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Cognition, Technology & Work (2022) 24:291–315
https://doi.org/10.1007/s10111-021-00681-0
ORIGINAL ARTICLE
Power law model forsubjective mental workload andvalidation
throughair traffic control human‑in‑the‑loop simulation
NorbertFürstenau1 · TheaRadüntz2
Received: 11 April 2020 / Accepted: 30 March 2021 / Published online: 14 May 2021
© The Author(s) 2021
Abstract
We provide evidence for a power law relationship between the subjective one-dimensional Instantaneous Self Assessment
workload measure (five-level ISA-WL scale) and the radio communication of air traffic controllers (ATCOs) as an objective
task load variable. It corresponds to Stevens’ classical psychophysics relationship between physical stimulus and subjective
response, with characteristic power law exponent γ of the order of 1. The theoretical model was validated in a human-in-
the loop air traffic control simulation experiment with traffic flow as environmental stimulus that correlates positively with
ATCOs frequency and duration of radio calls (task load, RC-TL) and their reported ISA-WL. The theoretical predictions
together with nonlinear regression-based model parameter estimates expand previously published results that quantified the
formal logistic relationship between the subjective ISA measure and simulated air traffic flow (Fürstenau etal. in Theor Issues
Ergon Sci 21(6): 684–708, 2020). The present analysis refers to a psychophysics approach to mental workload suggested
by (Gopher and Braune in Hum Factors 26(5): 519–532, 1984) that was recently used by (Bachelder and Godfroy-Cooper
in Pilot workload esimation: synthesis of spectral requirements analysis and Weber’s law, SCL Tech, San Diego, 2019)
for pilot workload estimation, with a corresponding power law exponent in the typical range of Stevens’ exponents. Based
on the hypothesis of cognitive resource limitation, we derived the power law by combination of the two logistic models
for ISA-WL and communication TL characteristics, respectively. Despite large inter-individual variance, the theoretically
predicted logistic and power law parameter values exhibit surprisingly close agreement with the regression-based estimates
(for averages across participants). Significant differences between logistic ISA-WL and RC-TL scaling parameters and the
corresponding Stevens exponents as ratio of these parameters quantify the TL/WL dissociation with regard to traffic flow.
The sensitivity with regard to work conditions of the logistic WL-scaling parameter as well as the power law exponent was
revealed by traffic scenarios with a non-nominal event: WL sensitivity increased significantly for traffic flow larger than
a critical value. Initial analysis of a simultaneously measured new neurophysiological (EEG) load index (dual frequency
head maps, DFHM, (Radüntz in Front Physiol 8: 1–15, 2017)) provided evidence for the power law to be applicable to the
DFHM load measure as well.
Keywords Mental workload· Cognitive resource limitation· Psychophysics power law· Instantaneous self assessment·
Air traffic control simulation
1 Introduction
The concept of mental workload (WL) addresses the demand
a task (task load TL) imposes on the operators limited cogni-
tive resources (e.g. processing, memory; Wickens and Hol-
lands 2000; Wickens 2002)). According to these authors,
WL research may be viewed in the context of prediction
(e.g. multi-task performance), WL assessment imposed by
equipment, and WL subjectively experienced by operators.
A review on WL modeling and prediction in the complex
air traffic control (ATC) work system was provided in (Loft
* Norbert Fürstenau
norbert.fuerstenau@dlr.de
1 Institute ofFlight Guidance, German Aerospace Center
(DLR), Lilienthalplatz 7, 38108Braunschweig, Germany
2 Unit Mental Health andCognitive Capacity, Federal Institute
forOccupational Safety andHealth, Berlin, Germany
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292 Cognition, Technology & Work (2022) 24:291–315
1 3
etal. 2007). They took into account changing task priorities
for WL management and strategies of operators, and empha-
sized traffic density as indirect WL predictor due to task
demands such as identifying, monitoring, and instructing
aircraft (AC) via radio communication. Corver etal. showed
(Corver etal. 2016) that traffic conflict, moderated by tra-
jectory uncertainty, mediates the positive effect of traffic
density on WL. The specific question of ATC complexity as
WL driver was investigated, e.g. in (Djokic etal. 2010) who
confirmed subjective WL to correlate strongly with traffic
count and ATCO’s communication load. In the present work,
we used a simulated approach sector ATC work environ-
ment of a medium size German airport (with reduced risk
of separation conflict), with variable traffic flow and online
radio communication between ATCOs and (pseudo) pilots
to provide evidence for the potential of the psychophysics
approach (e.g. (Stevens 1975)) to workload for deriving
quantitative WL-sensitivity parameters.
For this purpose, we validate the theoretically derived
power law relationship between the quasi real-time one-
dimensional subjective Instantaneous Self Assessment
workload measure (ISA-WL) (Kirwan etal. 1997; Jordan
1992; Brennan 1992; Tattersall and Foord 1996) and the
objective communication task load variable (frequency of
ATCO’s radio calls, RC-TL) as mediator between WL and
environmental traffic load, by means of a human-in-the-
loop (HitL) ATC-simulation experiment (Mühlhausen etal.
2018). Thereby, we formally combine the logistic RC-TL
model with the recently published logistic WL model
that was used for the analysis of the subjective ratings of
operators (Fürstenau etal. 2020). In that previous work, we
recorded during execution of the simulated ATC task, the
periodic reporting of the subjectively experienced WL level
as dependent on environmental load variable traffic flow n
(aircraft per hour, AC/h), by means of the online five-level
ISA questionnaire.
The ISA-WL and radio calls RC-TL data used for the
present work represent only part of the complete set of sub-
jective and objective WL measures (including expert ratings,
NASA-TLX, cardiovascular (heart rate and HR variation),
neurophysiological (EEG); for details see Sect.2) that were
registered online during the experiment and which required
the least pre-processing effort for the analysis. The HitL
ATC-simulation experiment was performed with a homog-
enous sample of experienced domain experts (ATCos) who
also provided prior information on realistic traffic param-
eters for the selected airport approach sector.
The experiment within a realistic ATC approach radar
and radio communication (between ATCos and (pseudo)
pilots) work environment was primarily designed to vali-
date the new robust neurophysiological real-time method of
Dual Frequency Headmaps, (DFHM) for quantifying men-
tal workload by means of the electroencephalogram (EEG)
(Radüntz 2017) (see Sect.2.4). Initial ANOVA-based data
analysis was published recently (Radüntz etal. 2019). The
successful use in that work of the logistic ISA(n) model
for validation of the objective EEG-based DFHM index
(Radüntz etal. 2020a, b) provided the motivation for inves-
tigating in more detail nonlinear correlations between differ-
ent WL and TL measures of our experimental data. Logistic
dependencies of subjective workload on traffic count were
reported before by (Lee 2005) who obtained significant fit
parameters from ATC-simulation WL data with the seven-
level ATWIT scale (Air Traffic WL Input (Stein 1985),
see Sect.2.2). A logistic model (comparable to our ISA(n)
characteristic) was used also by (Averty etal. 2008) for the
analysis of air traffic controllers decision-making in conflict
risk detection.
One advantage of HitL simulations with highly trained
domain experts is the online monitoring of different real-
time data such as traffic flow and communication times and
duration as environmental and TL variables, respectively,
to be used as independent physical stimuli for subjective
response within the psychophysics approach to WL. Moreo-
ver, a minimization of inter-individual variance is achieved
through a homogenous sample of highly trained participants
(Abich etal. 2013; Brookings etal. 1996).
Basic assumption for the derivation of our theoreti-
cal model was the cognitive resource or capacity limita-
tion hypothesis (Kahnemann 1973; Wickens and Hollands
2000). All the above-mentioned subjective and objective-
dependent WL and TL measures were correlated with traffic
flow n (AC/h) as independent external load variable under
nominal and non-nominal conditions (priority event e = 0,
1, two factor design). The derived power law in the pre-
sent work corresponds to the classical stimulus–response
relationship of Stevens (e.g., Stevens 1975; Link 1992). As
proposed originally by Gopher etal. (1984, 1985), and as
recently reported by Bachelder etal. (2019) the psychophys-
ics approach suggests the power law application also to the
relationship between objective task load as stimulus and
subjective workload measures as response. In fact, Lehrer
suggested in (Lehrer etal. 2010) the combined use of dif-
ferent measures due to well-known large inter-individual
differences in sensitivities, because “it is known that some
individuals respond more sensitively to task load changes in
self-report measures, others in specific physiological meas-
ures”. In the present context, the power law allows to predict
theoretically and to estimate through (nonlinear) regression
of experimental data the characteristic exponent that relates
subjective WL (as response) to objective TL measures (as
physical stimulus).
In the present work, the psychophysical power law is
derived through combination of the logistic functions ISA(n)
as dependent WL measure (subjective response) and RC(n)
as communication TL load variable (physical stimulus; see
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293Cognition, Technology & Work (2022) 24:291–315
1 3
Sect.4 and Appendix2) which are interrelated through the
independent environmental traffic flow variable n. The mag-
nitude of the theoretically derived power law exponent γ
is shown to correspond to the order of magnitude (≈1) of
(Stevens’) slope values of the generalized linear (log–log
transformed) representation of the (subjective) response
vs. (physical) stimulus. They were shown to be characteris-
tic for a large number of sensory modalities (e.g., Stevens
1957). A theoretical basis for the psychophysical laws was
derived by (Link 1992) (see Sect.2.5). An information theo-
retic approach was provided by Norwich (e.g., Norwich and
Wong 1997).
In what follows, we continue in Sect.2 with a brief over-
view on aspects of different WL measures relevant for the
present work. We introduce our study design in Sect.3 and
describe in Sect.4 the theoretical background of our mental
workload model with logistic and power law WL/TL char-
acteristics for parameter prediction and regression-based
parameter estimates. In Sect.5, we present our experimen-
tal results which are discussed in Sect.6 with regard to the
theoretical predictions. Finally, in Sect.7, we draw conclu-
sions and outline further research. In Appendix A1, we pro-
vide tables with the detailed experimental (pre-processed)
data for each participant, separated for experimental (traffic)
scenarios, followed by A2 with mathematical details for the
derivation of the theoretical model equations.
2 Mental workload andmeasures for(quasi)
real‑time applications
For the discussion of our results in Sect.6, we will briefly
address some aspects of mental workload, different subjec-
tive and objective WL measures which are relevant for the
present work with focus on online (real-time) capabilities
(ISA, ATWIT/WAK, SWAT, HR/HRV, EEG-DFHM), and
the psychophysics (power law) approach.
2.1 Mental workload
Quantification of mental workload constitutes one of the
main issues in cognitive ergonomics and human-factors
research. Like many concepts in psychology, there is no
singular agreed-upon definition or method for measuring
mental workload. Much more, it is assumed that successful
performance on a task or test requires cognitive resources,
which can be seen as mental workload. In other words, men-
tal workload is a theoretical construct referred to as “the
cost incurred by the human operator to achieve a particu-
lar level of performance (Hart and Staveland 1988). Simi-
lar definitions were given by (Kahnemann 1973; Wickens
and Hollands 2000), and (Xie and Salvendy 2000). Never-
theless, its quantification contributes to the evaluation of
human–machine systems, estimation of the appropriate-
ness of automation levels, and enhancement of interface
design. A good overview on different theoretical and practi-
cal aspects of workload with focus on transportation as our
major field of interest is given in (Hancock and Desmond
2001).
As mentioned above, for measuring mental workload
there are several methods available that can be categorized
in two groups: objective and subjective methods. Objective
methods rely upon quantification of performance or bio-
physiological data while the subjective methods consider
the subjective rating given by the performer. Although all
measurement methods aim to describe the relation between
task demands and subject’s ability to cope with them, sev-
eral investigations reported dissociations among methods’
results. A possible explanation might be that mental work-
load is a multidimensional concept that cannot be captured
in all its facets by a single method. Apart from the task
requirements, mental workload variations are caused by
individual characteristics such as habituation, actual pre-
condition, and coping styles (ISO-10075, 1991, 1996, 2004).
2.2 Subjective quasi real‑time measures
Several researchers suggested that the subjectively expe-
rienced workload is of particular importance when evalu-
ating subject’s state (Yeh and Wickens 1984; Sheridan
1980). Johannsen etal. (1979) stated that “if an operator
feels effortful and loaded, he is effortful and loaded”. The
most accepted subjective measure in ATC appears to be the
multidimensional NASA task load index (TLX) based on
questionnaires for capturing the different aspects constitut-
ing the experienced WL (Hart and Staveland 1988). NASA-
TLX data together with expert ratings and ISA self reports
(see below) were evaluated in a preliminary analysis of the
present experiment to study the WL effect of a non-nominal
event (Radüntz etal. 2019, see Sect.3.2) during the HitL
simulations. The main advantages of subjective methods are
the relatively low data acquisition effort and the high user
acceptance. Their main drawback is that they suffer from
subjective distortion. They are influenced by memory lapses
as the experienced workload took place at some time in the
past (NASA-TLX) and they are subject to social desirability
bias (Lehrer etal. 2010; Radüntz 2017). The questionnaire’s
items may not be readily understood or participants may lack
the ability to introspect. What is more, they do not allow for
fine-grained temporal sampling on the time scale of seconds
and can alter the current workload state (Radüntz 2017).
In the present work, our interest was focused on the com-
bination of the objective online communication TL measure
with a subjective WL measure appropriate for (quasi) real-
time data analysis of the simulator experiments. An early
subjective quasi real-time WL-assessment technique was
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294 Cognition, Technology & Work (2022) 24:291–315
1 3
introduced by (Stein 1985): the Air Traffic Workload Input
Technique (ATWIT) using the seven-level WL Assessment
Keypad (WAK). Lee etal. in (2005) reported on analysis
of ATC-simulation ATWIT-WL data with nonlinear (sig-
moid) dependency of WL on traffic count (see below and
Sect.6.1).
