Components of task switching: A closer look at task switching and cue ...

Acta Psychologica 151 (2014) 184–196

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Components of task switching: A closer look at task switching and cue switching Florian Schmitz a,⁎, Andreas Voss b a b

Institute of Psychology, Ulm University, Ulm, Germany Institute of Psychology, University of Heidelberg, Heidelberg, Germany

a r t i c l e

i n f o

Article history: Received 8 November 2013 Received in revised form 13 June 2014 Accepted 16 June 2014 Available online xxxx PsychINFO classification: 2320 Sensory Perception 2330 Motor Processes 2340 Cognitive Processes 2240 Statistics & Mathematics Keywords: Task switching Cue switching Reconfiguration Inertia Repetition priming Diffusion model

a b s t r a c t Research using the diffusion model to decompose task-switching effects has contributed to a better understanding of the processes underlying the observed effect in the explicit task cueing paradigm: Previous findings could be reconciled with multiple component models of task switching or with an account on compound-cue retrieval/ repetition priming. In the present study, we used two cues for each task in order to decompose task-switch and cue-switch effects. Response time data support previous findings that comparable parts of the switching effect can be attributed to cue-switching and task-switching. A diffusion model analysis of the data confirmed that non-decision time is increased and drift rates are decreased in unpredicted task-switches. Importantly, it was shown that non-decision time was selectively increased in task-switching trials but not in cue-switching trials. Results of the present study specifically support the notion of additional processes in task-switches and can be reconciled with broader multiple component accounts. © 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

1. Introduction People usually find it more difficult to shift between mental tasks rather than to repeat the same task. In laboratory experiments, taskswitches are usually accompanied by increased response times (RTs) and decreased accuracy compared to task-repetitions. This difference may be accounted for by task-switching costs and/or by benefits of repetition priming. We will briefly address both possibilities below. More extensive reviews of task switching accounts can be found elsewhere (e.g., see Kiesel et al., 2010; Vandierendonck, Liefooghe, & Verbruggen, 2010, and Schmitz & Voss, 2012, for a discussion on how theoretical accounts could be mapped onto parameters of the diffusion model). 1.1. Task-switching costs According to models of task switching costs, task-set reconfiguration (TSR) and task-set inertia (TSI) are assumed to contribute to taskswitching effects. The classical task-set reconfiguration account (Rogers & Monsell, 1995) assumes that in task-switch trials a new task set ⁎ Corresponding author at: Institute of Psychology, Ulm University, 89081 Ulm, Germany. E-mail address: fl[email protected] (F. Schmitz).

must be activated in working memory before stimulus classification can proceed. Hence, switching costs are interpreted as the time it takes to prepare the new task-set, whereas increased error rates are explained by failures to accomplish the task-set reconfiguration in time. Theories postulating an additional processing phase (or multiple phases) for task-switch trials have been referred to as additional process accounts (De Jong, 2000; Rogers & Monsell, 1995; Rubinstein, Meyer, & Evans, 2001). According to this perspective, preparing for the next task is considered a top-down mechanism exerted by (executive) control functions (Baddeley, Chincotta, & Adlam, 2001; Miyake, Friedman, Emerson, Witzki, & Howerter, 2000), and a number of ideas or metaphors are postulated on how task-set preparation could proceed. These include re-programming the system (Ruthruff, Remington, & Johnston, 2001), top-down biasing the relevant taskdemand units (Gilbert & Shallice, 2002), retrieving the new task set from long-term memory and loading it into working memory (Goschke, 2000; Mayr & Kliegl, 2000, 2003) or loading it into the region of direct access of procedural working memory (Risse & Oberauer, 2010). It was argued that the well-known cue–stimulus interval (CSI) effect denoting reduced switching costs when the CSI is long (e.g., Meiran, 1996) supports the view of task-set reconfiguration prior to stimulus onset.

http://dx.doi.org/10.1016/j.actpsy.2014.06.009 0001-6918/© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

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Differently from this view, the classical account on task-set inertia (TSI; Allport, Styles, & Hsieh, 1994; Allport & Wylie, 2000) assumes that switching costs result from proactive interference from previous processing of a competing task. In support of the TSI account, it was shown that the adverse effects of proactive interference in the explicit cueing paradigm are largest on trials directly succeeding task switches, but that they are not confined to these switch trials (e.g., Monsell, 2003). Further support of the TSI account stems from the observation of switch-cost asymmetries, i.e. larger costs when switching away from a less dominant to a more dominant task compared with the other direction (Allport & Wylie, 2000; Allport et al., 1994; Yeung & Monsell, 2003). Additionally, n-2 repetition costs observed in paradigms with 3 tasks support the view of task set inertia: Larger switch costs in the sequence ABA as compared with the sequence CBA have been accounted for by inertia of “backward inhibition” (Koch, Gade, Schuch, & Philipp, 2010; Mayr & Keele, 2000). Finally, neuroimaging data show that switching costs correlate with increases in task-irrelevant neural activity after a switch (Yeung, Nystrom, Aronson, & Cohen, 2006), supporting the notion of interference from recently performed tasks as a major determinant of task-switching costs. To conclude, task-set reconfiguration rather emphasizes top-down control, whereas task-set inertia points to bottom-up processes. However, more reactive forms of control may come into play as well when response conflicts are detected. Both views have been supported in the literature, and were integrate into multiple-component models of task switching (e.g., Mayr & Kliegl, 2003; Ruthruff et al., 2001). These models typically specify several phases of a task switch: In an earlier phase, task-set preparation is considered to take place, whereas in a later phase, stimulus information is used to select the response along the constraints (e.g., S–R links) set in the first phase. As the level of activation of S–R links is also affected by inertia effects, the latter will also contribute to the efficiency of response selection in the later phase. 1.2. Compound-cue retrieval/repetition priming An elegant alternative account of the observed difference in task performance between task-switch and task-repetition trials in the absence of executive control was offered by Logan and colleagues (Logan & Bundesen, 2003; Schneider & Logan, 2005, 2007). According to this approach, participants use a compound-cue retrieval strategy in the explicit cueing paradigm. Specifically, it is assumed that both task-cue and target stimulus need to be encoded to form a compound stimulus which uniquely determines the correct response. Importantly, cue encoding is predicted to follow a horse-race process which is terminated as soon as a representation of the cue is found. In all trials there should be a longterm memory representation, but in cue-repetition trials there should be as well a short-term memory representation which will easily win the race, resulting in short encoding time. Hence, cue-encoding benefits in task-repetition trials are made responsible for the observed differences in response times between task-repeat and task-switch trials. After encoding of cue and target stimulus, the compound cue is formed and response selection takes place, which is assumed to follow a random walk process with only a few steps, in line with retrieval theories of binary decision tasks (cf. Nosofsky & Palmeri, 1997). As all processes are stimulus driven, this account does not draw on concepts such as task sets, task switches, and task-set reconfiguration. Hence, it does not require an endogenous act of control. Effects of task-switching and cue switching cannot be teased apart with the classical explicit cueing paradigm, because both cue-switches sand task-switches are perfectly confounded. Hence, the repetition priming account was tested using a task-switching paradigm with a 2:1 cue-to-task mapping (Logan & Bundesen, 2003; Mayr & Kliegl, 2003): Employing two different cues for each task allows one to dissociate effects of cue switching from those of task switching. Cue-switching effects denote the difference in

