Running head: DISTINGUISHING SERIAL AND PARALLEL PROCESSES. Distinguishing Parallel and Serial Processes Using Variations of the Simultaneous-

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1 Running head: DISTINGUISHING SERIAL AND PARALLEL PROCESSES Distinguishing Parallel and Serial Processes Using Variations of the Simultaneous- Sequential Paradigm Alec Scharff and John Palmer Department of Psychology, University of Washington Last edit: 10/26/09 Intended for Journal of Experimental Psychology: Human Perception and Performance Please do not circulate without permission. 1

2 Abstract Alternative models of divided attention in visual search can be distinguished by the simultaneous-sequential paradigm. This paradigm compares accuracy performance between simultaneous and sequential presentations of otherwise equivalent stimuli. When processing capacity is limited, performance is better in the sequential presentation condition. When processing capacity is unlimited, performance in the two conditions is equivalent. Thus, the method can identify cases of unlimited-capacity, parallel processes. Our goal is to develop variations of this method to distinguish between other alternatives, including various limited-capacity parallel models and serial models. We describe one variant that can distinguish a fixed-capacity process and another that can distinguish a serial process. These methods are applied to two test cases: contrast increment detection and semantic word categorization. The results indicate that contrast increment detection is an unlimited-capacity process, while semantic word categorization is a serial process. These findings are consistent with previous findings using other methods that show that simple features are processed in parallel and without capacity limitations, but that some word judgments require serial processing. 2

3 Our goal is to characterize how divided attention affects visual perception. As a starting point, we assume that that the effect of divided attention may depend on what kind of information an observer is trying to extract. For example, try to read all the sentences on this page at the same time. You may find the idea of doing so is ridiculous. It seems impossible to extract meaning from many sentences at once. In contrast, imagine you are driving on a highway and glance at the dashboard to determine whether any warning lights are lit. This perceptual task, which amounts to extracting luminance information from multiple locations, is easily within our capabilities. Reading sentences and checking warning lights both depend on visual perception, but reading relies on more elaborate interpretation of the physical stimulus than does checking warning lights. It is reasonable that there is a strict limitation on the number of words that can be processed at once, but not on the number of warning lights. The contrast between these two tasks illustrates our question: how does divided attention limit processing, and under what conditions? In the present study, we test the cases of semantic word categorization and contrast increment detection, which are related to the two examples given above. Models of Divided Attention Properties Considered in this Study. We consider two major properties that distinguish models of divided attention: serial-parallel architecture and capacity limitations. Serial-parallel architecture addresses whether perception is achieved by processing items one-at-a-time or many-at-a-time. Capacity limitations describe defined as the relationship between the number of items being processed and the efficiency of processing for each item (Broadbent, 1957; 3

4 Kahneman, 1973; Townsend, 1974; Townsend & Ashby, 1983). If efficiency of processing is independent of the number items, capacity is unlimited. If efficiency declines as a function of the number of items, capacity is limited. There are other relevant properties that can define a model of divided attention, and correspondingly many more potential models than the three under consideration here. However, our goal is to distinguish between serial and parallel processes in perception. In doing so, we also consider capacity limitations because serial and parallel models cannot otherwise be distinguished. Models Considered in this Study In the present study, we consider three models defined by these properties: the standard serial model; the unlimited-capacity, parallel model; and the fixed-capacity, parallel model. The standard serial model and unlimited-capacity, parallel model are commonly discussed prototypes of divided attention (e.g., Sternberg, 1969; Gardner, 1973). The fixed-capacity, parallel model describes a special case of limited-capacity, parallel processing. It is a particularly interesting special case because it often mimics the predictions of a serial model. The fact that fixed-capacity, parallel models and serial models often make similar predictions can make serial and parallel processes difficult to distinguish experimentally. In Appendix A, we derive the predictions of each model. Unlimited-capacity, parallel model. The unlimited-capacity, parallel model describes a system with no capacity limit. Multiple stimuli can be processed concurrently and the processing of each one is independent of the others. In other words, the 4

5 processing of each individual stimulus proceeds identically no matter how many stimuli are processed at once. There is ample evidence that perception of simple features under ideal conditions has unlimited-capacity, parallel processing (e.g., Gardner, 1973; Palmer, 1994; Dosher et al, 2004; Thornton & Gilden, 2007). In ideal conditions, stimuli are wellspaced to prevent crowding, are isoeccentric, and there are no configural cues across stimuli. Standard serial model. A serial system processes one stimulus at a time. In order to perceive multiple items, a serial perceptual system must process them in sequence. In the standard serial model, each stimulus is processed independently, so that processing of any one stimulus is not affected by the number of other stimuli scheduled for processing. The clearest examples of serial processing exist in memory retrieval for order information (e.g, Dosher & McElree, 1993) and response selection (e.g, the psychological refractory period, see Pashler, 1998). Fixed-capacity, parallel model. A fixed-capacity, parallel system processes multiple stimuli simultaneously, but the total information processing efficiency is fixed. If a fixed-capacity, parallel system can process a single stimulus at rate r, then the average rate of processing for each of n stimuli being processed in parallel is r / n. One instantiation of a fixed-capacity, parallel model is the sample-size model (Shaw, 1980; Palmer, 1990). This model assumes that the observer collects discrete samples of information at a fixed rate. The observer allocates the samples to different objects in 5