Two established subjective self-report measures suitable for
near real-time application are SWAT (Subjective Workload
Assessment Technique) and the above-mentioned one-dimen-
sional ISA method. SWAT measures the three load dimen-
sions, time, effort, and stress, each with three levels (Reid
etal. 1989), while ISA monitors the experienced WL on a
one-dimensional five-level scale via online self reports in fixed
time intervals of a couple of minutes. In contrast to SWAT,
it minimizes possible additional WL (due to the reporting)
by not discriminating load dimensions (Brennan 1992; Jor-
dan 1992; Kirwan etal. 1997; Tattersall and Foord 1996). The
latter authors reported significant correlations of ISA ratings
with cardiovascular HRV and task performance, although the
primary task performance on a tracking task turned out poorer
during periods when ISA responses were required. Of course
this distortion certainly depends on the details of task and
reporting method (verbal, keypad, touchscreen). Girard etal.
adapted online ISA to a professional car driving simulator and
reported significant correlation of ISA-WL with dynamic traf-
fic density variation (Girard etal. 2005). The characterization
of the reported subjective load levels is listed in Table1:
Because the scale levels represent the subjective decision
of participants on the experienced load during task execution
the level differences may not be assumed to be equidistant. In
the theoretical model of Sect.4.1, we assume an equidistant
ISA scale so that any deviation from linearity is included in
the nonlinearities of the model equations. In a recent pub-
lication, we provided evidence for the logistic dependence
of ISA-WL on the environmental traffic load variable n and
derived a linearized ISA-WL-sensitivity index for subject
clustering (Fürstenau etal. 2020). The subjective index was
successfully applied for the validation of the neurophysio-
logical DFHM WL index (Sect.2.4; Radüntz etal. 2020a, b).
Our logistic model-based data analysis agreed with results of
Lee etal. (2005) and Lee (2005) based on ATC simulation
with dynamic traffic variation. They reported on results of
logistic WL data fits based on a heuristic sigmoid function
dependent on aircraft count within en-route sectors, with
significant four-parameter estimates of ATWIT-based sub-
jective WL measurements using the seven-level scale of the
WL Assessment Keypad (WAK, see above (Stein 1985)).
2.3 Psycho‑physiological measures heart rate
andhr‑variation
The analysis of bio-signals as objective measures (see also
Sect.2.4) offers the possibility to continuously determine
mental workload. They do not interfere with participant’s
current workload state as they can be obtained on-the-fly
during task execution. Their main issue is that user accept-
ance may be impaired because of the complexity of the reg-
istration system. However, recent developments in mobile
sensor technology promise small, lightweight, and wireless
systems (Radüntz 2017). Bio-physiological data include,
among others, cardiovascular biomarkers which are easy to
assess and were frequently used to analyse cardiovascular
activity under a wide range of experimental conditions (Kar-
avidas etal. 2006; Lehrer etal. 2010). The heart rate (HR)
and the heart rate variability (HRV) are the most prominent
biomarkers. Recently, Vanderhaegen et.al (2020) reported
on an experiment that showed synchronization between
dynamic events with heart beats and its impact on non-con-
scious errors in control.
In most cases, HRV is characterized in the frequency
domain by means of various spectral features. According to
the definitions by (Mulder etal. 2004), the frequency range
can be categorized in three bands: the low-frequency (LF:
0.02–0.06Hz), mid-frequency (MF: 0.07–0.14Hz), and
high-frequency (HF: 0.15–0.4Hz) bands. It was observed
that under mental load the total spectral power decreased,
whereby the spectral power between 0.02 and 0.20Hz was
particularly affected and contributed about 80% to the total
spectral energy (Mulder and Mulder 1981).
Basic research on HRV as WL measure for adaptive
automation was investigated by (Prinzel etal. 2003) with
a tracking task, together with EEG (see Sect.2.4) and
event-related potentials. Lehrer etal. (Lehrer etal. 2010)
reported an increase of association between self-report
Table 1 ISA workload categories after (Kirwan etal. 1997)
Level WL Heading Spare Mental
Resources Description
5 Excessive None Behind on task; loosing track of the full picture
4 High Very Little None essential tasks suffering. Could not work at this level very long
3 Comfortable busy pace Some All tasks well at hand. Busy but stimulating pace. Could keep going continuously at this level
2 Relaxed Ample More than enough time for all tasks. Active on tasks less than 50% of the time available
1 Underutilized Very Much Nothing to do. Rather boring
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295Cognition, Technology & Work (2022) 24:291–315
1 3
scale (using NASA-TLX) given immediately after each
5min task and both expert ratings of task load and task
performance in a flight simulator by means of cardiac
data. We recently reported on analysis of HR and HRV
measures within the present simulator experiment where
we aimed at clarifying their inherent timescales (Radüntz
etal. 2020a, b).
2.4 Neurophysiological (EEG‑based) measures
The spectral power of EEG oscillations in different fre-
quency bands (specifically α (4–7Hz), β (8–13Hz), θ
(14–30Hz) may be linked to different levels of workload
by means of analysis of variance (ANOVA) (e.g. Lei and
Roetting 2011; Aricó etal 2018). The potential of an EEG-
based task engagement-index (based on the power ratio β/
(α+θ) recorded from four scalp sites, 40s moving average,
2s clock rate) within the context of adaptive automation was
demonstrated by Prinzel etal. by means of a laboratory type
multi-attribute cockpit-instrument tracking-task simulator
experiment, using ANOVA for quantifying the significance
of the engagement level (Prinzel etal. 2003). The impor-
tant artifact rejection was based on a pre-set threshold volt-
age which for real-world applications of course would not
be sufficient. Meanwhile, classifiers are increasingly used
for the separation of workload levels. In previous publica-
tions, we have described the development and validation of
the new DFHM WL index using a support vector machine
classifier (based on frontal α-band and parietal θ-band pow-
ers), performed under laboratory conditions with standard
task load batteries. Once calibrated for discriminating low,
medium, and high WL levels, it was shown to require no
retraining of the machine learning algorithm, neither for
new subjects nor for new tasks (Radüntz 2016, 2017). For
the present experiment, we used a commercial 25 active-
electrode system (g.tec Ladybird) with 500Hz sample rate
and 0.5–50Hz bandpass. The corresponding data from the
present model-based data analysis showed the objective
DFHM index to provide significant correlation with control-
ler’s subjectively experienced self rating ISA-WL measure
under traffic load variation (Radüntz etal. 2020a, b). For
testing the DFHM-WL index sensitivity, the participants in
this analysis were separated into two groups (low and high
WL sensitivity) according to their individual linearized WL-
sensitivity parameters that were formally derived from the
logistic ISA characteristic of the subjective self-report meas-
ures. Fürstenau etal. (2020). In Sects. 6.2, 6.3, we briefly
address the potential of extending the resource limitation-
based logistic and power law model approach to the new
DFHM-WL index measure, by means of regression-based
parameter estimates.
2.5 Psychophysics ofMental Workload
Despite the fact that subjective WL measures are widely
accepted and used, there have been very few studies exam-
ining their methodological viewpoint. Based on laboratory
experiments with standardized cognitive tasks (Gopher
and Braune 1984; Gopher etal. 1985) proposed a scal-
ing approach that can be traced back to the psychophysi-
cal measurement theory of Stevens (1975). Psychophysical
research aims to describe the relationship between changes
in the amplitude of a physical stimulus (e.g. brightness,
loudness) and the subjective perception of these variations.
The classical Weber–Fechner law assumes a logarithmic
relation between physical stimulus
S
and subjective percep-
tion
P
=cln
(
S∕S
t)
, with an experimentally determined con-
stant
c
and a stimulus threshold
St
that denotes the intensity
of the stimulus at a state with no perception (Buntain 2012).
An improvement was introduced by Stevens (1975). In Ste-
vens’ law, the sensation magnitude is a power function of
stimulus intensity and the corresponding generalized linear
curve (double logarithmic scale) is described by the constant
b and Steven’s exponent γ (slope or sensitivity in log–log
scale) that is characteristic for the type of stimulus.
It is valid also for the stimulus–response transfer between
sensor input (stimulus amplitude) and sensor neurons firing
rate (action potential) (e.g. Birbaumer and Schmidt 2010). The
power law exponent γ with a typically magnitude of the order
of 1 was determined for a large number of different modalities
(e.g. brightness, loudness, apparent length) to adjust the curve
to the different psychophysical functions. Steven’s law was
derived from an information theoretic approach with P ~ per-
ceived sensory (Shannon) information by Norwich etal. (1987),
Norwich and Wong (1997). Within this context, it represents
an approximation for lower amplitude stimuli with prolonged
sampling time, while the Fechner law represents an approxima-
tion for the large amplitude brief stimulus duration. With regard
to workload, Gopher etal. in (1984) argued that, … “if the
human information processing system can be assumed to invest
… hypothetical processing facilities to enable the performance
of tasks then subjective measures can be thought to represent
the perceived magnitude of this investment, in much the same
way that the perception of …” a physical stimulus is changed
with variation of its magnitude. Gopher etal. based their formal
power law relationship on the measured average values across
the sample of 55 participants of perceived load for each of 21
single and dual-task conditions of a task load battery, with tasks
guided by Wickens’ multiple resources paradigm (e.g. Wickens
and Hollands 2000). In contrast to a standard psychophysical
(stimulus–response) experiment, in their WL experiment, there
existed no a priori physical quantity (e.g. brightness or sound
(1)
ln (P)=ln (b)+𝛾ln (S)
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296 Cognition, Technology & Work (2022) 24:291–315
1 3
pressure) that induced the subjective judgement, and that would
allow to derive the two parameters (γ, b) of the power law func-
tion (1) through regression analysis. Instead, they derived a
physical stimulus scale by means of the amount of (Shannon)
information attributed to the task load battery.
Recently, Bachelder and Godfroy-Cooper (2019) reported
on the application of the psychophysics power law to the
analysis of a pilot workload estimation simulator experi-
ment. They used a flight compensatory tracking task with
Bedford hierarchical unidimensional WL scale (a modified
Cooper-Harper rating scale) designed to identify opera-
tors spare mental capacity, while completing the task. The
physical stimulus S determining WL in Eq.(1) was derived
from the measured standard deviations of control error rates.
Theoretically predicted Stevens exponents of different tasks
were in the range 0.24 ≤ γ ≤ 0.41 and compared favorably
with those obtained from regressions of the data using the
power law (1): 0.21 ≤ γ ≤ 0.37, i.e. the order of magnitude
was comparable with those of the classical psychophysics
experiments.
A basic theoretical foundation for the power law was pro-
vided by Link with the stochastic brain wave discrimina-
tion theory (Link 1992) that allows for formal derivation
of psychophysical laws. Starting point was the probability
for reaching a decision threshold through random sam-
pling of the difference between stimulus and referent waves
that defined a logistic response function with exponential
dependence on wave amplitude difference and threshold.
Stevens’ power law was derived from sensation matching
by combining the two corresponding logistic functions. The
ratio of two normalized subjective response thresholds AS/AP
relate two simultaneously measured sensations with logis-
tic response probability functions. The product of this ratio
with the log(normalized sensation of physical stimulus S/S0)
equals the log(normalized subjective response P/P0) in the
generalized linear form of Stevens law (Eq.(1); S0 = stand-
ard stimulus). Based on the cognitive resource limitation
hypothesis as our theoretical starting point (Sect.4), we
use a comparable formal procedure for the derivation of
the ISA(RC) power law, however, in the present approach
through combination of the discrete logistic ISA(n)-WL
response and objective RC(n) task load stimulus charac-
teristics, with the variables assumed as statistical means
from averages across a sufficiently large random sample of
participants.
3 Experiment
Details of our experimental setup and procedures together
with initial results were provided in previous publications
on the validation of the new neurophysiological DFHM WL
index, with different subjective and objective WL measures
as reference (see Sect.2) (Mühlhausen etal. 2018; Radüntz
etal. 2019). Here, we give a brief overview with details
relevant for the validation of the power law WL index
only, based on the combination of ISA-workload data and
the ATCo’s frequency of radio calls with pilots (RC, calls
/ h). The experiment was designed within a collaboration
between the Federal Institute for Occupational Safety and
Health (BAuA) in Berlin and the Institute of Flight Guid-
ance of the German Aerospace Center (DLR) in Braunsch-
weig. Simulation experiments with data acquisition were
performed at the Air Traffic Management and Operations
Simulator (ATMOS) of the DLR. The investigation was
approved by the local review board of the BAuA and all
procedures were carried out with the adequate understanding
and written consent of the participants.
3.1 Procedure andsubjects
Every subject completed eight simulation scenarios in rand-
omized order within two consecutive half days and commu-
nicated online with pseudo-pilots who simulated the cockpit
crews, each one responsible for several aircraft (AC).
Our sample consisted of 13 approach controllers, 3 tower
controllers, and 5 employees of the DLR that exhibited ade-
quate expertise to handle the arrival management simulation
and interact with the pseudo-pilots. In total, we had N = 21
subjects between the ages of 22 and 64years (2 female, 19
male, mean age 38 ± 11) with different work experience who
came from different airports and were familiar with different
work positions.
3.2 Experimental design andworkload assessment
The experiment was conducted for investigation of workload
effects under different task-load levels j = 1,…,8 in a stand-
ard approach sector radar work environment. The load levels
were realized through four different traffic flow conditions
nj (25, 35, 45, and 55 aircraft AC/h) and a dichotomous pri-
ority-flight request event e = 0, 1. The combination of both
independent variables led to eight simulation scenarios (8
scenarios: j = 1, …, 4 without event e = 0 and j = 5, …, 8 with
priority event e = 1).
