A Chain-Retrieval Model for Voluntary Task Switching
André Vandierendonck1, Jelle Demanet2, Baptist Liefooghe3,
1 Department of Experimental Psychology, Ghent University, Henri Dunantlaan 2, B-9000 Gent,
Belgium, E-mail: Andre.Vandierendonck@UGent.be
2 Department of Experimental Psychology, Ghent University, Henri Dunantlaan 2, B-9000 Gent,
Belgium, E-mail: Jelle.Demanet@UGent.be
3 Department of Experimental-Clinical and Health Psychology, Ghent University, Henri Dunantlaan
2, B-9000 Gent, Belgium, E-mail: Baptist.Liefooghe@UGent.be
4 Psychology, University of Exeter, Washington Singer Building, Perry Road, Exeter EX4 4QG, UK,
Running Head: Modeling Voluntary Task Switching
Address Corresponding Author: André Vandierendonck,
Department of Experimental Psychology, Ghent University,
Henri Dunantlaan 2,
Phone: +32-(0)9-264 64 37, Fax: +32-(0)9-264 64 96
? Modeling Voluntary Task Switching 1
To account for the findings obtained in voluntary task switching, this article describes and
tests the chain-retrieval model. This model postulates that voluntary task selection involves
retrieval of task information from long-term memory, which is then used to guide task selection and
task execution. The model assumes that the retrieved information consists of acquired sequences
(or chains) of tasks, that selection may be biased towards chains containing more task repetitions
and that bottom-up triggered repetitions may overrule the intended task. To test this model, four
experiments are reported. In Studies 1 and 2, sequences of task choices and the corresponding
transition sequences (task repetitions or switches) were analyzed with the help of dependency
statistics. The free parameters of the chain-retrieval model were estimated on the observed task
sequences and these estimates were used to predict autocorrelations of tasks and transitions. In
Studies 3 and 4, sequences of hand choices and their transitions were analyzed similarly. In all
studies, the chain-retrieval model yielded better fits and predictions than statistical models of event
choice. In applications to voluntary task switching (Studies 1 and 2), all three parameters of the
model were needed to account for the data. When no task switching was required (Studies 3 and
4), the chain-retrieval could account for the data with one or two parameters clamped to a neutral
value. Implications for our understanding of voluntary task selection and broader theoretical
implications are discussed.
Keywords: voluntary task switching, task selection, cognitive control, random generation
? Modeling Voluntary Task Switching 2
A Chain-Retrieval Model for Voluntary Task Switching
Goal-directed behavior relies on a determination to achieve the current goal, whilst being
adaptive to changes in the environment. This balance between goal persistence, on the one hand,
and goal flexibility, on the other hand, is achieved by means of executive control (e.g., Logan &
Gordon, 2001; Norman & Shallice, 1986). Executive control processes are required to ensure that
behavior consistent with the goal is produced, while remaining sensitive to changes in the
environment. Such changes may require a shift to another goal, in which case the system must be
flexible enough to release the previous goal and install a new one adapted to the environmental
change. The present study focuses on an essential feature of such cognitive flexibility, namely the
processes underlying the behavioral choice between different goals or tasks.
Task switching has been the preferred paradigm to study such flexible changes in the
laboratory (see Kiesel et al., 2010; Monsell, 2003; Vandierendonck, Liefooghe, & Verbruggen,
2010, for reviews). Many studies have shown that task switching comes with a cost, which is
indicated by longer reaction times and higher error rates on task-switch trials than on task-repeat
trials. This switch cost has been attributed to task-set reconfiguration processes, interference, or
both (e.g., Allport, Styles, & Hsieh, 1994; Mayr & Kliegl, 2000; Meiran, 1996, 2008; Rogers &
Monsell, 1995; Waszak, Hommel, & Allport, 2003). In most task-switching studies, participants are
instructed when to repeat and when to switch. Behavioral flexibility, however, also involves the
possibility to choose voluntarily for a particular course of action or for a particular goal. Such goal-
selection or task-selection processes can be examined with the voluntary task switching (VTS)
procedure (Arrington & Logan, 2004, 2005). This procedure seems to tap into the same behavioral
mechanisms that are studied in the traditional task-switching procedures as testified by the finding
that task switch costs are driven by top-down (Arrington & Logan, 2005; Mayr & Bell, 2006) as well
as bottom-up factors (Arrington, 2008; Arrington & Rhodes, 2010; Mayr & Bell, 2006; Yeung, 2010).
Importantly, the VTS procedure was designed to allow voluntary choice or selection of a particular
task or task goal by giving subjects the freedom to select and execute the task of their choice on
every trial. This makes it an interesting procedure because it not only provides the usual task-
? Modeling Voluntary Task Switching 3
performance measures, such as the switch cost, but also enables investigation of the
characteristics of the task choices made over a series of trials. The latter feature allows for the
study of processes involved in choosing or selecting a task to be performed. Understanding the
processes underlying voluntary choice of courses of action is of paramount theoretical importance.
even though thus far not so much is known about these processes.
Previous research using the VTS procedure has shown that people tend to repeat the same
task more often than expected on the basis of chance (Arrington & Logan, 2004). This is known as
the task-repetition bias, and consists of a preference of task repetitions over task switches. This
repetition bias becomes weaker when more time is available to select a task (i.e., when the
response-stimulus interval (RSI) increases; Arrington & Logan, 2005). This observation suggests
that choosing to perform a task depends—at least to some extent—on endogenous (top-down)
time-consuming task selection processes (see also e.g., Arrington & Logan, 2005; Arrington &
Yates, 2009; Liefooghe, Demanet, & Vandierendonck, 2009). However, bottom-up factors clearly
play an important role as well, especially when the amount of time available to select a task is
small. This has been shown in several ways: the task repetition bias is affected by priming due to
stimulus repetitions (Mayr & Bell, 2006), stimulus availability (Arrington, 2008), previously learned
stimulus-task associations (Arrington, Weaver, & Pauker, 2010; Demanet, Verbruggen, Liefooghe,
& Vandierendonck, 2010), processing efficiency (Arrington & Rhodes, 2010), and differences in
task difficulty (Liefooghe, Demanet, & Vandierendonck, 2010; Yeung, 2010). Furthermore, the
presence of a working memory load that interferes with endogenous control processes increases
the task-repetition bias in the presence of stimulus repetitions (Demanet et al., 2010). In sum, task
choice seems to result from an interaction between exogenous (bottom-up) influences and
endogenous (top-down) cognitive control operations.
The present study aims to contribute to our understanding of the processes involved in task
choice by elaborating and testing a model of task choice in a context in which participants
voluntarily and frequently switch between tasks. When participants are free to select any of a
range of tasks for execution, in the absence of specific constraints, task switches are rarely
selected. In a study of Kessler, Shencar and Meiran (2009), for example, proportions of
? Modeling Voluntary Task Switching 4
spontaneously selected switches were rather low (between 0.04 and 0.13). Because the switch
cost is measured by comparing repetition and switch trials, for statistical reasons a less skewed
distribution of switches and repetitions is needed. In order to approach a more balanced
distribution, instructions in VTS typically stress that the tasks should be selected (and performed)
about equally often, and that tasks should be selected independently from the previous trial so as
to form an unpredictable series of choices. This is usually clarified by the coin-tossing metaphor
(imagine that on each trial, a coin with one task name on one side and the other task name on the
other side is flipped to decide which task to perform). These instructions have been frequently
used in VTS and result in distributions with 20 to 50 percent switches.
Under the constrained instructions, voluntary task choice seems similar to free and
independent selection of events and responses, which has traditionally been studied in the human
random-generation literature. In a typical random-generation experiment, participants are
requested to generate sequences of outcomes that would be obtained by repeatedly tossing a coin
or throwing dice. Furthermore, the heuristics of availability (Baddeley, 1996) and
representativeness (Rapoport & Budescu, 1997) seem to underlie random event-generation and
voluntary task choice (see Arrington & Logan, 2005, p. 699). This suggests that independent
selection of tasks and independent selection of events are based on common underlying
In view of all this, our purpose was to develop a model that not only accounts for voluntary
task choice behavior but also for independent event selection. Although this may seem
straightforward, the model must be able to account for some differences between voluntary task
selection and independent event generation. A first important difference between the selection of
tasks and the selection of events is that patterns of task selection show a repetition bias (see
above), whereas other patterns of event selection typically show an alternation bias which is a
tendency to generate more alternations than repetitions and to produce more frequently short runs
of repetitions of the same event (Lopes, 1982; Lopes & Oden, 1987; Neuringer & Allen, 1986;
Rapoport & Budescu, 1992; Treisman & Faulkner, 1987; Wagenaar, 1972). A second difference
concerns the context in which selection of tasks and events takes place. In VTS, the selected task
? Modeling Voluntary Task Switching 5
has to be applied, whereas no further consequences are associated with the selection of an event.
Because in VTS the selected task has to be executed, bottom-up factors that are external to the
process of task selection, such as stimulus repetition priming, may directly influence the selection
process (Demanet et al., 2010; Mayr & Bell, 2006; Orr & Weissman, 2011). Furthermore,
variations in the demands placed on top-down control during the execution of a task may equally
affect the selection process. If the selected task is likely to result in a performance error or takes
more time to execute, the probability of selecting this particular task may become smaller. The
experience that a particular task is easier (i.e., fewer errors and shorter RTs) than another one,
may affect the decision to repeat that task or to switch to the other task (Liefooghe et al., 2010).
Similarly, task repetitions are likely to be perceived as easier than task switches (also in VTS,
Arrington & Logan, 2005), which could explain the preference for task repetitions over task
switches. The preference for selecting the easier tasks or choosing task repetitions is a case of
going for the least mental effort (Botvinick, 2007; Hull, 1943), which can be defined as “a
preference for activities or strategies that minimize cognitive demand” (Botvinick & Rosen, 2009, p.
Thus, our model assumes that task selection and event selection are based on the same
selection mechanism. However, these mechanisms are biased by events external to selection,
namely experience-based preferences for particular tasks or events, and bottom-up triggered
intrusions leading to repetition of the same task or event. Such biases are more likely to occur in
VTS than in event selection, because participants have to execute the selected tasks in the VTS
2. Modeling task selection processes
For the mechanism common to task and event selection, already existing models of event
selection may provide a useful basis. Some of these models use Markov chains to account for
biased event selection (Budescu, 1987; Vandierendonck, 2000a). Such models—in particular
those that can predict perseverations—could be useful to describe what goes on in VTS.
However, it will become clear later in this article that these models are insufficient to account for
task-choice processes. A completely different type of model was proposed by Rapoport and
? Modeling Voluntary Task Switching 6
Budescu (1997). It is most explicit in specifying the processes involved in selection. According to
this model, people do not represent all possible combinations of events that may occur when
generating sequences of independent events, but only the events that are believed to be
representative of the situation at hand. This particular feature may be considered as a bias in
conceptualization of the situation; it can also be interpreted as a consequence of limited cognitive
capacity to overview all possibilities. This model further assumes that events are selected while a
monitoring process follows retrospectively within a window of a particular width (capacity
constraint) whether the part of the generated sequence within that window deviates from
independence. When the monitoring process detects a deviation from this subjective idea of
independence, the next event will be selected so as to restore the personʼs conception of statistical
independence within the window. This model provides an excellently fitting description of some
forms of independent event selection. However, because task selection in VTS has been shown to
be susceptible to bottom-up intrusions (e.g., Arrington et al., 2010; Demanet et al., 2010; Mayr &
Bell, 2006), application of the model of Rapoport and Budescu to task choice in VTS leads to some
difficulties. The intruding bottom-up events, which essentially consist of repetitions of the same
task executed on the previous trial, should show up in the window of generated tasks, and
consequently, the monitoring mechanism should detect these occurrences as deviations from the
personʼs conception of statistical independence (representativeness). This should then result in a
corrective action, namely the selection of as many task switches as needed to restore the
sequence, and an alternation tendency would still be present in the end. Clearly, the problem
arises because the monitoring mechanism of the model inspects retrospectively the recent window
of the sequence. The problem does not arise, however, if the generation mechanism would elicit
prospectively the presence of repetitions by selecting series of independent events in advance. In
what follows, we describe a model of task selection in VTS that replaces the retrospective
monitoring mechanism in the Rapoport and Budescu (1997) model by a mechanism that selects a
sequence of events in advance.
