Sequential sampling models of the flanker task: Model comparison and parameter validation
Dissertation
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By
Corey Nathan White, M. A. Graduate Program in Psychology The Ohio State University 2010
Dissertation Committee:
Roger Ratcliff, Advisor Gail McKoon Simon Dennis Alex Petrov Copyright
Corey Nathan White
2010 Abstract
The present study tests sequential sampling models of processing in the flanker task. In a standard flanker task, participants must identify a central target that is flanked by items that indicate the same response (congruent) or the opposite response
(incongruent). The standard finding is that incongruent flankers produce interference that primarily affects the early component of the decision process. The goal was to contrast different mechanisms of visual attention and to identify a simple processing model that can be used to augment analyses of this task. Several models were contrasted in their ability to account for flanker data from experiments that manipulated response bias, speed/accuracy tradeoffs, attentional focus, and stimulus configuration. Models that assume dynamic focusing of attention provided the best overall account of the behavioral data. Among the dynamic models, a spotlight model that assumes gradual narrowing of attention provided the best balance of fit and parsimony, though dual process models that assume discrete selection of the target also captured the main trends in the data when only standard congruent and incongruent conditions were used. Importantly, the
ii experimental manipulations were reflected by the appropriate model parameters, supporting future use of these models to decompose behavioral data from the flanker task into meaningful psychological constructs. The results of this study also indicate that standard flanker experiments do not provide strong evidence for contrasting gradual versus discrete target selection, and consequently they do not provide strong evidence to support or refute theories of the underlying mechanisms of dynamic attention models.
iii Dedication
To Heather and our animals, for keeping me happy.
iv Acknowledgments
I would like to thank Roger and Gail for their support and guidance, my labmates for their assistance and their empathy, and Scott Brown for assistance with the Bayesian models.
v Vita
May 1982...... Born in Missouri 2001...... Winnetonka High School 2005...... B.S. Psychology, Truman State University 2006...... M.A. Psychology, The Ohio State University 20062008...... Research Assistant, Ratcliff and McKoon Cognition and Language Lab 20072009...... Teaching Assistant, Department of Psychology, The Ohio State University
Publications
Vittengl, J. R., White, C. N., Mcgovern, R. J., Morton, B. J. (2006). Comparative validity of seven scoring systems for the Instrumental Activities of Daily Living scale in rural elders. Aging & Mental Health, 10, 3339. Leite, F. P., Ratcliff, R., White, C. N. (2007). Individual differences on speeded cognitive tasks: Comment on Chen, Hale, and Myerson (2008). Psychonomic Bulletin and Review, 14, 10071009. White, C. N., Ratcliff, R., Vasey, M. W., McKoon, G. (2009). Dysphoria and memory for emotional material: A diffusion model analysis. Cognition & Emotion, 23, 181205. White, C. N., Ratcliff, R., Vasey, M. W., McKoon, G. (2010). Anxiety enhances threat processing without competition among multiple inputs: A Diffusion model analysis. Emotion. in press. White, C. N., Ratcliff, R., Vasey, M. W., McKoon, G. (2010). Using diffusion models to understand clinical disorders. Journal of Mathematical Psychology, 54, 3953.
Field of Study Major Field: Psychology Area of Emphasis: Cognitive Modeling
vi Table of Contents
Abstract...... ii
Dedication...... iv
Acknowledgments...... v
Vita...... vi
List of Tables...... viii
List of Figures...... ix
1. Introduction...... 1
2. Models of processing in the flanker task...... 6
3. Model details...... 18
4. Experiments...... 33
5. Discussion...... 70
References...... 83
Appendix A: Mathematical Details of the Diffusion Model ...... 88
Appendix B: Validation of Simulation Method...... 89
vii List of Tables
Table 1: Behavioral data from Experiments 1 and 2...... 36
Table 2: Bestfitting model parameters from Experiment 1...... 40
Table 3: Bestfitting model parameters from Experiment 2...... 47
Table 4: Behavioral data from Experiments 3 and 4...... 53
Table 5: Bestfitting model parameters from Experiment 3...... 54
Table 6: Bestfitting model parameters from Experiment 4...... 58
Table 7: Behavioral data from Experiment 5...... 65
Table 8: Bestfitting model parameters from Experiment 5...... 66
viii List of Figures
Figure 1: Theoretical conditional accuracy function...... 5
Figure 2: Schematic of the standard diffusion model...... 14
Figure 3: Dynamic attention models...... 24
Figure 4: Bayesian perceptual models...... 29
Figure 5: Observed and predicted data from dynamic models in Experiment 1...... 38
Figure 6: Observed and predicted data from perceptual models in Experiment 1...... 41
Figure 7: Observed and predicted QPFs from Experiment 2...... 46
Figure 8: Observed and predicted CAFs from Experiment 2...... 48
Figure 9: Schematic of model behavior across response bias...... 50
Figure 10: Observed and predicted QPFs from Experiment 3...... 55
Figure 11: Observed and predicted CAFs from Experiment 3...... 56
Figure 12: Observed and predicted QPFs from Experiment 4...... 59
Figure 13: Observed and predicted CAFs from Experiment 4...... 60
Figure 14: Conditions in Experiment 5...... 63
ix Figure 15: Observed and predicted QPFs from Experiment 5...... 67
Figure 16: Observed and predicted CAFs from Experiment 5...... 68
Figure B1: Generated and recovered parameter values from the diffusion model...... 89
x 1. Introduction
The Eriksen flanker task has been extensively used to investigate the mechanisms underlying visual attention (Eriksen & Eriksen, 1974). In the standard task, participants must discriminate a single item (e.g., letter or arrow) that is surrounded, or flanked, by items that indicate the same or opposite response. For example, if participants had to decide whether the central arrow in a display faced right or left, a congruent trial would include flankers that faced the same direction as the central target (e.g., > > > > >), whereas an incongruent trial would include flankers that faced the opposite direction
(e.g., < < > < <). The standard finding is that the incongruent flankers produce interference that leads to slower and less accurate responses compared to the congruent condition, known as the flanker congruency effect (FCE).
The present study compares the ability of several response time (RT) models to account for data from the flanker task. The primary aim was to determine which processing mechanisms can best account for data from a range of experimental manipulations. The secondary aim was to identify a simple model that can be subsequently used as a tool for analyzing data from flanker tasks. RT models have been
1 used to augment analyses of data from a range of twochoice tasks by decomposing the data into relevant psychological constructs (Ratcliff & McKoon, 2008), and researchers using the flanker task to investigate spatial attention could likewise benefit from such modelbased analyses.
The next section reviews critical findings from the flanker task, placing particular focus on how the decision evidence varies over time. This is followed by a brief review of models of visual attention and perceptual processing that have been developed to account for data from the flanker task. Then the diffusion model is introduced with focus on how it can be augmented to incorporate principles from theories of flanker processing.
Two simple diffusion models are developed, one based on a singleprocess and one based on dualprocesses. These models are then tested with an extant dualprocess model
(Hübner, Steinhauser, & Lehle, 2010) and two Bayesian perceptual models (Yu, Dayan,
& Cohen, 2009) against a series of experiments where different components of the decision process are manipulated. This battery of experimental data allows us to determine first, whether the models can adequately fit the data, and second, whether the model parameters appropriately reflect the manipulations. To the extent that the models provide a parsimonious account of the data with appropriate parameter changes, there is support for their future use as analytical tools.
2 Processing in the Flanker Task
This study focuses on standard flanker tasks in which a stationary target with known location is flanked by congruent or incongruent items. Early studies involving flanker stimuli demonstrated several key aspects of visual processing and attention. First, the visual system is not capable of infinitely fine selectivity. Even when participants knew the upcoming location of the target, they could not effectively constrain their attention to eliminate effects from the flankers (Eriksen & Eriksen, 1974). This finding also implies that the FCE is not due to inefficient search for the target or uncertainty about its location. Indeed, the FCE has been shown with flankers over 2 degrees from the target, meaning they were unlikely to be confused with the target location (Eriksen &
Schultz, 1979). Second, the FCE is dependent on spacing, with more interference from near as opposed to far flankers (Eriksen & Eriksen, 1974; Eriksen & Schultz, 1979).
However, distant flankers still produce some degree of interference, suggesting that visual attention has a gradient edge with a gradual drop off in attentional resources
(Eriksen & St. James, 1986). Spacing has also been shown to affect overall processing of the stimuli, with tightlyspaced items leading to slower overall responses due to crowding or local attentional interference (Eriksen & Eriksen, 1974; McCarley & Mounts, 2008).
Third, the FCE is not the result of local perceptual interactions. Eriksen & Eriksen
(1974) found that congruent flankers produced similar effects whether they were
3 perceptually identical to the target (i.e., the same letter) or different (i.e., different letter but same response set). This suggests that the primary locus of the FCE is due to the mapping of the flankers onto the appropriate response, rather than lowlevel perceptual interactions. However, effects of perceptual contrast have been demonstrated in the flanker task when degraded stimuli are presented (see Rouder & King, 2003).