Radio communication between ATCo’s and pilots rep-
resents a major contribution to the total task load, besides
monitoring the traffic on the radar display (traffic count
n) for anticipating possible separation conflicts (Manning
etal. 2001; Averty etal. 2004; Djokic etal. 2010; Cor-
ver etal. 2016). Because communication (task) load under
nominal conditions increases with traffic count, it seemed
appropriate to use a one-dimensional WL measure for the
experiment. Generation of traffic was realized by means of
well-trained pseudo-pilots in a separate room with com-
puter systems for controlling the simulated pre-defined air
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297Cognition, Technology & Work (2022) 24:291–315
1 3
traffic according to the clearances via the simulated radio
connection to the ATCo at the approach radar work place.
Registration of the start and stop times of ATCo’s radio
calls provided time series that allowed to derive for the
eight scenarios the average radio call duration (RD/sec-
onds) and moving averages as well as the average across
the whole scenario of the frequency of radio calls (RC /
calls per hour).
Participants periodically judged their subjectively expe-
rienced WL in fixed time intervals of 5min by means of
the Instantaneous Self Assessment (ISA) self-report method
(Brennan 1992; Jordan 1992; Kirwan etal. 1997). Their
judgement based on an one-dimensional five-level integer
scale with values corresponding to (1) under-utilized, (2)
relaxed, (3) comfortable, (4) high, and (5) excessive (for
details see Sect.2.2). The realization by means of a touch-
screen for selecting the experienced scale level allowed for
minimum distortion (Tattersall and Foord 1996, Sect.2.2).
According to prior information from experts familiar with
the selected approach sector, for n ≤ n1 = 25 AC/h subjects
were expected to experience low load, while n2 < n ≤ n3 = 45
AC/h was the standard operating range with n3 = nc = sector
capacity) with high load. nc as prior knowledge was also
derived theoretically from the average separation minimum
of given traffic mix (3.1nm/AC) and average approach
speed of 140 kts. The highest traffic flow (n4 = 55 AC/h)
exceeded the realistic maximum traffic nc and served for
driving the load over the acceptable limit according to
experts comments.
Scenarios without priority event (e = 0) had a duration of
20min with four ISA reports, whereas scenarios including
the priority event “sick passenger on board” (e = 1) at simu-
lation time tS = 10min took 25min and contained five ISA
reports. For the theoretical modeling and data analysis, we
used as dependent variables the scenario means < ISA > (nj)
and < RC > (nj) calculated over the whole time series as WL-
rating and TL-value estimate, respectively, for each partici-
pant in the eight scenarios. Tables with pre-processed raw
data and results for individual participants are provided
in our previous publication (Fürstenau etal. 2020) and in
Appendix1 for completeness. In what follows, we restrict
the theoretical predictions, regression analysis and discus-
sion to the means across the 21 subjects.
4 Theory
In this section, we derive a theoretical psychophysical power
law ISA(RC) with exponent γ from the parametric represen-
tation of communication load RC(n), and WL self-report
ISA(n), with asymptotic upper limits ISAu, RCu as prior
information. With suitable normalization and transforma-
tions (S(RC), P(ISA)) into a generalized linear relationship
yP ~ γ yS we obtain a formal equivalence to Stevens’ law
(Eq.(1)), with yS ~ ln(S), yP ~ ln(P).
Starting point for our theoretical model was the assump-
tion of cognitive resource limitation (Kahnemann 1973;
Wickens 2002). The dynamics of growth of a population
or magnitude of a corresponding continuous variable that
increases with time t through consumption of a limited
resource may be formalized through the Verhulst differential
equation with the logistic (sigmoid) function as solution (see
Appendix A2). By replacing the usual time variable by the
independent environmental traffic load variable n, we used
this function as theoretical model for the characteristics of
the measured averages < < > > across the participant sam-
ple of the scenario means of subjective < ISA > (n) and rate
of radio calls < RC > (n) [calls/h]. The reported subjective
value ISA(n) WL level is assumed to measure the fraction
of limited overall cognitive resources (attention, processing,
memory) required for the specific task RC(n). In what fol-
lows, we will use I(n), R(n) where appropriate.
4.1 Logistic ISA(n) model
The logistic resource limitation approach for prediction and
regression-based estimates of ISA(n) model parameters
was used for deriving a linearized WL-sensitivity index
in Fürstenau etal. (2020). It allowed for subject cluster-
ing within the neurophysiological DFHM index validation
(Sect.2.4, Radüntz etal. 2020a, b). A comparable logistic
model approach was used also by Lee etal. for analysis of
ATC-simulation WL data using the 7-level ATWIT method
(Lee 2005) and by (Averty etal. 2008) for formalizing
ATCo’s decision analysis in the context of collision risk
judgement. Main feature is the asymptotic approach to an
upper WL boundary.
For analyzing the measured ISA data, we used prior infor-
mation for the detailed design of the logistic workload char-
acteristic to be fitted to the experimental data (see Sects. 2.2,
3.2). On one hand, prior knowledge concerns the selected
traffic flow range 25 ≤ n ≤ 55 (AC/h) to be handled by the
controllers and on the other hand, the ISA scale. The latter
by definition is limited to the range between ISA: = Id = 1
and Iu = 5 with five integer values 1 ≤ I(n) ≤ 5. In the most
simple approach, this leads to the assumption of constant
minimum and maximum ISA levels of ISA(n): = I(n) = Id = 1
for 0 ≤ n ≤ 25 = underload, and Iu = 5 for n ≥ 55 = excessive
load. If a linear increase is assumed in between, with slope
a ≤ (5–1)/(55–25) = 0.13 (AC/h)−1, this yields as intersection
I(n = 0) = 1–25 a = − 2.33. In reality, an idealized linear I(n |
a, b) characteristic would be different for different individu-
als because of inter-individual variation of task load sensitiv-
ity and transition to underload and overload (see Fürstenau
etal. 2020). Consequently, a random sample of participants
would generate distributions with density functions for slope
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298 Cognition, Technology & Work (2022) 24:291–315
1 3
a (sensitivity > 0) and intersection b (both negative and posi-
tive values possible). By assuming a variable n0 ≤ n1 = 25 for
the underload transition, we get b ≥ − 2.33, and with a > 0,
n0 > 0: − 2.33 ≤ b < 1.
For a more realistic model of the average ISA ratings,
we refer to the above-mentioned standard formalism for
resource limited growth and assume asymptotic conver-
gence of lim I(n > > n4 = 55) = Iu to be modeled by a logis-
tic (sigmoid) function:
With shift parameter μ = ν ln(k), k = Iu/Id − 1 and scal-
ing coefficient ν for the convergence towards the upper
and lower asymptote. ν also characterizes as sensitivity
index the maximum slope I′ = dI/dn = Iu/4ν at inversion
point n = μ with I(μ) = Iu/2). For the nominal traffic (e = 0),
we have k = 4 and μ = ln(4) ν (for mathematical details see
Appendix A2). As initial guess, we select for e = 0 the shift
parameter value μ: = μt: = 35 AC/h, because according to
domain experts a priori information, it corresponds to
the center between underload n1 = 25 and sector capacity
limit n3 = nc = 45, representing the optimum (nearly linear)
operational range for the given conditions, sufficiently far
away from the nonlinear sections (see Fig.1). A reason-
able uncertainty value may be selected as |δμt|= 5, i.e. half
the distance to the boundaries. As shown in Fig.1, the
characteristic features for the nominal case (e = 0, solid
curve) are the predicted effective ISA range between
approximately 2 and 3.5 and the only weak nonlinearity
for the given load variable range 25 ≤ n ≤ 55 AC/h, with
slope value I’(n = μ: = 35) ≈ Iu /4ν = 0.0495 (AC/h)−1(i.e.
significantly smaller than the initial rough estimate) For
comparison with other WL measures and derivation of
the power law, we define the normalized ISA metric
pI = I(n)/Iu through division by the upper asymptotic
value Iu. Via definition of the transformed ISA variable
P = p/(1–p) = I(n) / (Iu–I(n)), we arrive at the exponen-
tial dependence P(n) = 1/k exp(n/ν). Taking the logarithm
transforms this exponential characteristic into the general-
ized linear model y(n) = ag n + bg with parameters ag = 1/ν,
bg = − ln(k):
For the nominal case (e = 0) with μt = 35 AC/h (= n2,
operational traffic), the theoretically predicted slope
value (WL sensitivity) is obtained as agt = 1/νt = ln(4) /
μt = 0.0396 (AC/h)−1 or μt = 25.25 AC/h, and intersection
bgt = − μt/νt = − ln(4) = − 1.3863.
(2)
I
(n)=
I
u
1+exp
{
−n−𝜇
𝜈
}
=5
1+kexp
{
−n
𝜈
}
(3)
y
p(I(n)) =ln[P]=
n−𝜇
𝜈
=
1
𝜈
n−ln(k
)
We expect any effect of the priority request in simula-
tion runs j = 5–8 to generate an increase of slope of yp(n,
e = 1) from the nominal value (1/νe > 1/ν or νe < ν), how-
ever, only for traffic load larger than a threshold value nx,
i.e. n ≥ nx > underload traffic n1. This generates an intersec-
tion between the e = 0 and e = 1 sigmoids at nx defining a
critical threshold for onset of the priority effect (bifurca-
tion of e = 0 into separate e = 0, e = 1 characteristics for
nx > n1, with Ix > I1 and I(e = 1) > I(e = 0) for n > nx).
Basically, for the non-nominal (e = 1) simulations,
parameter estimates (μe, νe) have to be determined by two-
parameter (ke, νe) regression of the experimental data using
model Eqs. (2) or (3) due to lack of prior knowledge on the
magnitude of the WL effect of the priority event (in contrast
to e = 0). However, a one-parameter model (like for e = 0)
may be derived by means of a plausibility argument (prior
knowledge) for the intersection coordinate (nx, Ix) between
e = 0 and 1 characteristics that in turn allows for deriving a
relation between μe (or ke) and νe: μe(νe, nx) or ke(νe, nx). For
the non-nominal scenarios (e = 1), the shift parameter μe is
derived as (for details see Appendix A2)
A prior estimate of nx may be obtained with reference to
the multiple resources theory (Wickens 2002). The nomi-
nal traffic management task and the major part of addi-
tional decision-making due to priority request are both
(4)
𝜇
e=nx
(
1−
𝜈
e
𝜈)
+𝜈eln(4
)
Fig. 1 Theoretical ISA(n) characteristics (Eq.(2)) for nominal (solid
curve, e = 0: μ = 35, ν(μ) = 25.2) and non-nominal scenarios (dashed
line, priority event, e = 1: νe = 20, nx: = 30, μe (νe, nx) = 33.9). Intersec-
tion point (nx, Ix) = (30, 2.3). Abscissa: independent traffic load vari-
able 0 ≤ n ≤ 100 / AC/h. Ordinate: ISA-WL with ISA(n = 0): = Id = 1
for e = 0, asymptotic limit I(n) = Iu = 5 for n > > nc = 45; for details see
text and Appendix2
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299Cognition, Technology & Work (2022) 24:291–315
1 3
perception–cognition–communication tasks. The additional
task due to the priority request consists in checking for the
possibility of a direct route to final approach depending
on the traffic situation. Both nominal and priority flights
require traffic monitoring on the approach radar display and
communication with pilots, that use overlapping mental
resources. However, we may argue that the task of exclud-
ing a potential separation conflict for the changed routing
option generates additional WL only under higher traffic
load (n > nx), with nx between underload and operational
traffic (n1 = 25 < nx < n2 = 35 AC/h). Only small additional
mental resources and corresponding neglectable WL change
is expected for n < nx. Consequently, a plausible prior value
is nx:≈ 30 AC/h with (plausible) maximum nx-uncertainty
given by the n2–n1 interval: δnx: = (n2–n1)/2 = ± 5, yielding
δIx = ± 0.5 (error propagation including the independent
shift parameter (μ: = 35) uncertainty δμ: = ± 5). These esti-
mates based on domain experts prior knowledge allows for
deriving a plausible a-priori estimate for the bifurcation
point {nx, Ix} = {30, 2.25}. Figure1 depicts two theoreti-
cally predicted ISA(n) characteristics according to Eq. (2)
for the extended traffic flow interval 0 ≤ n ≤ 100 AC/h.
The solid line represents the nominal scenarios (e = 0)
with μ: = μt = 35 (ν(μt) = 25.2) (μt = inversion point, center
of the nearly linear range between underload n1 and nc = n4).
The dashed sigmoid shows an example with increased WL
sensitivity 1/νe (e = 1: νe = 20) with intersection at nx: = 30
and μe(νe, nx) = 33.9 < μ = 35, according to Eq.(4). For
n > nx, the sigmoid exhibits the predicted subjective WL
increase for e = 1, whereas for n < nx (underload range), the
priority scenarios are expected to follow the e = 0 cur ve (i.e.
dashed continuation to be ignored). The simulated traffic
range 25 ≤ n ≤ 55 covers the nearly linear section of the sig-
moid curves. This predicted quasi linearity was used in our
previous ISA data analysis (Fürstenau etal. 2020) for deriva-
tion of a linearized WL-sensitivity index (see Appendix2)
that was successfully applied to the analysis of the simul-
taneously monitored neurophysiological DFHM index (see
Sects. 2.4, 6.2, 6.3, Radüntz etal. 2020a, b) with regard to
participant clustering.
4.2 Logistic RC(n) model
Assuming a nearly linear increase of radio communica-
tion between ATCo and pilots with traffic flow n for small
RC (calls/h) (i.e. for small n < n1 R(n) ~ n with asymptotic
approach to the maximum Ru), the logistic R(n) character-
istic is given by
(5)
R
(n)=Ru
[
2
1+exp{−n∕𝜌}−1
]
With n/2ρ: = x, the normalized rate of radio calls R(n)
/ Ru: = s(n) may be written in short as tanh(x) (for math-
ematical details see Appendix A2). It is easily verified that
for n > > nc the dimensionless variable R(n)/Ru: = s(n) = 1.
If we introduce as prior knowledge an estimate of aver-
age radio call duration of TD ≈ 4s (see Sect.5.2), an
estimate for the asymptotic maximum number of calls
per hour may be obtained by Ru: = 3600 / (TD(ATCo) +
TD(Pilot) + TD(Pause)) ≈3600 / (4 + 4 + 1) = 400 calls / h.