? Modeling Voluntary Task Switching 7
2.1. Modeling assumptions
The model proposed here builds on the idea of Rapoport and Budescu (1997) that when
humans freely select independent events, they do not consider all possible combinations of events;
instead they rely on the representativeness of the sequence, which means that they prefer
sequences that are balanced (equiprobable events) and that the order in which events appear is
representative of randomness (unsystematic orderings). In line with the considerations regarding
the differences between event and task selection, we further adopt the hypotheses that (1) task
performance difficulty affects the outcomes of the task-selection mechanism (Liefooghe et al.,
2010; Yeung, 2010) and (2) that bottom-up events such as repetition priming compete with the top-
down task-selection processes (Demanet et al., 2010; Mayr & Bell, 2006). First, we describe the
event/task selection mechanism. Next, we discuss how the outcome of this mechanism can be
modified by task difficulty and bottom-up events.
In line with the view of Arrington and Logan (2005), the instantiation of the task-selection
mechanism calls upon the availability heuristic. This heuristic was used in studies of independent
event selection and entails that the selected tasks are retrieved from long-term memory (cf.
Baddeley, 1996). Because most studies of task switching use only two tasks, here we restrict the
development of the model to situations in which there are two tasks. Selection of one task (say
task A) on one trial may prime retrieval of the same task again on the next trial (which affects again
availability; e.g., Baddeley, Emslie, Kolodny, & Duncan, 1998). In order not to end up in long and
dependent series of the same task, the tendency to select task A must occasionally be suppressed
in order to allow selection of the other task (B) and achieve more ore less independent events (see
Mayr & Bell, 2006). Soon, participants could then learn that they could work with short series of
tasks, such as AABB, ABBA, and ABAB, to generate task sequences. Because of the large
amount of mental effort required on each trial to select a new task independently and to avoid
repetitions by priming while being engaged in executing the tasks, participants will be motivated to
learn action sequences that are less effortful. Working with short sequences diminishes the
burden of checking independence of the selected tasks on each trial. We propose, therefore, that
during the early stages of the experiment, the participants learn to select such short sequences by
? Modeling Voluntary Task Switching 8
retrieving a sequence from long-term memory (LTM) on the basis of the task names (cf. Schneider
& Logan, 2007). Initially, it may be necessary to produce the sequences one element at a time. As
practice progresses, the sequences are stored in LTM as a chain of task names and participants
would soon use such sequences to guide task-selection behavior.
These chains are continuously applied throughout the experimental session, but their
selection from LTM is biased by a number of factors. The following assumptions are central to the
development of the model.
(1) Subset of chains. In accordance with the central hypothesis of Rapoport and Budescu
(1997), only a subset of all possible two-task sequences of a certain length are stored (the
example above shows the subset for sequences of length 4): each of these sequences is
balanced so that each of the two tasks is equally frequent. Due to these characteristics, these
sequences are biased towards an alternation bias as predicted by the model of Rapoport and
(2) Preference for ʻeasierʼ chains. The application of the generated chain of tasks to the
targets may have implications for preferences among chains such that tasks that elicit fewer
errors and that are executed more swiftly (easier tasks) are preferred to tasks that lead to
more errors or that require longer response times, i.e., tasks that are more difficult (Liefooghe
et al., 2010). Similarly, some transitions between tasks are easier in that they take less time
and evoke less errors; these ʻeasierʼ transitions are preferred to difficult transitions. As task
switches are generally more difficult than task repetitions, we assume that the experience of
differences in difficulty between tasks or between task transitions will bias the selection of
chains from LTM towards chains that are on average easier. Because the switch cost is a
general phenomenon in task switching, in the model we will only consider the preference for
chains that can be executed with less effort, namely the chains with less switches (cf. the law
of least mental effort; Botvinick, 2007; Botvinick & Rosen, 2009). This principle leads to a
preference of chains like AAABB (only one switch) over AABAB (three switches). It is also
possible to further distinguish between switches that are easier and those that are more
? Modeling Voluntary Task Switching 9
difficult (Kessler et al., 2009; Yeung, 2010), but in the present paper we will ignore these as
well as the difficulty differences at the level of tasks.
(3) Limited working memory capacity. In order to monitor the produced sequence of events,
the model of Rapoport and Budescu (1997) assumes that a window of the sequence is
maintained in short-term memory. Our model does not call on monitoring processes, but there
is still a window of generated events that has to be maintained in working memory to guide
task selection over the next trials. We assume, therefore, that the size of this window is
constrained by working memory capacity (e.g., Baddeley et al., 1998; Baddeley, 1966). Such
constraint is also evident in the experiment on the task span reported by Logan (2004b) which
revealed that the task span (the number of tasks that could be remembered and correctly
executed) was essentially not different from the memory span (the number of tasks that could
be remembered). This proposal is also consistent with the findings reported by Bryck and
Mayr (2005) which suggest that verbalization of sequential information in task switching is
critical for maintenance of sequential courses of actions or sequential plans. These and other
studies (Baddeley, Chincotta, & Adlam, 2001; Emerson & Miyake, 2003; Miyake, Emerson,
Padilla, & Ahn, 2004; Saeki & Saito, 2009) strongly indicate that working memory forms a
basis for task switching without implying that shortage of working memory capacity would
completely disrupt task-switching performance.
(4) Coupling chains. Thus far we propose that short chains of tasks are retrieved from LTM.
After using a chain to guide task execution on the series of targets presented, a new chain
must be retrieved. The issue here is how the new chain is coupled to the previous one.
Given that priming may violate representativeness of the produced sequence (cf. supra), this
can be achieved in three different ways. One possibility is that the next chain is concatenated
to the previous one (e.g., AABB followed by ABBA leads to AABBABBA). In the example, the
end (B) of the first chain is coupled to the head (A) of the second chain, resulting in a task
switch. If the second chain started with a B, a repetition would have occurred. Ideally, each
time a new chain is selected, this would occur independently taking into account the
preference for the easier chains. Thus repetitions and switches would be added in a balanced
? Modeling Voluntary Task Switching 10
way. However, such a balance is difficult to achieve because the last task executed may
prime chains that begin with the same task (cf. supra). To ensure a balance of repetitions and
switches, top-down control over such repetition priming would be needed. A second
possibility avoids the need for top-down control, so that the last element of the chain primes
chains that begin with the same element. Combined with simple concatenation, this would
result in the addition of a repetition at every chain junction. This systematic addition of a
repetition between every pair of chains would result in an overall sequence that violates the
representativeness principle that lays at the basis of each individual chain (Rapoport &
Budescu, 1997). There is a third possibility that does not have the drawback of the previous
one. It shares the priming assumption of the previous possibility, but when the last task of the
chain is reached (the last task to be executed), this task is not executed but instead primes
the chains starting with the same task. In other words, arriving at the last task (A), this task
primes chains that start with an A, and thus chains such as ABBA, ABAB and AABB may
become active. This simplifies selection of a chain as only half of the chains stored in LTM are
activated. One of these is selected, and used to guide behavior in the following trials. For
example, at the end of the chain BBAA, the chain ABAB may now be selected, resulting in the
execution of BBA(AA)BAB, where the task between parentheses is executed only once as it
was used to prime the next chain. Because only the last of the three possible chain coupling
methods guarantees maintenance of the representativeness of the composite sequence
without the need for extra assumptions about top-down control, this coupling is preferred in
the present modeling.
These assumptions are now elaborated in a framework that defines a family of formal
models. We will define three parameters, m, b, and r, which express quantitatively the operation of
The first parameter, m, expresses the chain length that the person prefers. As explained
above, the chain length is partly constrained by available working memory capacity, but as will
become clear later on, other factors also affect chain length. Nevertheless, based on assumption 3
? Modeling Voluntary Task Switching 11
above (limited working memory capacity), the parameter m specifies the length of the chains
stored in and retrieved from LTM. We assume that the minimum length is 3. Chains of two
elements are of course possible, but chains of this length do not allow enough variability: only AB
and BA are balanced sequences, and because such sequences do not contain repetitions, a
repetition bias based on experience with the easier repetitions cannot develop (see assumption 2:
preference for easier chains). Therefore, we adopted 3 as the lower limit. Furthermore, because
many studies show that working memory span is limited to about 5-6 elements (e.g., Atkinson &
Shiffrin, 1968; Baddeley, 2007; Baddeley & Hitch, 1974), we adopted 6 as the maximum limit;
Table 1 displays the sequences that are possible at each of the lengths 3-6. The sequences
considered are balanced in the sense that the two tasks occur m / 2 times in the sequences with
an even length and occur minimally m / 2 times and maximally m / 2 +1 times in odd numbered
sequences. This way, both tasks will be selected approximately equally often. At the same time,
sequences show, on average, a tendency to alternate, which is shown in Table 1 in the column
labeled “# Rep”. We assume that all the sequences of the personʼs preferred length m are stored
---- Table 1 about here ----
The second parameter, b, is related to the strength of a chain in accordance with assumption
2 (preference for easier chains). In order to model the variability in the retrieval of chains, each
chain is supposed to have an associative strength that specifies its likelihood of being retrieved.
The associative strength is a common way to represent effects of learning (e.g., Anderson &
Lebiere, 1998; Bush & Estes, 1959; McClelland & Rumelhart, 1986; Rescorla, 1988; Rumelhart &
McClelland, 1986). In situations where feedback is explicitly provided, the associative strength of a
memory trace is changed in accordance with the feedback. If the memory retrieval is “correct” or
leads to a positive outcome, its strength is increased; if the retrieval is “incorrect” or leads to a
negative outcome, the strength is decreased. In task selection no explicit feedback is provided.
However, implicit feedback still remains present because behavioral consequences of task
execution may be used to strengthen a chain. Thus chains with more task repetitions may be
? Modeling Voluntary Task Switching 12
experienced as easier so that these chains are strengthened more frequently, and the degree of
strengthening will be larger the more repetitions the chain contains.
To implement the idea of chain strength or chain weight, w, we define a parameter 0 < b < 1,
which is the bias of a repetition in the stored sequence (1 - b is the likelihood of a switch). The
value of b determines the weight (w) of the chain by the simple multiplicative rule w = bR(1− b)S,
where R and S are respectively the numbers of repetitions and switches in the chain. With a given
value of b, the larger the proportion of repetitions in the chain, the larger the weight of the chain
becomes. With a fixed number of repetitions in the chain, the larger b, the larger the weight of the
chain will be. In other words, b represents an acquired bias towards more repetitions. This way,
the consideration is implemented that experience with execution of the tasks (repetitions are easier
than switches) will influence the retrieval of task chains. The parameter b indicates the probability
of a single element within a sequence of tasks and w defines the overall strength of a sequence.
The sequence AABB, for example, contains two repetitions and one switch: thus w = b2(1− b).
Each time a new chain has to be selected, the chains compete for being selected by producing an
amount of activation that is proportional to their strength. Using Luceʼs choice rule (Luce, 1959),
the probability that a particular sequence i with weight wiwill be selected, depends on the relative
degree of activation with respect to the total amount of activation of all chains. Let W be the sum
of the weights w of all the sequences in the set, the probability to select a particular sequence i
with weight wiis then wi/W . The underlying process is thus simply based on competition of the
alternatives for selection.