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task performance between task-repetition trials with either a cue switch or a cue repeat. Pure task-switch effects denote the difference between cue-switch trials with either a task-switch or a task repeat. Evidence obtained with the 2:1 mapping paradigm has supported the view that cue-switches contribute substantially to the task-switch effect (Logan & Bundesen, 2003), or that both, cueswitches and task-switches, contribute to the task-switch effect (Arrington, Logan, & Schneider, 2007; Logan & Bundesen, 2004; Mayr & Kliegl, 2003; Schneider & Logan, 2005). Some of the divergent findings concerning the relative magnitudes of the taskswitch and cue-switch effects may be attributed to the conditional probability of a task switch given a cue switch (Mayr, 2006), as will be discussed later. However, some findings challenge the compound-cue retrieval/repetition priming account, and seem to support the existence of task sets instead. For instance, there are n-2 repetition costs in paradigms with a 2:1 cue to task mapping irrespective of whether the cue switches or not (Altmann, 2007; Gade & Koch, 2008), suggesting inhibition on the level of task-sets. Similarly, task-performance in implicit sequence learning with a 2:1 cue to task mapping was found not to be affected by cue switches or cue repetitions (Gotler, Meiran, & Tzelgov, 2003). Finally, repetition priming cannot easily account for comparably fast responses in trials with complete switches, i.e. when neither the task, nor the stimulus, nor the response repeats from one trial to the next. Conversely, increased response times in trials in which some but not all features repeat (“partial repetition costs”, e.g., Hommel, 2004) have been explained by the bottom-up activation of competing task sets or according responses. 1.3. Further evidence of separable cue-switch and task-switch effects The above reviewed studies with a 2:1 cue-to-task mapping paradigm may help estimate the quantitative contribution of cueswitch and task-switch processes to the overall switching effect. Other studies offered more direct evidence that different processes may contribute to both effects (see Jost, De Baene, Koch, & Brass, 2013, for an excellent review). For instance, it was shown that only cue-switch costs were affected by CSI and practice, whereas the task-switch effect was shown to depend on response priming and task-set inhibition (Mayr & Kliegl, 2003). Similarly, a response cue interval (RCI) variation was found to affect the cue-switch effect (suggesting a decay of the cue-representation), but not the task-switch effect (Horoufchin, Philipp, & Koch, 2011). Additionally, physiological data support that cue switches and task switches are governed by distinct processes, separated both by time and the neural basis. In an EEG study using the 2:1 mapping paradigm, two dissociable ERP components were found, one associated with cue-switches at around 300 ms after cue onset, the other one associated with task switches 400 ms after onset of the target stimulus with a different topography (Jost, Mayr, & Rösler, 2008). Additionally, a number of neuro-imaging studies revealed that cue-switches and task-switches recruit different neural networks (Brass & Cramon, 2004; De Baene, Kuhn, & Brass, 2012; Wylie, Javitt, & Foxe, 2006). Hence, cue (-switch) processes and taskswitch processes seem to be separable. In fact, some sort of cueprocessing may be required in all trials (although this may take longer in case of cue switches), whereas task-switch specific processes may be confined to true switches of the task set. Preparation also may comprise several phases. For instance, Mayr and Kliegl (2003) suggest that in a first phase (corresponding with the cue-switch effect), the task set is retrieved from long-term memory. In a second phase (corresponding with the task-switch effect), it is installed in a task-appropriate attentional configuration (see also Risse & Oberauer, 2010). This may correspond, e.g., with the biasing of relevant task-demand units (Gilbert & Shallice, 2002; see also Ruthruff et al., 2001).

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1.5. Task-switching research using the diffusion model

Fig. 1. The diffusion model decomposes a decision process into non-decision time and the actual decision process. Non-decision time captures processes and activity before and after the actual decision phase. In turn, the actual decision process (displayed right of the rounded brace) is characterized by a continuous sampling of information. A decision counter, originating from starting point z, fluctuates over time (horizontal axes) as a function of systematic stimulus information and random noise (see gray sample path). When it hits the lower or upper response thresholds (at 0 or a, respectively), the according response is elicited. The mean slope of the counter across trials denotes the drift rate (v).

1.4. The diffusion model The diffusion model for response time data (Ratcliff, 1978; Ratcliff & Rouder, 1998; Voss, Rothermund, & Voss, 2004) can be used to decompose a binary decision process into a set of psychologically meaningful parameters. A graphical representation of the model is given in Fig. 1. The basic assumption of the diffusion model is a continuous accumulation of information over time. This information accumulation process is assumed to follow a stochastic diffusion process, which is a process that is driven by systematic stimulus based information and by random noise. In Fig. 1, an example of such an information process is displayed as the gray ragged line between the two thresholds. Outside the thresholds, predicted response time distributions for both alternate responses are displayed. In a diffusion model analysis, response times are decomposed into decision-time and non-decision time. The non-decision parameter (t0) is conventionally interpreted as capturing basic stimulus encoding and, particularly, motor time (Ratcliff, Thapar, & McKoon, 2006; Voss et al., 2004). Decision time is determined by the duration of the continuous sampling process, that is, by the time the diffusion process takes to reach an upper or lower threshold. This diffusion process is characterized by its starting point (z), threshold separation (a), and the mean slope (drift rate: v). As a diffusion process is noisy, it fluctuates randomly around its mean gradient, resulting in different response-times, and sometimes in hitting the wrong threshold. The drift rate (v) is determined by the efficiency with which stimulus information is used to select a response. Large drift rates will correspond with fast and accurate responses. Drift rates can be affected by task difficulty (Voss et al., 2004) as well as by individual differences in intelligence and working memory capacity (Schmiedek, Oberauer, Wilhelm, Suess, & Wittmann, 2007). The distance of response thresholds (a) is considered to indicate response caution. A high response criterion (large values of a) corresponds to slow but accurate responses, while small values of a denote an impulsive decisional style that is characterized by fast responses and low accuracy. The a parameter can reflect experimentally induced speed–accuracy settings (Voss et al., 2004). Additionally, it has been shown that a more cautious setting of the response criterion contributes notably to slow responding in elderly people (e.g., Ratcliff et al., 2006). Additional parameters in the full diffusion model (Ratcliff & Rouder, 1998) include inter-trial variability of the starting point, drift rate (sv), and non-decision parameter (sz, sv, and st0, respectively).

The full diffusion model or simplified variants were used in previous studies to analyze task-switching data (Karayanidis et al., 2009, 2010; Klauer, Voss, Schmitz, & Teige-Mocigemba, 2007; Madden et al., 2009; Mansfield, Karayanidis, Jamadar, Heathcote, & Forstmann, 2011; Schmitz & Voss, 2012). This has shed light on how its parameters are associated with taskswitching processes. Consider first the non-decision parameter (t0 ) which captures processes outside the actual decision phase. Across all studies, it was found that the non-decision parameter (t 0 ) is increased in task-switch trials relative to task-repeat trials. In line with additional process theories of task switching, it was suggested that this increase corresponds with the requirement to prepare the new task in these trials (Karayanidis et al., 2009; Schmitz & Voss, 2012). This interpretation was further corroborated by the observation that the non-decision time was increased only when participants could not prepare the new task set prior to stimulus presentation; otherwise it was reduced to the level as in taskrepetition trials (Karayanidis et al., 2009; Schmitz & Voss, 2012). These findings can be reconciled with the view that task-preparation is largely possible prior to stimulus onset. However, when the time does not suffice to prepare the up-coming task sufficiently, preparation has to continue after stimulus presentation before stimulus information can be efficiently used to select the response. Note that the reported findings of an increase in non-decision time in task-switch trials relative to task-repeat trials were interpreted to reflect high-level control processes. This does not question the conventional interpretation that the baseline level of this parameter (as indexed by the magnitude of t 0 in task repeat trials) captures basic perceptual processes and motor execution. However, the increase in the non-decision parameter (t 0 ) is not an unequivocal evidence of task-set preparation (see also Schmitz & Voss, 2012). In fact, increased non-decision time in task-switch trials might be also predicted by the compound-cue retrieval/repetition priming account (Logan & Bundesen, 2003; Schneider & Logan, 2005, 2007). Recall that cue-encoding should be particularly fast in cue-repetition trials because of a transient trace of the task-cue in working memory. This should save time in compound cue formation in these trials relative to cue-switch trials. Hence, the difference in the non-decision parameter (t 0 ) in task-switch and task-repeat trials can be easily reconciled with the repetition priming account. Consider next parameters of the response-selection phase which may correspond with processes at a later phase when stimulus information is used to select the response in line with settings in the prepared task set (Mayr & Kliegl, 2003; Ruthruff et al., 2001). It was argued that drift rates are affected by the state of task readiness (Schmitz & Voss, 2012) which can be influenced by both controlled and automatic factors (Koch & Allport, 2006; Rubinstein et al., 2001). If there is not sufficient time for preparation (at short CSI), drift rates should primarily reflect inertia effects, which can be beneficial in the case of task-repetition trials or adverse in the case of task-switches. However, if there is sufficient time to prepare (at long CSI), preparation can be used additionally to adjust task readiness. Accordingly, it was shown that drift rates are influenced by both task sequence and task predictability (Karayanidis et al., 2009; Schmitz & Voss, 2012). Finally, previous research has offered evidence that the response criterion is increased in mixed blocks as compared to taskpure blocks (Schmitz & Voss, 2012). Additionally, some studies have shown that the response criterion may vary from trial to trial within one block as a function of the task–cue, or as a function of the anticipated difficulty or error-proneness of the transition (Karayanidis et al., 2009; Mansfield et al., 2011; Schmitz & Voss,