Perhaps the most salient finding from flanker experiments is that the time course of the FCE is not constant. Gratton and colleagues (1988) showed this by plotting the condition accuracy function (CAF) for responses. The CAF shows the proportion of correct and error responses that have terminated by certain times, which provides an indirect measure of how the decision evidence varies over time. Figure 1 shows an idealized CAF from a standard flanker task. The critical portion of the CAF involves the fastest responses, where accuracy for incongruent trials dips below chance. Thus for early processing the flankers overwhelm the target, resulting in a higher likelihood of selecting the incorrect response. Recordings of lateralized readiness potentials (LRPs) support this account, showing partial activation of the incorrect response for incongruent trials, even if the correct response was given (Gratton et al., 1988). Crucially, though, accuracy rises rapidly for the longer responses in the incongruent condition.
4 Figure 1. Idealized conditional accuracy function for the flanker task.
The time course of the FCE suggests that accurate processing of the target takes time to build. This is typically conceptualized by assuming that visual attention is broad and nonselective at stimulus onset, but focused and selective after some period of time.
Thus the flankers have a large influence early in processing that is reduced as attention is focused. Conversely, the time course of the FCE might arise as a byproduct of perceptual processing, with relatively impoverished perceptual information that improves over time without any role for visual attention. The next section presents several models that implement these various assumptions to account for the impact of flanking stimuli over time.
5 2. Models of Processing in the Flanker Task
At a broad level, models of flanker processing can be grouped into two categories: dynamic attention models that assume visual attention is employed to focus on the target, and perceptual models that assume that flanker interference is a result of perceptual processing that improves over time independently of attention. Several of each class of model will be explored in this study.
Models with Dynamic Attention
The primary assumption of dynamic attention models is that attention can be applied during a trial to focus on the target and reduce or eliminate the influence of the flankers. There are two main types of model within this class: Singleprocess models that assume gradual narrowing of attention, and dualprocess models that assume discrete selection of the target. It is important to note that there can be considerable mimcry between these two classes of model, and it has been argued that dualprocess models simply represent poles on the continuum of attentional selectivity (Eriksen & St. James,
1986). In fact, if variability were added to the selection time in a dual process model it
6 would produce data similar to single process models. As the results of the experiments later show, these models are not easily differentiated.
Continuous narrowing of attention
Singleprocess models of attentional narrowing assume that the extent of spatial attention can be gradually reduced over time (Eriksen & Schultz, 1979; Eriksen & St.
James, 1986). These models often conceptualize visual attention as a spotlight in space, where any items within the spotlight feed later processing. The spotlight is diffuse at stimulus onset, allowing influence from the flankers, but gradually narrows in on the target as more attention is engaged. Within this framework, the size of the spotlight can be decreased before stimulus onset by perceptual inputs, such as precues (Eriksen & St.
James, 1986), and after stimulus onset by engaging attention. Thus visual attention effectively shrinks the spotlight (analogous to a zoom lens) to focus in on the target.
Cohen, ServanSchreiber, & McClelland (1992) showed that principles of a shrinking spotlight model could arise from a fairly simple neural network model in which attention serves to boost the perceptual input from the target. Combined with lateral inhibition of
7 nearby items, the attentional boost gradually reduces the influence of the flankers as more attention is applied.
Models in this class are labeled singleprocess because the decision process receives the same stream of evidence throughout the course of the decision, with visual attention moderating the inputs that determine the evidence. Crucially, since there is only one stream of evidence that feeds the decision process, the quality of evidence can improve during the narrowing of attention, even before the focus is entirely on the target.
Discrete selection of target
Dualprocess models differ from continuous models by assuming some form of allornone processing of the flankers (e.g., Hübner et al., 2010; Jonides, 1983;
Ridderinkhof, 1997). During the first process, attention is diffuse and thus the flankers have a significant influence on the decision evidence. The second process does not occur until the target has been selected, at which point only it drives the response. This is conceptually similar to saying that the narrowing of attention does not influence the decision evidence until it has completely narrowed in on the target. Models in this class
8 have also been referred to as dualstage models, assuming that selection of the target engages the second stage of processing that then feeds the decision evidence.
In this framework, the selection of the target and subsequent change in decision evidence are conceptualized as a separate process from the automatic response activation that occurs during early processing. The second process does not provide any decision evidence until it has selected the target. The primary difference between single and dualprocess models is how attention affects the evidence that feeds the decision. Single process models assume that decision evidence gradually changes as the flankers are suppressed (or the target is boosted), whereas dualprocess models assume an abrupt change in evidence once the target has been selected.
Perceptual Models of Flanker Processing
Recently, Yu and colleagues developed two Bayesian models to account for processing in the flanker task (Yu et al., 2009). These models are distinguished from the attentional models because they assume that the FCE is driven by perceptual processes rather than attention. Both of the models assume that the stimulus produces noisy inputs
9 to the perceptual system that are sampled over time. Because of the noise in the inputs and the sampling process, the participant's belief about the identity of each item improves as more information is sampled from the noisy inputs.
The first perceptual model, the compatibility bias model, attributes the FCE to a perceptual bias in processing. The bias model assumes that people have a bias to process nearby items as if they are compatible or congruent with each other. When presented with an incongruent stimulus, the participant is biased to believe that all of the items are congruent, which increases the probability of incorrectly indicating that the target faces the same direction as the flankers. This effect is most pronounced early in the trial when the sampled perceptual input is relatively poor. As more of the stimulus is sampled, the perceptual input eventually outweighs the bias and leads to the correct response. Thus there is no dynamic attentional selection or narrowing.
The second perceptual model, the overlap model, attributes the FCE to overlapping perceptual representations. The underlying assumption is that since neural receptive fields can cover large areas of the visual field (up to 20 degrees, Desimone &
Gross, 1979), the input for each item is contaminated by its neighbor. Variations of this
10 concept has been proposed previously in this domain (e.g., Logan, 1996) and in others
(e.g., Gomez, Ratcliff, & Perea, 2008; Ratcliff, 1981). With overlapping representations, the input for each item is a combination of the item itself and some portion of the neighboring items. On incongruent trials, the overlap from the flankers decreases the input strength of the target, leading to early errors that are reduced as more input is sampled. Similar to the bias model described above, the overlap model does not include a dynamic attentional mechanism.
Yu et al. (2009) showed that both the compatibility bias and overlap models can produce the belowchance dip for incongruent trials in the CAF. Unfortunately, neither model was ever fit to actual data, and the data for the CAFs are noisy and highly variable, because relatively few responses have terminated by the earliest bins (e.g., 300325ms).
Further, fitting the CAF does not guarantee good accord with all of the behavioral data.
For example, Yu et al. (2009) chose a particular set of parameter values for the compatibility bias model that produced a CAF similar to the one obtained empirically by
Gratton et al. (1988), and used this accordance as support for the viability of the model.
However, those same parameter values predict accuracy near 60% for incongruent trials,
11 which is substantially lower than the 80%90% reported by Gratton et al. (1988). Thus although those parameter values produced the correct form of the CAF, they significantly misfit the remainder of the data. In light of this, it is argued that neither of the perceptual models presented by Yu et al. (2009) have been shown to account for data from the flanker task. The next section discusses how different theories of flanker processing can be implemented in sequential sampling models.
Sequential Sampling Models
Each of the theories described above were implemented in the framework of a sequential sampling model. These are models of simple, twochoice decisions that share the basic assumption that information is sampled and accumulated over time until a criterial amount is reached. The Ratcliff diffusion model (Ratcliff, 1978; Ratcliff &
McKoon, 2008; Ratcliff, Van Zandt, & McKoon, 1999) was chosen as the framework to implement the dynamic models described above because it provides a successful theoretical account of the fast, twochoice decisions that are required in the flanker task.
Other sequential sampling models include the Linear Ballistic Accumulator Model
12 (Brown & Heathcote, 2008); the leaky competing accumulator model (Usher &
McClelland, 2001), and the poisson counter model (Smith & Van Zandt, 1995). The perceptual models were implemented in the Bayesian framework originally proposed by the authors that includes a sequential sampling mechanisms that mimics the diffusion process (Yu et al., 2009), and will be described shortly.
In the diffusion model, noisy evidence is accumulated over time until a criterial amount has been reached, at which point the corresponding response is initiated. The basic model is shown in Figure 2. The decision process starts at some point, z, between the two boundaries, and evidence is sampled over time and accumulated until one of the two boundaries is reached. Each boundary represents one of the two possible choices
(e.g., right or left) and the time taken to reach the boundary is the response time for the decision. Processes outside of the decision process, like stimulus encoding and response execution, are combined into one parameter with mean time, Ter. The overall response time for a trial equals the decision time plus the nondecision time.
13 Figure 2. Schematic of the standard diffusion model.
In addition to Ter, there are three other primary components to the model, each with a straightforward psychological interpretation. The starting point, z, indexes response bias. If participants were biased toward one of the two responses (e.g., through proportion manipulation), they would move their starting point closer to that boundary.
This produces faster and more probable responses at that boundary since less evidence is needed to reach it. The separation of the two boundaries, a, indexes response caution or speed/accuracy tradeoffs. If the boundary separation is relatively small, responses will
14 take less time to reach a boundary, leading to faster responses, but they will also be more likely to reach the wrong boundary due to noise in the process, leading to more errors.
The drift rate, v, is the average rate of approach to a boundary, and indexes the quality or strength of evidence extracted from the stimulus. A large value of drift will approach the appropriate boundary quickly, leading to fast and accurate responses. The withintrial noise in the accumulation of evidence is represented by s, which is a scaling parameter that was fixed at .1.