Taking Ru: = 400 as prior knowledge, Eq.(5) turns into a
one-parametric model. A rough theoretical estimate for
the scaling parameter ρ may be obtained from a linear
extrapolation of the maximum slope at n = 0 as Δs /Δn = 1 /
2ρt yielding
𝜌t∶≈
nc/2 = 22.5 (see Appendix2, Eq.A2.11;
with Δs = 1, and Δn: = capacity limit nc = n3). The slope
at the inversion point (linearized sensitivity) is predicted
as 1 / 2ρ = 0.02 > 1 / 2ν = 0.01, i.e. larger than the WL
sensitivity.
Through normalization and logarithmic transformation,
the nonlinear characteristic (5) may be transformed into
a generalized linear model, comparable to yp(n) (Eq.3).
With the normalized and transformed radio calls variable
S = (1 + s) / (1 – s) = (Ru + R(n)) / (Ru – R(n), we arrive at
the exponential dependence S(n) = exp(n/ρ). Taking the
logarithm transforms this exponential characteristic into
the generalized linear form of the radio calls sigmoid char-
acteristic ys(n) = ln(S) = asg n + bsg or
with slope 1/ρ: = asg as RC task load sensitivity param-
eter and bsg = 0 (see Appendix A2 for details). The choice
of variable name S and index s indicates the usage of
the transformed RC variable as physical stimulus for the
(transformed) subjective ISA-WL variable P (for report of
subjective perception of the physical stimulus) according
to Eq.(1) (see following Sect.4.3).
In contrast to the ISA(n, e) curves with prediction (for
n > nx) of the non-nominal scenarios ISA(n, e = 1) > ISA(n,
e = 0), we may expect for RC(n) the inverse behavior: R(n,
e = 1) < R(n, e = 0). According to (Sperandio 1978) approach,
controllers under (suddenly) increased traffic load (in our
case the occurrence of a priority request as non-nominal
event with increased task load) prefer switching of control
strategy to standard procedures with global routing for most
AC, i.e. global approach sequence with pilots responsible for
controlling the standard separation distance. Consequently
for ATCos, control of the first AC in the AC sequence will be
sufficient, resulting in decreased RC(n) with corresponding
decrease of ISA-WL, and attention resources free for focus
on the priority event (see discussion in Sect.6). Because
our initial model assumption, RC(n = 0)) = 0 should be true
(6)
y
s(n)=ln(S)=
1
𝜌
n
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300 Cognition, Technology & Work (2022) 24:291–315
1 3
for both e = 0 and 1 the intersection of both characteristics
is predicted at {nsx, Rx} = {0, 0}. Fig.2 depicts predicted
theoretical radio call rate (calls/h) characteristics for nominal
traffic (e = 0) and scenarios with non-nominal event (e = 1
with somewhat decreased TL sensitivity, i.e. increased ρe,
value selected as example):
4.3 Power law model forISA(RC)
The power law for the ISA(RC) characteristic may be
derived from the parametric representation [ISA(n), R(n)]
by introducing n(R) as obtained from Eq.(5), into Eq. (2)
(for details see Appendix A2). Using prior information on
upper (asymptotic) limits Iu = 5, RCu = 400 calls/h and lower
limits I(n = 0) = Id = 1, RC(n = 0) = 0 the normalized nonlin-
ear ISA(RC) characteristic p(s) with p = I/Iu, s = RC/Ru is
obtained as a two-parametric model (γ, k) with γ = ρ/ν and
μ/ν = ln(k) = − bg (see EQ. (3))
With k = 4 for the nominal case (e = 0), this is reduced
to a model with power γ as the single free parameter and
a theoretical estimate obtained from the stimulus–response
ratio γt: = ρt/νt = 22.5/25.25 = 0.89, i.e γ is predicted to be of
the order 1 as usually observed for psychophysics power law
exponents measured in classical stimulus–response experi-
ments ( e.g. (Stevens 1957; Link 1992; Bachelder and God-
froy-Cooper 2019) and references therein). As expected and
shown in the following Fig.3 for three examples (γ = 0.8,
(7)
p
(s)=
1
1+k
[
1−s
1+s]
𝛾
1.0 and 1.2), all characteristics converge independently from
the single parameter γ to p = 0.2 (ISA = 1) for s = 0 (RC = 0),
and to p = 1 (ISA = 5) for s = 1 (RC = RCu = 400).
Like for ISA(n) the nonlinear power law Eq. (7) may
be transferred into a generalized linear relationship that is
obtained after transformation of p and s into the dimension-
less variables P(p) = p / (1–p) and S(s) = (1 + s) / (1 – s),
respectively, (for details see Appendix2):
with bs = bgt = − ln(4) and γ defining the slope of the
generalized linear (log–log) form yp(ys) of Stevens law
(Eq.1) corresponding to
P=bS𝛾
in linear coordinates with
b: = 1/k = exp(− μ/ν).
For our experimental scenarios j = 5…8 with additional
task load due to the non-nominal event e = 1, the inter-
cept bs = bgt = − ln(4) of the e = 0 scenarios is replaced by
the second free model parameter bse. The unknown shift
parameter μe (< μ) of the generalized linear characteris-
tic together with γe > γ defines a two-parameter power law
model with offset change bse < bgt = − 1.386 and an inter-
section with the nominal characteristic at Rx or sx, respec-
tively (for details see e.g. Eq. (A2.16) in Appendix2).
Again, like for ISA(n), it appears plausible that for e = 1
additional task load leads to ISA(RC)-WL increase only for
radio call frequencies RC > Rx = R(nx) (corresponding to
p > px and yp > ypx for n > nx). Rx characterizes the commu-
nication underload threshold. Based on our prior numeri-
cal prediction of parameters nx: = 30, ρt ≈ 22.5, νt = 25.3,
bgt = − ln(4), we may derive a rough theoretical prediction
(8)
yp=ln (P)=
𝛾
ln (S)−ln(k)=
𝛾
ys+bs
Fig. 2 Plot of theoretical radio calls rate R(n) (Eq.5) with sensitiv-
ity parameter ρ = 23 (solid line: nominal traffic e = 0) and ρe = 25
(dashed line: priority event, e = 1,). Maximum slope with linear
increase at origin R(n = nx = 0) = 0. Asymptotic limit of calls per hour
for n > > nc, Ru: = 400 as prior information (for details, see text and
Appendix2)
Fig. 3 Theoretical power law characteristics p(s) for nominal case
(e = 0: k = 4) with normalized variables using Eq.(7), with γ = 0.8,
1.0, 1.2, from top to down. Abscissa: normalized radio calls rate
RC/Ru; ordinate: normalized WL ISA/Iu. For details see text and
Appendix2
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301Cognition, Technology & Work (2022) 24:291–315
1 3
for the hypothesized intersection coordinates of the gener-
alized linear power law (8) of nominal (e = 0) and priority
event (e = 1) characteristics, to be compared with the exper-
imental results in Sect. 5.4 (for details see Appendix2):
{
R
x
,I
x}
=
{
R
u
tanh
(
n
x
∕2𝜌
)
,I
u
∕
(
1+4 exp
[
−n
x
∕𝜈
])}
or
{ysx, ypx} ≈ {nx/ρt, nx/νt – ln(4)} = {1.3, − 0.2}.
Like for non-nominal (e = 1) scenarios of ISA(n) in Sect.
4.1, we can also derive for the ISA(RC) power law charac-
teristic a generalized linear regression model with only one
free parameter by utilizing the prior estimate of the e = 0,
e = 1 intersection coordinate nx: = 30. Introducing into Eq.
(8), an expression for the offset bs(e = 1): = bse(γe) = ypx – γe
ysx yields for the one-parametric non-nominal model
leaving γe as single free parameter of the non-nominal
model equation that is valid for ys ≥ ysx (for details see Appen-
dix2). This means that corresponding to Fig.1 for the logis-
tic ISA(n) characteristics in linear coordinates, also the power
law characteristic exhibits a bifurcation of ISA(RC) at inter-
section coordinate Rx into separate branches for the nomi-
nal and non-nominal scenarios (i.e. for RC > Rx): ISA(RC |
e = 1) > ISA(RC | e = 0)). It should be kept in mind that all the
above theoretical predictions are valid only for the means of
a sufficiently large statistical sample of participants.
5 Experimental results
In what follows, we use the above theoretical characteristics
and numerical predictions for (nonlinear) regression analysis
of the experimental subjective ISA-WL and objective radio
calls communication (RC-TL) data with logistic and power
law models. This analysis is based on the set of scenario
means averaged across the 21 participants (< < ISA(nj) > > ,
< < RC(nj) > > (j = 1,…,8; see Appendix1for complete pre-
processed dataset). In contrast to the traffic flow n (AC/h)
as independent environmental load variable the measured
time series of radio calls between controller and pilots rep-
resents a resource limited controller activity with upper limit
Ru, well defined by simple considerations of available and
required communication time (see Sect. 4.2).
After presenting the experimental ISA(n) and RC(n)
results with regression analysis for scaling parameter esti-
mates ν and ρ in Sect.5.1 and 5.3, respectively, we focus
in Sect.5.4 on the correlation between ISA-WL and RC-TL
data. In what follows (where not mentioned otherwise), we
include as uncertainties for parameter regression estimates
(ν, ρ, γ) standard errors ε of means (= standard deviation /
√N), with 95% confidence intervals CI = ε t, and Student’s
t(95%) ≈ 2.1 for N − 1 = 20 degrees of freedom. Linear and
nonlinear (iterative) regressions were performed with the
Matlab® statistics toolbox using “fitlm” and “nlinfit”.
(9)
ype(
y
se)
=𝛾
e(
y
se
−y
sx)
+y
px
5.1 Logistic < < ISA > > (n) characteristic
For the present purpose, we analyze the means across par-
ticipants with the generalized linear version of the logis-
tic model (Eq. (3)). We quantify the scaling (sensitivity)
parameter ν for the nominal (e = 0) scenarios through appli-
cation of the one-parameter model, using the theoretical
intercept bgt = − ln(4) = − 1.3863. The lower ISA scale
limit Id = 1 allowed for deriving the dependency between
slope and shift parameter μ = ν ln(4). The non-nominal
(e = 1) case with increased slope 1/νe (and consequently
ke, μe) requires a two-parameter estimate (νe, ke) due to
the a-priori unknown intersection Ide < Id(n = 0 | e = 0) = 1
of the non-nominal sigmoid. Both regressions provide an
experimental estimate for the predicted intersection at (nx,
Ix) between the e = 0 and e = 1 curves. The logistic fit model
for e = 1 neglects the small deviation originating from the
(expected) merging of the e = 0 and e = 1 characteristics for
n < nx. Figure4 depicts in semi-log coordinates the result of
fitting transformed ISA variable yp(I) = ln(p(n)/(1 – p(n))),
p = I(n)/Iu, with Eq. (3).
The slope parameter (± stderr) for e = 0 is estimated
as ag = 1/ν = ln(4)/μ = 0.0380 (± 0.0004) with T test
p(T = 110) = 1.7 10–6. It corresponds to ν = 26.32 (± 0.3) and
μ = 36.49. This result provides evidence that the theoretical
offset bgt = − ln(4) derived for the generalized linear logistic
model is in fact a good approximation for the e = 0 scenarios.
As expected, the two-parameter regression of the e = 1
group of simulations (with priority event) yields less pre-
cise parameter estimates (stderr): age = 0.0471 (0.0013)
or νe = 1/age = 21.231, wit h p(|T|= 37) = 0.0007; bge = − μe
/νe = − 1.670 (0.05), with p(|T|= 32) = 0.001. Nevertheless,
the CI(95%) in Fig.4 clearly separate the transformed logis-
tic < < yI > > (n) fits for the two factor-2 groups.
Through the inclusion of the theoretical intercept
bg: = bgt = − ln(4) as prior knowledge for e = 0, and two-
parameter regression (age, bge) for e = 1 the crossing coor-
dinates of the generalized linear fits confirm (for the par-
ticipant sample means) the minimum traffic flow n = nx as
underload threshold:
nx,ypx
=
(
bge−bg
)
(
a
g
−a
ge)
,agnx+bg
={31.2, −0.201
}
, and
through back-transformation Ix = Iu/(1 + exp(− ypx)) = 2.25,
in agreement with the theoretical predictions within the
given uncertainty (for details see Appendix2). Estimates
of uncertainty (sterr.) may be derived from those of the
above parameters through error propagation yielding: {δnx,
δypx} = {0.6, 0.03} and δIx = 0.013. i.e. the experimental
uncertainty δnx/nx ≈ 2% is an order of magnitude smaller
than the prior estimate (5/30 ≈ 0.2). So for the average
across participants, the experimental results confirm the
theoretical prediction that below threshold nx (see Sect.
4.1) the priority event induced additional task load does
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302 Cognition, Technology & Work (2022) 24:291–315
1 3
not generate reporting of any additional workload, of
course with large inter-individual variation (as detailed in
(Fürstenau etal. 2020)).
5.2 Radio Call Duration < < RD > > (n)
Figure5 depicts the observed linear decrease of radio call
duration (RD, as mean over the 21 subject sample) with
increasing traffic flow. This result is in agreement with findings
of Djokic etal. (2010). According to these authors, the radio
frequency occupation time as determined by radio call rate
RC (frequency of radio calls, not to be confused with physi-
cal radio transmission frequency) and radio call duration RD
represents the communication load as significant factor deter-
mining the workload. They report an increase of perceived WL
with increasing overall frequency occupation time and with
decreasing RD. We will show below that this agrees with our
results with regard to RC(n) and ISA(RC). In terms of control-
ler strategy, reduction of call duration may be understood as
a method to reduce or stabilize workload in case of task load
increase, e.g. through increase of traffic (Sperandio, 1978).