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2012). In the study conducted by Karayanidis et al. (2009), it was found that participants increased the response criterion only when a difficult task-switch was cued with certainty, whereas in task-switch trials cued with non-informative cues, the responsecriterion was comparable to that in task-repetition trials. Similarly, Schmitz and Voss (2012) report that participants generally increased the response criterion in mixed blocks, but, they slightly reduced the response criterion when an easy task-repeat trial could be predicted with certainty. Additionally, results suggest that adjusting the response criterion requires time (or cognitive capacity) and is apparently only completed when the nature of the next task transition is known early enough prior to stimulus onset (Schmitz & Voss, 2012). Neuropsychological evidence further suggests that setting the response criterion is exerted within a frontostriatal network, with the pre-supplementary motor area biasing the striatum towards lower response thresholds (Mansfield et al., 2011). 1.6. Aims of the current study and predictions In previous research using the diffusion model to decompose task switching costs, task switches and cue switches were confounded. Hence, one of the aims of this study was to tease apart effects of task switching from those of cue switching. Further, the previously obtained findings in the parameters of the diffusion model could be reconciled both with multiple component models of task-switching and the compound-cue retrieval/repetition priming account. Consequently, we intended to test both theoretical accounts more specifically on the level of parameters. To this end we employed a task-switching paradigm with a 2:1 cue-to-task mapping (Logan & Bundesen, 2003; Mayr & Kliegl, 2003) while keeping the procedure largely comparable with that in the previous studies (Schmitz & Voss, 2012). We expected the usually found pure taskswitch and cue-switch effects in the observable performance data (see Jost et al., 2013, for a review). More importantly, a number of specific predictions could be derived from the theoretical accounts that go beyond mean response times and errors that have been previously investigated. These predictions could be tested with the diffusion model. Multiple components models motivated a number of predictions. Given that additional processes are required to prepare the upcoming task in task-switch trials, there should be an increase in non-decision time in task-switch trials relative to task-repeat trials when there is not sufficient time to prepare for the upcoming task prior to stimulus onset (i.e. at short CSI). One can speculate that the non-decision time is also slightly increased in task-repeat trials with cue switches, as the participant needs to realize that the same task repeats. But, such an orienting reaction would be expected to be rather small compared with the time required for actual task preparation. When the CSI is short, drift rates should largely reflect task set inertia that can be adverse (in the case of task-switches) or beneficial (in the case of task-repetitions). If the CSI is long, task set preparation can be used more efficiently to adjust task readiness. Then, drift rates can be expected to be additionally affected by preparation. According to the compound-cue retrieval/repetition-priming account, the task-switch (or actually cue-switch) effect is predicted to be largely caused by differences in non-decision time. The latter should be decreased in cue-repeat trials relative to both other trial types, as the task cue repeats only in these trials. Consequently, the beneficial effect of a transient representation from the last trial will only exceed an effect in cue-repeat trials. Conversely, non-decision time would be increased in task-switch/cue-switch trials and taskrepeat/cue-switch trials, but no substantial difference between these trial types is predicted. Given that response selection follows a random walk process with only a few steps, drift rates can be

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predicted to be generally steep, but comparable across different trial types. Neither theoretical account makes specific predictions concerning the adjustment of response caution, although classical definitions of task set have incorporated the “degree of caution to set one's criterion for response” (Monsell, Yeung, & Azuma, 2000, p. 252). Based on previous results obtained in some studies using the diffusion model, we predicted that participants may adjust response caution in a trial-to-trial fashion given that the nature of the next trial can be predicted and there is sufficient time prior to stimulus onset. 2. Method 2.1. Sample Sixty-two participants volunteered in the study (26 female). Most of them were students, 8 of which with a major in Psychology. On average they were 23.9 years old (SD = 2.5; range 18–30). All participants reported normal or corrected to normal vision. They received 7 Euros or partial course credit as compensation. 2.2. Procedure After signing informed consent, participants were individually seated in a dimly-lit room. All tasks and instructions were presented on a 19 in., 100 Hz CRT monitor. The session started with the task-switching paradigm and was completed by a few demographic questions that were also presented on the screen. After that, participants were thanked and debriefed. The total duration of the experiment was about 45 min. 2.3. Paradigm and materials We used a variant of the classical number–letter task-switching paradigm (Rogers & Monsell, 1995) with a 2:1 cue-to-task mapping (Arbuthnott & Woodward, 2002; Logan & Bundesen, 2004). Digit– letter pairs were used as bivalent stimuli that could be either classified according to the parity of the digit (odd vs. even) or according to the type of letter (vowel vs. consonant). All stimuli appeared in a rectangle in the center of a light gray screen (RGB = 220, 220, 220). The background color of the rectangle served as a task cue. Four different background colors (red, blue, green, and yellow) were randomly assigned the two tasks so that each task was cued by two colors. The cue–stimulus interval (CSI) was manipulated in two groups: in the short CSI group, the background color of the rectangle was changed from neutral gray to the respective task–cue color 150 ms prior to stimulus onset, whereas in the long CSI group, the new task–cue color appeared 800 ms prior to stimulus onset. The response–stimulus interval (RSI) was fixed at 800 ms in both groups. Stimuli were identical to the ones presented by Rogers and Monsell: Odd digits were drawn from the set 3, 5, 7, 9, even digits were drawn from the set 2, 4, 6, 8, consonants were drawn from the set G, K, M, R and vowels were drawn from the set A, E, I, U. All stimuli were presented in black Courier font. The digit–letter pair was approximately 2 cm wide and 1.2 cm high. The rectangle in which the stimulus pair appeared was ca. 7.5 cm × 7.5 cm and surrounded by a thin black line. The background colors used as task cues were red (RGB = 221, 97, 97), blue (RGB = 109, 179, 255), green (RGB = 94, 224, 76), and yellow (RGB = 228, 224, 76). They were randomly assigned to the two tasks (digit task vs. letter task) in a 2:1 manner for each participant and their assignment did not change throughout the experiment. In case of a classification error, a gray ‘X’ appeared below the stimulus pair. All stimuli remained on screen until the correct response was given. Then the stimulus was removed from the screen and the rectangle

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was set to either neutral gray (in the short CSI group) or directly to the task–cue of the next trial (in the long CSI group). The experiment comprised 5 runs of blocks, each composed of two task-pure blocks (one for each task, each block comprised 32 trials plus 4 warm-up trials) and a mixed block comprising both tasks (128 trials plus 8 warm-up trials). Participants could make a short pause after completing each block and re-starting the next block was self-paced. Individual trial lists were generated prior to the experiment for each participant. Half of all trials were taskswitches and half were task repetitions; half of the latter with cue-switches and half with cue-repetitions. In the task pure blocks also, there were cue-repetition and cue-switch trials. Within each of these trial types, there were equal numbers of trials for the combinations of task, response congruency of the relevant and irrelevant stimulus features, and response side. As in the Rogers and Monsell (1995) study, there were no direct stimulus repetitions, avoiding trial-to-trial stimulus-specific effects of negative priming or competitor priming. Additionally, there were no sequences of more than 4 trials affording the classification with the same key. Trials from the first sequence of blocks were discarded because of strong training effects, as done in the Schmitz and Voss (2012) study. The analyzed four sequences of blocks consisted of 128 cue-repeat and 128 cue-switch trials from the task-pure blocks, and of 256 taskswitch/cue-switch trials and 128 task-repeat/cue-switch and 128 taskrepeat/cue-repeat trials from the mixed blocks.