The diffusion model has been shown to fit data from a range of twochoice tasks
(see Ratcliff & McKoon, 2008; White, Ratcliff, Vasey, & McKoon, 2010a). The model accounts for the entire set of behavioral data from these tasks, namely the proportions of responses and the distributions of correct and error RTs. Importantly, the parameters of the model have been shown to accurately reflect the constructs which they represent.
Manipulations of speed/accuracy tradeoffs affect boundary separation (and sometimes nondecision time, manipulations of proportion or payoff affect starting point, manipulations of stimulus difficulty affect drift rates, and manipulations of response execution affect nondecision time (Voss, Rothermund, & Voss, 2004; White et al.,
15 2010a). Thus the parameters of the diffusion model have been empirically validated.
This validation lends support for use of the model to decompose behavioral data into the constituent processes. By separating out the different decision components, the model allows researchers to perform direct comparisons for each of the components. Further, since individual differences in criterion settings are separated into the respective model components, the drift rates extracted by the model provide a more direct measure of stimulus evidence compared to RTs or accuracy values (White, Ratcliff, Vasey, &
McKoon, 2009; White et al., 2010a).
Previous work with the diffusion model has demonstrated that the model can lead to different conclusions than traditional comparisons of RTs and/or accuracy. In studies of aging, for example, Ratcliff and colleagues have shown that for certain tasks such as lexical decision and singleitem recognition memory, slower RTs in older (6590) participants are due to longer nondecision time and wider boundary separation, not poorer stimulus evidence (Ratcliff, Thapar, Gomez, & McKoon, 2004; Ratcliff, Thapar,
& McKoon, 2006). In studies of anxiety, White, Ratcliff, Vasey, and McKoon (2010b) have shown that drift rates are more sensitive to small differences between groups than
16 RTs or accuracy values, and the increase in sensitivity can show reliable differences that are not significant in comparisons of behavioral data. These results illustrate the importance of accounting for the different components of the decision process, as differences in RTs or accuracy could result from different underlying processes.
Sequential sampling models like the diffusion model provide a theoretically principled method to accomplish this.
The diffusion model has proved to be a powerful theoretical and analytical tool, supporting its use in the present study. However, the standard model does not account for data from the flanker task. This is due to the fact that the drift rate is fixed over the course of the decision. While this assumption sufficiently captures data from many two choice tasks, it cannot account for the timevarying evidence that drives the flanker decision. By incorporating concepts from models of flanker processing, this shortcoming can be overcome. The next section presents the details of each flanker model to be tested.
17 3. Model Details
This section present the details of the flanker models that were tested in the present study. Among the models, one is based on a singleprocess spotlight model, one is based on a simple dualprocess model, one is an extant model involving two stages and two phases (DSTP; Hübner et al., 2010), and two are Bayesian models involving perceptual bias or overlapping spatial representations (Liu, Yu, & Holmes, 2009; Yu et al., 2009). For the first two models, the standard framework of the diffusion model was used with the drift rate allowed to vary over time. In this framework, the spotlight and dualprocess components determine the relative contribution of the flankers and target to provide the value of drift rate at each time step.
This is not the first study to implement a singleprocess diffusion model of the flanker task. Liu, Holmes, and Cohen (2008) showed that the neural network model of
Cohen et al. (1992) could be reduced to a drift diffusion model with a 2term exponential function to describe drift rates for congruent trials, and a 3term exponential function to describe drift rates for congruent trials. While this work demonstrates the agreement
18 between certain neural network models and diffusion models, the resulting model is not readily interpretable in terms of psychological processes, which limits its use as an analytical tool. In other words, there is no principled link between the different exponential terms, the stimuli, and psychological processes of interest. Further, Hübner et al. (2010) showed that the Liu model provided poorer fits than their Dual Stage Two
Phase model (presented shortly), so it is not included in the present comparisons.
The aim in developing the spotlight and dualprocess models was to produce simple models with psychologically meaningful components. Each model contained the following standard components of the diffusion model: boundary separation, nondecision time, perceptual input, and starting point. To avoid confusion, the evidence provided by each item is referred to as perceptual input, and the overall evidence as drift rate.
Previous applications of the diffusion model allow for acrosstrial variability in some components based on the assumption that the values fluctuate across the course of the experiment. In support of this, Gratton et al. (1988) found prestimulus response activation in EMG recordings that varied across trials and was associated with fast responses. Their findings are consistent with a decision process with a fixed threshold
19 and a varying starting point, which is implemented in applications of the diffusion model
(see Ratcliff & McKoon, 2008). Nonetheless, the variability parameters were excluded to allow more direct comparison of the models' primary components. Further, no variability components were incoporated in the original versions of DSTP, the bias model, or the overlap model, thus excluding them in the newer models provides a fairer comparison. Still, it is important to note that the fit quality for each of the models can be improved through the addition of acrosstrial variability in starting point, nondecision time, and drift rate.
Dynamic Attention Models
Each of the dynamic attention models was developed in a manner that reflects how attention interacts with the stimuli to produce the timevarying decision evidence.
Each arrow in the display has a perceptual input value that is either positive or negative, depending on which direction it faces. In this regard, the models will be constrained by the stimuli, since no other parameters will be allowed to vary as a function of the stimulus. The attentional component of the models determine how each item is weighted
20 to determine the total drift rate that drives the decision. Figure 3 shows each of the dynamic models for a 5item display. In the following experiments, the arrows are always symmetric around the target (though the models are easily adjusted for when this does not hold). Because of this symmetry, the stimulus component of the drift rate can be represented by two outer flankers, two inner flankers, and one central target. For the spotlight and dual process models, the contribution of each item to the drift rate is governed by the perceptual input strength of the item times the proportion of attention allocated to it. Both models represent the initial allocation of attention (i.e., at stimulus
onset) as a normal distribution centered over the target, where sda reflects the width of the attentional distribution and a reflects the proportion of the spotlight that falls within the range of each item in the display. A normal distribution was chosen because previous work supports a gradual dropoff in attentional resources rather than a sharp boundary
(e.g., Eriksen & St. James, 1986; Laberge, Brown, Carter, & Bash, 1991). For simplicity, we define each item's region as one unit wide with the target centered at 0. To ensure that the total attentional area always summed to 1, any portion of the attentional distribution that exceeds the outer flanker range is allocated to the outer flanker. This
21 also allows the models to capture potential edge effects, because a wide spotlight results in more attentional capacity allocated to the outer compared to the inner items. The original allocation of attention for the target, inner flankers, and outer flankers, is given as:
∞ 1.5 −.5
aouter t =∫ [0,sda t]; ainner t =∫ [0,sd at]; atarget t =∫ [0,sd at] (1) 1.5 .5 .5
In equation 1, is the density function for a normal distribution with mean 0 and
standard deviation sda. The primary difference between the spotlight and dual process model involves how attention is narrowed to select the target. The dual process model assumes discrete target selection where only the target feeds the response once it is selected. The drift rate, v(t), is a piecewise function given as:
vt =2 a1 pouter 2 a2 pinner a3 ptarget ; if tt s a1 ,a2=0; a3=1 (2)
Where p is the perceptual input strength of each item (target, inner flanker, or outer flanker), which can take a positive or negative value depending on the direction of
22 the arrow, ts is the time taken to select the target and engage process 2, a1 and a2
represent the initial attention allocated to the outer and inner flankers respectively, and a3 is the allocated attention for the target. The absolute value of p was the same for each item in the display. Thus v(t) is the sum of the perceptual strength of each item weighted by the amount of attention allocated to it. All of the items exert some influence on the drift rate until the target has been selected, at which point the influence of the flankers is eliminated and the drift rate is entirely determined by the input strength of the target.
The spotlight model is very similar, except that attentional narrowing is
implemented as a gradual reduction in sda rather than discrete selection of the target. The
rate parameter, rd , determines how quickly sda decreases, and thus how quickly attention narrows in on the target. For the spotlight model, v(t) is given as:
vt =2 pouter a1 t2 pinner a2 t ptarget a3 t (3)
The width of the spotlight, sda(t), decreases over time from the original width at a linear
rate, rd:
sd a t = sd a 0 − r d t (4)
23 Several different functions were explored for the decrease in sda, and the linear function provided a good balance between simplicity and fit quality. Thus as the
spotlight shrinks over time, a1 and a2 approach 0 until attention is solely focused on the
target. A lower bound of .001 was implemented for sda to prevent negative values.
Figure 3. Dynamic models of flanker processing. See text for details.
24 The third model is the DualStage TwoPhase model (DSTP) of Hübner et al.
(2010). The model can be considered a more complex version of the dual process model presented above. In Phase 1 of the model, attention is diffusely distributed and the drift rate is determined by the early stage of stimulus selection, which simply sums the inputs
from the target, tar, and all of the flankers combined, fl. In this phase, the flankers and target have different rates to allow for some degree of initial attentional selectivity,
conceptually similar to sda in the previous models. In contrast to the simpler dual process model presented above, DSTP specifies the process underlying target selection. The model assumes a separate diffusion process that begins concurrently with Phase 1 and selects the target (see Figure 3). This process is governed by three parameters: a drift
rate for the target selection, ss, a boundary separation for the diffusion process, c, and a starting point for evidence accumulation, which is fixed at c/2 for the current study.
