dropout (break-
off) of participants. Although we tried to reduce the potential that could
lead to dropouts (Howell, nD) (e. g., by asking personal information first),
we still had a high dropout rate. In the following, we discuss the impact of
dropouts (as suggested for example by Crump et al. (2013)).

Each participant who did not complete the study had at least some experience
with the experiment. Among the different groups, $3$ participants left the
control group, $8$ Group $1$, $13$ Group $2$, and $12$ Group $3$. Four
participants answered all pairs of numbers but not the final overall repulsion
question.

The reasons for participants to leave the study cannot be determined. Time and
environment could be important factors (Dandurand et al., 2008). Since the
number of dropouts is higher in the experimental groups than in the control
group, the repulsion could be too high due to the long system response times,
so that participants left the study in frustration. If some participants left
the study frustrated, this could support hypothesis 3. However, there is no
additional evidence or data to support this assumption. Since hypothesis 3 was
not supported by the study results, it seems unlikely that the frustration was
high enough that participants left the study.

Some of the participants who left the study may have participated (and
completed) the study again at a later time. This could have influenced their
behavior and, because of differences in system response times between both
participations, could have betrayed the goal of the study. However, it is
generally difficult to avoid double participation in anonymous online studies.
Future studies could include a _Continue later_ mechanism, but this could also
influence the results if there is a large break in the experiment.

###  5.4 Key Presses as Indicator of Repulsion

Participants had to use either key _D_ or _L_ to answer which of the two
presented numbers was the greatest. Once they had given the answer, it was not
possible for the participants to influence the duration of the delay. However,
it could be possible that some participants became impatient due to the long
system response times and used some keys to check whether the answer was
registered by the system or the system itself needed another input to present
the next pair of numbers. Thus, the keys that were pressed impatiently between
the answer for one pair of numbers and the presentation of the next pair were
blocked. Whether and how much keys were pressed could be an alternative
indicator for repulsion.

Impatiently pressed keys can be separated into two groups: _useful_ , i. e.,
_D_ and _L_ to give an extra response attempt, and _non-useful_ keys (all
others). Useful keys were used between $3$ and $19$ times per participant
($M=8.96,SD=2.69$), while non-useful keys were pressed between $0$ and $3$
times ($M=0.47,SD=0.717$). Figure 3 takes a closer look at the amount of
useful and non-useful key presses between the different experimental groups.
In addition, an one-factorial Anova was used to compare all group means. Since
Levene’s test for homogeneity of variance was not significant
($F(3,96)=1.843,p=0.145$), we assumed that there were no influences from
unbalanced group sizes (Grace-Martin, 2020). The Anova itself also yielded no
significant results ($F(3,96)=1.811$, $p=0.150$). As a conclusion, the key
presses do not significantly differ between the groups. This is also supported
by Figure 3. In other words: if impatient key presses are an alternative
indicator of repulsion, then repulsion did not differ significantly between
groups. In this study, there was no correlation between the self-reported
repulsion score and the number of impatient key presses.

Figure 3: Key presses in dependence of the experimental group.

###  5.5 Limitations and suggestions for further studies

Although this study led to interesting results, it has some limitations that
are explained in detail below.

####  5.5.1 Sample Sizes and Task Choice

The task was explicitly designed for this study by taking into account the
time constraints of participants in online studies, typical reaction time
tasks, and an easy implementation, measurement, and influence of the delays
(due to Covid-19 restrictions, we could not use external measures in a
laboratory setting (Woods et al., 2015; Reimers and Stewart, 2015; Schmidt,
2001)). Using a custom task, however, complicates the comparison of results
with previous studies and makes it even more complex to determine the pre-hoc
effect and the required sample sizes, especially due to the lack of
information on expected effect sizes (Segieth et al., 2004). Therefore, the
minimum sample size of $100$ participants was determined with regard to other
studies on this topic and by our own assumptions about what would be possible
within the given time limit.

Power calculations help determine sample sizes to find effects of different
intensities. To find measurable minimal effects, higher sample sizes are
necessary in contrast to finding high effects, where smaller samples are
sufficient. Table 4 shows different sample sizes for our hypotheses 1-3 and
alternative analyses 1 and 2 with models 1 and 3. The sample sizes in the
table were calculated using G*Power (Faul et al., 2007, 2009).

Table 4 suggests that our samples allow us to find measurably high effects
with an acceptable power ($>$ .80). Confirming of our hypotheses failed as
explained. For this reason, our hypotheses do not appear to have a high
effect. Previous research suggests that the effect, if present at all, is
rather small. Further research on this topic should consider higher sample
sizes in order to find smaller effects and to reexamine hypotheses 1-3 against
this backdrop. At this stage, we cannot definitively exclude effects of delay.
The conclusions of previous studies (Szameitat et al., 2009; Butler, 1983;
Schleifer and Amick, 1989) should in any case be treated with caution, as
their sample sizes were also rather small.

The power results of Table 4 look more promising regarding our additional
analyses. The sample sizes allow acceptable conclusions for even moderate
effect sizes with acceptable power above 0.80.