2.4. Analyses Parameters of the diffusion model were estimated with fast-dm (Voss & Voss, 2007, 2008) for each person and condition separately (see Schmitz & Voss, 2012 for details)1. The full set of parameters of the diffusion model was estimated without constraints using the information in the joint cumulative distributions of erroneous and correct responses minimizing the vertical distance of the empirical and model-implied functions according to the Kolmogorov– Smirnov statistic. In case of a low number of error trials, parameter estimation is largely based on the information in the distribution of correct responses. Parameter estimation is nevertheless possible, because responses were coded as correct or erroneous and the starting point was fixed in the middle between both response thresholds. Conventional RT and error data as well as parameters of the diffusion model were submitted a number of analyses of variance (ANOVAs) with the trial type as a within participants factor and the CSI length as a between participants factor. Mixing costs are addressed first. To this end, an ANOVA was conducted across task repeat trials from the task pure block and from the mixed task block. Contrasting trials of the same type across the task pure and mixed blocks may index block-wise differences in working memory load, specific processing strategies, global response caution, and other settings independent of the local task-switching effect. Next, two separate ANOVAs were conducted for each block. Most important for the purpose of the current study are the results of the ANOVA across trials of the mixed block which was complemented by theoretically informative contrasts capturing pure task-switch and cue-switch effects for each group (cf. Horoufchin et al., 2011, Exp. 4; Koch, Lawo, Fels, & Vorlander, 2011, Exp. 3). Specifically, the taskswitch contrast denoted the difference between task-switch/cueswitch and task-repeat/cue-switch trials, whereas the cue-switch contrast was composed of task-repeat/cue-switch and task-repeat/ cue-repeat trials.

1 We counterchecked parameter estimates using the EZ program (Wagenmakers, van der Maas, & Grasman, 2007; Wagenmakers et al., 2008), and findings were highly comparable.

3. Results 3.1. Data pre-treatment As done in the Schmitz and Voss (2012) study, trials directly following an error (7.6%) were removed for two reasons: First, there may be confounding error-related cognitive processing unrelated to the demands of the actual task. Second, the nature of the next trial is not unequivocal, as task-confusion errors will invert the type of trial transition (i.e., task-switch trials become task-repetition trials, and vice versa; see Steinhauser & Hübner, 2006). Additionally, we excluded outlier latencies according to individual box-plot criteria (Tukey, 1977). Response times were considered to be outliers when they were longer than 3 inter-quartile ranges on top of the 75% percentile of the RT distribution, or 3 inter-quartile ranges below the 25% percentile or shorter than 200 ms (3.3% excluded trials, 0.1% as fast, 3.2% as slow outlier values). We tested whether the background color of the rectangle had any effect on response times or errors. There was neither a main effect nor an interaction effect with trial type (all p N .21). Parameter estimates of the diffusion model were found to closely reproduce empirical RT distributions. A graphical display of the model fit is displayed Appendix A. Additionally, model-implied response times and error rates are displayed in Fig. 2. 3.2. Behavioral data 3.2.1. Latencies As can be seen in the upper panel of Fig. 2, median response times (RTs) from the task pure blocks were comparable in both CSI groups, but the increase in response times in the mixed block was more pronounced in the short CSI group relative to the long CSI group. To test for group effects in mixing costs, we conducted an ANOVA across task repeat trials from the task pure block and from the mixed task block, collapsing cue-switches and cue-repetitions. This 2 (task pure vs. mixed task block) by 2 (short vs. long CSI) ANOVA yielded a main effect of trial type (F[1,60] = 107.53, p b .001, η p 2 = .64), a main effect of the CSI group (F[1,60] = 10.25, p b .01, ηp2 = .15), and an interaction of both factors (F[1,60] = 32.12, p b .001, ηp2 = .35). The latter indicated that the increase in response times from the task pure block to the mixed task block was more pronounced in the short CSI group (ΔRT = 198 ms) than in the long CSI group (ΔRT = 58 ms). Next, separate ANOVAs were conducted for each block. For the task-pure blocks, a 2 (trial type) by 2 (CSI) ANOVA confirmed that latencies did not differ between trial types or groups (all p N .26). Differently, in the mixed block, a 3 (trial type) by 2 (CSI) ANOVA revealed main effects of trial type (F[2, 120] = 77.56, p b .001, ηp2 = .56) and of CSI group (F[1, 60] = 24.59, p b .001, ηp2 = .29), indicating generally slower RTs in the short CSI group across all trial types (all p b .001). Additionally, there was an interaction of both factors (F[2, 120] = 18.16, p b .001, ηp2 = .23). This pattern was characterized by a significant task-switch contrast (ΔRT = 158 ms) as well as a cue-switch contrast (ΔRT = 98 ms) in the short CSI group (F[1, 30] = 75.47, p b .001, ηp2 = .72 and F[1, 30] = 23.68, p b .001, ηp2 = .44, respectively). In the long CSI group, only the task switch contrast (ΔRT = 73 ms; F[1, 30] = 22.20, p b .001, ηp2 = .43) reached significance, but not the cue-switch contrast (ΔRT = 14 ms; F[1, 30] = 3.47, p = .07, ηp2 = .10). 3.2.2. Accuracy As can be seen in the lower panel of Fig. 2, the long CSI group tended to commit more errors across all trial types. But there was not much of a difference between the task pure and the mixed task block (Δerrors = − 0.4% and Δerrors = 0.9%, for the short and long CSI groups, respectively). This was confirmed in an ANOVA across task repeat trials with block type and CSI group as factors.

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Fig. 2. Response times and error rates as a function of trial type (TR/CS = task repetition/cue switch, TR/CR = task repetition/cue repetition, TS/CS = task switch/cue switch, TR/CS = task repetition/cue switch, TR/CR = task repetition/cue repetition) and cue–stimulus interval (CSI: long vs. short). Columns display the observed median RT and the proportion of errors, respectively. Centered error bars represent the 95% confidence interval (CI) of these statistics. The “+” sign and its corresponding 95% CI shifted to the right of each column represent the model implied prediction.

Only the main effect of CSI group was significant (F[1, 60] = 5.36, p b .05, ηp2 =.08), but neither the main effect of block (F[1, 60] = 0.07, p = .79, ηp2 b .01), nor the interaction of both factors (F[1, 60] =0.58, p = .45, η p2 = .01). Testing separately in the task-pure block, a 2 (trial type) by 2 (CSI) ANOVA confirmed that there was only a main effect of the CSI group (F[1, 60] = 6.31, p b .05, ηp2 = .10). Neither the main effect of trial type nor its interaction with the CSI group was statistically significant (both p N .44). Consider next the mixed block, a 3 (trial type) by 2 (CSI group) ANOVA only yielded a significant main effect of the trial type (F[2, 120] = 18.46, p b .001, ηp2 = .24), whereas the main effect of the CSI group (F[1, 60] = 3.73, p = .06, ηp2 = .06), and an interaction effect of both factors (F[2,120] = 3.05, p = .051, ηp2 = .05) missed significance. In the short CSI group, there was a significant task-switch contrast (Δerrors = 1.9%, F[1, 30] = 11.22, p b .01, η p2 = .27) and a cue-switch contrast (Δerrors = 1.6%, F[1, 30] = 10.18, p b .01, ηp2 = .25). In the long CSI group, only the task-switch contrast (Δerrors = 3.5%, F[1, 30] = 15.74, p b .001, ηp2 = .34) was significant, whereas the small cueswitch contrast in the non-predicted direction was not significant (Δerrors = − 1.0%, F[1, 30] = 1.81, p = .19, ηp2 = .06). 3.3. Diffusion model analyses 3.3.1. Response criterion (a) Estimates for the a parameter are displayed in the upper panel of Fig. 3. In the task-pure blocks, a comparable response criterion was estimated for both CSI groups. In the mixed blocks, the response criterion was generally increased, but the increase was more pronounced in the short CSI group (Δa = 0.71) as compared to the long CSI group (Δa = 0.27). The ANOVA testing mixing costs across trials of the task pure and mixed task block yielded a main effect of block (F[1,60] = 145.83, p b .001, η p 2 = .71), a main effect of CSI group (F[1,60] = 11.96, p b .001, η p2 = .17), and an interaction of both factors (F[1,60] = 28.95, p b .001, η p 2 = .33). An ANOVA across trials of the task pure blocks with trial type (2) and CSI group (2) as factors neither indicated a main effect of trial type, nor of the CSI group, nor an interaction effect of both factors (all p N .26). An ANOVA across trial types of the mixed block with trial type (3) and CSI group (2) as factors revealed a main effect of trial type (F[2, 120] = 3.13, p b .05, η p 2 = .05), a main effect of CSI

group (F[1, 60] = 17.03, p b .001, ηp2 = .22), but no interaction effect (F[2, 120] = 1.42, p = .24, η p2 = .02). In the short CSI group, there was neither a task-switch contrast (Δa = 0.04, F[1, 30] = 0.78, p = .39, η p 2 = .03) nor a cue-switch contrast (Δa = 0.07, F[1, 30] = 0.91, p = .35, η p 2 = .03). In the long CSI group, the task-switch contrast missed significance (Δa = 0.06, F[1, 30] = 3.62, p = .07, ηp2 = .11), but the cue-switch contrast was significant (Δa = 0.08, F[1, 30] = 4.62, p b .05, ηp2 = .13), and so was the total switching effect denoting the difference between task-switch/cueswitch and task-repeat/cue-repeat trials (Δa = 0.14, F[1, 30] = 12.42, p b .01, ηp2 = .29).