Once the stimulus selection process reaches a boundary, either the target or a flanker has been selected and Phase 2 begins. In Phase 2, the diffusion process from Phase 1 is now
driven with a new drift rate based on the selected item, rs2, that is either positive or negative depending on which item was selected as the target. In this framework, some of
25 the trials will terminate in Phase 1 before the target has been selected, leading to a larger influence of the flankers for early trials. For this model, the drift rate that drives the decision, v(t), is given by a piecewise function:
Phase 1: v t = fltar ; Phase 2: vt = ±rs2 (5)
The DSTP model includes the same allornone principle as the simple dual process model, but it allows much more flexibility in how the target is selected and how the drift rate is affected after target selection. By using a separate diffusion process for target selection, the model allows for a flanker to be selected as the target on some trials, and it also allows a range of selection times that are governed by the stimulus selection parameters. Further, by allowing a separate drift rate after stimulus selection, the impact of the selected stimulus is relatively unconstrained. In other words, there is no constraint
on rs2. In contrast, the spotlight and dual process models require that the maximum contribution of the target is equal to the perceptual input of any item in the display (i.e., when attention is completely focused on the target).
26 Bayesian Perceptual Models
The remaining two models were the models of Yu et al. (2009) that are implemented within a Bayesian framework that is similar to a diffusion process. Both models assume that the stimulus produces a noisy series of inputs that are sampled over time by the perceptual system. As more samples are taken, an increasingly accurate representation of the stimulus is formed. Each item in the display produces noisy output,
si, that is represented as a normal distribution with mean = 1 (if left arrow) or = 1 (if
2 right arrow) and variance . This produces a series of perceptual samples, xt, which are corrupted by noise that is independent at each time step (Figure 4, top). This series of inputs is used by the observer to infer the generating stimulus and produce a decision based upon their belief at any given time, and the observer's belief about the identity of the stimulus is updated with each new sample of perceptual input. The compatibility of a trial is denoted by M, where M=1 if the flankers are compatible and M=0 if they are not.
The posterior probability in Equation 6 gives the ideal observer's belief about the target
identity (star) and stimulus compatibility based on the sampled inputs (see Yu et al., 2009
27 for details).
p xt∣star ,M P star , M∣Xt −1 Ps , M∣X = (6) tar t p x s' ,M ' P s ' ,M ' X ' ∑s' m' t∣ tar tar ∣ t−1 2
In this equation, Xt indicates the vector of total inputs accumulated until time t.
To make a perceptual decision based on the posterior probability, the observer computes
the marginal probability of star being a left facing arrow (L) by summing over the uncertainty over compatibility.
P s = L ∣X =.5[ Ps = L , M=C∣X P s = L , M=I∣X ] tar t tar t tar t (7)
In Equation 7, .5 indicates the prior probability of the target facing left, which was 50%
for the first experiment. Because the probabilities have to sum to 1, P(star = R|Xt) is given
as 1P(star = L|Xt). A response is triggered whenever one of the two marginal probabilities exceeds a threshold value, q.
28 Figure 4. Bayesian perceptual models of flanker processing. M = compatibility of stimulus; B = bias for compatibility; a2 = influence of neighboring item; s = stimulus for outer flanker (o), inner flanker (i), and target (t); x = perceptual inputs for outer flanker (o), inner flanker (i), and target (t).
In essence, the observer uses the accumulated perceptual information to perform
29 pattern matching and determine if the overall pattern of activity is more likely from a congruent stimulus with a left target, a congruent stimulus with a right target, an incongruent stimulus with a left target, or an incongruent stimulus with a right target (see
Figure 4, top). This pattern matching is influenced by the prior expectation of compatibility, . To implement the compatibility bias model, it is assumed that observer's are biased to expect compatible inputs ( > .5), meaning that more weight is given to the compatible stimulus patterns. For incongruent trials, a above .5 leads observers to believe that the identity of the target is the same as the flankers, leading to incorrect responses. However, as more perceptual inputs are sampled they begin to outweigh the compatibility bias, leading slower responses to be more accurate. Thus the rise in accuracy for the incongruent condition is due to improved perceptual information rather than attentional focusing.
The overlap model uses the same framework, but assumes that each item in the display partially affects its neighbors due to overlapping spatial representations. For this model, compatibility bias is set at .5, and parameters are added for the contribution of the
neighboring item to the mean and standard deviation of xt. Equation 8 gives the
30 perceptual inputs for the overlap model.
2 2 xouter t ~ N aI outera2 inner , I2 (8)
2 2 xinner t ~ N aI innera2outera2 target , I 2
2 2 xtarget t ~ N aI targeta2 innera2inner , I 2
Thus aI and I determine the relative contribution of the item itself, and a2 and 2 determine the contribution of its neighbor. The perceptual decision is calculated in the same way as for the compatibility bias model. Similar to the bias model, early errors in the incongruent condition arise because early perceptual input is relative poor and influenced by neighboring items, but later responses are less influenced because of more accurate perceptual information has been accumulated.
Although the two perceptual models are presented in a different framework as the diffusion models, they all share the same primary components. Boundary separation is reflected by the response threshold, q, the input of each item is driven by the perceptual
input ( and )weighted by attention (aI and a2), and starting point is reflected by the .5 in Equation 7. Further, both Bayesian models rely on sequential sampling of noisy
31 information until a criteria has been reached, similar to the diffusion models. To allow the model to fit the data and provide a direct comparison with the other models, a scaling parameter was added to translate the model steps into RT, and nondecision time was added to account for processing time outside of the decision process. The scaling parameter was allowed to freely vary and set at a starting value of 2 based on a rough estimate from the data. The original models included a fastguess parameter, , to account for the effects of a response deadline. Since none of the experiments in this study employed such a deadline, was not included in the models.
32 4. Experiments
A series of experiments was conducted to contrast the models and provide empirical validation for their components. The first experiment was a simple flanker task which will demonstrate the relative ability of each model to account for the basic data patterns. Experiment 2 varied the proportion of left and right targets to bias participants' responses, which should be reflected by the starting point (z) of the models. Experiment
3 instructed participants to emphasize speed or accuracy on alternate blocks, which should be reflected by boundary separation (a or q) and nondecision time (Ter).
Experiment 4 manipulated the proportion of congruent and incongruent trials across blocks, which should be reflected by differences in attention. Finally, Experiment 5 used more complex flanker stimuli where the inner and outer flankers could vary independently.
This battery of experimental manipulations will provide strong constraints for contrasting the models. Any successful model must account for the data across each experimental manipulation. To the extent that any model can accommodate the results from each experiment with the appropriate parameter changes, there can be confidence
33 that the extracted parameter values accurately reflect the underlying psychological processes. This latter point is crucial if the models are to be used as tools to interpret data.
Experiment 1
Method
Procedure
Experiment 1 was a standard flanker experiment. Participants were instructed to decide if the central arrow in a display of five arrows faced left or right, and to respond quickly and accurately. They were informed that the surrounding arrows might face the same or opposite direction as the target, but they were supposed to base their responses on the target only. Responses were collected from the keyboard, with the "/" key indicating a rightfacing target, and the "z" key indicating a leftfacing target. No other feedback was provided. The stimuli were presented uncued and remained on screen until a response was given, followed by 350 ms of blank screen until the next trial.
Participants completed 48 practice trials followed by 8 blocks of 96 trials. Each block of trials had an equal number of left and right targets, and congruent and incongruent stimuli. The entire experiment lasted approximately 40 minutes.
34 Stimuli
Each stimulus array contained five arrows (>) displayed vertically, with the central (target) arrow in the center of the screen. Each arrow subtended .7 degree of visual angle, with .4 degree separation between the arrows. For congruent trials, the flankers faced the same direction as the target, whereas for incongruent trials the flankers faced the opposite direction. The flanking arrows were always symmetrically displayed around the target.
Participants
Twentyfive Ohio State undergraduates participated for credit in an introductory psychology course.
Results
Responses shorter than 300 ms or longer than 1500 ms were excluded from analyses (less than .9% of the data). Data were collapsed across trials with right and left facing targets. The accuracy values, mean RTs for correct and error responses, and number of observations for each condition are shown in Table 1. As expected, accuracy was lower and RTs were longer for incongruent compared to congruent trials.
35 Note. Standard deviations are shown in parenthesis. Con = congruent trials; Incon = incongruent trials; N = number of observations; Away = biased away from the correct response; Toward = biased toward the correct response.
For each experiment, the data will be graphically displayed in two ways. Figure 5
(top) shows the quantile probability function (QPF) from Experiment 1. The QPF displays the accuracy values and the correct and error RT distributions simultaneously.
The position on the xaxis indicates the probability of a response, with correct responses on the right and errors on the left. The points in the figure are the quantiles (.1, .3, .5, .7,
.9) of the RT distributions, which provide a summary of the distribution shape. Thus the lowest point for a condition represents the fastest 10% of responses, or leading edge of
36 the distribution, and the highest point represents the slowest 10% of responses, or tail of the distribution. For all of the QPFs presented in this study, data from congruent trials are represented by the column of circles nearest 1 for correct responses and the column nearest 0 for error responses. Due to the low number of errors for congruent trials, only the median quantile is presented in the graphs. Incongruent trials are represented by the columns nearer the center of the graphs. The QPFs show that the incongruent condition led to slower and less accurate responses, and the errors for incongruent trials were substantially faster than correct responses.
The data are also displayed as CAFs with 25ms bins (e.g., 300325ms, Figure 5 bottom). The CAF provides useful information about the relative speed of correct and error responses for each condition, but it does not provide overall accuracy information since the relative proportion of trials for each bin is not displayed. In other words, while early responses for incongruent trials are below chance, overall accuracy for that condition is still around 90% because relatively few trials had terminated by 350 ms.