From underload (25 AC/h) to overload (55 AC/h) RD
reduces from ca. 4 to 3.6s/call, i.e. a decrease of 10%,
independent of factor 2 (e = 0 or 1). This is consistent with
(Manning etal. 2001) who measured for en-route sector
radar control an average (± sterr) of 3 (± 1) s. Assuming the
same duration for the pilot response, the duration of com-
munication events (e.g. for pilots clearing request) is 7–8s.
From this number, we may derive an asymptotic upper limit
of radio call frequency as a rough estimate when we add a
minimum average interruption between ATCos calls of 1s.
With 2 × 4 + 1 = 9s, we obtain as maximum RCu = 3600 / 9
≈ 400 calls/h.
5.3 Logistic radio call‑frequency
characteristic < < RC > > (n)
The iterative logistic two-parameter fit (RCu, ρ) with Eq.(5)
of ATCO’s frequency of radio call (RC / calls/h) for both
factor-2 cases e = 0, 1 is presented in Fig.6.
The regressions exhibit a common quasi exponential
convergence of (e = 0, 1) towards RCu ≈ 400 h−1, precisely
(± sterr): RCu(e = 0) = 388 (± 10) and RCu(e = 1) = 401
(± 12), that agrees with the theoretical prediction in the pre-
vious section. Within standard errors, parameter estimates
Ru (± 3%) are the same. Also, scaling parameter estimates
ρ = 19.6 (± 0.98) ρe = 21.9 (± 1.1) are reasonably close to
the linearized theoretical prediction (ρt:≈ 22.5) in Sect. 4.2.
Only weak evidence is observed for a difference of scaling
parameters ρ, ρe between nominal and non-nominal sce-
narios (e = 0, 1, respectively) with measured relative sterr.
of ± 5%.
The evidence for a common asymptotic limit (400 h−1) is
tested with the generalized linear one-parameter (ρ) model
(6) using nor malized variables < < RC > > /Ru: = s, and
transformation S(s) (see Appendix2, Eq.A2.12) for a linear
regression as depicted in Fig.7.
The slope estimates (with sterr) with linear regression
are 1/ρ: = as = 0.0479 (0.0006) with p = 5 10–6 (|T|= 77),
ase = 0.0459 (0.0004) with p = 2 10–6 (|T|= 105). Standard
Fig. 4 Transformed ISA measurements (participant sample means
of the four scenario averages for e = 0 (j = 1–4: circles) and for e = 1
(with priority event, j = 5–8, crosses). Abscissa: traffic load n (air-
craft / hour); ordinate left: log(natural) of transformed ISA, right: ISA
scale. Solid lines: linear regressions with 95% confidence intervals
(dashed) using generalized linear logistic model with one-parame-
ter regression (ν, Eq.3) for e = 0 scenarios, and with two-parameter
regression (μ, ν) for e = 1. Intersection of e = 0, 1 lines observed at
(nx, yIx) ≈ (31, − 0.2)
Fig. 5 Radio call duration RD(n) (ordinate) as dependent on traffic
flow n (Abscissa). Measured scenario mean values j = 1–8, each aver-
aged over the 21 subjects sample, separated for factor 2 (e = 0: crosses
j = 1–4, nominal traffic; e = 1: squares j = 5–8, with priority event).
Least squares fits: solid/dashed lines for e = 0/1
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303Cognition, Technology & Work (2022) 24:291–315
1 3
errors ≤ 1% and T test suggest to reject the 0-hypothesis, i.e.
within the 95% CI a significant effect of the priority event
leading to reduced radio call rate is observed.
5.4 Power Law Characteristic
of < < ISA > > (< < RC > >)
The ISA(RC) power law (Eqs.7, 8) was derived from the
parametric representation {R(n | ρ), I(n | ν)} (for details see
Appendix2). By eliminating n, it provides the direct depend-
ence of the subjective ISA-WL response on objective com-
munication task load (RC-TL) through the exponent γ. With
ν and ρ estimates in Sects. 5.1 and 5.3, we can predict the
power law exponent via the theoretically derived ratio γ =
ρ/ν of the measured logistic parameter values for compari-
son with the direct estimate of γ (independent of n) using
Eqs. (7, 8), and with the theoretical estimate in Sect. 4.3.
The numerical estimates are collected in Table2, separated
for nominal/non-nominal scenarios, γ and γe, respectively.
A significant increase is observed for the priority event sce-
narios γe (e = 1) as compared to γ(e = 0). This means that the
sensitivity (= slope) ag = 1/ν of the transformed subjective
ISA(n)-WL characteristics increases significantly more for
the priority scenarios than slope asg = 1/ρ of the transformed
objective communication variable R(n).
The nonlinear (iterative) LSQ
fit < < ISA > > (< < RC > >) with model Eq. (7) of the
eight measured scenario means, each averaged over the par-
ticipants and separated for factor 2 (e = 0, 1) is depicted in
Fig.8.
In normalized coordinates, the s(n) = R(n)/Ru measure-
ment range covers 0.5 ≤ s ≤ 0.9 with p(n) = ISA/Iu range
0.38 ≤ p ≤ 0.68 which may be compared with the theoreti-
cal prediction in Fig.3. Like ISA(n) also ISA(RC) exhib-
its a significant increase for priority event scenarios (e = 1,
dashed line) as compared to the nominal case (solid line),
partly due to the inverse behavior of the calls vs. traffic flow
RC(n) in Fig.6 and 7. With the (k, γ) parameter estimates
(e = 0: {4, 0.792}; e = 1: {5.28, 1.025}), the measured cross-
ing coordinates are calculated as {Rx, Ix} = {214, 1.96} with
standard errors of the order 2%. These values are close to
the low traffic underload region, in reasonable agreement
with the theoretical prediction {Rx, Ix} ≈{233, 2.3} (see
Sect.5.1 and Appendix2, Eq.A2.18). For comparison, the
linear regression of the transformed yP(p) vs. yS(s) data (with
the 95% confidence intervals) with the generalized linear
model Eq. (8) is depicted in Fig.9.
The one-parameter fit estimate of slope γ (± stderr)
for the nominal scenarios e = 0 (with theoretical value
k: = kt = 4, intercept bs: = − ln(k) = − 1.386) is obtained as
γ = 0.7933 (0.011; t = 69.9, p = 6.4 10–6). For the non-nomi-
nal scenarios (e = 1), the two-parameter fit yields: γe = 1.025
(0.069); t = 14.7, p = 0.046; bse = − ln(ke) = − 1.668 (0.132);
t = − 12.6, p = 0.006, i.e. ke = 5.302, with somewhat reduced
confidence as depicted in Fig.9. The values confirm the
above results of the iterative NL regression with sufficient
significance according to t tests within 95% CI. Introducing
Fig. 6 Frequency of ATCos’ radio calls for the eight scenarios
RC(nj), j = 1–8, separated for factor 2 (e = 0: circles, j = 1–4; e = 1:
crosses, j = 5–8) represented by scenario means averaged across the
21 subjects sample. Nonlinear regressions based on two-parametric
(RCu, ρ) logistic Eq.(5) with RC(n = 0) = 0
Fig. 7 Frequency of ATCos’ radio contacts for the eight scenarios nj,
j = 1–8, separated for factor 2 (e = 0, j = 1–4, circles); e = 1, j = 5–8,
crosses)). Ordinate ys(n): transformed RC variables of normalized
radio call rates R(nj)/Ru (scenario means averaged across partici-
pants). Linear regressions (solid lines) based on one-parametric (scal-
ing ρ) generalized linear (log-lin) form of logistic model (Eq.6) with
Ru: = 400 as prior knowledge. Dashed/dotted lines: 95% CI for e = 0/1
condition
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304 Cognition, Technology & Work (2022) 24:291–315
1 3
these parameter estimates into the intersection equations
for the fit parameters (Appendix2, Eqs.A2.18, A2.19) pro-
vides the bifurcation coordinates in transformed double-log
coordinates {ysx, ypx} = {1.216, − 0.422} corresponding to
{Rx, Ix} = {217, 2.0}, in agreement (within given uncertain-
ties of 2%) with the above results of the nonlinear fit. It
also agrees reasonably well with the rough theoretical esti-
mates obtained with the a priori assumptions in Sect.4.3
(nx: = 30, ρ: = ρt = 22.5), yielding {ysx, ypx}t = {1.33, − 0.20}.
By means of the experimental estimate nx = 31.2 (± 0.6) in
Sect.5.1 (i.e. uncertainty reduced by factor 10 as com-
pared to the prior guess) and a relationship for the inter-
cept bse(ν, νe, ρe, nx) (Eq.A2.20, Appendix2), we obtain a
one-parameter fit with Eq. (9) also for the priority scenarios
(e = 1), yielding γe = 1.058 (0.016); t = 65.6, p = 7.8 10–6, i.e.
Table 2 Comparison of theoretical predictions with experimental parameter estimates
Column 2, 3: Logistic parameter estimates for yp and ys vs. traffic flow n (from separate one- and two-parameter regressions with generalized lin-
ear (log-lin) model for e = 0, 1, respectively). Column 4, 5: Comparison of power law exponent γ = ρ/ν (from logistic parameters) with indepen-
dently estimated γ using nonlinear and generalized linear model yp(ys) (rel.sterr. < 2% for 1/ν and ≈ 1% for ρ). Last row: intersection coordinates
of e = 0, 1 characteristics. For details, see text
Parameter estimate
(sterr) ν
log-lin model Equ.(3)ρ
log-lin model Equ.(6)γ
power lawmodel
Equ.(7) nonlin.
regression
γ
log–log power law
model Equ. (8), lin.
regression
γ
T-testp(T)
Theory (e = 0) ≈ μt/ln(4) ≈ nc/2 ρt/νt (ypx + ln(4))/ysx
μt: = 35, nc = n3= 25.25 = 22.5 = 0.89 = 0.89
Nominal Scenarios (e = 0) 1-param. fit 26.3 (0.3) 20.9 (0.3) 0.79 (0.01) 0.79 (0.01) 6.4 10–6 (70)
Priority Scenarios (e = 1)
2-parameter fit 21.2 (0.6) 1.03 (0.03) 1.03 (0.07) 0.046 (15)
1-parameter fit 21.8 (0.2) 1.06 (0.02) 7.8 10–6 (66)
e = 0/1-Intersection {nx, Ix} = {nx, Rx} {Rx, Ix} = {ysx, ypx} =
Theory {30, 2.3} := {0,0} {233.1, 2.3} {1.33, − 0.2}
Experiment (sterr) {31.2 (0.6), 2.3 (0.3)} {214 (4), 1.96
(0.04)} {1.2, − 0.42}
Fig. 8 < < ISA > > vs. < < RC > > scenario means (j = 1–8) averaged
across participant sample together with model-based nonlinear fit
(Matlab NLINFIT, using Eq.7). Separated for e = 0 (circles, j = 1–4,
solid regression line) and e = 1 (crosses, j = 5–8, dashed regression
line). Power law fit parameters (k, γ) with standard errors. Intersec-
tion coordinate of e = 0, 1 curves observed at {Rx, Ix} ≈ {215, 2}. For
details, see text and Appendix2
Fig. 9 Normalized, transformed (yP(p) vs. yS(s)) scenario means (j = 1
–8) < < ISA > > (< < RC > >) averaged across the 21 participant sam-
ple together with model-based generalized linear (log–log) fit (using
Eq. (8), solid lines), separated for factor 2. (e = 0: j = 1–4, circles,
1-parameter fit (γ), dashed lines: 95% CI; e = 1: j = 5–8, crosses, two-
parameter fit (γe, bse) dotted lines: 95% CI). For details see text and
Appendix2
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305Cognition, Technology & Work (2022) 24:291–315
1 3
a significantly improved confidence. Table2 summarizes
the results of the model-parameter estimates based on the
nonlinear and generalized linear characteristics fitted to the
scenario means averaged across the 21 subject samples.
6 Discussion
Based on the cognitive resource limitation hypothesis and
prior knowledge (domain experts, WL, and TL scales),
logistic two-parameter models (shift, scaling) were designed
for theoretical parameter prediction and regression-based
parameter estimates of the experimental subjective ISA-WL
and objective communication (RC-TL) data. Traffic flow n
(aircraft per hour entering the approach sector) served as
(environmental) independent load parameter (four levels of
traffic load) that defined the scenarios in the human-in-the-
loop simulation experiments of approach traffic control. A
non-nominal event (priority request, e = 1) was hypothesized
to increase WL measures at higher load levels (n > nx > under-
load n1). For nominal traffic load scenarios (e = 0) pr ior
knowledge and simple communication time considerations
allowed to derive asymptotic upper scale limits for normali-
zation of dependent variables (ISA, RC) and simplification
of model equations to one free (scaling) parameter (νt and ρt)
for ISA-WL and RC-TL, respectively. As a consequence the
transformation (from linear scales) of logistic ISA and RC
characteristics into generalized linear characteristics (log–log
scales) yp(ISA), ys(RC) allowed to derive a yp(ys)) power law
relationship, independent of environmental load n, with theo-
retical exponent γt = ρt/νt as formal equivalent to the classical
stimulus–response (Stevens) law of psychophysics.
6.1 Subjective Workload Measure ISA(n)
A covariance and nonlinear regression analysis of the com-
plete set of individual ISA(n) data (see TableA1.1 in Appen-
dix1) in a previous article (Fürstenau etal. 2020) provided
initial evidence for the validity of the logistic model. In
fact a normalized ISA-WL-sensitivity index derived from
a linear approximation to Eq. (2) was successfully applied
to clustering of the participants with regard to a new neu-
rophysiological WL index (DFHM, see Sect. 2.4) that was
measured simultaneously with the ISA and communication
data within the present experiment (Radüntz etal. 2020a, b).