Whereas b expresses a bias towards chains with more repetitions, its value determines the
probability (p) of top-down selection of a repetition. For example, for chains of length 3 as
described in Table 1, if b = 0.5 (i.e., repetitions and switches have an equal weight), the overall
likelihood of a repetition, p = .33, because 1/3 of the transitions within this set are repetitions. If the
value of b were 0.7, the weights of the three chains would be respectively 0.21 (0.7 × 0.3), 0.21
(0.7 × 0.3), and 0.09 (0.32). Given the weight of the first chain (0.21) and the fact that half of the
events in that chain are repetitions, the weight of a repetition in that chain is 0.21 × 0.5. The
?Modeling Voluntary Task Switching 13
overall probability of a repetition, p, would thus be (0.21 × 0.5 + 0.21 × 0.5)/(0.21 + 0.21 + 0.09) =
0.41. The set of chains intrinsically constrain the maximum value of p; for example, in the set of
chains with size 3 (2 transitions), p can never become larger than 0.5 (when b approaches 1)
because each chain always contains at least one switch. The longer the chain, the larger p can
The third parameter, r, is related to bottom-up priming. This parameter is introduced because
several studies have shown that bottom-up events, such as stimulus repetitions (Demanet et al.,
2010; Mayr & Bell, 2006), stimulus availability (Arrington, 2008), and processing efficiency
(Arrington & Rhodes, 2010) affect task choice in voluntary task switching. The probability r , then,
represents the likelihood of a bottom-up event overruling planned or intended task selections. We
assume that with an unknown probability events occur that activate a tendency to repeat the same
task. On trials on which a repetition is intended, this parameter has no effect, but on trials on
which a switch is intended, this leads to a conflict between the tendency to switch to another task
and the tendency to repeat the previous task. On some trials this conflict will be resolved in favor
of the intended action (switch), on other trials, the triggered repetition will win the competition; we
assume the latter occurs with a probability r. Importantly, in combination with the b parameter, this
r parameter could account for the task-repetition bias typically observed in VTS. Several choices
as to what happens with the already selected chain are possible; in the present model, the primed
repetition overrules the current event in the selected chain, and execution of the retrieved chain
continues after the intrusion. The reason for implementing an overruling mechanism in the model
is based on the rationale that the intended task (i.e., the task specified in the chain) has been
selected, but then loses the competition with the primed task repetition. In testing the model, other
possible choices will also be considered.
Thus far this chain-retrieval model specifies that short balanced sequences (chains) avoiding
too many repetitions are stored in memory, and have a strength that determines the probability of
being selected. This model has three important distinctive characteristics. First, the model
specifies a selection mechanism that is based on retrieval of short chains of events from LTM (see
Table 1). Because the chains stored in LTM are in agreement with the representation bias
? Modeling Voluntary Task Switching 14
specified by Rapoport and Budescu (1997), the produced sequences deviate from statistical
independence without the need for a retrospective monitoring mechanism to guarantee this.
Second, because the model allows the selection mechanism to be sensitive to the difficulty of the
transitions (the b parameter coupled to strengths of chains on the basis of the relative difficulty of
the task transitions), it may be useful to predict the characteristics of the sequence of task names
as well as the characteristics of the sequence of transitions1. The advantage of this feature will
become clear in the application of the model to the data. The third characteristic of the model
allows that bottom-up events in the form of task repetitions could gain control over task selection
(with a probability r). This is in line with the nature of the bottom-up events reported in the
literature (Demanet et al., 2010; Mayr & Bell, 2006; Vandamme, Szmalec, Liefooghe, &
The fundamental assumptions adopted allow for some freedom of implementation in a
formalized model, which suggests that it is possible to build similar models with slightly different
implementations of some assumptions. Appendix B contains an overview of variations in the
implementation and how they fare when used to fit actual data. As these variations do not result in
better fits and predictions, we do not consider them in the main text. The purpose of the present
investigation was to test the usefulness of this framework and its formalization. Because the
framework generates a series of hypotheses about processes involved in selection of independent
events as well as voluntary task selection, an exhaustive test of all these hypotheses is beyond the
scope of this article. For that reason, we focus on testing how well the presented model can
account for voluntary task selection and independent-event selection performed by participants in
experimental settings. In a first study, the model is applied to data of 17 participants in voluntary
task switching based on the standard procedure (Arrington & Logan, 2004). In a second study, the
model is applied to data of 80 participants in a voluntary task switching study based on the double
registration procedure (Arrington & Logan, 2005). This second study is also used to test the
assumption that the length of the sequences stored in LTM depends on experience over trials, and
the simplifying assumption that task selection over an experimental session is quite stable. The
? Modeling Voluntary Task Switching 15
1 For this reason the model could also be framed as a transition-chain retrieval model.
third study presents a fit of the model to a situation where a response hand is selected for
execution of one single task, and the fourth study presents a fit of the model when participants are
required to independently generate events instead of tasks. In each case, we describe the
experiment and its results, and then we apply the model to the reported data.
3. Study 1
As a first test of the modeling, we focus on the task choice data of an experiment that has
been published before (Liefooghe et al., 2009). This experiment used the standard method of
voluntary task switching, where one single key-press informs about the task selected and the
categorization of the target (Arrington & Logan, 2004, 2005). This version of the procedure has
been shown to be very sensitive to bottom-up triggered repetitions (Demanet et al., 2010; Mayr &
Bell, 2006). Because the study has been published, we briefly describe the methods and only
report the results that are relevant for the present purpose.
Eighteen first-year psychology students of Ghent University participated for course
requirements and credit. They were assigned to one of two between-subjects conditions, a
condition with an RSI of 100 ms or a condition with an RSI of 1000 ms. Nine other students in the
original study were assigned to a within-subjects condition. These participants were not included
in the present study, because the RSIs were varied randomly from trial to trial, leading to
confounding of the manipulation with other factors affecting task choice. Data for one subject (in
the RSI-100 condition) were not included in the present analysis, because the proportion of
selections of one of the tasks amounted to .85.
Stimuli consisted of letters (A, E, B, or D) surrounded by a geometric figure (circle, ellipse,
square, or rectangle) to which either a letter categorization task (consonant vs. vowel) or a form
categorization task (quadrangle or ellipsoid) could be applied. A keyboard was used to register the
responses. The form task was performed with the left hand (keys f or g) and the letter task was
performed with the right hand (keys j or k).
Participants were tested individually on a PC with a 17-inch color monitor running Tscope
(Stevens, Lammertyn, Verbruggen, & Vandierendonck, 2006). Instructions of Arrington and Logan
? Modeling Voluntary Task Switching 16
(2004) were used. In each condition, participants received one practice block and four blocks of 64
test trials each. A trial started with presentation of the stimulus in the center of the screen until a
response was given within a response deadline of 3000 ms. Immediately after the response or
when the response deadline was attained, the stimulus disappeared. After an incorrect response,
the screen turned red for 200 ms before the RSI started (100 ms in one condition; 1000 ms in the
3.2. Data analysis
The analysis focuses on the series of task choices and task transitions. As the task choices
and the transitions within the sequence of tasks are binary events, the sequence of events can be
expressed as a series of binary digits (0 or 1). A sequence of tasks is a series of task names;
using the letters L (letter categorization) and F (form categorization), an example of a series of
selected tasks may be LLFLFFL. Similarly, the letters R (repetition) and S (switch) can be used to
describe a sequence of transitions (RSSSRS in the example). To convert the sequence to binary
values, L can be recoded as 1 and F as 0 (or vice versa) and R can be coded as 1 and S as 0 (or
Such sequences of random events are often summarized by using a runs statistic, which
yields a proportion of the runs of the same event at a series of lengths. The runs statistic
(Sternberg, 1959a; see also Vandierendonck, 2000a), can be defined as follows
N − k +1
where N is the number of events in the complete sequence and rkis the proportion of runs with
length k in which xk= 1. Consider a sequence like “0110111010100011”, where the target outcome
is coded 1. The number of runs of length 1 equals the number of 1s in the sequence, which is 9
(code 1 occurs 9 times). Runs of length 2 consist of two consecutive 1s; there are four such
groups of 1s in the sequence. Runs of length 3 consist of three consecutive 1s; there is only one
such group. By dividing these counts by the number of possible runs of a particular length, a
proportion is obtained for each length. Although the runs statistic captures large deviations from
? Modeling Voluntary Task Switching 17
independence, it is not particularly sensitive in detecting very small deviations from independence
because the values become smaller as the run length increases (see example). However, the
statistic is useful because it captures deviations from statistical independence in both directions:
when there are more repetitions, there will be fewer short and more long runs, and when there are
more alternations, the opposite pattern will occur (relatively more short and less long runs).
In addition to the runs statistic, we used the autocorrelation statistic (Sternberg, 1959a; see
also Vandierendonck, 2000a), which is very sensitive to deviations from independence, and
therefore useful to make a more fine-grained analysis of the data. This statistic expresses the
tendency for pairs of events in the sequence to correlate with each other. The pairs of events that
are considered can be close together or further apart (short or long lag). The autocorrelation
statistic lag k (ck) is defined as
N − k
The correlation expresses the probability that both elements in the pair (separated by lag k)
are the same. When lag is 1, for example, the autocorrelation expresses the probability that the
current event is the same as the previous one. Considering the example we had before
“0110111010100011”, the autocorrelation lag 1 looks at all occurrences of two consecutive 1s;
there are 4 of these. Actually, by definition this is the same as runs of length 2. The
autocorrelation lag 2 looks at two occurrences of a 1 separated by another (not relevant) outcome.
The triplets to consider are 011, 110, 101, 011, 111, 110, 101, 010, 101, 010, 100, 000, 001, 011
and there are only four cases out of these 14 where the first and the third element are both 1. The
autocorrelation statistic is a measure that is sensitive to statistical dependencies based on learning
to repeat an event: the autocorrelation will tend to be larger when a learning process governs the
production of the events in the series (Sternberg, 1959a, 1959b). Application of the statistic
requires that one task is coded 1 and the other 0. By applying the statistic twice to the sequence
of task choices, once with Letter coded as 1 (Form 0) and once with Form coded as 1 (Letter 0),
the joint outcome specifies all correlations in the data. This joint outcome is complementary to all
?Modeling Voluntary Task Switching 18
tendencies to alternate instead of to repeat. Hence, the statistic applied in this way is sufficient to
describe all deviations from independence.
The data analysis will be performed in two steps. In the first step, the analysis focuses on
the sequence of task choices; in the second step, on sequences of transitions. Although the
transitions are derived from the sequence of task choices, due to the regrouping of successive
events in terms of repetition or switch, the second analysis may detect different statistical
properties within the sequence. Inspection of the sequence of transitions may, for example, clarify
whether a bias to repeat tasks as seen in the task analysis is a local phenomenon or a more global
phenomenon. If the repetition bias is local, task repetitions will occur in only one or a few
subsequent transitions, soon interrupted by a switch. On the contrary, when the repetition bias is
more generalized, long as well as short series of task repetitions will occur. Performing these two
analyses, then, may reveal information about the sequence of tasks that would otherwise remain
undetected. Within each type of analysis, the results of the runs and the autocorrelation statistic
are reported separately.
In the task-based analysis, both tasks were selected equally often and the proportions of
runs decreased at the same rate in both tasks. Run proportions were on average smaller (i.e., less
repetition bias) in the condition with long RSI (see Figure 1, top-left). In the autocorrelation data,
the size of the correlation varied with lag with lower values at lags 2-4 than at the other lags (see
Figure 1, bottom-left). Even though, the autocorrelations depended only marginally on RSI, the
effect on lags 2-4 was stronger in the long RSI condition. The autocorrelation findings confirm a
repetition tendency at lag 1 followed by a tendency to alternate at lags 2-4; as this alternation
tendency was stronger in the long RSI condition, the tendency to repeat tasks was stronger in the
short RSI condition.