3.3.2. Drift rates (v) Drift rates were generally higher in trials of the task-pure blocks compared with those of the mixed blocks, whereas effects of the CSI group seemed to depend on trial type: Drift rates were higher in the short CSI group relative to the log CSI group, but they were lower in the mixed block. This resulted in a more pronounced decrease from the task pure to the mixed task block in the short (Δν = 1.16) relative to the long CSI group (Δν = 0.58). This was confirmed in an ANOVA across task repeat trials of the task pure and of the mixed task block. There was a main effect of the block type (F[1, 60] = 67.14, p b .001, ηp2 = .53), no main effect of the CSI group (F[1, 60] = 1.42, p = .24, ηp2 = .02), but an interaction of both factors (F[1, 60] = 7.52, p b .01, ηp2 = .11). Testing across trial types of the task pure blocks, the 2 (trial type) by 2 (CSI) ANOVA only revealed a main effect of the CSI group (F[1, 60] = 10.81, p b .01, ηp2 = .15), but neither a main effect of trial type (F[1, 60] = 0.56, p = .46, ηp2 b .01) nor an interaction of both factors (F[1, 60] = 0.46, p = .50, ηp2 b .01). Across trials of the mixed blocks, the 3 (trial type) by 2 (CSI) ANOVA yielded a main effect of trial type (F[1, 120] = 62.73, p b .001, ηp2 = .51), no main effect of the CSI group (F[1, 60] = 0.80, p = .38, ηp2 = .01), but an interaction of both factors (F[1, 120] = 11.11, p b .001, ηp2 = .15). In the short CSI group, there was a significant task-switch contrast (Δν = 0.52, F[1, 30] = 18.64, p = .001, η p2 = .38) and a cue-switch contrast (Δν = 0.97, F[1, 30] = 49.48, p = .001, η p2 = .62). Differently in the long CSI group, there was only a significant task-switch contrast (Δν = 0.71, F[1, 30] = 19.79, p = .001, η p 2 = .40), but no cueswitch contrast (Δν = 0.07, F[1, 30] = 0.16, p = .69, ηp2 = .01).

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comparable in both conditions of the cue-switch contrast in the short CSI group (Δt0 = .017, F[1, 30] = 4.00, p = 06., ηp2 = .12). In the long CSI group, there was virtually no total switching effect in this parameter (Δt0 =.003, F[1, 30] = 0.23, p = .64, ηp2 b .01), as indicated by the highly comparable non-decision time in task-switch/cue-switch and task-repeat/cue-repeat trials. This suggests that the constituents of the total switching effect, namely the pure task-switching and cueswitching effects, are also absent. The observation of a small decrease in non-decision time in task-repeat/cue-switch trials relative to both other trial types (Δt0 = .017, F[1, 30] = 4.68, p = .04, ηp2 = .14, and Δt0 = − .014, F[1, 30] = 5.14, p = .03, ηp 2 = .15, for task-switch/ cue-switch and task-repeat/cue-repeat trials, respectively) violates the logic of the 2:1 mapping paradigm, indicating that the specific contrasts should not be computed in this case. 4. Additional analyses

Fig. 3. Parameters of the diffusion model as a function of trial type (TR/CS = task repetition/cue switch, TR/CR = task repetition/cue repetition, TS/CS = task switch/cue switch, TR/CS = task repetition/cue switch, TR/CR = task repetition/cue repetition) and cue– stimulus interval (CSI: long vs. short). Error bars represent the 95% confidence interval of the mean.

3.3.3. The non-decision parameter (t0) As can be seen in the bottom panel of Fig. 3, estimates of the nondecision parameter were comparable across all trial types with the exception of a marked increase in the non-decision parameter in taskswitch trials in the short CSI group. In task-repeat trials, there was not much of a difference between the task pure and mixed task blocks, neither in the short (Δt0 = .007) nor in the long CSI group (Δt0 = −.008). This was confirmed in the ANOVA which yielded neither a main effect of block type (F[1, 60] b 0.01, p = .96, ηp2 b .01), nor a main effect of CSI group (F[1, 60] = 1.30, p = .26, ηp2 = .02), nor an interaction of both factors (F[1, 60] = 0.12, p = .73, ηp2 b .01). In the ANOVA across trials of the task pure block, none of the main or interaction effects was significant (all p N .61). The ANOVA across trial types of the mixed block yielded a main effect of trial type (F[2,120] = 44.73, p b .001, ηp2 = .43), a main effect of CSI (F[1,60] = 11.85, p b .001, ηp2 = .16), and an interaction of both factors (F[2,120] = 28.88, p b .001, ηp2 = .33). The only contrast that indicated a substantial effect was the taskswitch contrast in the short CSI group (Δt0 = .117, F[1, 30] = 56.16, p b .001, η p 2 = .65). Differently, the non-decision time was

Small unpredicted effects such as the latter provoke the question whether these effects are substantial or whether parameters could be constrained to be equal across conditions. As formal model comparisons are not possible using the KS method, we estimated parameters again for each participant using maximum likelihood (ML) on the basis of individual responses. Additionally to the model without parameter constraints, a number of alternative models were fitted in which parameters were systematically constrained to be equal across conditions. Appendix B summarizes results obtained for 28 models with different combinations of parameter constraints, separate for the short and long CSI groups. All models were formally tested against the baseline model (#1) without parameter constraints using the likelihood-ratio chi-squared (LR-χ2 ) test. In case of a non-significant difference, the Akaike information criterion (AIC) and the Bayesian information criterion (BIC) were used to determine the best model in terms of fit and parsimony. In terms of the LR-χ2 test alone, the response criterion (a) could be constrained across conditions in the short and long CSI groups without a significant increase in model misfit. Differently, constraining drift rates (ν) across all conditions was not possible, either alone (in the short CSI group), or after constraining the response criterion (in the long CSI group). In the short CSI group, the markedly increased non-decision parameter (t0) in task-switch/cue-switch trials could not be constrained to be equal with any of the other conditions. Differently, non-decision time could be constrained across all conditions in the long CSI group, alone and after constraining the response criterion. When comparing all models that did not differ significantly in LR-χ2, the AIC and BIC model comparison indices converged in their recommendation of the unconstrained model in both groups. Accordingly, parameter estimates of this model were submitted ANOVAs to test for task-switch and cue-switch effects. Generally, the pattern of parameter estimates across conditions was very similar to the ones of the KS analyses (see Appendix C for descriptives of the ML parameter estimates). Only the difference between trial types in the response criterion in the long CSI group did not reach significance in the ML analyses anymore (p = .12, for the cue-switch contrast), however, a lack of statistical power cannot be ruled out either. (Actually, this contrast was also significant in the EZ analyses in the long CSI group). Apart from that, the short CSI group generally displayed increased response caution relative to the long CSI group (p b .001). In the short CSI group, drift rates were lowest in task-switch/cue-switch trials and highest in taskrepeat/cue-repeat trials, and all conditions differed from each other (all p b .05). Differently in the long CSI group, drift rates were reduced in task-switch/cue-switch trials relative to both other trial types (both p b .001), but they were comparable in task-repeat/cue-switch and task-repeat/cue-repeat trials (p = .60). Importantly, the pronounced increase in the non-decision parameter selectively occurred in the short CSI group relative to both other trial types (both p b .001). Differently in the long CSI group, task-switch/cue-switch trials did not differ