6 view. The number of samples devoted to a given stimulus determines the quality of processing for that stimulus. Distinguishing Among the Models Several experimental approaches exist for distinguishing among these models, as reviewed in the discussion section. Here we focus on the simultaneous-sequential paradigm, which can specifically rule out the unlimited-capacity, parallel model for a given task, but cannot rule out other models. To establish serial processing, we develop two variations on this paradigm that can rule out other alternative models. Together, the three variants on the simultaneous-sequential paradigm constitute a method for distinguishing between serial and parallel processes. Each model makes a distinct pattern of predictions across the three proposed experiments (see Table 1). The Simultaneous-Sequential Paradigm The simultaneous-sequential paradigm was first developed by Eriksen and Spencer (1969) and has been used to address a variety of questions about capacity limitations in visual perception (Shiffrin and Gardner, 1972; Hoffman, 1978; Duncan, 1980; Prinzmetal & Banks, 1983; Fisher, 1984; Kleiss & Lane, 1986; Harris, Pashler, & Coburn, 2004; Huang & Pashler, 2005). The paradigm distinguishes between unlimitedcapacity and limited-capacity processing. In previous applications, it has most often been applied to visual searches through displays of alphanumeric characters. In a simultaneous-sequential experiment, observers make judgments about briefly presented stimuli. The stimuli are presented in two different conditions: simultaneous and 6

7 sequential. In the simultaneous condition, multiple stimuli are presented concurrently. In the sequential condition, sub-sets of the display are presented one-at-a-time. The simultaneous and sequential displays used in the present study are illustrated in Figure 1. Panel A depicts the simultaneous condition, in which four stimuli are briefly displayed. Panel B shows the sequential condition, in which two stimuli are displayed in one frame, and then the remaining two stimuli are displayed in a second frame. Predictions. All limited-capacity models predict better performance in the sequential condition than in the simultaneous condition. When only a limited amount of information can be processed within a given time window, it is beneficial to have the stimuli displayed at different times. The predicted magnitude of the sequential-condition advantage depends on specific assumptions (see Appendix A). In contrast, the unlimited-capacity, parallel model predicts that performance must be equivalent between the simultaneous and sequential conditions. Under an unlimitedcapacity, parallel model, performance is solely determined by the display duration of each stimulus. Stimuli do not compete for processing. Thus, the simultaneous-sequential manipulation has no effect on performance. Comparison with Set-Size Experiments A commonly used method to address questions of processing architecture and capacity is to manipulate display set-size in visual search experiments and measure the effect on response time (e.g., Neisser, 1967; Treisman & Gelade, 1980; Wolfe, 1998). In most response time, set-size experiments, the observer s task is to indicate whether or not 7

8 a target is present in a display of multiple distractors. The mean response time is measured for several different display set sizes. Generally, the mean response time increases as a function of set size, but the magnitude of this increase varies with the task. In visual search literature, the emphasis is on identifying which tasks yield small set-size effects and which yield large set-size effects. A crude summary of this body of research is that set-size effects are small for easy feature searches and some letter tasks, while set-size effects are large for difficult feature searches and semantic word tasks (e.g., Karlin & Bower, 1976; Treisman & Gormican, 1988). In the early analysis of these experiments, the magnitude of set-size effects were interpreted as evidence for serial or parallel processing. Small set-size effects are consistent with parallel processing, while large set-effects are consistent with serial processing. It is now clear that this analysis is problematic: In fact, any set-size effect can be consistent with either parallel or serial processing. The Interpretation of the Response-Time, Set-Size Experiment The traditional interpretation of the response-time, set-size experiment fails when one considers limited-capacity, parallel models. Townsend and colleagues proposed such models and demonstrated that they can generate any magnitude of set-size effect (e.g., Townsend, 1971, 1974, 1990, Townsend & Ashby 1983; Townsend & Wenger, 2004). Subsequently, many have reported search data that is consistent with limited-capacity, parallel processing (van der Heijden, 1975; Shaw & Shaw, 1977; Dosher, Han & Lu, 8