In the transition-based analysis, repetitions were reliably more frequent than switches and
this difference was maintained at all run lengths (see Figure 1, top-right). Repetitions were also
more frequent at short than at long RSI. This clearly confirms the presence of a repetition bias
which is stronger at short RSI. The autocorrelations were lower at lag 1 than at the other lags and
? Modeling Voluntary Task Switching 19
were stable from lag 2 on (see Figure 1, bottom-right). This difference did not interact with
transition type but it did interact with RSI. The fact that autocorrelations tend to be lower at lag 1
indicates that although repetitions are more frequent overall, an immediate repetition of the same
transition is lower than on average at other lags.
These findings are described with more statistical detail in the following paragraphs 3.3.1 and
3.3.2. Readers who prefer to skip these details can do so and move on directly to the discussion
of the results in section 3.4.
3.3.1. Focus on tasks.
For the analysis with focus on the letter task, the 4 x 64 task selections of each participant
were coded 1 when the letter task was selected and 0 otherwise (form or no selection). Similarly,
for the form task, the selections were coded 1 when form was selected and 0 otherwise. In only
one participant, about 1.5% of the trials were non-selections; in the other participants, no non-
selections occurred. Anyway, due to the coding, non-selections did not contribute to the run length
or the autocorrelation data.
These binary data were used to calculate the proportion of runs of lengths 1-8 and
autocorrelations at lags 1-82. Per statistic, the multivariate general linear model was applied to the
data on the basis of a 2 (RSI: 100 or 1000 ms) x 2 (Task: letter vs. form) x 8 (Lengths or Lags)
factorial design with repeated measures on the last two factors. For all analyses, α = .05, unless
22.214.171.124. Runs. On average, the selection proportions of letter and form were both .50. Run
proportions depended on RSI and run length only. Proportions of runs decreased with run length,
F(7,10) = 870.46, ηp2 = 1.0. Run proportions were smaller when RSI was long (.10 for long
versus .16 for short RSI). Neither task (.14 vs. 13), nor any of the interactions attained significance
(smallest p > .20). This shows that there was a tendency to repeat tasks more at short than at long
RSI. Figure 1 (top-left) displays the runs proportions as a function of RSI, task, and run length.
---- Figure 1 about here ----
? Modeling Voluntary Task Switching 20
2 We calculated lengths and lags 1-10, but in order not to violate the assumptions of the analytic
tool, we restricted the analysis to 8 lengths or lags.
126.96.36.199. Autocorrelations. The autocorrelations are shown in Figure 1 (bottom-left) as a
function of RSI, task, and lag. Only the effect of lag was reliable: F(7,10) = 4.10, ηp2 = 0.74. The
effect of RSI was marginally significant, F(1,16) = 3.10, ηp2 = 0.16. Figure 1 shows that the
correlations start high, drop off at lags 2-4 and stabilize from lag 5 on. This pattern is more clearly
present at longer RSI, though. Analysis confirmed that overall autocorrelations were higher at the
first than at the other lags, F(1,16) = 27.68, ηp2 = 0.63. A contrast between the autocorrelations at
lags 2-4 and lags 6-8 was not reliable overall (F < 1), but it interacted with RSI, F(1,16) = 4.69, ηp2
= 0.23, such that the difference between lags 2-4 and lags 6-8 was larger in the condition with the
long RSI. These findings show that there is a rather strong repetition tendency (autocorrelation) at
lag 1, but that in lags 2-4 in the long RSI condition, the autocorrelation is rather weak. This
suggests that the repetition bias is stronger at short RSI.
3.3.2. Focus on Transitions.
In the sequential analysis of the transitions, repeating the same task was coded 1 and
changing the task or failing to select a task was coded 0; in the calculation targeting on switches,
changing tasks was coded 1 and repeating the same task or failing to select a task was coded 0.
In all other respects, the same data-analytic method was used as for the analysis focusing on task
selections. In order not to overload the report with an enumeration of statistical tests, the
outcomes of this analysis are presented in Table 2; only the effects that are central to our main
purpose are reported in the text.
---- Table 2 about here ----
188.8.131.52. Runs. Transition runs are shown in Figure 1 (top-right) as a function of RSI,
transition and length. Clearly, repetitions (M = 0.63) were more frequent than switches (M = 0.37)
and they also differed (.17 vs. .06) at all lengths. Run proportions were larger on short (M = 0.14)
than on long RSI (M = 0.10). Interactions of transition and length and RSI and length were reliable
(see Table 2). The dominant presence of repetitions confirms the repetition bias already observed
at the level of tasks. Repetitions were repeated more often than switches especially at short RSI.
---- Figure 1 about here ----
?Modeling Voluntary Task Switching 21
184.108.40.206. Autocorrelations. Figure 1 (bottom-right) displays the transition-based
autocorrelations as a function of RSI, transition, and lag. Lag correlations were larger for
repetitions (M = .41) than for switches (M = .15). They also varied over lags. Transition and lag
did not interact, but transition was involved in an interaction with RSI, as displayed in Figure 1.
Finally, also the interaction of RSI and lag was significant (Table 2). In contrast to the task-based
autocorrelations, the transition-based autocorrelations seem quite stable, except that they were
lower at lag 1 (M = .26) than at other lags (M = .29), F(1,16) = 13.00, ηp2 = 0.45. This contrast
explains the major part of the variance among the means per lag: r2 = .82. Repetitions are
selected more often than switches and therefore repetitions also tend to be repeated more than
switches. However, for both, repetitions and switches, this tendency is smaller at lag 1 than at
longer lags, where transitions rather show a pattern of independence, but repetitions are still
repeated more than switches.
These analyses confirm the repetition bias that has been reported in the literature (e.g.,
Arrington & Logan, 2004; Mayr & Bell, 2006), as well as the finding that this bias is stronger at
shorter RSI (Arrington & Logan, 2005). The autocorrelations of the task choices show that the
repetition bias is strong at all lags except lags 2-4. Transition autocorrelations are stable from lag
2 on and indicate in fact that the repetition tendency of both repetitions and switches is a global
phenomenon, except for lag 1 which suggests a local tendency to repeat the previous transition
less often than average. This pattern of findings is not consistent with existing perseveration
models (e.g., Vandierendonck, 2000a), as they predict a decrease of autocorrelation over lags.
3.5. Model testing
In this section, we report the results of the model tests performed on the task-selection data
of this experiment. First, we will present the results of fitting the chain-retrieval model to the runs
proportions obtained in the experiment. In order to show that the chain-retrieval model yields a
better account than simpler models, also the results of the statistical independence (Bernoulli)
model and two statistical dependency models are presented. One of these statistical dependency
models is the perseveration model (Vandierendonck, 2000a). It assumes that with probability a the
? Modeling Voluntary Task Switching 22
previous event is repeated and if no such repetition occurs, with probability q an event is sampled
independently from the previous event:
Pr(xi) = a + (1− a)q (3),
where Pr(xi) refers to the probability that a certain event (x) occurs at time i, a represents the
probability that the previous event is repeated (perseverates) and q is the probability that the
present event is sampled independently from the previous trial. The second dependency model is
the alternation model (Vandierendonck, 2000a). This model is also based on equation (3), but now
parameter a represents the probability that an alternation occurs (i.e., the previous event is not
repeated) and if no alternation occurs, with a probability q an event is sampled independently from
the previous trial.
In a next step, the parameter estimations obtained in these fits of the four models will be
used to predict the autocorrelations in sequences of tasks and sequences of transitions. After this
phase of global model testing, more specific tests of the chain-retrieval model will be reported. To
that end, each of the free parameters will in turn be clamped to a particular value and the other
parameters will be estimated resulting in new fits and new predictions.
3.5.1. Model fitting and parameter estimation.
First, we used the run proportions of the task-based data analysis to estimate the parameters
of the task-chain retrieval model and the three comparison models. Because the data of each
individual participant are sufficient to fit the models, the runs data of each subject were used to
? Modeling Voluntary Task Switching 23
estimate the free parameters of the best fitting model for that subject. The fits per subject could
then be entered in statistical analyses comparing the merits of the models3.
All model fits were based on maximum likelihood estimation. The parameters were
estimated jointly for the letter task and the form task data on the basis of the observed runs
proportions length 1-10. The likelihood function to estimate the correspondence between the
model and the data is:
N − i
ni(1− pi)N−ni−i? (4)
where L is the likelihood function, K is the number of lengths considered, pi is the estimated
proportion of runs of length i, N is the total number of observations (number of trials) and niis the
number of observed runs of length i. In order to find the maximum of such function, it is easier to
take = lnL , and to minimize − :
N − i
⎠⎟niln pi− (N − ni− i)ln(1− pi) (5)
This minimization was performed by means of a univariate search method (Brent, 1973). On
each step of the iteration, the current parameter values were used to generate a sequence of
50,000 task choices. On the basis of this generated sequence, estimated runs for the letter task
?Modeling Voluntary Task Switching 24
3 This procedure has several advantages over the alternative procedure based on fitting the
models on the between-subject average of runs. First, the processes described in the chain-
retrieval model and also those of the other dependency models are constrained by the skills and
capacities of each subject. This variability is given the best chance by using individual data.
Second, the between-subject average of the runs statistics does not adequately represent the
processes that resulted in the generated random sequences. It is easy to imagine that the
average runs of a subject generating a sequence with a repetition bias and another subject
generating a sequence with an alternation bias will show either a very small bias or no bias at all.
were calculated. Next the sequence was converted (0 was recoded to 1 and 1 to 0) to calculate
the estimated runs for the form task. The estimated proportions were weighted as specified in
formula (5) to yield − . The search procedure would then sample new parameter values and start
a new step. This continues until a minimum is obtained. By performing the estimation jointly for
the two tasks, the characteristic of random succession of the two tasks is captured in the
parameter fit. The search procedure finds a local minimum in a very efficient way. In order to
maximize the chance of finding the global minimum, the search procedure was applied ten times
with random starting values. In the application for the chain retrieval model, this estimation
procedure was repeated for each of the values 3-6 of m. Per participant, the value of the three
parameters of the best fitting model was then selected.
The top panel of Table 3 displays the fit and the estimated parameters of the four models.
This table shows that the averaged minimization over all participants was quite good for the chain-
retrieval model but was much worse for the three statistical models (Bernouilli, perseveration and
alternation). In order to test whether the difference between the fit of the chain-retrieval model and
the other models was statistically reliable, Akaikeʼs Information Criterion (Akaike, 1974) with
correction for the number of observations (AICc, Hurvich & Tsai, 1989) was calculated per subject
and per model. The proportion of subjects in which the AICc was smaller for the chain-retrieval
model than for the comparison model is displayed in Table 3 on the row labeled “Prop(AICc)”. To
test whether this proportion was significant, a z-score was calculated in the Gaussian
approximation to the binomial distribution of proportions; this z-score is also shown in the table
together with the probability that this would occur under the null hypothesis that both models are
equivalent. These tests show that the chain-retrieval model yielded a better fit than each of the
other three models, except the perseveration model. In view of this, further tests will have to clarify
how well the chain-retrieval model captures the data.