Accordingly, data for the fastest bins (e.g., 300325ms) are sparse and more variable.
Further, not all participants had responses that terminated in the earliest bins. Together, these aspects of the CAF suggest that it should be used primarily to judge qualitative, rather than quantitative, patterns. It is also important to note that since CAFs display the relative proportions of trials that have terminated at different time points, they only
37 provide an indirect display of the timevarying evidence that is driving the responses.
Figure 5. Observed and predicted quantile probability functions (top) and conditional accuracy functions from Experiment 1 for the dynamic models of attention. Bars and ellipses represent 95% confidence intervals.
Model Fitting
Each model was fit to the data using a SIMPLEX routine for 2 minimization.
Fits were performed to both individual and grouped data with similar results, so results are presented for the averaged data. The data entered into the fitting routine were the
38 accuracy values, RT quantiles (.1, .3, .5, .7, .9) for correct and error responses, and the number of observations for each condition (see Ratcliff & Teurlinckx, 2002; White et al.
2010). For each model, a set of parameter values was chosen, simulated data were generated (see Appendix B), and the predicted quantile proportions were compared against the data to produce a 2 value. This 2 value was then minimized by the
SIMPLEX routine. Each model was simulated several times with different starting values to ensure the results were not due to local minima in the parameter space.
This method of model fitting ensured that the models had to account for the entire data set from the experiment, including the accuracy values and RT distributions for correct and error responses. The 2 values from the fitting routine provide a measure of model fit. However, the 2 value is sensitive to the number of observations. The conditions in Experiment 1 have nearly 10,000 observations each, meaning that even a small misfit from the model would produce a significant 2 value. Further, because group data were fit rather than individuals, the resulting 2 values do not follow a typical
2 distribution (see Ratcliff & Smith, 2004). In light of this, the 2 values are used as a measure of relative fit quality among the models, and augmented with graphical displays to assess potential misfits. Because the 2 measure does not penalize for model complexity and thus does not account for differential complexity among the models, BIC
39 values (Schwarz, 1978) were also calculated.
No parameters were allowed to freely vary between conditions, so the models would need to account for data from the congruent and incongruent conditions with only the perceptual input values differing (which were constrained by the stimuli to be +/ p depending on the direction of the arrow). The starting point was fixed at a/2 (or .5 for the perceptual models) since there were an equal number of right and left facing targets.
The best fitting parameters for each model are shown in Table 2 (ts is given in ms). All three dynamic attention models provided adequate fits, though the spotlight and dual process models provided a numerically superior fits compared to DSTP. In contrast, the
40 fit quality was significantly poorer for the Bayesian models.
Figure 6. Observed and predicted quantile probability functions (top) and conditional accuracy functions (bottom) from Experiment 1 for the Bayesian perceptual models. Ellipses and bars represent 95% confidence intervals.
The model predictions of the dynamic models are shown in Figure 5. The figure shows that all three dynamic models captured the main patterns of data, namely the differences in accuracy and RT distributions between congruent and incongruent conditions. The predictions of each model fall within or near the 95% confidence
41 ellipses, showing good accord with the data. The CAF plots also show that each model captured the main trends in the data, showing belowchance accuracy for the fastest responses and a sharp rise to asymptote for the incongruent conditions. Note that the models are not fit to the CAF but rather the data in the QPFs (along with the number of observations in each condition).
The fit quality was much poorer for the Bayesian perceptual models, particularly the overlap model (see Figure 6). The Bayesian models fail primarily because the mechanism for improving evidence, sampling increasingly accurate perceptual information, is relatively slow compared to the dynamic attention mechanisms in the competing models. If the perceptual models assumed a large influence of the flankers early on (from a strong compatibility bias or large overlap), they capture the early dip in the CAF, but predict incongruent responses that are too inaccurate and slow because the perceptual sampling mechanism takes too long to overcome the early interference.
Conversely, if the models assume a small influence of the flankers early on, the accuracy values are better predicted, but the models fail to produce the belowchance dip in the
CAF, and they fail to predict significantly faster errors than correct responses for incongruent trials.
As Figure 6 shows, the best fitting parameters from both Bayesian models predict
42 very little influence from the flankers. For the compatibility bias model, was estimated at .69, which is slightly greater than the value necessary (.675) to produce a below chance dip in the posterior odds (i.e., decision evidence) for a 5 item display (see Yu et al. 2009 for details). However, as Figure 6 shows, this slight dip in the posterior odds was not sufficient to produce belowchance responses for the incongruent trials. For the
overlap model, the overlap parameter (a2) was relatively low as well, predicting under
15% overlap from the neighboring item that also fails to produce a belowchance dip in the CAF. Both of these models fail to account for the accuracy values and the relative speed of correct and error responses for congruent and incongruent trials.
The results of Experiment 1 show that each of the diffusionbased models captured the behavioral data. Although the BIC values favor the simpler spotlight and dual process models, there was only a small difference in fit quality amongst the three dynamic models that is magnified by the large number of observations, implying that each provides a viable account of flanker processing. The Bayesian perceptual models, on the other hand, failed to capture the trends in the data because the method for improving perceptual information, gathering more samples of noisy input, is too slow to allow a large swing in decision evidence. Yu et al. (2009) did discuss the possibility of adding a dynamic, conflictbased, attentional mechanism to their Bayesian models. This would no doubt improve the fit quality by incorporating a faster mechanism for
43 improving the decision evidence. However, adding such a mechanism renders the overlap and compatibility bias assumptions unnecessary, resulting in model components that are theoretically similar to the dual process models explored in this study.
Accordingly, there is little supporting evidence for the processing components in the
Bayesian models.
Experiment 2
Since neither of the Bayesian perceptual models were able account for the data from the basic flanker task, they are not included in the remainder of the study. For the dynamic attention models, although the quality of fit was slightly superior for the spotlight model, each model captured the main trends in the data, suggesting that each provides an adequate account of the underlying processes. The remainder of the experiments are meant to further contrast these models through experimental manipulations that target one or more of the parameters. In Experiment 2, response bias was manipulated by presenting blocks of trials with different proportions of left and right targets. This manipulation should be reflected by changes in the starting point of the models. This manipulation provides further constraint because a successful model must produce a timevarying drift rate that captures the data when the diffusion process starts
44 near one boundary, in between both boundaries, and near the other boundary.
Method
Procedure
The same stimuli and presentation were employed as in Experiment 1. The only difference was that the proportion of left and right targets was varied between blocks.
Each block consisted of 96 total trials with an equal number of congruent and incongruent trials. After completing a block of practice trials, participants were informed for each subsequent block whether it had mostly left facing targets, mostly rightfacing targets, or an equal number of each. The mostlyleft and mostlyright blocks had 64 trials in the biased direction and 32 in the other. Each participant completed 5 blocks of each type in random order.
Participants
Twentyone participants from the same pool as Experiment 1 contributed data.
Results
All responses shorter than 300 ms or longer than 1500 ms were excluded (less than .9% of the data). For display purposes, the data have been collapsed over left and
45 right trials according to whether the participants were biased toward the response, away from the response, or not biased (equal left and right targets). The behavioral data are shown in Table 1. The speed and proportion of responses increased as participants were biased toward the response, suggesting that the manipulation was effective.
Figure 7. Observed and predicted QPFs from Experiment 2. Ellipses represent 95% confidence intervals.
This is apparent in the QPFs shown in Figure 7. The accuracy values, particularly for the incongruent condition, decreased as the bias shifted from the correct response to the
46 incorrect response. The CAFs in Figure 8 show that the bias manipulation was reflected in the earliest responses, and that biasing participants toward the correct response mitigated, but did not eliminate, the incongruent "dip".
Model Fitting
The models were fit in the same manner as in Experiment 1, except that an
additional parameter was included to account for the differential bias across blocks, z1.
Thus each model was fit simultaneously to all six conditions, with the assumption that z =
a/2 for the unbiased condition, z = a/2 + z1 when biased toward the right response, and z
= a/2 z1 when biased toward the left response. The best fitting parameters from the models are shown in Table 3.
The 2 and BIC values in Table 3 show that the bias manipulation was more effective at differentiating the models compared to the previous experiments. The
47 spotlight model provided the best fit, followed by the dual process model and DSTP.
Each model accounted for the general pattern of decreasing accuracy and speed as the starting point shifted to the incorrect response. However, the spotlight model was better able to account for the accuracy of the incongruent condition across each level of bias.
Figure 8. Observed and predicted CAFs from Experiment 2. Error bars represent 95% confidence intervals.
As Figure 7 shows, both discreteselection models predicted too many errors for the incongruent, biasedaway condition. The reason has to do with the primary
48 difference between the spotlight model and both dual process models, the gradual
narrowing of attention. Because only z1 was allowed to vary across bias conditions, the same attentional narrowing or selection must account for the data when the diffusion process starts near the correct response, midway between each response, and near the incorrect response. Figure 9 demonstrates why this is a bigger problem for the dual process and DSTP models than the spotlight model. The figure shows a theoretical drift rate for the incongruent condition across each level of bias. For this example, suppose that the attention parameters in each model produce the drift rates shown in the figure that provide a good fit for the "biased toward" condition. When participants are in the
"no bias" condition, each model predicts a decrease in accuracy and an increase in RT, consistent with the data. The problem for the dual process and DSDP models comes in the "biased away" condition. Because the drift rates do not improve until the target is selected, many of the processes would terminate at the incorrect boundary before selection occurs, leading the models to predict too many errors in that condition, which can be seen in the bottom panels of Figure 7. Shortening the selection time would improve accuracy for that condition, but the models would consequently overpredict accuracy for the other bias conditions. The fact that the DSTP model fails in the same way as the simpler dual process model suggests that this shortcoming cannot be overcome by using a separate diffusion process that produces a range of selection times.