9 2004, Thornton & Gilden, 2007). Thus, large set-size effects do not necessarily indicate serial processing. A second problem for the traditional analysis of set-size experiments is that it ignores the possibility of error: Observers can mistake a distractor for a target, or vice versa. This issue is most intuitively understood when one considers accuracy performance for search tasks. Imagine that in a given search task, an observer can correctly identify a single stimulus, target or distractor, with 95% accuracy. At set-size 1, accuracy is 95%. At set-size 10, the observer correctly identifies all 10 stimuli only about 60% (.95^10) of the time (Shaw, 1980; Palmer, 1990, 1994, 1995; Palmer, et al, 2000). When errors are possible, set-size inevitably determines accuracy, even under unlimitedcapacity, parallel models. Palmer and McLean (1995) generalized this analysis to response time, set-size search experiments. In short, as the set-size increases, observers avoid making more errors and increase their response time. Thus, if errors are considered in the analysis, large-set size effects do not to rule out unlimited-capacity parallel processing. When capacity limitations and error are taken into account, it is clear that both parallel and serial models can generate any magnitude of set-size effect. Thus, the magnitude of the response-time, set-size effect cannot distinguish serial and parallel processes. Direct Comparison of Simultaneous-Sequential and Set-Size Experiments Huang and Pashler (2005) directly compared the simultaneous-sequential paradigm with the response time, set-size paradigm in three different difficult search 9

10 tasks. The tasks were a simple feature search (detect the presence of a slightly smaller square among other squares), a feature conjunction search (detect a large, vertical rectangle among large, horizontal; small, horizontal; and small, vertical rectangles), and a spatial configuration task (detect a rotated T among rotated L distractors). In the set-size experiments, all three types of search tasks produced substantial set-size effects (about 40 ms/item). In the simultaneous-sequential experiment, however, only the spatial configuration search showed an advantage in the sequential condition. This study provides an empirical example that response time, set-size effects do not indicate the capacity limits revealed by the simultaneous-sequential paradigm. Advantages and Disadvantages of the Simultaneous-Sequential Paradigm The simultaneous-sequential paradigm avoids the problem of error in visual search. The problem of error arises whenever an experimenter makes comparisons between tasks with different numbers of stimuli. In the simultaneous-sequential experiment, a comparison is made between two conditions with the same number of stimuli, so error is not a concern. The simultaneous-sequential paradigm tests for the presence of capacity limits. These limits can arise from either a serial process or a limited-capacity, parallel process. The simultaneous-sequential paradigm is limited in that it cannot distinguish between serial models and limited-capacity, parallel models. In the present study, we used variations on the simultaneous-sequential paradigm to order to distinguish among a greater variety of alternative attention models. These variations are similar to ones introduced by Townsend (1990, p. 50) and discussed more 10

11 generally in Sagi and Julesz (1985, p. 147). In keeping with the spirit of the original paradigm, the two variations make comparisons between two display conditions. The set size does not vary between conditions. In each experiment, each model predicts either equivalent performance between two conditions, or it predicts a non-equivalence in a specified direction. Search Tasks Used in this Study In this study, we assessed two different perceptual tasks. In the first task, observers indicated the location of a higher-contrast disc among lower-contrast discs. We refer to this task as the contrast increment detection task. Figure 3 shows a sample display: four luminance discs are presented, embedded in dynamic random visual noise. The observer s task is to indicate the location of the target disc, which has a higher contrast to the background than the other discs. There is considerable evidence that such a simple feature discrimination has unlimited-capacity, parallel processing (e.g., Palmer, 1995; Palmer, et al, 2000; Huang & Pashler, 2005; Davis, et al, 2006; but see Posner, 1980). In the second task, observers locate a word from a specified semantic category. For example, if the target category is animals, the target word might be dog and distractors might be car, belt, and poet. We refer to this task as the word categorization task. Figure 4 shows a sample display from this task. We chose this task because it relies on word reading, a process that many argue is serial (Karlin & Bower, 1976; Starr & Rayner, 2001; but see Brown, Gore and Carr, 2002). Outline of this Study 11

12 We employed a property-testing approach to distinguish the models, in which each experiment has the potential to reject some, but not all of the models. Across the three experiments in this study, the pattern of predicted results is distinct for each of the three models we consider. Table 1 demonstrates this pattern of predicted results. The models we consider are represented in the rows of the table. In addition, the last row shows the predictions of a limited-capacity, parallel model with a non-fixed capacity. The columns represent the three experimental comparisons conducted in this study. Each cell gives the predicted outcome of a given model for a given experiment. For example, the top-left cell indicates that the unlimited-capacity, parallel model predicts equivalent performance in the simultaneous and sequential conditions. Accepting the predicted equivalences as evidence requires acceptance of the null hypothesis. In contrast, the predicted nonequivalences represent rejections of the null hypothesis generally agreed to be a stronger form of evidence. The three comparisons are the simultaneous-sequential comparison and two new variations: the repeated-sequential comparison and the repeated-new comparison. The first two comparisons were conducted in Experiment 1. The repeated-new comparison was conducted in Experiment 2. Experiment 1 This experiment includes the simultaneous-sequential and repeated-sequential comparisons for both the contrast increment detection and word categorization tasks. The simultaneous-sequential comparison can distinguish between unlimited-capacity, parallel 12