---- Table 3 about here ----
3.5.2. Model predictions.
The estimated parameters of these four models were used to predict the autocorrelation
statistic for the task data. Predictions of the models were calculated on a sequence of 50,000
?Modeling Voluntary Task Switching 25
generated events on which the likelihood function was calculated given the parameters estimated
on the runs data. In order to measure the degree of correspondence between the predictions and
the data, the root mean squared deviation was calculated:
where oirefers to the data points and eirefers to the predicted values. This measure obtained for
each of the models is displayed in the middle panel of Table 3, and shows that the prediction of
chain-retrieval model is better than that of the three other models. The chain-retrieval modelʼs
prediction accuracy was compared pairwise with that of the other models. Table 3 shows the
proportion of participants performing better on the chain-retrieval model than on each of the other
models. The probability of this proportion was assessed by calculating a z-score (also shown in
the table). Figure 2 (top-left panel) illustrates that the task-chain retrieval model yielded the best
correspondence between predictions and data. When applied to data that are more sensitive to
deviations from statistical independence than the runs statistic, it seems that the chain retrieval
model significantly better accounts for these deviations than the other dependence models.
---- Figure 2 about here ----
The estimated parameters of these four models were also used to predict the autocorrelation
statistic for the transition data. In order to keep task and transition statistics equivalent in terms of
number of events, only lags 1-9 were included for the transitions. The sequence generated for the
prediction of the task autocorrelations was converted into a sequence of transitions, once with
focus on repetitions and once with focus on switches. The rmsd was calculated per subject. Table
3 (bottom panel) displays the rmsd and the proportion of participants having smaller rmsd on the
chain-retrieval model than on the comparison models. Again, the probability of this observation
under the null hypothesis is derived by calculating z-scores. Figure 2 (bottom-left panel) illustrates
that the task-chain retrieval model yielded the best correspondence between predictions and data,
and this was confirmed by the significant difference with each of the other models.
? Modeling Voluntary Task Switching 26
3.5.3. Model validation.
As the model has three free parameters, it is also important to know whether each of the
parameters is indispensable. To that end, each of the parameters in turn was clamped to a neutral
value, and with one parameter fixed, new estimations of the other two parameters were obtained.
A neutral value for the free parameters was considered to be a value at which the effect of the
corresponding process was the lowest; so for m and r the lowest value, respectively 3 and 0, was
taken, and for b, a value of 0.5 (no bias in either direction) was considered to be neutral. Table 4
shows that -ln L yielded a lower value for the full than for the restricted models (see also, Figure 2,
right panels). For testing the significance of the differences between these nested models, the
likelihood ratio test was used (Buse, 1982). Let Arefer to the log likelihood of the full model and
Bto the log likelihood of the nested model, then the statistic W, which is defined as,
is distributed according to a χ2distribution with the difference in the number of parameters
between the two models as the number of degrees of freedom. Per subject, W was calculated for
all pairs consisting of the full (three-parameter) model and a model where one of the parameters
was fixed. In order to test whether the majority of subjects was in favor of the full model, a normal
approximation to the binomial test for proportions was performed. On the line labeled “P(LR)” the
proportion of subjects with a significantly smaller likelihood of the full model is displayed, and on
the next line the z-score with the probability of the observation under the null hypothesis is listed.
---- Table 4 about here ----
It appears that the fit of the full model is better than the fit of the nested models, but only
significantly for the variant with parameter r fixed at 0. The other two nested models (m = 3 and b
= 0.5) could not be rejected. However, when also the predictions of the full and the nested models
are taken into account (see lower panels of Table 4), it appears that in the task autocorrelation
? Modeling Voluntary Task Switching 27
predictions only the nested model with b fixed is equivalent with the full model, and in the
predictions of the transition autocorrelations, the full parameter model was significantly better than
each of the nested variants. These findings suggest that the m and the r parameter may play an
important role, whereas the evidence in favor of the b parameter is not convincing. Possibly the
statistical power of the present study is not strong enough for supporting strong conclusions.
Table 4 also shows the estimate of the proportion of repetitions due to top-down control and
the estimate of the proportion of bottom-up triggered repetitions. Together these two proportions
account for the observed proportion of repetitions. Clearly, the proportion of top-down triggered
repetitions is much smaller than the value of the bias parameter (b). Also the proportion of bottom-
up triggered repetitions is smaller than expressed in parameter r. This is because some bottom-up
repetitions coincide with top-down repetitions and do not change the top-down controlled actions.
Comparison of these proportions across the different model variants shows that when r is 0, there
are no bottom-up events whatsoever, with as a result that the other parameters (m and b) are
adapted to account as much as possible for the observed proportion of repetitions. Similarly, when
b is clamped to a neutral value, m and r become larger to achieve an acceptable estimation. The
observation that m increases to compensate for the lack of repetitions, suggests that this
parameter is not only constrained by the length of the chains that can be handled but also by the
frequency with which top-down repetitions do occur. Finally, even when m is fixed, the estimated
value of the two other parameters is adapted (increased), again suggesting that m affects the
number of top-down repetitions.
The chain-retrieval model seems quite promising when fitted to the run length proportions in
sequences of tasks. For a significant majority of the participants, the obtained fit was better than
the fits obtained for the statistical independence model and existing statistical dependence models.
Also the predictions of the autocorrelations in sequences of tasks as well as in sequences of
transitions were better for a significant majority of the participants than the predictions of the
? Modeling Voluntary Task Switching 28
In a final test designed to investigate whether the processes underlying the three free
parameters are all involved in achieving the very good correspondence of the chain-retrieval model
and the data, each of the parameters in turn was clamped to a neutral value so as to exclude or to
minimize the role of the underlying process. These analyses indicated that the m and r parameters
are important: even when corrected for the number of estimated parameters, the full parameter
model still yielded significantly better fits to the runs proportions and significantly better predictions
of the task and transition autocorrelations. The case for the b parameter was less convincing in
the present study, although the findings clearly indicate that this parameter also affects the top-
down repetition tendency.
4. Study 2
Although the application of the chain-retrieval model to the task choice data of Study 1 shows
that the model yields a better account of task choices in VTS than statistical independence and
dependence models, some concerns may be raised about the generality of these findings. First,
the size of the subject sample and the number of choices per subject were both rather small.
Second, the data were collected with the single-registration procedure which is known to be very
sensitive to bottom-up intrusions. As the occurrence of such intrusions is a central assumption of
the model, the test in Study 1 may be biased in favor of the model. In order to counter the latter
criticism, in Study 2 the double registration procedure (Arrington & Logan, 2005) was used. In this
procedure, two responses have to be emitted in each trial. First, a probe appears to which the
subject responds by indicating the task that will be used in the current trial. Next, the target
stimulus is presented and the selected task is applied to this stimulus. The concern regarding the
amount of data was countered by having more subjects (80) and longer sequences of task choices
(2 blocks of 256 trials).
This design has the additional advantage that it becomes possible to include the effect of
practice in the study. If task-choice behavior changes with practice, then the pattern of choices
might be different between the first and the second block of 256 trials. In the same vein, this
design allows to investigate whether the best fitting model parameters change with practice. The
rationale for this is that in developing the chain-retrieval model, it was assumed that the chains in
? Modeling Voluntary Task Switching 29
LTM are learned from experience within the constraints of working memory because once
retrieved, the chain has to be unfolded in working memory in order to access the next element in
the sequence on consecutive trials. In order to be able to fit the model parameters, it was further
assumed that learning was confined to the practice trials so that chain length could be considered
constant over the experimental session. A secondary aim of this study is to test whether this
simplified assumption is tenable: are changes in the m parameter negligible or are there important
changes with further practice?
4.1.1. Participants and design
Eighty first-year psychology students at Ghent University participated for course
requirements and credit. All participants had normal or corrected-to-normal vision, were right-
handed, and all were naïve to the purpose of the experiment. Participants were randomly
assigned to two conditions (forty participants per condition) that differed in the stimulus onset
asynchrony (SOA) of the stimulus and the probe (see below).
Stimuli were the digits 1-9, excluding 5. Participants were required to classify the digits
either on the basis of their magnitude (smaller or larger than five) or their parity (odd or even).
Responses were registered by means of the numeric pad of a standard keyboard. One hand was
used for pressing the task-selection keys; the other hand was used for pressing the task-execution
Although we only need the task-selection data, we describe the complete procedure of data
collection. However, for completeness and to show that the experiment replicates typical task-
switching results, the main findings regarding the task-execution data obtained in this experiment
are briefly presented in Appendix A. Pentium III personal computers with a 17-inch color monitor
running the Tscope C/C++ library (Stevens et al., 2006) were used. Each session lasted for
approximately 45 minutes. After participants signed an informed consent, instructions were
presented on screen and paraphrased if necessary. The instructions concerning unpredictability of
? Modeling Voluntary Task Switching 30
voluntary task switches were the same as those used by Arrington and Logan (2005), namely that
each task should be performed about equally often and that the sequence should be as
unpredictable as in coin tossing.
On each trial, a probe (“?”) was presented in a square 5 mm above the centre of the screen.
This probe disappeared when participants pressed one of the task-selection keys. This was
followed 400 ms later by the appearance of the target stimulus, 5 mm below the centre of the
screen. The target remained on screen until participants responded on the basis of the previously
selected task or until a maximal response time of 2500 ms elapsed. The probe of trial n appeared
either 50 ms (short SOA condition) or 1500 ms (long SOA condition) after the presentation of the
stimulus of trial n-14.
After two practice blocks of 64 trials, participants performed two experimental blocks of 256
trials. In the first practice block, the emphasis was on familiarization of the procedure of selecting
and executing the different tasks. In the second practice block, the emphasis was on
unpredictable selection of the tasks. In order to increase participantsʼ awareness of their selection
behavior, the warning “do not forget to switch tasks” was shown for 1000 ms whenever the
participants selected the same task four times in a row. When the participants switched between
tasks four times in a row, the warning “do not forget to repeat tasks” appeared for 1000 ms. This
task-selection feedback was presented in the second practice block only.
? Modeling Voluntary Task Switching 31
4 This variation of SOA between the probe and the target is reminiscent of the PRP (psychological
refractory period) paradigm. However, unlike in the PRP paradigm, in the present procedure, task
selection does not have to wait for the probe to start. In fact, task selection can start at any time,
even at an earlier trial. The function of the probe is to indicate that a task-indication response must
be emitted. On the trials where the task choice has not yet been made, the task-selection
response may result in postponement of the task execution response. Appendix A shows,
however, that probe RTs are fairly short suggesting that on average task selection has been
completed before presentation of the probe. For all these reasons, strictly speaking, the present
design is not a PRP paradigm, although it may bear similarities to it and it may be the case that
task choice processes compete with task execution.
During the entire experiment, participants received on-line feedback about their performance.
A red screen appeared for 50 ms when they made an error on the target. When they were too
slow to select a task (RT > 2500 ms), the message “no task selected” was displayed for 1500 ms.
Following each block (practice and experimental), a general summary about the performance
during that block was shown. This feedback included the mean reaction times on the targets, the
percentage of errors, the selection percentage of each task, the percentage of failures to select a
task, and the percentage of task repetitions and task switches. If necessary, participants were
corrected: they were urged to switch more or to repeat more when the proportion of repetitions or
switches was above .70, to make fewer errors when percentage of errors was above 15%, to
respond faster when mean task-execution reaction time was above 1200 ms or when the
proportion of trials without a task-indication response was above 10%, and to be more random
when they selected a particular task on more than 75% of the trials.
As in Study 1, the data analysis was performed in two steps. In the first step, the analysis
focused on the sequence of task choices; in the second step, on sequences of transitions. The
analyses of the runs and the autocorrelation statistic are reported separately. As for Study 1, we
first provide an overall summary of the findings before presenting the statistical details.
In the task-based analyses, runs proportions were lager in the parity task, in the second
block and in the short SOA condition (Figure 3, top-left). The drop in proportions with run length
was steeper in the short SOA condition and also in the second block. The autocorrelations varied
with task and lag (see Figure 3, bottom-left). At lag 1, the autocorrelations were rather larger and
then showed a dip at lags 2-4. The size of the dip was larger at long SOA. The size of the dip was
also larger in the second block than in the first.