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from task-repeat/cue-repeat trials (p = .62). Again, there was an unpredicted slight decrease in task-repeat/cue-switch trials relative to the other conditions in the long CSI group (both p b .05). This small, unpredicted effect could have been avoided when more constrained models would have been selected, such as model 20 for the short CSI group and model 24 for the long CSI group. The pattern in the other parameters would have been identical in this case. 5. Discussion The aim of the current study was to dissociate processes of cue switching and task switching with the diffusion model. To this end, we employed an explicit cueing paradigm with a 2:1 cue-to-task mapping to tease apart both effects and decomposed the performance data with the diffusion model. By this means, we intended to investigate predictions derived from multiple component models of task switching and the compound-cue retrieval/repetition priming account. In the following, we will, first, address effects in the behavioral data. Next, findings for the parameters of the diffusion model will be presented. 5.1. Findings in the behavioral data Replicating previous studies on mixing costs (Rogers & Monsell, 1995; Ruthruff et al., 2001), we found that the response times were generally increased in mixed blocks compared to task-pure blocks across all trial types. More importantly, across trial types of the mixed blocks, there was evidence of costs related to both cue-switch and taskswitch. Particularly in the short CSI group, RTs were longer in cueswitch trials relative to cue-repeat trials, indicative of a cue-switch effect (or repetition priming; Logan & Bundesen, 2003). However, RTs in the task-switch trials were further increased by about the same magnitude in task-switch relative to cue-switch trials, indicative of a pure task-switch effect. RTs were generally shorter in the long CSI group relative to the short CSI group and they were only slightly increased in the task-switch trials relative to the other trial types in the mixed block. This pattern was predicted by accounts which posit an extra phase of task-set preparation in task-switch trials (e.g., Rogers & Monsell, 1995). However, findings also corroborate that preparation can be largely completed prior to stimulus onset given the predictability of the next task set and sufficient time to complete preparation. The absence of a cue-switch contrast in the long CSI group, suggests that 800 ms was sufficient time to arrive at a comparable state of task readiness for these participants. Note that reduced task-switch costs in the classical 1:1 mapping procedure at long CSI have been previously reported in studies using a within-participants design, but not necessarily in ones using a between-participants design (e.g., Altmann, 2004). It was suggested that this could reflect strategic factors to engage in task-set preparation (Koch, 2001). One may speculate that participants in the current study considered it relevant to prepare for the up-coming task because of the 2:1 cue-to-task mapping. Additionally, participants in the long CSI group might have been confident enough to believe that they will be successful in doing so. In fact, generally fast response times and increased error rates relative to the short CSI group suggest a confident processing style in the long CSI group (which was also corroborated by the lower response criterion setting, see below). Note that the behavioral results can be generally reconciled with the compound-cue retrieval/repetition priming account. Previous studies have shown that the effects of CSI on mean response times can be modeled in the absence of reconfiguration (Logan & Bundesen, 2003; Schneider & Logan, 2005). Additionally, the absence of a major difference in RTs in cue-switch and cue-repeat trials in the long CSI group could indicate that cue-encoding has been finished after 800 ms in both conditions. However, the further increase in task-switch relative to cue-switch trials seems to be a challenge to explain in the absence of task-switch specific processes.

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For both groups, the pattern of effects in the error scores across trial types largely paralleled those obtained in the RTs. In the short CSI group, there was a task-switch as well as a cueswitch effect, whereas in the long CSI group there was only a task-switch effect but no cue-switch effect. However, the general performance level of both groups was inverted: The long CSI group tended to commit more errors across all trial types compared with the short CSI group. The pattern of shorter RTs and increased errors suggests that the long CSI encouraged participants to respond less cautiously. This was tested next more directly with the diffusion model. 5.2. Findings in the parameters of the diffusion model 5.2.1. Non-decision parameter (t0) According to multiple component models that also specify an extra phase of task-set preparation, non-decision time was predicted to be increased in trials affording task-switches, particularly, if the time prior to stimulus onset does not suffice to finish preparation in advance. The finding of a selective increase in the non-decision parameter in task-switch trials in the short CSI group seemed to support this prediction. Additionally, the absence of a notable task-switch contrast in the long CSI group implies that whatever processes take place during this phase, they can be largely completed prior to stimulus onset given the informative task cues and sufficient time, supporting previous findings (cf. Karayanidis et al., 2010, and Karayanidis et al., 2009, for the role of cue-informativeness; cf. Schmitz & Voss, 2012, for the role of preparation time). The absence of a cue-switch effect implies that there is no pronounce orienting phase, or, that this phase has been finished in the 150 ms prior to stimulus onset. Additionally, it constrains the interpretation of the t0 effect: Task-set retrieval from long-term memory (Goschke, 2000; Mayr & Kliegl, 2003) cannot account for the increase in the non-decision parameter, as these retrieval processes should also take place on taskrepeat/cue-switch trials. Plausibly, these earlier processes have been completed in the available CSI, anyway. Most likely, the task-switch specific increase in the non-decision parameter indexes processes at the later stage of getting the task-set ready in procedural working memory (Risse & Oberauer, 2010; see also Goschke, 2000; Mayr & Kliegl, 2000, 2003), which may correspond with the biasing of its task-demand units (Gilbert & Shallice, 2002; see also Ruthruff et al., 2001). In any case, the selective effect obtained in the current study implies that the increase in non-decision time may be used as a relatively specific indicator of the pure task-switch effect. Hence, the current finding contributes to existing behavioral and physiological evidence that task switching requires distinct processes that go beyond those in cue switching (e.g., Horoufchin et al., 2011; Mayr & Kliegl, 2003; see Jost et al., 2013, for a review of cue-processing in task switching). From the perspective of the compound-cue retrieval/repetition priming account, the pattern of results in the non-decision parameter is more difficult to explain. The decreased non-decision time in taskrepeat/cue-repeat trials relative to task-switch/cue-switch trials makes sense. However, non-decision time was not predicted to be decreased in task-repeat/cue-switch trials, as the task-cue does not repeat in these trials. Of course, one could argue that cue-encoding is largely completed by the end of the 150 ms CSI. However, even then, the pronounced increase in task-switch/cue-switch trials seems to be difficult to explain. 5.2.2. Drift rate (v) Multiple component models predict that task readiness can be affected by inertia effects and task preparation (Koch & Allport, 2006). Consequently, drift rates should differ across trial types as a function of trial transition and the time available for preparation. This was confirmed in the present study. To start with mixing effects, drift rates

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were generally higher in the task-pure relative to the mixed blocks. This makes sense given that all previous trials in the task pure block required the same task set. Consequently, beneficial inertia effects would lead to a particular high level of task readiness. Additionally, drift rates were somewhat higher in the short CSI group relative to the long CSI group, which may reflect the dissipation of cue-driven task-set priming with increasing CSI. Further, the task-cue may function as an alerting signal shortly before stimulus presentation. Both factors would contribute to an optimal state of task-readiness at stimulus presentation in the short CSI group, accordingly, evidence accumulation should be steep. To continue with the mixed block, drift rates in the short CSI group had a pattern that would be predicted if drift rates were largely determined by inertia effects. Drift rates were lowest in task-switch trials, as would be expected given the adverse inertia effects or incomplete activation or biasing of the relevant task-set relative to the competing one just switched away from. Drift rates were at an intermediate level in cue-switch trials, which would be expected given that the relevant task-demand units are still activated. Finally, drift rates were highest in cue-repeat trials, possibly reflecting additional cue-driven task-set biasing. Differently in the long CSI group, drift rates were reduced in taskswitch trials only, but comparably high in task-repetitions with cueswitches and with cue-repetitions. This pattern suggests additional effects of controlled task-set activation, particularly in the cue-switch trials. It seems to be reasonable to assume that the long CSI left sufficient time to realize that the same task will be repeated and, hence, reactivate it, so that comparable levels of task-readiness can be achieved as in cue-repeat trials. Differently, the account on compound-cue retrieval/repetition priming does not specifically predict substantial differences in drift rates across trial types. However, if one permits that repetition priming does not only affect compound cue encoding/formation but also affect response retrieval in case the compound-cue repeats, then faster drift rates would be expected in task-repeat/cue-repeat trials. However, this would not easily explain the difference obtained between task-switch/ cue-switch and task-repeat/cue-switch trials.