With a plausibility argument for a numerical estimate of μ
(: = (n1 + n3)/2 = 35 = operational traffic n2) also the theoretical
prediction for the logistic scaling parameter ν (theory: ν =
μ/ln(4)) exhibited surprising agreement with the correspond-
ing regression parameter estimate (see Table2, Sect. 5.4).
The experimental results confirmed the theoretically predicted
subjective ISA range (for means across participant sample) for
the given range of traffic flow levels n1 ≤ n ≤ n4 which were
selected by domain experts as realistic a-priori values for the
specific approach area (≤ n3), with n4 as excessive load.
For the non-nominal scenarios (including a priority event
e = 1), a theoretical estimate of the hypothesized sensitivity
increase (1/νe > 1/ν, for n > nx) did not exist so that also the
second model parameter had to be estimated through regres-
sion. The numerical parameter estimates are included in
Table2 (Sect. 5.4). For the predicted intersection between
nominal and priority event scenarios, an experimental value
nx = 31.2 (± 0.6) was obtained with Eq.A2.9 (Appendix2)
using parameter estimates (ag, age, bge). This is reasonably
close to the initial theoretical guess nx = 30 (± 5), however,
with an uncertainty reduction by an order of magnitude.
It should be pointed out that the significance of the ISA-WL
increase of the non-nominal scenarios was based on a con-
servative analysis of data, because we averaged the ISA values
across the full scenario times, whereas the (pseudo) pilots’
priority request during the e = 1 scenarios was introduced after
simulation time ts = 10min. It means that any influence on
workload could be effective only during ts > 10min so that
the restriction on the ISA averaging for ts > 10min should
increase the scaling parameter differences between e = 0, 1,
however, at the cost of increased uncertainty. This successful
validation of the logistic ISA(n) model, including the bifurca-
tion coordinates (nx, Ix) provided the initial evidence for the
basic hypothesis of cognitive resource limitation.
Our logistic modeling approach may be compared with
results of a simulated ATC HITL-simulation experiment
reported by Lee etal. (2005). They used a heuristic sigmoid
function based nonlinear 4-parameter regression for ATWIT-
WL data analysis (see Sect. 2.2) which exhibited significant
parameter estimates.
6.2 Objective task load measure radio calls rate
RC(n)
The theoretical predictions in Sect. 4.1 and 4.2 showed
that within the traffic load range n1 ≤ n ≤ n4, in contrast to
the nearly linear increase of the subjective workload ISA(n)
the radio calls rate RC(n) as task load (TL) exhibits a clear
nonlinear resource limitation behavior (see Fig.2). This
correlates (nonlinearly) with the decreasing mean call
duration from 4 to 3.6s and it agrees with the experimen-
tal results (Fig.6). It is interesting (and counter-intuitive)
that the RC(n)-TL appears systematically higher for the
nominal traffic case (e = 0) as compared to the case of a
non-nominal event (e = 1), in contrast to the ISA behavior
where just the opposite is observed. Although the decrease
of slope ase = 1/ρe amounts to only ≈ 4% (≈ decrease of
call rate at n = nc = 45 the difference is significantly larger
than the standard error of 1% for 1/ρ. Therefore, at the same
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306 Cognition, Technology & Work (2022) 24:291–315
1 3
traffic level n, the average of controllers basic communica-
tion activity (< < RC > > calls/h) is estimated significantly
lower with the additional task (e = 1 scenar ios, j = 5–8), with
regard to sterr. and predicted by theory.
As explanation, the radio calls rate is thought to be
decreased during a priority event (reducing the respective
part of task load) as a strategy to be able to spend cognitive
resources for solving the additional priority task, because
under high traffic volume both the nominal traffic and the
priority event contribute to communication load i.e. the
same processing modalities (Wickens 2002). Sperandio in
(Sperandio 1978) categorized the strategy change under
increasing load as workload homeostasis. For approach
controllers, he found that under low traffic they preferred a
direct approach strategy with extensive use of aircraft (AC)
performance data for individual control (direction, altitude,
speed,…), whereas under high traffic they switched to stand-
ard procedures with global routing for most AC, i.e. global
approach sequence with standard separation distance so that
control of the first AC in the sequence would be sufficient
(see also Sect. 4.2). Nevertheless, under higher traffic vol-
ume despite reduced communication load with e = 1, the
complete subjectively experienced workload due to audi-
tory and visual processing increases due to the additional
task and obviously is included into the ISA(e = 1) WL rating.
The approach to the asymptotic limit Ru of RC com-
munication load as objective TL measure (see Fig.6) is
reflected by the comparable asymptotic behavior of the
objective neurophysiological DFHM(n) workload measure
(dual frequency head map index DF(n), see Sect. 2.4) which
was measured simultaneously (Radüntz etal. 2020a, b). The
following Fig.10 depicts the measured data of all eight sce-
narios for the DF(n) scenario means averaged across par-
ticipants together with a nonlinear regression analysis using
model Eq.(5) (originally designed for communication task
load RC(n) with RC(n = 0) = 0). To highlight the similar-
ity to the RC(n) characteristic (Fig.6, Sect. 5.3), the fit
was not separated for the two conditions e = 0, e = 1. The
default mode EEG activity during the rest measurements
was reflected in a corresponding offset (average across par-
ticipants (DF0 = 25 ± 5) that was measured separately for
each participant before and after each simulation run and
(as average of bias runs) subtracted from DF(n) data before
regression.
Analyzed data for j = 1–8 were limited to the simulation
time interval following the priority event (e = 1 scenarios)
at ts = 10min, within 0.5–3min after ts. Because no prior
information was available for deriving an estimate of the
effective asymptote a two-parameter fit had to be used for
estimating DFu (= 62.8 ( ± 0.2)) and scaling parameter δ
(=11.8( ± 0.2), t = 37, p = 3 10–9). This result indicates that
the neurophysiological DFHM index derived from the EEG
as response to the ATCO’s activity appears to reflect the task
load, measured through the communication load as a kind of
physical stimulus (DF(RC)), corresponding to the ISA(RC)
power law. The load sensitivity of the new DFHM WL index
relative to the RC(n)-TL sensitivity may be estimated from
the ratio of the logistic function slopes at n = 0 (inversion
point), i.e. the logistic scaling parameter ratio ρ/δ = 20.9
/11.6 = 1.80 (± 0.04). This ratio is exactly the prediction
for a power law (psychophysical) stimulus (RC)–response
(DF) exponent γd according to Eq.A2.16 (with ms = mp = 0,
Appendix2). It appears of interest that in contrast to sub-
jective ISA(n)-WL sensitivity (1/ν), the DFHM index with
1/δ > 1/ρ exhibits a higher sensitivity than RC-TL with
regard to dependence on traffic flow (for low n).
6.3 Power law stimulus–response relationship
ISA(RC)
The subjective ISA report on a one-dimensional online
WL measure reflects the load due to the different limited
resources attributed to perceptual input (visual, auditory),
response demand (vocal), and cognitive (judgement, deci-
sions, strategy change) modalities with corresponding
processing stages (Wickens 2002). That is why ATCO’s
ISA report was not expected to correlate linearly with the
radio communication activity RC (calls/h) derived from the
logged simulation data (time series of traffic flow, pilots
clearance request, ATCO’s communication times and dura-
tion). Another reason is given by the communication time
Fig. 10 Neurophysiological DF(n) WL index (analyzed simula-
tion time interval ts = 10.5–13 min, immediately following priority
request of e = 1 scenarios at ts = 10min, as dependent on traffic load
nj, j = 1–8, for both e = 0 and e = 1 scenarios (circles: average across
participants for scenario means). Solid curve: 2-parameter nonlinear
iterative lsq. regression using logistic Eq.(5) after offset subtraction.
Inset: Parameter estimates, offset DF0, asymptote DFu, scaling param-
eter δ ( ± sterr). Dashed curves: 95% confidence interval
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307Cognition, Technology & Work (2022) 24:291–315
1 3
restrictions (required time / available time for radio calls < 1)
with upper limit Ru of calls frequency (see Sect. 4.2). The
predicted theoretical RC(n) characteristic contrasts with the
nearly linear ISA correlation with n (for the given simulated
traffic range; compare Fig.1 and Fig.2). This is theoretically
reflected by the ISA(RC) power law relationship formalized
by Eqs. (7) and (8) (see also Appendix2).
The average frequencies of radio calls between control-
ler and pilot (< < RC > > j (calls/h) as mean per scenario
j = 1…8, averaged across the participants) represent as a
response demand a more direct measure of task load than
the traffic flow nj as environmental load variable, visualized
in the radar display. The ISA(RC) and transformed yp(ys)
model functions used for the regression-based model param-
eter estimates for e = 0, 1 (γ, bs), and the derived intersection
coordinates (Rx, Ix) provided experimental evidence (Figs.8
and 9) for the theoretically predicted power law character-
istic (Eqs.1, 7, 8, 9). The numerical predictions were
achieved with a plausible assumption on the logistic shift
parameter μt: = 35 AC/h = operational traffic n2 (together
with theoretically predicted relationship νt = μt/ln(4)), and
with a RC(n)-slope linearization, respectively (yielding
ρt ≈ nc/2 = 22.5 AC/h). The value of the theoretically pre-
dicted dimensionless power law exponent γt = ρt/νt = 0.89
corresponds to the order of magnitude of typical psycho-
physics (Stevens) exponents (e.g. Link 1992). Considering
the uncertainty of the theoretical prediction with δμt/μt:≈
15%, γt compares well with the experimental regression-
based estimates in Table2 (Sect. 5.4).
ISA(RC) dependence on frequency of radio calls (Figs.8,
9) like ISA(n) in Fig.4, exhibits a significant sensitivity
increase with priority event scenarios (e = 1) as compared
to the nominal case (e = 0), i.e. 1/νe > 1/ν, despite the inverse
behavior of RC(n) (Figs.6, 7) as compared to ISA(n). This
factor-2 effect again exhibits a clear onset at the intersec-
tion point {Rx, Ix} and {ysx, ypx}of the e = 0, e = 1 curves in
Figs.8 and 9, respectively. The (k, γ) parameter estimates
obtained with an iterative regression procedure (Fig.8, Sect.
5.4) are consistent with those obtained with the generalized
linear model (Fig.9, Table2). The crossing coordinate Ix of
the e = 0, 1 bifurcation obtained with the power law analysis,
is in reasonable agreement with that one obtained from the
logistic ISA(n) characteristics (Ix = 2.3) in Sect. 5.1 (Fig.4).
Our results for the RC(n) characteristics in Sect. 5.3 indi-
cated the strategy change of ATCOs radio communication
in e = 1 scenarios for n > nx (see Sects. 4.2, 6.2) to stabi-
lize the communication TL. The result exhibited an inverse
behavior of the TL sensitivity (1/ρe < 1/ρ) as compared to
the ISA-WL sensitivity (1/νe > 1/ν). That is why the theoreti-
cally derived relationship γ = ρ/ν could explain the (slightly)
improved significance of the observed bifurcation of the
power law characteristic at Rx with slope increase (log–log
coordinates) γe/γ = 1.03 / 0.79 = 1.30 (± 0.04) as compared
to the logistic slope increase of the ISA-WL-sensitivity ratio
1/νe / 1/ν = 26.3/21.3 = 1.23 (± 0.04) (see Table2 in Sect.
5.4). The measured exponent γ < 1 for the nominal traffic
condition is consistent with the result of Bachelder etal.
0.2 < γ < 0.4 < 1 (see Sect. 2.5) (Bachelder and Godfroy-
Cooper 2019).
For the non-nominal (e = 1) scenarios, a theoretical esti-
mate of the offset parameter bse = − ln(ke) due to the pre-
dicted sensitivity increase (γe > γ for RC > Rx) did not exist
a priori (in contrast to ISA(n) with nx: = 30 by plausibility)
so that for e = 1 also the second model parameter ke or bse
had to be estimated through regression that increased the
uncertainty of the e = 1 parameter estimates as compared to
the nominal case, e = 0. However, using the experimental
estimates of nx, ν, ρ, νe, ρe derived from the logistic ISA(n)
and RC(n) models as prior information (Sects. 5.1, 5.3), an
estimate for bse was obtained via Eq. (A2.20) in Appendix2.
This allowed for a one-parameter (γe) model also for the
e = 1 condition (Eq.8), valid for RC ≥ Rx, with a reduction
of uncertainty δγe by a factor 4 (see Table2).
To summarize, our predicted and measured power law
exponents γ are of the order of 1, in agreement with the
classical Stevens exponents (e.g. Link 1992; Stevens 1957).
Moreover, for the nominal condition (e = 0), we obtained
γ < 1 (i.e. subjective response sensitivity 1/ν < objective TL
(stimulus) sensitivity 1/ρ), in agreement with the recent
result of (Bachelder and Godfroy-Cooper 2019) (see Sect.
2.5).
The preliminary regression analysis of the new neuro-
physiological EEG-DFHM index in the previous Sect. 6.2
(Fig.10) with regard to the logistic dependence on traf-
fic load n suggested the generalizability of the power law
hypothesis also to the DF(RC) correlation. Figure11 depicts
the linear one-parameter regression of the normalized trans-
formed DF(RC) data (after subtraction of offset DF0 = 25)
for the estimate of exponent γd. Asymptote DFu = 63.6 as
prior information required for normalization (with slight
increase due to algorithmic requirements) was derived from
the nonlinear logistic DF(n) regression.
The estimate of the exponent γd = 1.66 ± 0.05 was
obtained independently of the environmental load param-
eter n. It confirms the theoretical prediction in the previous
section where the exponent was calculated based on the
theoretically derived ratio of the logistic scaling param-
eters γδ = ρ/δ = 1.80 ± 0.04. Consequently, the power law
fit supports the initial result that the neurophysiological
DFHM index exhibits a higher sensitivity (~ 1/δ) to envi-
ronmental load (for low n) than RC-TL (1/ρ) and than
ISA-WL as well.