49 Figure 9. Model behavior across different levels of bias.
In contrast to the dual process and DSTP models, the spotlight model can better account for the data across each level of bias because of the assumption of gradually improving drift rates (see Figure 9). Since the narrowing of attention, and thus the improvement in drift rate, starts concurrently with the decision process, the spotlight model does not predict as sharp a decline in drift rate as the dual process models do, and consequently produces better accord with the data.
50
Experiment 3
The results of Experiment 2 favor the spotlight model with gradual focusing of attention over the two models with discrete target selection. However, since the two discreteselection models still captured the main trends in the data, they will be included for the remaining two experiments. In Experiment 3 speed/accuracy criteria were manipulated across blocks of trials. This should be reflected by differences in boundary separation and nondecision time in the models (e.g., White et al., 2010).
Method
Procedure
The same stimuli and presentation were used as in the previous experiments.
After completing a practice block of 48 trials, participants were informed that for each subsequent block of 96 trials they were to emphasize either speed or accuracy. For speed blocks, participants were told to respond more quickly than they normally would, even if it meant making more errors. For accuracy blocks, they were told to try and minimize their errors, even if that meant responding more slowly. Each participant completed 8 of each block type in random order.
51 Participants
Twentytwo participants contributed data from the same pool as the other experiments.
Results
The same cutoffs were used as before to trim outlier responses (3001500ms).
The percentage of excluded trials was only slightly higher in the speed (1.3%) compared to accuracy blocks (.8%). Using lower cutoffs for the speed condition did not change the overall conclusions, so the standard cutoffs were used. The behavioral data are shown in
Table 4. Responses were faster and less accurate in the speed blocks, indicating that the manipulation was effective. The FCE effect for accuracy was greatly affected by the manipulation, with 10% decrease between speed and accuracy blocks. In contrast, the
FCE in RTs did not vary (73ms vs. 70ms).
52 Note. Standard deviations are shown in parenthesis. Speed = blocks where speed was emphasized; Acc = blocks where accuracy was emphasized; 66% con = blocks with mostly congruent trials; 50% con = blocks with equal congruent and incongruent trials; 33% con = blocks with mostly incongruent trials.
Model Fitting
The models were fit to the data in the same manner as the previous experiments.
To account for the speed/accuracy instructions, boundary separation and nondecision
time were allowed to vary between speed (as, Ters) and accuracy (aa, Tera) blocks. No other parameters were allowed to vary between the blocks. The best fitting parameters are shown in Table 5, and the predicted and observed data are shown in Figures 10 and
53 11.
The results were similar to Experiment 2. Each model captured the main trend of shorter RTs and lower accuracy in speed compared to accuracy blocks, and each model roughly captured the shapes of the RT distributions. Also consistent with Experiment 2, the fit quality was numerically superior for the spotlight model. The poorer fit quality for the dual process and DSTP models stems from the discrete nature of the target selection, consistent with Experiment 2. Both of the discreteselection models predict relatively fast target selection (see Table 5), which avoids the problem of overpredicting errors in the speed blocks that occurred in Experiment 3. However, this fast selection time produces errors that are too fast for the incongruent condition in the speed blocks. This can be seen in Figure 10, where the quantiles for incongruent errors are too fast and condensed for the discreteselection models.
54 Figure 10. Observed and predicted data from Experiment 3. Ellipses represent 95% confidence intervals.
Similar to Experiments 1 and 2, each of the dynamic models captured the main trends from the speed/accuracy manipulation with only a and Ter varying, but the spotlight model produced a superior fit. This was again due to the difference in attentional narrowing, with the data favoring a gradual improvement in decision evidence.
55 Figure 11. Conditional accuracy functions from Experiment 3. Error bars represent 95% confidence intervals.
Experiment 4
In Experiment 4 the proportion of congruent and incongruent trials was manipulated across blocks. Previous research with this type of proportion manipulation has shown that attention is more focused when there are a higher proportion of incongruent trials (e.g., Hübner et al., 2010). Thus this manipulation should be reflected
56 by the attention parameters in the models.
Method
Procedure
The same stimuli and presentation were used as in the previous experiments. In contrast to Experiment 2, the proportion of right and left targets was held equal while the proportion of congruent and incongruent trials was varied across blocks. After a practice block of 48 trials, participants were informed at the beginning of each block of 96 trials whether it had mostly incongruent trials (64/32), mostly congruent trials (64/32), or an equal number of both. Participants completed 5 of each block in random order.
Results
Responses faster than 300ms or slower than 1500ms were excluded (less than
1.1% of the data). The behavioral data are shown in Table 4 and suggest that the proportion manipulation was effective. The FCE was larger when there were fewer incongruent trials in a block, consistent with the claim that attention was less focused in those blocks. This was due almost entirely to improvement for the incongruent condition as the proportion of incongruent trials increased. As Figure 12 shows, the speed and accuracy of incongruent trials increased as the proportion of those trials increased in a
57 block.
Model Fitting
The models were fit to all 6 conditions simultaneously with only the attention parameters allowed to vary. For each model, there could be differences in initial focus
(sda or ufl) or the rate of target selection (rd, ts, ss) to account for the effects of the proportion manipulation. Inspection of the CAFs for each condition (Figure 13) shows that as the proportion of incongruent trials increased, there was an upward shift in the accuracy for the fastest responses on incongruent trials. This is consistent with increased
attentional focus at stimulus onset, which is reflected by smaller sda in the spotlight and
dual process models, and by smaller value of fl in DSTP. If the manipulation affected the rate of target selection, it would be reflected by steeper slopes for the incongruent
CAFs. Accordingly, a separate value of sda or fl was calculated for each block of
58 congruency.
Figure 12. Observed and predicted QPFs from Experiment 4. Ellipses represent 95% confidence intervals.
59 Figure 13. Observed and predicted CAFs from Experiment 4. Error bars represent 95% confidence intervals.
The best fitting parameters are shown in Table 6 and the predicted values are shown in Figures 12 and 13. Each model captured the data well, accounting for the improvement for incongruent trials as the proportion of incongruent trials increased.
Similar to the previous experiments, the fit quality was superior for the spotlight model, but the differences in fit quality among the models were fairly small. In contrast to
Experiments 2 and 3, there were no systematic misses for either of the discreteselection models.
The results of this experiment speak to the issue of parsimony and model
60 flexibility. Each model accounts for the effects of the proportion manipulation with one parameter that affords a simple psychological account. When blocks of trials contain many incongruent stimuli, participants increase their prestimulus attentional focus so as to reduce the effect of the flankers. We can contrast this to Hübner et al. (2010), who performed a similar proportion manipulation and fit a separate DSTP model to each proportion condition, thus allowing each parameter to vary. Their results showed that a
low proportion of incongruent trials led to low early selectivity (i.e., high fl), consistent with the results presented here. In contrast, a high proportion of incongruent trials led to
better spatial filtering (i.e., lower fl), stronger stimulus selection (ss), stronger response
selection after the target had been identified (rs2), and a higher threshold for target selection (c). Their model provided a satisfactory account of the data, but we argue that the resulting interpretation is unnecessarily complex, especially when the data can be accounted for with the simpler assumption that congruency proportion affects the initial focus of attention.
Experiment 5
Application to more complex stimuli
The spotlight model and dual process model are formulated in fairly general
61 terms, which allows them to be readily augmented to account for more complex stimuli.
Thus the models can be tested against flanker stimuli where, for example, the inner and outer flankers are varied independently, or the flankers are asymmetric around the target.
For the spotlight and dual process models, the adjustment is straightforward: all that needs to change is the perceptual input of each item in the display. The adjustment is much less straightforward, however, for DSTP because it implements the target selection process as a standard diffusion process. Take for example the trials in which the stimulus selection process terminates at a flanker instead of the target. For the standard flanker stimuli, like presented in the present study, this does not present a problem because the inner and outer flankers are the same, as are the right and left flankers. However, when this is not the case, DSTP will need to specify the likelihood of choosing the left, right, inner, or outer flankers. This more complex selection scheme would not fit well with the assumed twochoice diffusion process.
Experiment 5 incorporated a more complex stimulus set to further contrast the models. Due to the mimicry between the dual process and DSTP models, and the limitations to applying DSTP to more complex stimuli, only the spotlight and dual process models were tested in Experiment 5. Since the standard flanker conditions did not strongly differentiate between gradual and discrete target selection, more complex stimuli were added. The stimuli are shown in Figure 14. Two new conditions were
62 added, one in which only the inside flankers were incongruent, and one in which only the outside flankers were incongruent. The conditions are labeled according to which flanker is incongruent with the target.
Figure 14. Conditions for Experiment 5.
These two conditions should help contrast the spotlight and dual process models.