The transition-based analyses revealed that runs proportions were higher for repetitions than
for switches, and higher for short than for long SOA (see Figure 3, top-right), and higher in the
second block as compared to the first block . At short SOA repetitions were repeated more often
than switches and in the second block runs proportions tended to be larger than in the first block at
all lengths. The autocorrelations were characterized by a lower value at lag 1 than at the other
? Modeling Voluntary Task Switching 32
lags and overall there was not much variation from lag 2 on (see Figure 3, bottom-right). The
autocorrelations were lager for repetitions than for switches and were larger in block 2 than in
In the following paragraphs, these findings are substantiated with the statistical details.
Readers who prefer to skip these details can continue at section 4.3 (Discussion).
4.2.1. Focus on tasks
For the magnitude task, the task selections of each participant were coded 1 when
magnitude was selected and 0 otherwise (parity or no selection). Similarly, for the parity task, the
selections were coded 1 when parity was selected and 0 otherwise. About 1% of the trials were
non-selections. Due to the coding, non-selections did not contribute to the run length or the
These data were used to calculate the proportion of runs of lengths 1-10 and
autocorrelations at lags 1-10 separately in each block. Per statistic, the multivariate general linear
model was applied to the data on the basis of a 2 (SOA: 50 or 1500 ms) x 2 (Task: magnitude vs.
parity) x 2 (Blocks) x 10 (Lengths or Lags) factorial design with repeated measures on the last two
factors. For all analyses, α = .05, unless otherwise mentioned. In order not to overload the report
with an enumeration of statistical tests, the outcomes of the complete analyses are presented in
Table 5; only the effects that are central to our main purpose are reported in the text.
---- Table 5 about here ----
220.127.116.11. Runs. Figure 3 (top-left) displays the runs proportions as a function of SOA, task,
and run length. On average, the selection proportions of magnitude and parity were 0.48 and 0.51
respectively. As can be seen in Table 5, run proportions depended on task, blocks, length, and
SOA. Proportions of runs were higher for the parity task (M = .13) than for the magnitude task (M
= .11), and they decreased with run length. Run proportions were larger in the second block (M = .
12) than in the first block (M = .11), and they were smaller when SOA was long (.10 for long
versus .13 for short SOA). Length interacted with task and with SOA. The drop in the proportions
was less steep for the parity task and for short SOA. This shows that there was a tendency to
? Modeling Voluntary Task Switching 33
repeat the parity task more than the magnitude task and also a tendency to repeat tasks more at
short than at long SOA. Blocks interacted with length, SOA and length x SOA.
---- Figure 3 about here ----
In the second block, more short run lengths were observed and the proportions of runs
decreased over lengths at a slower rate than in the first block. This is confirmed in the interaction
of the contrast between short (1-2) and medium lengths (4-5) by blocks, F(1,78) = 16.58, ηp2 = .18
and in the interaction of this contrast, SOA and blocks, F(1,78) = 7.13, ηp2 = .08.
18.104.22.168. Autocorrelations. The autocorrelations are shown in Figure 3 (bottom-left) as a
function of SOA, task, and lag. Only the effects of task and lag were reliable (see Table 5). Figure
3 shows that the correlations start high and then quickly drop off and stabilize from lag 5 on.
Overall, autocorrelations were lower at lags 2-4 (M = .22) than at lags 7-9 (M = .24), F(1,78) =
23.76, ηp2 = 0.23. This contrast interacted with SOA, F(1,78) = 6.47, ηp2 = 0.08. The contrast was
smaller at short (.22 vs. .23) than at long SOA (.21 vs. .25). These findings show that there was a
rather strong repetition tendency (autocorrelation) at lag 1, but that at lags 2-4, the autocorrelation
was rather weak. This suggests that a tendency to immediately repeat the task is soon followed by
one or more switches.
There was no main effect of blocks, and blocks interacted only with lags. Autocorrelations at
short lags (1-2) tended to be larger in the second block (.269 vs. .258), whereas from lags 4-5 on,
the difference tended to vanish (.228 vs. 225). This was confirmed in an interaction of the contrast
between short (1-2) and medium (4-5) lags, F(1,78) = 6.04, ηp2 = .07.
4.2.2. Focus on transitions
In the sequential analysis of the transitions, repeating the same task was coded 1 and
changing the task or failing to select a task was coded 0; in the calculation targeting on switches,
changing tasks was coded 1 and repeating the same task or failing to select a task was coded 0.
In all other respects, the same data-analytic method was used as for the analysis focusing on task
selections. The statistical analyses are reported in Table 6.
---- Table 6 about here ----
?Modeling Voluntary Task Switching 34
22.214.171.124. Runs. Transition runs are shown in Figure 3 (top-right) as a function of SOA,
transition, and length. Overall, runs of repetitions (M = 0.14) were more frequent than runs of
switches (M = 0.07). Run proportions were only slightly but reliably larger on short (M = 0.103)
than on long SOA (M = 0.102). All interactions were reliable (see Table 6). The dominant
presence of repetitions confirms the repetition bias already observed at the level of tasks.
Repetitions were repeated more often than switches especially at short SOA.
In the first block, run proportions were smaller (M = .10) than in the second block (M = .11),
and this effect interacted with all other factors except transition; all higher-level interactions with
blocks were significant. The contrast between the first (1-5) and the second half of the lengths
(6-10) revealed an average drop of .17 in the first block and a drop of .18 in the second block.
Although values started higher in the second block and dropped more, at length 10 they were still
higher than in the first block. This contrast accounts to a large extent for the interactions involving
run length and blocks.
126.96.36.199. Autocorrelations. Figure 3 (bottom-right) displays the transition-based
autocorrelations as a function of SOA, transition, and lag. In contrast to the task-based
autocorrelations, the transition-based autocorrelations seem quite stable, except at lag 1. Overall,
autocorrelations were larger for repetitions (M = .38) than for switches (M = .17). They were larger
in the second block (M = .28) than in the first block (M = .27). They also varied over lags. In
particular, correlations were lower at lag 1 (M = .22) than at other lags (M = .28), F(1,78) = 146.54,
ηp2 = 0.65. This contrast explains most of the variance among the means per lag: r2 = .98.
Transition interacted with SOA and lag. The latter interaction basically corresponds to an
interaction of transition with the contrast between lag 1 and lags 2-10, F(1,78) = 72.16, ηp2 = 0.48.
Blocks interacted with SOA but not with transition or lag. Finally, the triple interactions of SOA,
transition and blocks and of SOA, transition and lag were significant, as well as the interaction of
all four factors.
Repetitions were selected more often than switches and therefore repetitions also tended to
be repeated more than switches. However, for both, repetitions and switches, this tendency was
? Modeling Voluntary Task Switching 35
smaller at lag 1 than at longer lags, where transitions rather showed a pattern of independence,
but repetitions were still repeated more than switches.
The pattern of findings replicates the pattern observed in Study 1: a repetition bias was
observed and this bias was stronger at short than at long SOA (Arrington & Logan, 2004, 2005;
Mayr & Bell, 2006). The pattern of autocorrelations for both the tasks and the transition was also
completely similar to that observed in the first study. Again, it may be concluded that the repetition
bias is global with the exception that immediate (lag 1) repetition of a transition is less frequent
than the global average.
The presence of more alternations in lags 2-4 is consistent with the observation of Lien and
Ruthruff (2008) that task repetitions at lag 2 (i.e., ABA and BAB) are avoided. These authors
attribute their results to persistence of the inhibition (e.g., Mayr & Keele, 2000) of the task at trial
n-2 on the present trial (n). However, in the present data, the alternation tendency is present in the
window at lags 2-4 so that even if persistence of inhibition could account for the findings at lag 2,
this explanation fails to account for the observed alternation tendency at lags 3 and 4.
Practice seems to result in a tendency to have more shorter runs of the same task, but this
effect depends on combinations of the other variables in the design such as SOA, lag and task.
The transition-based analysis showed evidence for an effect of practice with an increase in the
number of repetitions in the transition data. Basically, practice seems to enhance the repetition
4.4. Model testing
The same model-testing procedure was used as in Study 1. Additionally, the effect of
practice on the obtained parameter estimates was investigated by comparing model fits based on
the first versus the second block of task choices. Next, the chain-retrieval model was compared to
model variations were each parameter in turn is clamped to a neutral value. In each case, first the
tests are applied to the entire data sequence ignoring practice effects before presenting data that
show how practice modulates the estimated parameters, the fits and the predictions.
? Modeling Voluntary Task Switching 36
4.4.1. Model fitting and parameter estimation.
Table 7 (top panel) displays the fit and the estimated parameters of the four models to the
complete sequence of task choices per subject. This table shows that the average fit over all
participants was very good for the chain-retrieval model, while the fits of the statistical
independence and dependence models were rather poor. Based on the AICc criterion which
balances the goodness of fit and the number of free parameters, the chain-retrieval model yielded
the best fit for a large majority of the participants (73 of the 80 participants for the independence
and alternation models; 62 out of 80 participants for the perseveration model). This significantly
better fit to the task run proportions of the chain-retrieval model is interesting in that it suggests that
the model better captures the relevant information in the data.
---- Table 7 about here ----
As an additional investigation of the goodness of fit, we tested whether the estimated
parameter values captured differences between the subjects due to a short versus long SOA
between the previous target and the present probe stimulus in the experiment. The estimated
values of parameters m and b did not depend on the SOA, but the value of the r parameter did:
with short SOA the parameter value was larger (M = 0.31) than with long SOA (M = 0.18), F(1,78)
= 10.11, ηp2 =.11. This is consistent with the assumption that r represents the probability of
intrusions which occur more often at short SOA. Similarly, in the perseveration model, only the
perseveration parameter was sensitive to SOA, with a larger perseveration tendency at short (.19)
than at long (.07) SOA, F(1,78) = 8.34, ηp2 = .10. This is again consistent with the findings of a
larger repetition tendency at short SOA. In the alternation model, the alternation parameter was
sensitive to SOA with a smaller probability at short (.07) than at long (.13) SOA, F(1,78) = 4.92, ηp2
=.06. The smaller tendency to alternate at short SOA is also consistent with a larger repetition
tendency at short SOA. In neither of these two models, the q parameter varied with SOA, and
similarly the p parameter of the Bernoulli model did not depend on SOA. These findings show that
the repetition bias is something that is not captured in a general probability of selecting a task, but
that the chain-retrieval model and the statistical dependence models are able to capture its
?Modeling Voluntary Task Switching 37
The assumption that parameter b is related to the switch cost was tested by means of the
correlation between this parameter and the switch cost. Because the perception of the difference
in difficulty between repetitions and switches should be driving the bias, the correlation of the error
switch cost per participant with the individualʼs estimated value of b was calculated. The error
switch cost is the difference between the average proportion of incorrect repetitions and the
average proportion of incorrect switches. The product-moment correlation amounted to 0.28, t(78)
= 2.58, p < .05. This indicates that b is related to the difference in difficulty of repetitions and
4.4.2. Modulation of model fits due to practice
Next, the chain-retrieval model parameters were estimated for each block separately. The
results of these fits are also shown in Table 7. The minimized − did not significantly differ
between the two blocks, F(1,78) = 2.30, ηp2 = .03, p = .13, but more participants (47 out of 80)
obtained a lower minimum in the first block than in the second block. The average m and b
parameter values did not differ between the two blocks, respectively F(1,78) = 1.54, ηp2 = .02, p = .
22 and F < 1. Parameter r, on the contrary, was smaller in the first block than in the second block
2, F(1,78) = 5.34, ηp2 = .06, which would suggest that vulnerability to bottom-up intrusions
increases from the first to the second block, possibly due to fatigue or a related drop in attentional
focusing. That m did not change significantly over the two blocks, suggests that a short practice
period suffices to achieve stable performance.