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308 Cognition, Technology & Work (2022) 24:291–315
1 3
7 Conclusion
The goal of the present work was to provide evidence by
means of selected WL/TL measures of an ATC-simulation
experiment for a psychophysics (stimulus – response) power
law model with (Stevens) exponent as combined WL/TL
index (e.g. Stevens1957, 1975; Link 1992). It combines two
logistic model-based (WL, TL)-measures via a common
independent stress generating environmental load variable
(simulated traffic flow n (aircraft/h)). This goal means an
expansion of a previous publication (Fürstenau etal. 2020)
where we provided evidence for the logistic dependency of
air traffic controllers’ subjective quasi real-time ISA-WL
reports on the environmental traffic load n as a consequence
of the cognitive resource limitation hypothesis (e.g. Kahne-
mann 1973; Wickens and Hollands 2000). This result agrees
with previous reports of a comparable sigmoid dependency
of ATWIT-WL self-report on traffic count in (Lee 2005)
(see Sects. 2.2, 6.1). Moreover, our results support the early
psychophysics approach to workload by Gopher and Braune
(1984), and they are consistent with a recent application of
Stevens law to workload data analysis by (Bachelder and
Godfroy-Cooper 2019) (see Sects. 2.2, 2.5).
The present stimulus–response form of the power law
(Eq.1) was obtained by combination of logistic two-param-
eter models after transformation of (ISA, RC) measures into
normalized variables (P(ISA), S(RC)). The formal derivation
relied on the prior knowledge on asymptotes (Ru, Iu) of the
two-parametric {ISA(n | Iu, ν), RC(n | Ru, ρ)} characteris-
tics, with logistic sensitivity parameters (ν, ρ). Based on
the generalized linear (log–log) version of the power–law
model yp(P(ISA/Iu)) ~ γ ys(S(RC/Ru)), the experimental expo-
nent estimates γ and γe under nominal and non-nominal traf-
fic load conditions, respectively, could be derived directly
as slope parameters from the transformed ISA(RC) data,
independently of the environmental load variable n. Within
sterr. they agreed with the theoretically predicted ratio of the
logistic WL/TL sensitivities γt = (1/νt)/(1/ρt) = 0.8 (γte = 1.0)
With regard to formal aspects of model development, this
procedure exhibits some analogy to the derivation of Stevens
law within the wave discrimination theory of psychophys-
ics (Link 1992) (see Sect. 2.5). Specifically, our theoretical
prediction of the power law exponent γ = ρ/ν may be com-
pared with the ratio of two normalized subjective response
thresholds AS/AP that relate two simultaneously measured
stimulus and perception sensations to logistic response prob-
ability functions. This analogy of our deterministic resource
limitation-based workload model with Link’s stochastic
wave discrimination theory of psychophysics suggests both
approaches to explain complimentary (stochastic vs. deter-
ministic) aspects of (one-dimensional) subjective real-time
workload measures in terms of stimulus–response relation-
ships (see also Appendix2: “Resource Limitation Model”).
The subjective ISA-WL and objective communication
RC-TL data analyzed in the present work served as a ref-
erence for a new neurophysiological workload measure
(DFHM index) based on real-time EEG-data (Radüntz
2017; Radüntz etal. 2020a, b). Initial model-based analysis
of the simultaneously measured DFHM-WL index provided
evidence (see Fig.10 in Sect. 6.2) that also this new neu-
rophysiological measure follows the same logistic depend-
ence on the environmental traffic load n (with scaling δ)
as the communication task load variable RC(n) (scaling ρ).
Consequently, the data provided evidence also for a power
law relationship for DF(RC) with exponent γd = ρ/δ (Fig.11
in Sect. 6.3), like ISA(RC) with γ = ρ/ν. This suggests the
hypothesis that the neurophysiological DFHM WL index
may be treated as objective (bio-) physical stimulus induc-
ing the subjective ISA-WL response in a power law rela-
tionship with exponent γD = δ/ν, formally equivalent to the
RC-TL stimulus generating the ISA-response according to
ρ/ν. A causal sequence may thus be hypothesized, accord-
ing to operator activity driven by traffic count or flow: n
(AC/h) communication (RC(ρ)) neural activity DFHM(δ)
conscious response ISA(ν), quantified by logistic sensitivi-
ties and power law (stimulus – response) exponents γ, γd
(Sect.2.5) and γD. Based on the cognitive resource limitation
Fig. 11 Transformed normalized DFHM workload index (after off-
set DF0 subtraction) vs. transformed radio calls rate s = RC/Ru in
log–log scale (circles: average across participants for both e = 0, 1
scenario means j = 1–8; analyzed simulation time interval ts = 10.5–
13 min, following priority event at ts = 10 min for j = 5–8). Solid
line: 1-parameter (γ) lin. regression power law fit (gen. lin. model
Eq.(A2.16) with bs = 0, Δs = Δd = 1 and 95% confidence interval
(dashed). For details, see text and Appendix2
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309Cognition, Technology & Work (2022) 24:291–315
1 3
hypothesis, these parameters quantify the dissociation
between subjective and objective WL and TL measures,
respectively.
The psychophysical scaling offers a theoretically founded
approach for a WL parameter γ that combines subjective and
objective measures. It was shown to predict the (near real-
time) ISA-WL means across a sufficiently large sample of
well-trained domain experts in a work environment reason-
ably well described by one-dimensional TL/WL variables.
The ISA(RC) characteristic after transformation into nor-
malized variables P(S) represents a response or judgement
measure on the perceived workload, with normalized task
load S corresponding to the physical stimulus of classical
psychophysics experiments formalized by Stevens law.
Ongoing research addresses the more detailed (logistic
and power law) model-based analysis of the DFHM index
and HR/HRV WL data of the present experiment to clar-
ify the generalizability of the psychophysics approach to
workload. One potential major field of application of the
discussed predictive mental WL models for online WL
measures is adaptive automation for WL stabilization in
safety critical (aeronautic) human system interfaces (Paras-
uraman and Hancock 2001; Prinzel etal. 2003).
Appendix1: Individual ISA andRC averages
acrossscenarios
The following two tables list the 168 scenario
means < ISA > (njk) and < RC > (njk) of the eight scenarios
j = 1…8) for each of the participants k = 1–21, with j = 1…4
for nominal traffic (e = 0) and j = 5…8 for traffic with non-
nominal (priority) event (e = 1) after ca. 10min of simu-
lation time. The < ISA > jk (short < I > jk) provide the data
base for the regression analysis and discussion in Sectis.
5 and 6.
See Tables3, 4.
Table 3 ISA means averaged
over scenario time series
(< ISA > (njk), j = 1–4 for
nominal (e = 0), j = 5–8 for
non-nominal (priority event)
scenarios. Participants k = 1
… 21; subject code 14–34)
as used for the model-based
data analysis. Last three rows:
column means, standard
deviations, standard errors
Subject Scenario means < ISA > j = 1–4, e = 0 Scenar io means < ISA > j = 5–8, e = 1
Code | nj25.00 35.00 45.00 55.00 25.00 35.00 45.00 55.00
14 1.50 3.25 3.25 3.25 1.60 2.40 2.80 3.60
15 2.00 2.75 3.50 3.50 2.40 2.80 3.60 4.40
16 2.00 3.00 3.50 4.25 2.60 2.60 3.40 4.20
17 2.00 2.00 4.00 3.50 1.80 2.60 3.20 3.80
18 3.00 3.25 3.25 3.50 3.00 3.00 3.20 4.20
19 3.00 3.00 3.00 3.50 2.40 2.80 3.20 3.40
20 2.00 2.75 3.25 3.75 1.40 2.60 3.00 3.20
21 1.25 2.25 2.00 3.00 2.75 1.80 2.60 3.40
22 2.00 3.00 3.25 2.75 1.40 2.80 3.00 2.80
23 1.75 2.00 3.00 4.00 1.80 2.00 3.00 3.40
24 2.00 2.25 2.00 3.75 1.60 3.00 3.40 3.40
25 1.00 1.50 2.25 2.75 1.60 1.40 2.80 3.40
26 2.25 2.00 2.75 3.25 2.00 2.60 3.00 3.60
27 2.00 2.00 2.25 2.75 2.00 2.20 2.00 2.60
28 1.25 2.50 2.75 3.50 1.60 2.80 2.80 4.00
29 2.25 2.75 3.25 4.67 2.20 3.20 4.00 4.60
30 1.50 2.00 2.00 2.50 1.20 2.00 2.80 2.80
31 1.25 2.00 2.00 2.75 1.40 2.00 2.40 2.80
32 2.00 2.50 3.75 4.25 1.80 2.20 4.20 4.60
33 1.75 1.75 2.50 2.25 1.80 1.80 1.60 3.20
34 2.50 3.00 3.00 3.25 2.00 2.60 3.80 4.00
Mean μp1.92 2.45 2.88 3.37 1.92 2.44 3.04 3.59
stdev σ0.53 0.52 0.62 0.62 0.48 0.47 0.61 0.60
sterr σ/√N0.12 0.11 0.14 0.14 0.10 0.10 0.13 0.13
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310 Cognition, Technology & Work (2022) 24:291–315
1 3
Appendix2: Derivation oftheoretical model
equations
Resource limitation model
The most simple formal approach for growth dynamics
under limited resources (e.g. processing and memory capac-
ity) of a characteristic system variable (energy consuming
cognitive activity) is given by the nonlinear (1st order, non-
linear 2nd degree) logistic or Verhulst differential equation:
With normalized function f(t) = F(t) / ΔF, growth interval
ΔF = Fu – Fd between upper Fu and lower Fd limit and rate
constant κ. The well-known general solution is the logistic
or sigmoid function (2) and (A2.2) with Δ = 0) as solution
of (A2.1) that exponentially approaches the lower and upper
asymptotic limits with time constant τ=1/κ. Equation (A2.1)
is the most simple version of a Bernoulli equation with time
dependent coefficients κ1(t), κ2(t) for the linear and quadratic
terms that allows for arbitrary time varying growth κ > 0 and
decay rates κ < 0. An application for resource limited growth
(A2.1)
df ∕dt =̇
f(t)=𝜅f(t)(1−f(t))
under external disturbance was demonstrated in (Fürstenau
etal. 2016). A stochastic Bernoulli-Langevin equation is
obtained by adding a random noise term to (A2.1). By trans-
formation into the equivalent Fokker–Planck stochastic par-
tial differential equation (Risken 1988) with deterministic
(Bernoulli) drift and stochastic (e.g. Gaussian) diffusion
term it allows for modeling the time dependent dynamics
of the probability density with mean f(t). This procedure
possibly could provide an approach to formally connect the
stochastic wave discrimination theory of psychophysical
laws (Link 1992) with our resource limitation-based power
law that connects subjective ISA-WL with objective load
measures.
Logistic ISA(n)‑WL model
In what follows, we replace the time parameter of Eq. (A2.1)
by the environmental (traffic) load variable n. In a more gen-
eral form the logistic ISA(n, μ, ν)-WL Eq.(2) besides shift
parameter m: = μ and scaling parameter τ:= ν includes an
additional offset parameter Δ that allows for offset adap-
tation to different subjective and objective WL measures
which are assumed to follow the mental resource limitation
Table 4 RC means averaged
over scenario time series
(< RC > (njk), j = 1–4 for
nominal (e = 0), j = 5–8 for
non-nominal (priority event)
scenarios. Participants k = 1–21;
subject code 14–34) as used for
the model-based data analysis.
Last three rows: column means,
standard deviations, standard
errors
Subject Scen. means < RC(n) > / AC/h, j = 1–4, e = 0 Scen. means < RC(n) > / AC/h, j = 5–8,
e = 1
Code | nj25 35 45 55 25 35 45 55
14 180 249 294 309 172.8 187.2 276 319.2
15 246 309 336 336 235.2 300 340.8 333.6
16 198 303 297 339 225.6 259.2 288 343.2
17 210 243 297 318 182.4 264 273.6 321.6
18 234 270 321 330 204 247.2 290.4 300
19 219 231 255 288 187.2 228 261.6 264
20 240 330 327 423 196.8 304.8 388.8 360
21 222 273 276 306 300 280.8 319.2 328.8
22 234 357 297 384 225.6 314.4 333.6 364.8
23 198 240 291 273 187.2 259.2 283.2 331.2
24 237 303 315 381 196.8 295.2 307.2 350.4
25 270 285 378 408 256.8 300 384 403.2
26 222 267 327 366 204 268.8 290.4 357.6
27 204 237 321 363 206.4 264 331.2 314.4
28 174 270 282 336 180 247.2 292.8 340.8
29 195 237 219 294 160.8 211.2 290.4 297.6
30 243 387 414 405 220.8 292.8 381.6 386.4
31 252 330 384 399 213.6 314.4 360 393.6
32 201 276 339 354 184.8 278.4 316.8 376.8
33 195 297 360 378 189.6 297.6 333.6 381.6
34 180 219 261 258 146.4 225.6 230.4 240
μs(< RC >) 216.9 281.6 313.9 345.1 203.7 268.6 313.0 338.5
stdev σs26.33 44.00 45.95 46.85 33.71 34.94 42.18 41.44
σs / √N 5.75 9.60 10.03 10.22 7.36 7.62 9.20 9.04
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311Cognition, Technology & Work (2022) 24:291–315
1 3
hypothesis. In normalized form (for ISA-WL: division by the
asymptotic resource limit p(n): = I(n)/Iu), the logistic model
function is written as:
with k = exp{m/τ} = Iu (1 + Δ)/Id–1. It is easily verified
that the hypothesized ISA(n) characteristic in Sect.4.1 with
m: = μ is expressed by Δ = 0. RC(n) in Sect.4.2 is defined by
m = 0 and Δ = 1 so that (A2.2) equals tanh(x) with x = n/(2ρ)
(see below). For ISA(n), the constant k can be expressed by
the scale limits (Id = 1, Iu = 5):
i.e. k = Iu/Id − 1 = exp{μ/ν} yielding μ = ν ln(4). This
reduces Eq. (A2.2) and Eq.(2)) to a one-parametric model
for the nominal scenarios (e = 0). The asymptotic limits are
given as lim I(n→∞) = Iu and lim I(n→ − ∞) = 0 (with n < 0
as mathematical design aspect only, with no real meaning).
with maximum slope at n = μ: dI/dn=I’(n = μ) = Iu/4ν. For
the e = 1 scenarios, we set (with index e for e = 1) μ: = μe, i.e.
k: = ke and ν: = νe < ν (see Sect.4.1):
The different shift and scaling parameters (μe, νe) of Eqs.