For the spotlight model, the decision evidence will briefly decrease in condition 3 when the spotlight has narrowed past the outside flankers but not past the inside flankers. In essence, the benefit from the congruent, outside flankers is no longer present to offset the interference from the incongruent inside flankers. In contrast, the dual process model predicts no change in decision evidence until the target is selected because the attentional narrowing is allornone.
63 Method
Procedure
Experiment 5 used the same procedure as the standard flanker task in Experiment
1. After completing a block of 48 practice trials, participants completed 8 blocks of 96 trials each. Pilot work showed that the two new conditions were not well differentiated unless participants were encouraged to respond quickly. Accordingly, the stimuli were only shown for 120ms before being removed from the screen, and a TOO SLOW message appeared for responses longer than 850 ms.
Participants
Twentyfour participants contributed data from the same pool as the previous experiments.
Results
Responses faster than 300ms or longer than 1000ms were excluded from analysis
(less than .8% of the data). The behavioral data are shown in Table 7. The data show higher accuracy and shorter RTs for condition two compared to condition three, which was consistent with the expectation that the flankers that are nearer the target have a greater influence on the response (e.g., Eriksen & Eriksen, 1974). There was also a
64 difference in error RTs, with longer errors for the inside condition.
Figure 15 shows the QPFs from Experiment 5. The conditions are split into separate graphs for the purpose of clarity. Both of the models captured the main trends in the data, though the predicted error responses were too fast. The behavior the the models is better shown through the CAFs in Figure 16, which show all conditions presented together to better contrast the time course for these conditions. The standard conditions
(congruent, incongruent) showed the same patterns as in the previous experiments. As
Figure 16 shows, there was a small crossover in the CAFs for the two new conditions, with the inside condition starting higher but later dipping below the outside condition.
This difference in initial accuracy is likely due to edge effects (e.g., Merikle, Coltheart, &
Lowe, 1971; Gomez, Ratcliff, & Perea, 2008).
65 Model Fitting
The models were fit to the data in the same manner as Experiment 1, with only the basic model components. The model predictions are shown in Figures 15 and 16, and the best fitting parameters are given in Table 8. Since the models both had five parameters, BIC values are not presented. There was an advantage for the spotlight model over the dual process model, both in terms of fit quality and visual correspondence with the data. The CAFs in Figure 16 show that the dual process model is unable to produce the crossover between conditions two and three because both conditions show the same improvement in decision evidence (i.e., due to target selection). In contrast, the spotlight model better accounts for the data because the drift rate decreases for the inside condition when the spotlight is focused enough to eliminate the impact of the outside flankers.
66 Figure 15. Observed and predicted QPFs from Experiment 5. Ellipses represent 95% confidence intervals.
The bestfitting parameters in Table 8 show that the spotlight model assumed a wide initial spotlight. This captures the early advantage for the inside condition because more attentional weight is given to the congruent edge flankers. However, in the spotlight model, the decision evidence decreases some as attention is narrowed to the incongruent, inner flankers, so responses for the inside condition are still significantly slower and less accurate than for the outside condition. In essence, the outer flankers
67 have a greater influence early in the trial, but the inner flankers have a greater influence later in the trial. This aspect of the spotlight model produces reasonable accord with the data.
Figure 16. Observed and predicted CAFs from Experiment 5.
The dual process model can not account for the data as successfully because of the discrete target selection. The only mechanism in the model that produces change is the target selection process, which does not differentially affect the inside and outside conditions. Thus if the dual process model assumes a large spotlight at stimulus onset to capture the early influence of the outside flankers, it predicts that the inside condition is faster and more accurate than the outside condition, which is not borne out in the data.
68 The parameters in Table 8 show that the dual process model assumed a small role of the outside flankers, which allows the model to produce reasonable predictions for each condition, but fails to produce the crossover in the CAFs.
Experiment 5 provides even further support for a gradual narrowing of attention rather than discrete target selection. However, although the spotlight model provided the best fit to the data, the model still missed some aspects of the data. In particular, both models predict errors that are too fast for the inside and incongruent conditions.
Currently, neither model accounts for the potential role of stimulus conflict or grouping effects, which likely have some influence on perceptual processing and/or attentional narrowing with the more complex stimulus set. Nonetheless, the parsimonious spotlight model captures the primary trends in the data.
69 5. Discussion
Mechanisms of Flanker Processing
The results of the experiments in this study provide several insights into the underlying processes involved in the flanker task. At a broad level, the models that assume dynamic attentional selection greatly outperformed the Bayesian perceptual models. This suggests that bias for compatibility and overlapping spatial representations do not adequately account for the FCE, at least not as they are implemented by Yu et al.
(2009), and that some form of attentional selection or flanker suppression is necessary to account for the time course of the decision evidence in the task.
The Bayesian models fail primarily because the mechanism for improving evidence, sampling increasingly accurate perceptual information, is relatively slow compared to the dynamic attention mechanisms in the competing models. If the perceptual models assumed a large influence of the flankers early on (from a strong compatibility bias or large overlap), they capture the early dip in the CAF, but predict incongruent responses that are too inaccurate and slow because the perceptual sampling mechanism takes too long to overcome the early interference. Conversely, if the models
70 assume a small influence of the flankers early on, the accuracy values are better predicted, but the models fail to produce the belowchance dip in the CAF, and they fail to predict significantly faster errors than correct responses for incongruent trials. Yu et al. (2009) did discuss the possibility of adding a dynamic, conflictbased, attentional mechanism to their Bayesian models. As mentioned above, this would improve the models' ability to capture the relative speed of correct and error responses for incongruent trials by incorporating a faster mechanism for improving the decision evidence. However, if such a mechanisms were added, there would no longer be a need for the overlap and compatibility bias assumptions, and the models would reduce to variants of the dual process model presented in this study.
For the models of dynamic attention, the spotlight model provided the best fits overall, but there was considerable mimicry among the models for the experiments with standard flanker conditions (14). One conclusion from this study is that both discrete and gradual narrowing of attention provide adequate descriptive accounts of the basic flanker data. For the most part, each of the diffusionbased models captured the relevant behavioral data with the appropriate parameter changes, and the difference in fit quality
71 among the models was relatively small. However, manipulating the position of the starting point relative to the decision boundaries (Experiments 2 and 3) provided evidence in favor of a single process model with gradually narrowing attention. Neither the dual process model nor the DSTP model accounted for the data across each level of response bias or speed/accuracy settings as well as the spotlight model. Further, for models that do assume discrete selection of the target, the dual process model developed in this study provides a simpler, more constrained version of DSTP that produced comparable or superior fits. Subsequently, there is little support for a separate target selection diffusion process in standard flanker processing.
The use of more complex stimuli in Experiment 5 did provide differential support for the spotlight model, as the dual process model was unable to account for the qualitative pattern in the CAFs. Combined with the results of the other experiments, the present study suggests that flanker processing involves gradual narrowing of attention that can be approximated by discrete target selection for the basic flanker stimuli.
72 Flanker Models as Tools for Data Analysis
One aim of the present study was to identify a simple model of the flanker task that could be subsequently used as a tool for decomposing behavioral data into relevant psychological constructs. There were two primary criteria that needed to be satisfied for such a model. The model must provide an adequate account of the behavioral data, and each component of the model must reflect the appropriate underlying construct.
Experiments 14 showed that the spotlight model, and to a lesser degree the dual process and DSTP models, captured the data well and provided adequate descriptive accounts of the data. Further, the experimental manipulations were tracked by the appropriate model parameters. Manipulations of response bias were captured by changes in starting point
(Exp. 2), manipulations of speed/accuracy settings were captured by changes in boundary separation and nondecision time (Exp. 3), and manipulations of congruency were captured by changes in attentional parameters (Exp. 4). This series of experimental manipulations provides empirical support for the primary components of the dynamic attention models. Importantly, this also shows that incorporating a timevarying drift rate into the standard diffusion model does not adversely affect the other components of the
73 model.
Based on its performance in the current study, the spotlight model can be used to augment analyses of data from flanker tasks. The task has been discussed primarily in terms of the mechanisms involved in visual/selective attention, but it has also been employed to provide a measure of attentional differences between groups. Heitz and
Engle (2007) showed that higher working memory span was associated with a sharper rise in the CAF for incongruent trials. This would likely be reflected by a higher value of
rd in the spotlight model, which would be consistent with the authors claim that working memory span is correlated with attentional processing. Another study manipulated mood by playing either happy or sad music, and found that the FCE was larger for participants in the positive mood condition (Rowe, Hirsh, Anderson, & Smith, 2007). The authors claimed that positive mood is associated with broader, less restricted attention, thus accounting for the larger flanker interference for happy participants. However, their data also show that happy participants were slightly slower than sad participants overall. Thus positive mood might be associated with a wider boundary separation, in addition to, or in place of, less focused attention. Tieges, Snel, Kok, and Ridderinkhof (2009) used a
74 flanker task to determine if caffeine modulates inhibitory control. They found that participants who were given caffeine were faster overall, but showed no difference in the
FCE compared to controls. The spotlight model could be used in this case to determine whether the speed difference (possibly due to narrower boundary separation or stronger perceptual input) possibly masks any differences in attentional processing.
By decomposing the behavioral data into the relevant psychological constructs, the spotlight model can aide analysis of the flanker task. Recent work has shown that the standard diffusion model produces more direct measures of the underlying constructs than RTs or accuracy values by separating the effects of each component into the relevant parameter (White et al., 2010). This allows researchers to compare processing across groups that have different response styles (e.g., older versus younger adults). Similar advantages can be obtained by using the spotlight model for future analyses of the flanker task.