5.2.3. Response criterion (a) Given that both theoretical perspectives do not make specific assumptions with respect to trial-to-trial adjustment in response caution, we based our predictions on previous studies that employed the diffusion model. Confirming previous findings, the response criterion was found to be generally increased in the mixed blocks relative to the task-pure blocks (Schmitz & Voss, 2012). Consequently, part of the mixing costs can be attributed to a more cautious response style in the mixed block compared with the task-pure block. This was the case for both CSI groups, however, the increase in the response criterion was more pronounced in the short CSI group compared with the long CSI group. This finding suggests that the long CSI group perceived the task as less difficult and chose a more confident processing style. Compared with the effects of block and of CSI group, the adjustment of the response criterion in a trial-to-trial fashion was of small magnitude only. Results obtained for this effect were less consistent across methods of parameter estimation, although a lack of statistical power might have prevented statistical significance of a descriptive effect in the ML analyses. If anything, participants in the long CSI group tended to reduce response caution in the case of an easy task-repeat/cue-repeat trial (as confirmed by the KS and EZ estimates). Differently in the short CSI group, the response criterion did not differ between trial types of the mixed blocks. This can be reconciled with previous findings that response criteria are only adjusted when there is enough time (Schmitz & Voss, 2012; see also Karayanidis et al., 2010; Karayanidis et al., 2009), possibly because fine-grained trial-to-trial adjustment of the response criterion consumes time, or is a resources-demanding process (cf. Mansfield et al., 2011).

5.2.4. Evaluating model predictions In the present study we used the diffusion model in order to test predictions derived from two important theoretical frameworks, (a) multiple component models of task switching that comprise task-set preparation and inertia effects, and (b) the compound-cue retrieval/ repetition priming account. The most important findings in this respect will be summarized and discussed below. Generally, it turned out that results in the parameter estimates can be reconciled with multiple component models, whereas some of the findings pose a challenge for the compound-cue retrieval account. As to the latter, the absence of an increase in non-decision time in taskrepeat/cue-switch trials relative to task-repeat/cue-repeat trials is particularly difficult to explain. As the task cue has changed, there should not be an advantage in the compound cue retrieval process. Additionally, some of the trial-type dependent effects in the drift rate were not specifically predicted. Possibly, one could argue that the response retrieval process is also facilitated in case the compound-cue repeats. However, this cannot easily explain the increase in drift rate in taskrepeat/cue-switch trials over task-switch/cue-switch trials that was observed in both groups. Conversely, predictions from multiple-component models of task switching (e.g., Mayr & Kliegl, 2003; Ruthruff et al., 2001) were generally confirmed. The selective increase in the non-decision parameter seems to correspond with the requirement to prepare for a new task set; and results of the present study extend previous findings that this effect is indeed task-switch (and not cue-switch) specific. However, the increase in the non-decision parameter does not elucidate how exactly task-set preparation takes place (but see e.g., Gilbert & Shallice, 2002; Goschke, 2000; Mayr & Kliegl, 2000, 2003; Risse & Oberauer, 2010; Ruthruff et al., 2001, for theoretical accounts). Nevertheless, one of the core components should be the activation of relevant S–R links (e.g., Allport et al., 1994; Monsell et al., 2000) that will allow using stimulus information to select the appropriate response in the next phase. The stronger the relevant S–R link (relative to competing ones), the higher should be the task readiness, corresponding with high drift rates. Drift rates seemed to largely reflect inertia effects when the CSI is short. However, when there is sufficient time to prepare, drift rates may be additionally affected by preparation (cf. Koch & Allport, 2006, for the joint effects on task readiness). 5.3. Scope and limitations 5.3.1. Advantages of modeling task switching data We conducted this study primarily with the intention to test predictions derived from important theoretical accounts. However, diffusion modeling may be generally an appealing technique for the analyses of task-switching data that yields certain advantages over conventional RT and error analyses. The increase in non-decision time may serve as a relatively specific indicator of task preparation, as shown in the current study. The drift rates in experimental conditions may offer a less biased performance measure of task-switching costs: They integrate the information in correct and erroneous responses into a common metric and simultaneously avoid a bias of response caution. The latter can be additionally quantified with the criterion parameter. 5.3.2. Suitability of the paradigm The explicit cueing paradigm with a 2:1 mapping of cues to tasks is appealing as it promises to tease apart the task-switch and cue-switch components. However, the paradigm was also criticized for manipulating conditional task-switch expectancies (Mayr, 2006; Schneider & Logan, 2006). In fact, some of the divergent results obtained with the 2:1 mapping procedure concerning the relative magnitudes of cueswitch and task-switch effects may, in part, depend on the conditional probability of a task switch given a cue switch (see Mayr, 2006, for a global; and Schneider & Logan, 2006, for a local account). Given a high conditional probability, participants may engage in task switch activity

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upon the detection of a cue switch so that the relevant task-set needs to be prepared again later. Additionally, an orienting reaction may take place in case the actual task set does not match with expectancies. Both would lead to increased response times in the task-repeat/cueswitch condition (cf. Forstmann, Brass, & Koch, 2007), simultaneously increasing estimates of the cue-switch effect and diminishing estimates of the task-switch effect. Both effects could have taken place in the current study, as a task switch occurred in 66% of all cue switches. Nevertheless, there was still a substantial task-switch effect in the RT data. Additionally, from a diffusion modeling perspective, one can predict that postponed task-set preparation and surprise/orienting would contribute to increased non-decision time in task-repeat/cue-switch trials. However, this was not observed in the present study. The problem associated with cue-switch based expectancies could have been circumvented by using the transition-cueing paradigm (Forstmann et al., 2007) in which cue switches and task switches can be combined orthogonally. However, this paradigm is characterized by particularly large switch costs which are not fully understood (Vandierendonck et al., 2010). Hence, we chose the 2:1 explicit cueing paradigm in the current study which is more frequently used in the literature, too. Additionally, some of the predictions derived from the compound cue/priming account could be directly tested using the 2:1 mapping paradigm.

5.3.3. Suitability of the diffusion model The diffusion model applied in the current study is relatively simple. Hence, its parameters will not suffice to describe all possible component processes discussed in the rich task switching literature in a one-to-one fashion. Nevertheless, a model may be considered useful, when it helps dissociating at least some of the processes of interest or when it helps testing meaningful predictions (Brown & Heathcote, 2008; Wagenmakers, van der Maas, Dolan, & Grasman, 2008). Following this rationale, we used the diffusion model in order to test predictions derived from multiple component models of task switching and the compound-cue retrieval/repetition priming account. Of course, this makes sense only, if the fitted model adequately captures important aspects of the theoretical account. This seemed to be relatively evident in the case of the compound-cue retrieval/repetition priming account that was developed within a random-walk framework that closely resembles the diffusion model. Hence, predictions appeared relatively straightforward. Differently, multiple component models are generally more complex and they were not explicitly developed with a continuous sampling model in mind. However, as previously argued (Schmitz & Voss, 2012), we believe that diffusion model parameters may help dissociate some of the core processes, namely the effects of task preparation from response selection, with the latter possibly more strongly affected by inertia effects. At least, previous research using the explicit cueing paradigm has supported this notion (cf., Karayanidis et al., 2009, 2010; Mansfield et al., 2011; Schmitz & Voss, 2012). Of course, the question arises as to whether this also holds for the currently used task-cueing paradigm with 2:1 mapping that was criticized for introducing additional affordances such as cue-disambiguation (Altmann, 2006; Kleinsorge, 2012). If the latter corresponds with an additional decision process per trial, i.e. cue-based task-set selection, the question arises as to whether the standard diffusion model is still suited to describe the data. However, in support of the applied model, it has to be noted that the pattern of results obtained in this study closely resembles the results in previous studies using the explicit-cueing paradigms with a 1:1 mapping (Karayanidis et al., 2009, 2010; Schmitz & Voss, 2012). Additionally, the model-implied RT distributions closely matched empirical RT distributions, supporting the descriptive adequacy of the model. Finally, the pattern observed in the estimated parameters seems to correspond well with theoretical predictions, as previously outlined.