(A2.4), (A2.5) for the scenarios with priority event generate
the intersection at coordinates (nx, Ix(nx)) between e = 0 and
e = 1 characteristics. Via a plausibility argument in Sect.4.1,
a theoretical estimate of nx was obtained from the prior infor-
mation on the traffic scenarios n1… n4: nx: = 30 (± 5), with
Ix = I(nx) = 2.25 (± 0.5). Setting equal Eqs. (A2.4) , (A2.5) at
n: = nx yields for k(e = 1): = ke, μ(e = 1): = μe
with ke > k(e = 0) = 4 and Ide < Id(e = 0) = 1. Introducing ke
into Eq. (A2.5) (or replacing m in Eq. (A2.2) by μe and ρ = 0)
yields
(A2.2)
p
(n)=
1+Δ
1+exp
{
−n−m
𝜏}
−Δ=
1+Δ
1+kexp
{
−n
𝜏}−Δ
(A2.3)
p
(n=0)=pd=
I
d
I
u
=1
1+k
(A2.4)
I
(n)=
5
1+4 exp
{
−n
𝜈}
(A2.5)
I
e(n)=
5
1+keexp
{
−n
𝜈e}
(A2.6)
k
e=
(
Iu∕Ide −1
)
=exp{me∕ne}=4 exp{nx(1∕ne−1∕n
)}
𝜇e
=n
x
+𝜈
e(
ln (4)−n
x
∕𝜈
)
(A2.7)
p
e(n)=
1
1+exp
{
−n−𝜇e
𝜈
e}
=
1
1+keexp
{
−n
𝜈
e}
Numerical estimates for ke and μe based on prior esti-
mates on intersection nx and nominal scaling parameter ν
provide a one-parameter equation also for the non-nominal
case (e = 1). It is depicted in Fig.1 (Sect.4.1) as dotted
curve with ν(μ: = 35) = 25.3, νe: = 20, μe = 34 and slope I’e
(νe) = 0.0625 together with the solid curve for the nominal
(e = 0) characteristic.
The generalized linear (semi-logarithmic) form of the
logistic characteristic yp(n) = 1/ν n – ln(4) = ag n + bg (Eq.(3))
utilizes the normalized ISA-WL p(n): = I(n)/Iu (with prior
knowledge Iu = 5) together with the semi-logarithmic scale of
the nonlinear transformation P(ISA): = p(ISA) / (1–p(ISA)).
The latter is obtained from Eq. (A2.2) by some basic algebraic
operations. For the non-nominal scenarios (e = 1), the general-
ized linear form is obtained from Eq.(3) by replacing ν: = νe
and k: = ke yielding (y = ln (P(ISA)):
With slope age = 1/νe and intercept
y
pe(n=0)=bge =nx
(
1
𝜈−1
𝜈
e)
−ln(4
)
, yielding ype(n = nx)
= yp(n = nx) = ypx = nx/νt – ln(4) = − 0.20.
Using the plausibility argument for nx as prior knowl-
edge (see Sect. 4.1), the intersection coordinates between
nominal (e = 0) and priority (e = 1) characteristics for the
generalized linear model are
{
n
x
,y
px}
=
{
n
x
,a
g
n
x
− ln(4)
}
= {30, − 0.20}, with uncertainty δμ: = ± 5 yielding {δnx,
δyIx} = {± 5, + 0.43/ − 0.32}. If for the regression analysis
of the priority event scenarios (e = 1) no a-priori knowl-
edge on nx is assumed and both slope age and intersec-
tion parameter bge have to be estimated from the data, the
intersection coordinates are obtained by combination of
Eqs. (3), (A2.8) as:
with bg: = bgt = − ln(4) = − 1.3863. With the regres-
sion-based parameter estimates (ag = 0.0380 (4 10–4),
age = 0.0471 (1.3 10–3), bge = − 1.670 (0.05)) we get
nx = 31.2 (0.56) that confirms the initial estimate (30 ± 5),
however, with uncertainty reduced by factor 10.
Linearized logistic ISA model
The results in Sects.4.1, 5.1 show that the nearly lin-
ear part of the logistic characteristic may be sufficiently
(A2.8)
y
pe(n)=ln
(
Pe
)
=
1
𝜈
e
(
n−nx
)
+
n
x
𝜈
−ln(4)=agen+b
ge
(A2.9)
nx,ypx=
bge −bg
ag−age,agnx−ln(4)
=
−𝜇e∕𝜈e+ln(4)
1∕𝜈−1∕𝜈
e
,agnx−ln(4)
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312 Cognition, Technology & Work (2022) 24:291–315
1 3
represented by a linearized model for the selected traf-
fic flow range 25 ≤ n ≤ 55. In the vicinity of the logis-
tic shift parameter μ with n = μ + Δn, |Δn| =|n–μ| < < μ,
I(n = μ) = Iu/2 the logistic characteristic is approximated by
where the slope in the vicinity of μ, dI/dn ≈ alt = Iu/(4
ν) = Iu ln(4)/(4 μ) is obtained through neglection of quad-
ratic and higher powers of the exponents (n–μ)/ν.
Within the linear approximation a dimensionless index
sak(ak, bk) can be defined for individual participants
k = 1…21 via slope and intercept (ak, bk) by normalization
of the independent and dependent variables (n, I(n))k
through division by the mean value of the traffic variable
n (n1 + n4)/2 and the individual ISA intervals (Imax + Imin)/2,
respectively. In what follows, we skip the participant index
k. Normalized variables are written with small letters or
index n, yielding
n
n=
n
(n1+n4)∕2
=
n
40
, and for the individual
ISA scenario means < ISA > jk:
where the individual maximum and minimum ISA val-
ues are taken from the linear regression predictions at the
maximum n4 and minimum n1 of the predictor variable.
This yields the dimensionless linear sensitivity index
with anticorrelation sb = 1–sa. sa allows for clustering
of participants k with individual sensitivities ak as used
in (Radüntz etal. 2020a, b) for discriminating between
subjects of low and high WL sensitivity.
Logistic radio calls frequency model RC(n)
The hypothesized logistic radio calls characteristic
R(n) is obtained from Eq. (A2.2) by setting the off-
set parameter Δ: = 1 yielding Eq. (5). With normal-
ized call rate s: = R(n)/Ru and x: = n/(2 ρ) Eq. (5)
can be written in the form s(x) = (1—exp(− 2x))/
(1 + exp(− 2x)) = (exp(x)−exp(− x))/(exp(x) + exp(− x))
which is the definition of the hyperbolic function tanh(x)
I
(n)≈blt +altn≈
I
u
21−𝜇
2𝜈+
I
u
4𝜈
n
=
Iu
2
1−ln (4)
2
+
Iuln (4)
4𝜇
n
⟨
isa
⟩
=
⟨ISA⟩
⟨
ISA
⟩
max+
⟨
ISA
⟩
min
2
=
ISA
40a+b=sb+san
n
s
a=
isa
max
−isa
min
nnmax −nnmin
=
1
1+b
40a
(A2.10)
s
(n)=
[
2
1+exp {−n∕𝜌}−1
]
=tanh
(
n
2𝜌
)
With turning point at the origin (n, s) = (0, 0) it starts
with slope s´(n) = ds/dn given by
Maximum slope (at n = 0) is
s�(n=0)≈Δs∕Δn=1∕2𝜌
.
With a linear extrapolation of the maximum slope at n = 0,
Δs = 1 and Δn: = capacity limit nc = n3 yield the theoreti-
cal estimate for the radio calls scaling parameter
𝜌t∶≈
nc/2 = 22.5.
Like for ISA-WL, the generalized linear form for
R(n) is obtained via nonlinear transformation S(s) of
the normalized TL-variable s(n) that is obtained from
Eq.(5) or (A2.10) by some basic algebraic opera-
tions: S(s) = (1 + s(n)) / (1–s(n)), followed by taking the
logarithm:
With slope ags = 1/ρ and intersection between nominal
and non-nominal characteristics at the origin nxs = 0.
Psychophysics power law model ISA(RC)
In the simulation experiment with external load variable
n [AC/h] defining simulation scenarios, the function pair
(R(n), I(n)) defines the parametric dependence of the sub-
jectively perceived and reported ISA(n) WL on objec-
tively measured frequency of radio calls (R(n)). To derive
the hypothesized psychophysical power law relationship
between subjective response P(ISA(n)) (representing frac-
tion of used cognitive processing resources) and objec-
tive stimulus measure (S(RC(n)), the ISA(n)-WL scale
(1 ≤ I(n) ≤ 5 and the metric TL RC(n) measure [number
of radio calls / h] have to be normalized and nonlinearly
transformed into (S, R) (see Eqs. (A2.8), (A2.12)). The
suitable transformations for the ISA-WL and RC-TL vari-
ables have been derived before for the generalized linear
forms of the logistic characteristics. The power law may be
derived by combining these linear versions. A more gen-
eral expression is obtained by transformation of Eq. (A2.2)
into the generalized linear form that includes a general
bias term Δ and the general scaling and shift parameters
τ and m, respectively:
Based on this general expression, we define separate
equations for stimulus (ys(n)) and perception (of mental
load) (yp(n))
(A2.11)
s
�=
1
2𝜌cosh
2
(n∕2𝜌)
(A2.12)
y
s(n)=ln(S)=
1
𝜌
n
(A2.13)
y
(n)=ln
(
Δ+p(n)
1−p(n)
)
=ln(P(n)) =1
𝜏
n−
m
𝜏
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313Cognition, Technology & Work (2022) 24:291–315
1 3
By solving Eq. (A2.14) for n and introducing the
expression into (A2.15), we obtain the generalized linear
form of the power law:
For the nominal scenarios (e = 0) with ms = 0 and
mp = μ we have bs = bgt = − ln(4) so that (A2.16) = Eq. (8)
as depicted in Fig.3 of Sect.4.3, i.e. a one-parameter
model for regression-based estimate of γ. For the non-
nominal scenarios due to the lack of prior information
on bse = bs(e = 1), a two-parameter fit is required. With
exponent γ of the power law for ISA(RC) defined by the
ratio of the scaling coefficients of stimulus ρ and percep-
tion ν: ρ/ν: = γ and ms = 0, mp = μ the linear equation in
log–log coordinates (A2.16) may be written in in linear
coordinates
In correspondence to the intersection {nx, Ix = ISA(nx)}
of the nominal and non-nominal ISA(n) characteristics the
intersection {sx, px} for the (normalized) e = 0 and e = 1
power law equations is derived from Eqs. (3, 5) as
For the generalized linear form in (log–log)-coordinates
of the power law (Eq. (A2.16)), the intersection coordinates
are obtained from Eqs. (3) and (A2.12) with n = nx:
Based on the combination of Eq. (A2.16) for e = 0 and
e = 1 at crossing coordinate n = nx (using yp(nx) = ype(nx)), a
theoretical relationship between the e = 1 intersection param-
eter bse and the unknown power law exponent γe in this case
is derived as:
With prior estimate nx = 31.2 (± 0.6) obtained from the
logistic ISA(n) regressions for e = 0, 1, we can use this rela-
tionship for a one-parameter regression estimate of γe (e = 1)
(A2.14)
y
s(n)=ln
(Δ
s
+s(n)
1−s(n))
=ln(S(n)) =1
𝜚
n−
ms
𝜚
(A2.15)
y
p(n)=ln
(Δ
p
+p(n)
1−p(n)
)
=ln(P(n)) =1
𝜈
n−
m
p
𝜈
(A2.16)
y
p
(
ys
)
=ln(P(n)) =𝛾ys+
m
s
−m
p
𝜈
=𝛾ys+b
s
(A2.17)
P
(S)=exp
{
−
𝜇
𝜈}
S𝛾=
1
k
S
𝛾
(A2.18)
sx,px
=
tanh
n
x
2
𝜌
,
1+4 exp
−nx∕𝜈
−1
(A2.19)
ysx,ypx
=
nx
𝜌
,nx
𝜈
−ln(4)
(A2.20)
y
pe
yse =0
=bse =
1
𝜈
−𝛾e
𝜌
nx−ln(4
)
with the generalized linear form (A2.16) of the power law
model:
with ysx = γe nx/ρ = ρe nx / (νe ρ) and ypx = nx/ν–ln(4), where
we use the theoretical relationship γe = ρe/νe and logistic
regression parameter estimates as prior information.
Vice versa, without prior estimate of nx, we may use the
parameter estimates (γ, γe, bse, and bs = − ln(4)) from the
one-parameter fit for e = 0 and the two-parameter fit for e = 1
to quantify the intersection of the linear e = 0, 1 characteris-
tics from the two linear Eq. (A2.16):
Acknowledgements We are indebted to Thorsten Mühlhausen and
André Tews of the ATMOS simulator team for support in the design,
setup and execution of the experiment. A.T. wrote the simulator data
acquisition code and was responsible for the data pre-processing. We
also acknowledge the contribution of Monika Mittendorf who wrote
most of the Matlab® and the ML statistics toolbox-based data analysis
code. We are indebted to the three anonymous reviewers who provided
valuable suggestions for significant improvements of the manuscript
for the revised version.
Funding Open Access funding enabled and organized by Projekt
DEAL.
Open Access This article is licensed under a Creative Commons Attri-
bution 4.0 International License, which permits use, sharing, adapta-
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