Table 4: Each section includes six lines. Lines (1) and (2) show the required effect size to achieve a power of .95 / .80 given the underlying sample size. Lines (3) and (4) show the achieved power given a large and medium effect size. Lines (5) and (6) present the required sample size for a power of .95, assuming a medium / small effect size. Alternative Analysis 2 M2 could not adequately be investigated in regards to its power due to the lack of information on power calculation in complex linear models with multiple interactions. Hypothesis | Sample Size | Effect Size | Power
---|---|---|---
Hypothesis 1, Effect Size $d$ | 20 | 24 | 1.013 | 0.950
20 | 24 | 0.765 | 0.800
20 | 24 | 0.800 | 0.830
20 | 24 | 0.500 | 0.491
88 | 88 | 0.500 | 0.950
542 | 542 | 0.200 | 0.950
Hypothesis 2, Effect Size $f$ | 100 | 0.423 | 0.950
100 | 0.330 | 0.800
100 | 0.400 | 0.920
100 | 0.250 | 0.518
280 | 0.250 | 0.950
1724 | 0.100 | 0.950
Hypothesis 3, Effect Size $f$ | 71 | 0.477 | 0.950
71 | 0.377 | 0.800
71 | 0.400 | 0.848
71 | 0.250 | 0.438
252 | 0.250 | 0.950
1548 | 0.100 | 0.950
Alternative Analysis 1, Effect Size $\rho$ | 28097 | 0.022 | 0.950
28097 | 0.017 | 0.800
28097 | 0.500 | 1.000
28097 | 0.300 | 1.000
134 | 0.300 | 0.950
1289 | 0.100 | 0.950
Alternative Analysis 2 M1, Effect Size $f^{2}$ | 95 | 0.168 | 0.950
95 | 0.105 | 0.800
95 | 0.350 | 1.000
95 | 0.150 | 0.924
107 | 0.150 | 0.950
776 | 0.020 | 0.950
Alternative Analysis 2 M3, Effect Size $f^{2}$ | 95 | 0.189 | 0.950
95 | 0.120 | 0.800
95 | 0.350 | 1.000
95 | 0.150 | 0.888
119 | 0.150 | 0.950
863 | 0.020 | 0.950

####  5.5.2 Repulsion and Dropout

If the participants who left the study before completing their tasks did so
because of frustration, then this higher repulsion does not appear in our data
and, therefore, not in our analyses. As a consequence, the missing data could
lead to strong interaction with the results.

Our simple measurement of repulsion with a single question after each
experimental block was used to keep the study short, as it was an online and
not a laboratory setting. As a consequence, no pre-existing emotion-
theoretical model can be applied (such as in Szameitat et al. (2009)) and an
accurate measurement of repulsion cannot be guaranteed.

####  5.5.3 Differences in Unintended System Response Times

As reported in Section 4, there were significant differences in unintended
system response times between the groups (i. e., the delay was not added on
purpose). Figure 4 illustrates these unintended response times. The largest
difference occurred between the control group $0$ and all experimental groups
$1$-$3$. Although the experiment was carefully implemented to avoid side-
effects, the overhead of adding a delay led to unintended system response
times in the experimental groups. Compared to the intended additional delays,
the unintended system response times are small. Furthermore, several studies
report that the threshold for perceiving visual changes is between $50$ and
$90$ Hz ($11$ to $20$ ms, respectively). Most of the values reported here are
not within this time-span and seem negligible. If further studies, such as
that of Davis et al. (2015), indicate that perceiving visual changes is faster
in some situations, the results of our study would need to be interpreted
further in terms of the unintended system response time. Since the variations
are mostly consistent with the experimental manipulation, we decided against
regrouping participants based on their distributions of unintended system
response times.

Figure 4: Distribution of unintended system response times and their variances
among groups.

##  6 Conclusion

Unintended system response times in reaction time studies may affect the
measurement accuracy and the participants’ behavior in the experiment. In the
worst case, such system times systematically interfere with the study results
and lead to false conclusions. For this reason, this paper presented a new
study investigating the relationship between variation in system response
time, especially in variability, participant’s performance, and repulsion. Our
three postulated main hypotheses were not supported by the data: reaction
times, repulsion, and error rates appear to be almost independent of computer
delays and the height of variability. However, the lack of dectable effects
might only be due to the tests comparatively small power. Our additional
analyses show only small positive influences of increased system response
times, primarily when including age and gender as additional factors. In
simple words: When system response time increases, reaction times and error
rates seem to decrease slightly. Two theories have been presented to explain
this effect. In general, the results suggest that participant’s behavior in
reaction time studies is relatively stable when it comes to delays of modern
computers. Therefore, such behavior changes do not seem to deliver an
explanation for frequently discovered historical trends and replication
difficulties (Holden et al., 2020; Plant, 2016; Woods et al., 2015)

Reaction time studies are popular in HCI. For this reason, the present paper
provides new insights into the effects of unintended system response times on
study results and user behavior. It supplements the rare palette of studies on
these effects and we hope that it will motivate further research on this topic
as well as the consideration of system response times in analyses,
evaluations, and discussions in future reaction time studies. Although this
paper has shown that reaction time studies seem to be robust in the presence
of computer delays, there are also some small effects that should be taken
into account.

In future studies on this topic, _un_ intentional system response times should
be stable across all conditional groups. Since humans seem to need some time
to switch from one task to the next, the control groups should not have almost
no delays. It might be interesting to add a default delay to all groups
(control and experimental). Finally, new studies dealing with repulsion
should, if possible, try to measure repulsion more accurately and to prevent
participants from leaving the experiment. Both seem to be only possible in a
laboratory setting. There, more accurate information about the effects on
repulsion can be measured than it was possible in the online study used.

While our results contribute to interruption research, further studies have to
be conducted to investigate the impact of task choice and other (computer)
systems, such as mobile phones or tablets.

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