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5.3.4. Learning effects Finally, it shall be discussed that task performance improved across all runs of blocks, although the rate of improvement diminished from block to block. To give numbers, we contrasted the RT in each block with the mean of the following blocks for the short CSI (long CSI) group. The RT in TS/CS trials improved by 319 ms (116 ms) from the first to the following blocks; the next improvement was only 150 ms (71 ms). Similarly, in TR/CS trials the first improvement was 272 ms (159 ms), and the next one was 118 ms (53 ms). In TR/CR trials, the first improvement was 203 ms (139 ms), and the next was 78 ms (49 ms). The last improvement from the fourth to the fifth block reached from 40 ms (TS/CS in the short CSI group) to 13 ms (TR/CR in the long CS group). Hence, some of the contrasts were still significant from the fourth to the fifth run. This non-stationary can inflate variance and may bias parameter estimates, as the model fit tends to be sensitive to variance. Generally, the effects of learning on parameter estimates deserve more attention. Future studies should circumvent this limitation by allowing enough trials to arrive at asymptotic performance. 5.4. Conclusion The previously observed increase in the non-decision parameter (t0) in task-switch trials was shown to be task-switch specific, whereas it was not increased in cue-switch trials in which the task was repeated. The selective increase in the non-decision parameter can be best reconciled with additional process accounts of task switching which assume that task-switches require additional preparation processes (Rogers & Monsell, 1995; Rubinstein et al., 2001). The task-switch specific increase further rules likely that preparatory processes such as loading the task-set into procedural working memory (Risse & Oberauer, 2010), transforming it into a task-appropriate attentional configuration (Mayr & Kliegl, 2000, 2003), for instance by biasing relevant taskdemand units (Gilbert & Shallice, 2002), may contribute to the effect. In the following phase, the task-set is applied to the stimulus to retrieve the response, according to multiple component models of task switching (Mayr & Kliegl, 2003; Ruthruff et al., 2001). The efficiency of this process, as indicated by the drift rate (v), was shown to depend on both task-predictability and task-sequence. The first points to the role of some sort of task-set preparation (e.g., Rogers & Monsell, 1995), whereas the latter supports the view of task-set inertia (e.g., Allport & Wylie, 2000), which jointly determine the state of task readiness (Koch & Allport, 2006). Finally, the response criterion (a) varied largely between task-pure and mixed blocks and as a function of CSI length. However, trial-totrial adjustment within blocks was of comparatively small magnitude. If there is trial-to-trial adjustment of the response criterion, it seems to require time or capacity to complete prior to stimulus onset. Appendix A. Graphical display of model fit The plots show overlaid predicted (parameter based) and empirical cumulative distribution functions (CDFs) for all trial types. Predicted and empirical CDFs were first computed for each participant and then averaged across participants. The plotted functions are joint CDFs of correct and erroneous responses, with latencies of erroneous responses plotted on the left side of the horizontal axis (with negative scale values) and latencies of correct trials plotted on the right side (with positive scale values). The intercept of the cumulative distribution function indicates the percentage of erroneous responses. The CDFs for the long CSI and short CSI group are displayed in the upper and lower panels, respectively. Trial types of the task-pure and mixed blocks are shown on the left and right sides, respectively. The order of trial types in the legend from top to bottom corresponds with trial difficulty, as indicated by the order of the steep increase in the CDF: from left to right task-repeat/cue-repeat (TR/CR), task-repeat/cue-switch (TR/CS), and task-switch/cue-switch (TS/CS) trials.

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Appendix B. Model comparison #

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Parameter constraints

No constraints btw. conditions a: TS = CS a: TS = CR a: CS = CR a: ALL ν: TS = CS ν: TS = CR ν: CS = CR ν: ALL t0: TS = CS t0: TS = CR t0: CS = CR t0: ALL a: ALL; ν: TS = CS a: ALL; ν: TS = CR a: ALL; ν: CS = CR a: ALL; ν: ALL a: ALL; t0: TS = CS a: ALL; t0: TS = CR a: ALL; t0: CS = CR a: ALL; t0: ALL a: ALL; t0: ALL; ν: TS = CS a: ALL; t0: ALL; ν: TS = CR a: ALL; t0: ALL; ν: CS = CR a: ALL; ν: ALL; t0: TS = CS a: ALL; ν: ALL; t0: TS = CR a: ALL; ν: ALL; t0: CS = CR a: ALL; ν: ALL; t0: ALL

Short CSI

Long CSI

LR-χ 2 (df)

AIC

BIC

LR-χ 2 (df)

AIC

BIC

– 2.09 (1) 2.89 (1) 0.66 (1) 4.02 (2) 2.52 (1) 6.40 (1)* 2.30 (1) 7.41 (2)* 7.08 (1)** 7.19 (1)** 0.89 (1) 10.69 (2)** 4.95 (3) 17.17 (3)*** 8.99 (3)* 18.87 (4)*** 7.99 (3)* 11.22 (3)* 3.41 (3) 13.84 (4)** 28.25 (5)*** 71.92 (5)*** 27.18 (5)*** 46.55 (5)*** 70.97 (5)*** 26.97 (5)*** 81.54 (6)***

165.80 167.90 168.69 166.46 169.82 168.32 172.20 168.10 173.21 172.88 172.99 166.69 176.49 170.75 182.97 174.79 184.67 173.79 177.02 169.21 179.64 194.05 237.72 192.98 212.35 236.78 192.77 247.34

240.94 243.04 243.83 241.60 244.96 243.47 247.34 243.24 248.35 248.03 248.14 241.83 251.63 245.90 258.12 249.93 259.81 248.93 252.16 244.35 254.79 269.19 312.86 268.12 287.49 311.92 267.92 322.48

– 0.51 (1) 0.15 (1) 0.06 (1) 1.25 (2) 2.71 (1) 2.97 (1) 0.89 (1) 5.04 (2) 1.25 (1) 1.21 (1) 1.12 (1) 2.44 (2) 5.84 (3) 9.99 (3)* 2.61 (3) 11.21 (4)* 2.38 (3) 3.47 (3) 2.22 (3) 4.14 (4) 15.48 (5)** 21.72 (5)*** 6.23 (5) 22.16 (5)*** 23.06 (5)*** 12.39 (5)*** 28.55 (6)***

−166.90 −166.39 −166.76 −166.85 −165.65 −164.19 −163.93 −166.01 −161.86 −165.66 −165.70 −165.78 −164.47 −161.07 −156.92 −164.29 −155.70 −164.53 −163.43 −164.68 −162.76 −151.43 −145.19 −160.67 −144.75 −143.85 −154.52 −138.35

−92.54 −92.03 −92.39 −92.48 −91.28 −89.82 −89.56 −91.64 −87.50 −91.29 −91.33 −91.41 −90.10 −86.70 −82.55 −89.92 −81.33 −90.16 −89.06 −90.32 −88.39 −77.06 −70.82 −86.30 −70.38 −69.48 −80.15 −63.98

Note. a = response criterion, ν = drift rate, t0 = non-decision parameter; TS = task switch/cue switch, CS = task repeat/cue switch, CR = task repeat/cue repeat, ALL = parameters of all conditions constrained to be equal; LR-χ 2 (df) = likelihood-ratio chi-squared test with df degrees of freedom, AIC = Akaike information criterion, BIC = Bayesian information criterion, the value of the model recommended by each information criterion is printed in boldface; *p ≤ .05, **p ≤ .01, ***p ≤ .001.

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Appendix C. Means (standard deviations) of the maximum likelihood parameter estimates Short CSI

a ν t0 sz sν st0

Long CSI

TS/CS

TR/CS

TR/CR

TS/CS

TR/CS

TR/CR

1.87 (0.60) 1.89 (0.54) .437 (.065) 0.50 (0.25) 0.13 (0.23) .164 (.111)

1.83 (0.50) 2.31 (0.88) .318 (.071) 0.18 (0.25) 0.77 (0.62) .075 (.068)

1.83 (0.58) 3.81 (2.23) .326 (.057) 0.17 (0.23) 1.63 (1.71) .091 (.070)

1.36 (0.47) 2.21 (1.09) .334 (.058) 0.22 (0.27) 0.55 (0.70) .125 (.104)

1.41 (0.50) 3.28 (1.45) .311 (.053) 0.25 (0.32) 1.28 (0.91) .116 (.069)

1.33 (0.33) 3.41 (1.33) .330 (.045) 0.51 (0.33) 1.03 (0.84) .133 (.071)

Note. Parameters of the diffusion model: a = response criterion, ν = drift rate, t0 = nondecision parameter, sz = inter-trial variability of the starting point, sν = inter-trial variability of the drift rate, st0 = inter-trial variability of the non-decision parameter. Trial types: TS/CS = task switch/cue switch, TR/CS = task repeat/cue switch, TR/CR = task repeat/cue repeat.

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