4.4.3. Model predictions.
The same procedure was followed as in Study 1 to obtain predictions of the autocorrelation
statistic for both the task data and the transition data. The outcomes of the statistical tests of this
correspondence are displayed in Table 7 (lower panels). The correspondence of the predictions
and the data is shown in Figure 4 for the predictions of the task autocorrelations (top-left) and the
transition autocorrelations (bottom-left). This figure illustrates that the task-chain retrieval model
yielded the best correspondence between predictions and data. The overview in Table 7 shows
that the chain-retrieval model yields better predictions for both the task autocorrelations and the
transition autocorrelations than the other models in a significant majority of the participants. The
?Modeling Voluntary Task Switching 38
present observations confirm that when applied to data that are more sensitive to deviations from
statistical independence than the runs statistic, it seems that the chain-retrieval model significantly
better accounts for these deviations than the other dependence models.
---- Figure 4 about here ----
The correspondence between the observed and the predicted autocorrelations was on
average very similar in the two blocks, as can be seen in the lower panels of Table 7. In the
predictions of the autocorrelations of tasks and transitions, the difference in the number of
participants with a smaller rmsd on the first than on the second block (respectively 43/80 and
32/80) was not significant.
4.4.4. Model validation.
As in Study 1, each of the parameters in turn was clamped to a neutral value and with one
parameter fixed, new estimations of the other two parameters were obtained. Table 8 and Figure 4
(right panels) show that except for the model with m = 3, each of these restricted models yielded a
significantly worse fit than the full model. However, because the m parameter varies stepwise in
units of 1, it is the case that for 15 subjects, the full model is exactly the same as the model with m
fixed. When these participants—for whom no distinction is possible between the full and the nested
model—are excluded from the test, the proportion of subjects for which the full model yielded a
better fit amounted to 0.64; this proportion has a z-score of 2.36 (p < .01). Moreover, in each of the
three nested models, the predictions of the autocorrelations of tasks were dramatically worse. In
the predictions of the transition autocorrelations, the results were similar except that here the full
model was not significantly better than the model with m = 3. When again, the 15 overlapping
cases were excluded from the test, the proportion of subjects with better predictions for the full
model amounted to .77 (z = 4.34, p < .001). Taken together, these findings support the conclusion
that all three parameters and the underlying processes are needed to account for the data.
---- Table 8 about here ----
The application of the chain-retrieval model in the present experiment confirms and
strengthens the findings reported in Study 1. Compared to the statistical independence model and
? Modeling Voluntary Task Switching 39
existing statistical dependence models, both the parameter fit to the runs proportions and the
predictions of the autocorrelation statistic of tasks and transitions yielded a better correspondence
to the data. In the validation test designed to investigate whether the processes underlying the
three free parameters are all involved in achieving this good correspondence, each of the
parameters in turn was clamped to a neutral value so as to exclude or to minimize the role of the
underlying process. These analyses indicated that all three parameters of the model and the
underlying processes are important in achieving the good correspondence with the data. Other
assumptions for the underlying processes of the model were also investigated; these are reported
in Appendix B. The variations considered relate to the way the chains are connected and the way
bottom-up events affect task choice. Instead of working with overlapping chains, the possibility to
work with concatenated chains was tested and instead of bottom-up repetitions that overrule the
intended action, other types of intrusions were also considered. More details are provided in the
appendix that shows that changing the present assumptions does not seem to improve the chain-
retrieval model fitted to task sequences.
Separate model fitting to each of the two blocks in the present study shows a modest effect
of practice, only the size of the r parameter changed with practice. These findings suggest that a
small amount of practice at the start of the experiment suffices to obtain stable behavior. Over
time, bottom-up events seem to become more important (increase in r), which suggests that top-
down control over task selection becomes less strict as practice proceeds. In view of the high
degree of similarity between the two blocks in both the degree of fit to the runs data and the
degree of correspondence with the autocorrelations, it seems that the model yields quite stable
results and is not jeopardized by accepting the simplifying assumption that the chain length is
stable over the session.
If the hypothesis is correct that the b parameter expresses the sensitivity to the difference in
difficulty between task repetitions and task switches, then as soon as task switching is involved, a
bias towards choosing more repetitions would be expected. The question raised here, is whether
such a bias would still be expected if the switching is irrelevant to the task choice. For example,
would the bias still exist when participants are required to randomly choose which hand they will
? Modeling Voluntary Task Switching 40
use to perform the task? Will there be a bias towards selecting more hand repetitions or may it be
expected that b would take a value close to neutral (close or equal to 0.5)? This question was
investigated in Studies 3 and 4.
5. Study 3
In study 3, participants were requested to perform only one task throughout the experimental
session (e.g., magnitude judgment) so that, strictly speaking, there were only task repetitions.
Thus, they did not have to select tasks but they had to select the hand with which they wanted to
execute the response. Hand switching is associated with a cost (Stelzel, Basten, & Fiebach,
2011), but this cost is due to associative carry-over in the hand repetition trials. Pashler and Baylis
(1991) proposed that during execution of a task, a transient association between the stimulus
identity and the executed response is formed, and if this association can be reused on the next
trial, a repetition advantage is observed as the normal response-selection stage is bypassed.
Applied to the present context and assuming that subjects represent their hand selection by means
of an internal label, this implies that a shortcut between this internal label and the selected hand
can be reused on hand repetition trials, bypassing the top-down controlled hand-selection stage.
No such advantage is present in the hand switching situation, where both the internal label and the
selected hand would be different. This account suggests that the difference between hand
repetitions and hand switches is not based on a difference in the amount of mental effort needed
for preparing and executing the tasks. As the bias towards repetitions is assumed to result from
such differences in mental effort (Botvinick, 2007), it is expected, therefore, that in the case of hand
switching no bias towards selecting more hand repetitions will develop. Hence, it is predicted that
in such a hand-switching design, a hand switch cost (or hand alternation cost) may be observed
(cf. Pashler & Baylis, 1991), but this will not result in a bias toward more frequent selection of hand
repetitions. In other words, it is expected that the model with b fixed to 0.5 will not yield a worse fit
and worse predictions than the full model. In addition, it should be noted that the present
procedure constitutes a dual-task situation in which random hand selection is required while
performing another task. Therefore, bottom-up events may be expected to play an important role.
? Modeling Voluntary Task Switching 41
5.1.1. Participants and design
Thirty-five first-year psychology students at Ghent University participated for course
requirements and credit. None of them had participated in one of the previous studies. They all had
normal or corrected-to-normal vision, were right-handed, and all were naïve to the purpose of the
experiment. Participants were randomly assigned to two conditions that differed in the task they
performed throughout the session (see below). One participant was excluded for not following the
instructions and producing errors on about half of the trials; a second participant was excluded for
producing a pattern of task choices that was completely deviant from that of the other participants,
so that as well the proportions of runs as the proportions of autocorrelations in this participantʼs
choice data deviated more than 2 standard deviations from the sample mean.
5.1.2. Materials and procedure
The same procedure (single registration) as in Study 1 was used in combination with the
stimuli and the tasks of Study 2. Participants were instructed to perform one single task
throughout the experimental session that consisted of two blocks of 65 trials, preceded by a
practice block of the same length. Half of the participants performed the magnitude task and
pressed the left key for smaller and the right key for larger than 5. The other half was requested to
execute the parity task and pressed the left key for odd and the right key for even. They were
further instructed to switch hands in an unpredictable manner with the restriction of using both
hands about equally often. During the practice block, they received feedback in the same way as
in Study 2. At the end of each block a feedback screen showing performance during the block was
presented, also completely similar as in Study 2.
For completeness, analyses of task performance are presented in Appendix C. Here we
focus on the sequential analysis. This data analysis was completely similar to that of Studies 1
and 2. The analysis was performed once with a focus on tasks (hands in the present study), and
once with a focus on transitions. Both runs and autocorrelations were analyzed on the basis of the
? Modeling Voluntary Task Switching 42
multivariate linear model in a 2 (Task: magnitude or parity) × 2 (Hand: left or right) × 10 (Length or
Lag) with repeated measures on the last two factors.
Previewing the findings, the hand-based analyses revealed larger runs proportions for the left
than for the right hand, but decrease of runs proportions over lengths did not differ across the two
hands. As in Studies 1 and 2, the autocorrelations showed a dip at lags 2-4, and although the
average autocorrelations were higher for the left than for the right hand, the size of the dip did not
differ across hands. In the transition-based analyses, proportion of hand repetitions was larger
than the proportion of hand switches. Decrease of runs proportions over lengths was faster for
repetitions than for switches. Autocorrelations were larger for repetitions than for switches and as
in the previous studies, the autocorrelation at lag 1 was smaller than at the other lags.
More statistical details about these findings are presented in the following paragraphs.
Readers who prefer to skip these details can move directly to session 5.3 (Discussion).
5.2.1. Focus on hands
Hand selection data were coded as binary series of events in the same way as in Studies 1
and 2. More specifically, the 2 × 64 hand choices was first coded 1 for left and 0 for right to
generate runs and autocorrelation statistics. The codes were then reversed to calculate the values
of the same statistics with the focus on the right hand (right = 1, left = 0). Also the transitions were
coded, namely first with hand repetition coded as 1 and hand switching as 0 and then with the
coding reversed. This way completely the same analysis was performed as in Studies 1 and 2,
with the understanding that in the present study, there was no task switching but only hand
The pattern of runs was completely similar to the one obtained in Studies 1 and 2. Selection
proportions of the left hand and the right hand were .52 and .48 respectively. Overall, runs
proportions were larger on the left hand (M = .11) than on the right hand (M = .10). This difference
was significant, F(1,31) = 7.59, ηp2 = 0.20. Furthermore, only the main effect of run length was
reliable, F(9,23) = 91434, ηp2 = 1.00. Importantly, hand did not interact with run length, F(9,23) =
1.97, p = .09.
?Modeling Voluntary Task Switching 43
Similarly, average autocorrelations were larger when focus was on the left hand (M = .26)
than when focus was on the right hand (M = .23), F(1,31) = 13.59, ηp2 = 0.30. Besides,
autocorrelations varied over lags, F(9,23) = 4.11, ηp2 = 0.62. As in Studies 1 and 2,
autocorrelations were lower in the window at lags 2-4 than at longer lags (7-9), F(1,31) = 15.17, ηp2
= 0.33. No other effects or interactions were reliable.
5.2.2. Focus on transitions
Proportions of hand repetitions (M = .53) were larger than proportions of hand switches (M
= .47), but these values were not significantly different from 0.5, t(32) = 1.37, p = 0.18. On average
runs proportions were larger for repetitions (M = .11) than for switches (M = .09), but this difference
was not reliable (F < 1). Run length was significant, F(9,23) = 23894, ηp2 = 1.00, as well as its
interaction with transition type, F(9,23) = 10.76, ηp2 = 0.81.
On average autocorrelations were larger for repetitions (M = .30) than for switches (M = .23),
but this difference was not reliable. Autocorrelations only varied over lags, F(9,23) = 10.70, ηp2 =
0.81. As in Studies 1 and 2, transition autocorrelations were smaller at lag 1 (M = .23) than at the
other lags (M = .27), F(1,31) = 72.22, ηp2 = 0.70. This contrast did not interact with the other
The present study differs from Studies 1 and 2 by not showing a significant repetition bias for
hand selection even though the hand switch cost was reliable (see Appendix C), and replicates
Studies 1 and 2 in revealing a similar pattern of findings both in the analysis of the hand choice
and in the analysis of the transitions between hands. These similarities are obtained
notwithstanding an important difference between the previous studies and the present one, namely
that in the present study no task switching is needed as there is only one task, so that participants
only have to select the hand they will use to perform the task.