Due to the significant mimicry between the spotlight and simple dual process model, either can reasonably be used to fit data and provide a descriptive account of the relevant processes for standard flanker conditions (e.g., Experiment 1). However,
75 researchers should be cautious when interpreting the results of the dual process model, particularly since it did not perform as well as the spotlight model. Dual process models allow interpretation in terms of early and late selection, depending on whether or not the target was selected before the response was executed. The original application of DSTP employed this logic to examine whether different manipulations selectively affected early or late selection (Hübner et al., 2010). Since the model assumes that Phase 1 and 2 are separate and driven by distinct processes, this interpretation is viable. In contrast, a spotlight model interpretation renders this distinction meaningless, as early and late selection merely reflect different points on the same function. However, if the model parameters are instead interpreted in terms of the rate or time for target selection, the theoretical interpretations can be consistent between the spotlight and dual process models.
Conflict Monitoring and Adaptivity
The flanker task falls into a broad category of conflict tasks. On incongruent trials, the flanker stimulus presents evidence for both the left and right responses, thus
76 there is conflict at the level of the response. This is conceptually similar to the Stroop task, where an incongruent trial with the word "red" presented in blue print presents evidence for both colors as responses. There is a rich literature focused on how conflict is processed and how it affects responses on subsequent trials. Botvinick and colleagues
(Botvinick, Braver, Barch, Carter, & Cohen, 2001) presented a simple, 3layer neural network of conflict adaptation that assumes that when conflict is detected on a trial, it leads to a boost of attention for the subsequent trial. This accounts for the standard finding of conflict adaptivity, that the FCE is smaller on trials that follow an incongruent versus a congruent trial. Davelaar (2008) took this concept one step further and implemented a neural network model in which conflict was monitored during the course of the trial and used to narrow attention during the trial itself. In this framework, attentional narrowing does not occur unless conflict has been detected. This is consistent with a conservation of effort policy, where visual attention is only engaged when it is needed (see Eriksen & St. James, 1986).
The models presented in the present study implicitly contain a basic assumption that conflict processing does not change any component of the decision process.
77 Attentional selection starts concurrently with the diffusiondecision process, meaning that any withintrial conflict detection must have occured during the nondecisional processing that precedes the accumulation of evidence in the decision process. Further, each of the diffusionbased models employs attentional selection for both congruent and incongruent trials, since attentional narrowing or selection occurred regardless of trial type. For the spotlight and dual process models, it is irrelevant whether or not attention is narrowed on congruent trials. This is because the total input is based on the items in the stimulus and normalized, meaning that processing five congruent arrows provides the same evidence as processing only one. In the DSTP model, the drift rate changes once the target has been selected, even for congruent trials, so the model will behave differently if attentional selection is assumed only to occur when conflict is present. Further work is required to disentangle how appropriate these assumptions are.
Each of the models could be augmented to include assumptions about conflict processing. Withintrial conflict adaptivity could be implemented by assuming that the narrowing of attention does not begin until a criterial amount of conflict has been reached, with the amount of conflict calculated from the stimuli (see Davelaar, 2008).
78 This would lead to a delay before attentional selection begins. Betweentrial conflict adaptivity could be implemented by assuming that attention is more focused following incongruent trials. This is roughly equivalent to assuming some amount of residual attentional focusing that carries over to the subsequent trial. Thus although the models were not designed to handle conflict adaptation, they provide a general framework that can be augmented to account for more complex findings.
There is some debate, however, over the findings of postconflict adaptivity.
Although Botvinick et al. (2001) couched postconflict adaptivity in terms of increased attention, others have argued that the postconflict decrease in FCE results from other causes. Nieuwenhuis and colleagues (Nieuwnhuis, Stins, Posthuma, Polderman,
Boomsma, & de Geus, 2006) found that postconflict adaptation only occurred when the incongruent stimulus was repeated, implying that the effect is more likely from stimulus priming than increased attention. In contrast, Ullsperger, Bylsma, and Botvinick (2005) did find evidence for conflict adaptation after controlling for repetition. The most recent entry into this debate comes from Davelaar and Stevens (2009), who conducted a flanker experiment where sequential effects were controlled so as to produce a sufficient number
79 trials with each combination of stimulus and congruency repetition. They found that the conflict adaptation effect was mostly due to stimulus priming, but there was an effect of the congruency of the previous trial. In addition, others have shown a difference between stimulus conflict, where different items or letters indicate the same response, and response conflict, where the different items indicate opposing responses (see Davelaar,
2008). Notebaert and Verguts (2006) found that adaptation was more sensitive to stimulus conflict than response conflict. There was likely unaccountedfor stimulus conflict in Experiment 5, particularly in the inside condition where every other item faced the opposite direction.
Thus there is still considerable debate over which types of conflict engender which types of adaptation. The spotlight model could potentially inform this debate by allowing the various assumptions to be tested against the data. If conflict adaptation occurs as predicted by Botvinick et al. (2001), data from postconflict trials should be
well fit by either or both a decrease in sda and an increase in rd. If the appearance of conflict adaptation is actually due to stimulus priming from repeat trials (e.g.,
Nieuwenhuis et al., 2006), data from those trials should be well fit by adjusting the
80 perceptual inputs of the items. Lastly, the appearance of conflict adaptation could be the result of other components of the decision process, such as increased response caution or biases in the starting point due to sequential effects. Each of these possibilities has corresponding parameters in the spotlight model, meaning that the particular assumptions can be tested against the behavioral data. Unfortunately, since the present study did not control sequential effects, the data are not ideal for adaptivity analyses.
Conclusion
The present study shows that a spotlight diffusion model that incorporates gradually narrowing attention provides a simple, flexible framework that captures data from the flanker task better than dual process models with discrete attentional selection.
The experimental manipulations suggest that gradual narrowing of attention more accurately captures flanker processing than allornone target selection, though the models with discrete selection did account for the primary patterns in the data when standard congruent and incongruent conditions were used. In contrast, the two versions of Bayesian perceptual models without dynamic attention were unable to fit the basic
81 flanker data.
This study also demonstrates a successful addition of a "frontend" to the diffusion model. The standard diffusion model provides only a descriptive account of the drift rate because there is no theory about how the drift rate is calculated. The spotlight and dualprocess components of the models explored in this study go one step further to elucidate the processes that determine the timevarying decision evidence that drives the diffusion process (see Smith & Ratcliff, 2009 for a related example). Although the time varying drift rate increased the flexibility of the models, the drift rate was still constrained by the stimuli and theoretical assumptions of the models of selective attention. The resulting models provided a significantly better fit to the flanker data than a standard diffusion model, and the components of the models were empirically validated by targeted manipulations that affected the appropriate model parameters. Consequently, these models can be used in future studies of flanker processing to decompose the behavioral data into meaningful psychological constructs.
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87 Appendix A: Mathematical Details of the Diffusion Model
The basic mathematical details of the standard diffusion model are presented in this section. Interested readers should see Ratcliff and Smith (2004) for further details about the diffusion model, Hübner et al. (2010) for further details about DSTP, and Yu et al. (2009) for further details about the bayesian models. Ratcliff (1980) also presents details about implementing timevarying drift rates into the diffusion model.
For the standard diffusion model, the probabilities of choosing response a or b will be given as P(a) and P(b), respectively, with response time T. T is identified with the time at which the diffusion process first exceeds a response threshold, known as the first passage time. Equations A1 and A2 give the first passage time probability density functions for each response (a or b) with drift v(t), starting point x(0) = z, and absorbing boundaries a and b. The timevarying drift rate, v(t), is given in the main text for the spotlight and dual process models.
2 2 2 2 s vta−z vt 2t ∞ −k s t ka−z g t exp x k exp sin a = 2 [ 2 − 2 ] ∑ [ 2 ] [ ] (A1) a−b s 2s k=1 2a−b a−b
2 2 2 2 s vtz−b vt2 t ∞ −k s t k z−b g t exp x k exp sin b = 2 [ 2 − 2 ] ∑ [ 2 ] [ ] (A2) a−b s 2s k=1 2z−b a−b
88 Appendix B: Validation of Simulation Method
Analytical solutions are available for fitting the standard diffusion model with a constant drift rate. However, the timevarying drift rate that was incorporated in the models explored in this study increased the complexity enough that the analytical solutions were intractable. Consequently, the models were simulated to produce predicted values of accuracy and response times. This section presents a validation of this simulation method. The goal was to show that the simulated diffusion model produces data that are consistent with the analytical solutions. To accomplish this, data were generated from the standard diffusion model (with analytical solutions) and then fit by the simulated model to assess parameter recovery. Data were generated for twenty parameter combinations (i.e., subjects), with the values for each combination drawn from uniform distributions centered on the following values (range in parenthesis): a = .12
(.02), Ter = .3 (.04), v = .2 (.03), z = .06 (.01). Then the simulated model was fit to each data set with a fixed value of v to match the generating model. The generated and recovered parameter values are shown in Figure B1. The figure shows that the simulated
89 model recovered the generated values accurately with no apparent biases in parameter recovery. This result verifies that the simulated diffusion model adequately mimics the analytical solutions from the original model.
Figure B1. Generated and recovered parameter values from the standard diffusion model. See text for details.
90