5.4. Model fits
The present study was designed to specifically test the validity of the chain-retrieval model.
In the experimental procedure of the present study, task switching was replaced by hand switching
while performing one single task. As explained in the introduction to this study, on the basis of the
? Modeling Voluntary Task Switching 44
considerations that a hand switch cost may be observed but that this switch cost is not driven by
task-set preparations, it was predicted that no bias would develop towards chains with more hand
repetitions. Because a hand switch cost was observed, as well in error data as in RT data (see
Appendix C for details),in order to verify the prediction, we must consider two possibilities. First,
due to the presence of a hand-switch cost, participants may learn from this performance difference
to prefer chains with more hand repetitions to chains with less hand repetitions. If that is the case,
the present model application should simply replicate the results of Studies 1 and 2. The second
possibility, is that notwithstanding the presence of a hand-switch cost, participants did not learn
from the performance difference between hand repetitions and hand switches and did not acquire
a bias towards more repetitions. Such failure to acquire a bias, given that a performance
difference is present, is probably due to the fact that switching hands does not involve mental effort
to prepare for using another hand, whereas in task switching, changing from one task to another
involves a set of preparatory processes that can be modulated by performance monitoring
(Botvinick, Nystrom, Fissell, Carter, & Cohen, 1999; Botvinick, 2007; Botvinick, Braver, Barch,
Carter, & Cohen, 2001; Botvinick, Cohen, & Carter, 2004). If participants did not acquire a bias
towards chains with more repetitions, then the present model application should not simply
replicate Studies 1 and 2. More particularly, it would be expected that the b parameter would take
a value close to neutral (0.5).
Because of this focused research question, the present section therefore reports only the
tests of the chain-retrieval model. The full parameter models, and the three variants with one of
the parameters clamped to a neutral value were tested. The questions addressed were whether
the fit of the model with b fixed would be close to the fit obtained for the full parameter version and
whether any of the other two-parameter models would be sufficient to explain the data.
Table 9 displays the fit and estimated parameters of the four model variants studied. The fit
of the full-parameter chain-retrieval model was quite good. The same goes for the variants with m
fixed and b fixed. In fact, the fit of the full model was not statistically better than the fit with either
parameter fixed. The fit of the model with r fixed was substantially worse, although the difference
fell short from statistical significance. When considering the predictions of the task
? Modeling Voluntary Task Switching 45
autocorrelations, the full parameter model was significantly better than the fixed-r model. The
difference between the full and the m-fixed model was also significant when the overlapping cases
were excluded (z = 3.67, p < .001). Only the difference with the b-fixed model was not statistically
significant. The predictions of the transition autocorrelations yielded all significant differences in
favor of the full-parameter model.
---- Table 9 about here ----
These findings show that the bias parameter (b) does not play a role when task difficulty
differences are not at issue: neither in the parameter estimation nor in the prediction of
autocorrelations, the full model was better than the model with b fixed. This observation is further
substantiated by the finding that the correlation between the value of the b in the full model and the
RT switch cost (r = -0.43, opposite direction than expected) and the correlation of b with the error
switch cost (r = 0.27, p = 0.13) are not reliable. Even though hand switching was more difficult
than hand repetition, this does not seem to intrude into the selection of a hand for executing a task.
That there is no bias towards chain with more repetitions does not mean that the repetition
bias is completely abolished. The difference between the r-fixed and the full model shows that
hand selection is still affected by bottom-up events. As well in the model fits as in the prediction of
autocorrelations, the full model was reliably better than the model with r fixed to 0. The estimated
proportion of bottom-up triggered repetitions seems to be smaller than in Studies 1 and 2. A test of
the difference showed a marginally significant difference, F(1,127) = 3.30, p = .07, ηp2 = 0.03.
Finally, in comparing the full model to the model with m fixed, the fits showed that the full
model was not better than the restricted model, but in the prediction of the autocorrelations, the
difference between both sets of predictions was significant. This observation indicates that the
preferred chain length is an important modulator of selection and that it plays an important role in
achieving the tendency to repeat present in this data set, the more so that bias does not seem to
be effective to augment proportions of repetitions and that fewer bottom-up repetitions do occur.
6. Study 4
Study 3 tested the chain-retrieval model in a dual-task context: it required hand switching
while performing another task. Although a hand-switch cost was observed, no bias was acquired
?Modeling Voluntary Task Switching 46
on the basis of this difference in difficulty between hand repetitions and hand switches. In that
context, the m and r parameters could not be discarded. The question addressed in Study 4 is
whether the processes underlying these parameters would still play an important role, when the
context is further simplified, i.e., when the dual-task characteristic is removed. In Study 4, the
situation was completely similar as in Study 3, except that no target-based task processing was
required. On every trial, a fixed probe was presented to which participants responded by pressing
a key either with the left or with the right hand. Hence, the task situation only required voluntary
hand selection. The research questions are (a) whether the chain-retrieval model would still be
better than the statistical independence and dependence models, and (b) whether the three
parameters would still be instrumental in obtaining good fits and predictions.
Twenty-four participants were selected from the same pool as Study 3. They all served in a
single condition in which they randomly switched between hands. One subject failed to emit a
response on about one third of the trials and was not included in the data analysis.
6.1.2. Materials and procedure
Participants were instructed to switch hands in an unpredictable manner with the restriction
of using both hands about equally often. Each trial consisted of a probe (exclamation mark) that
signaled that a response was required, namely to either press a left key with the left hand or to
press a right key with the right hand. Time allowed for responding was 3000 ms. During the
practice block, they received feedback. At the end of each block a feedback screen showing
performance during the block was presented completely similar as in Study 2. The practice block
contained 65 trials; this was followed by two experimental blocks of 65 trials each.
RTs were not faster for repetitions than for switches (Ms respectively 332 and 359 ms), F
(1,22) = 2.04, p = 0.17, ηp2 = 0.08. The sequence data were analyzed once with a focus on hands
and once with a focus on transitions between hands. Both runs and autocorrelations were
? Modeling Voluntary Task Switching 47
analyzed on the basis of the multivariate linear model in a 2 (Hand: left or right) × 10 (Length or
Lag) repeated measures design .
6.2.1. Focus on hands
Hand selection data and transition data were coded in the same way as in Study 3. This way
we could conduct completely the same analyses as in the previous studies, with the understanding
that in the present study, hand switching was the only task.
The pattern of runs was completely similar to the one obtained in the previous studies.
Selection proportions of the left hand and the right hand were .51 and .49 respectively. Overall,
runs proportions were not larger on the left hand (M = .10) than on the right hand (M = .10), F < 1.
The main effect of run length was reliable, F(9,14) = 6.16, ηp2 = 0.80. Importantly, hand did not
interact with run length, F(9,14) = 1.13, p = .40.
Average autocorrelations were slightly larger when focus was on the left hand (M = .25) than
when focus was on the right hand (M = .24), but this difference was not reliable, F < 1.
Autocorrelations varied over lags, but again not reliably, F(9,23) = 1.40, p = .28, ηp2 = 0.47. In
contrast to the previous studies, autocorrelations were not lower in the window at lags 2-4 than at
longer lags (7-9), F < 1, ηp2 = 0.04. No other effects or interactions were reliable.
6.2.2. Focus on transitions
Proportions of hand repetitions (M = .53) were larger than proportions of hand switches (M
= .47), but these values were not significantly different from 0.5, t(22) = 1.08, p = 0.29. On average
runs proportions were not larger for repetitions (M = .10) than for switches (M = .10), F < 1. Run
length was significant, F(9,14) = 4.81, ηp2 = 0.76, but its interaction with transition type was not, F
(9,14) = 2.02, p = .12, ηp2 = 0.57.
On average, autocorrelations were larger for repetitions (M = .27) than for switches (M = .25),
but this difference was not reliable, F < 1. Autocorrelations did not vary reliably over lags, F(9,14)
= 1.23, p = .35, ηp2 = 0.44.
Like Study 3, the present study did not show a repetition bias for hand selection. The pattern
in the autocorrelations was completely different from the pattern found in the three previous
? Modeling Voluntary Task Switching 48
studies, both in the hand choice and in the transitions. The most striking feature is that neither a
hand preference, nor a repetition bias is observed. On the basis of the random generation
literature, an alternation bias could have been expected. In comparison to typical random
generation experiments, the time allowed per trial was rather large in the present study (3 s), so
that there was enough time to overcome such bias. Note, however, that hand choice has so far
not been used in random generation tasks, and possibly hand selection is sensitive to bottom-up
intrusions. The model application will show that this may indeed have been the case.
6.4. Model fits
Like Study 3, the present study was designed to specifically test the validity of the chain-
retrieval model, presently in a situation where only random choice of hands is required. Because
this is the kind of situation that corresponds most closely to typical random generation tasks, first a
comparison is made between the chain-retrieval model and the three statistical models tested in
Studies 1 and 2, namely the independence model, and the perseveration and alternation models.
Next, an analysis like the one presented in Study 3 is performed to check whether the assumptions
of the chain-retrieval model all hold in this more simple kind of task setting.
Table 10 presents an overview of the findings comparing the chain-retrieval model to the
statistical models. Although the goodness-of-fit of the statistical models now seems to be slightly
better than in Studies 1 and 2, the chain-retrieval model still yields a better fit, even when the
number of free parameters is taken into account; this model yields a statistical significant better fit
than each of the other models. In terms of the predictions, all models seem to be doing equally
well. This is not very informative on the relative quality of the models, as the data analysis had
already shown that the autocorrelation data did not vary over the lags.
---- Table 10 about here ----
The comparison of the full and nested variants of the chain-retrieval model is summarized in
Table 11. The table shows that each of the nested models provided an equally good fit as the full
model. When considering also the predictions, only the model with r = 0, produced significantly
worse predictions than the full-parameter model. Similar to Study 3, the proportion of bottom-up
triggered repetitions tended to be smaller. Indeed, when contrasting the number of bottom-up
? Modeling Voluntary Task Switching 49
triggered repetitions in Studies 1 and 2 with the number in Studies 3 and 4, less such repetitions Download full-text
did occur in the latter two studies, F(1,149) = 4.92, p < .05, ηp2 = 0.03. As in Study 3, the full
model was not better than the model with b fixed; the correlation of b with the hand-switch cost
was close to zero (r = 0.09, p = 0.70).
---- Table 11 about here ----
By and large, this study shows that the chain model accounts well for random generation
data, while parameters m and b are fixed. A good fit was obtained with r fixed to 0, but predictions
of autocorrelations were better when r varied freely. Apparently, bottom-up repetitions (e.g., due to
repetition priming) did occur in hand switching. In a task context with less opportunities for bottom-
up repetitions, fixing r to 0 may be expected to yield good correspondence. That the chain-
retrieval model achieves a good account of random generation data with so many of the free
parameters fixed further supports the assumptions that go back to the model of Rapoport and
7. General Discussion
7.1. Overview of the findings
The objective of the present study was to develop a framework that specifies the processes
involved in voluntary task selection, to define a formal model on the basis of the framework, test
this model in situations involving task selection, and test the limits of the model by applying it also
to event selection. On the basis of functional similarities between VTS and the selection of
independent events as studied in the random generation experiments, we formulated the central
hypothesis of the framework, namely that in order to produce sequences of task choices in which
both tasks occur independently and about equally often, people quickly learn to retrieve short
balanced sequences (in line with the model of Rapoport & Budescu, 1997) of task names from
LTM to guide the trial-by-trial choice of tasks. We assumed that the length of the chains is
constrained by working memory capacity, although we noticed that chain length is also related to
the proportion of task repetitions in the chains. We further assumed that there is a bias towards
retrieval of chains with more repetitions, and that bottom-up triggered repetitions may overrule the
currently intended event. This view was specified in a model with three free parameters
? Modeling Voluntary Task Switching 50