13 models and all other alternatives described here. The repeated-sequential comparison can distinguish models that have a fixed-capacity and alternatives models with a larger or smaller capacity limit. Fixed-capacity models extract a constant amount of information over time. Both the standard serial model and the fixed-capacity, parallel model have a fixed capacity. The predictions for the fixed-capacity model in the repeated condition depend on assumptions about memory, as discussed below. Predictions We derive the formal predictions of the models in Appendix A. The predictions of each model for Experiment 1 are illustrated in Figures 2 and 5. In each panel, a grey bar represents the presence of a stimulus. Stimuli can appear in one of four locations, indicated by the numbers 1 through 4 arranged vertically. Stimuli can appear during the first and/or the second frame, with time progressing from left to right. The black arrows overlaying the grey bars represent the hypothesized course of processing for each condition and model. A solid black line represents a stimulus being processed to the fullest extent possible, as if it were the only stimulus present. Dashed lines indicate less complete processing. The coarser dashes indicate slower rates of processing. Simultaneous-sequential comparison. The simultaneous-sequential comparison can distinguish the unlimited-capacity, parallel model from other models. The two conditions being compared are illustrated in Panels A and B of Figures 1. In the simultaneous condition, four stimuli are briefly displayed at locations corresponding to four corners of a square. In the sequential condition, the presentation of stimuli was 13

14 divided across two frames. After each trial, the observer indicates which of the four locations held the stimulus. All stimuli are presented for 100 ms. The unlimited-capacity, parallel model predicts equivalent performance in the simultaneous and sequential conditions. The standard serial model and the fixed-capacity parallel model both predict superior performance in the sequential condition. In Figure 2, separate columns of panels show the predictions of each model. The unlimited-capacity, parallel model posits that each stimulus is processed at a constant rate regardless of the number of stimuli present. Thus, the extent of processing is determined solely by the stimulus duration. This fact is indicated in Panels A and B of Figure 2 by the solid black line overlaying each grey bar. In both the simultaneous and sequential conditions, each stimulus appears for a single frame and each stimulus is processed fully for the duration of that frame. Thus, the unlimited-capacity, parallel model predicts that performance in two conditions is equivalent. The standard serial model posits that stimuli are processed one-at-a-time. In order to process all of the stimuli, an attention mechanism switches processing from one stimulus to the next during the stimulus display. This switching of attention is indicated in Panels C and D of Figure 2 by the sequence of short, black arrows, representing the shorter amounts of time spent processing each stimulus. As indicated in the Figure, an observer can process twice as many stimuli in the sequential display as in the simultaneous display. The processing sequence need not occur in the manner shown in the figure: an observer could spend more or less time processing any given stimulus. Regardless, the average extent of processing for each stimulus is greater in the sequential 14

15 condition than in the simultaneous condition. For this reason, the standard serial model predicts superior performance in the sequential condition. The fixed-capacity, parallel model posits that processing capacity is divided between available stimuli. The more stimuli are concurrently displayed, the slower the rate of processing. This is indicated in Panels E and F of Figure 2 by the dashed lines indicating the processing of each stimulus. The dashes are coarser in the simultaneous condition than in the sequential condition, reflecting slower processing in the simultaneous condition, where the fixed capacity is split between four stimuli. In the sequential condition, capacity is split between only two stimuli at a time, so that each is processed faster. Each stimulus receives more processing in the sequential condition, so the fixed-capacity, parallel model predicts superior performance in the sequential condition. Sequential-repeated comparison. The simultaneous-repeated comparison can distinguish between fixed-capacity models and all other models. When capacity is fixed, the extent of processing of each stimulus decreases proportionally as more stimuli are added to the display. The two conditions in this comparison are shown in Panels B and C of Figure 1. In a sequential condition, the presentation of stimuli was divided across two time intervals. In a repeated condition, all four stimuli were shown in each of two frames. The repeated condition is designed to double the duration of the simultaneous display, both physically and in memory. The sequential-repeated experiment is similar in spirit to a paradigm employed by Townsend and colleagues (1990, p. 50). They compared a simultaneous display, with n 15

16 stimuli presented for time t, with a sequential display in which each of n stimuli was presented for time t / n. Similar to our sequential-repeated comparison, this experiment creates an equality prediction for the standard serial model. The sequential-repeated comparison can reject both the unlimited-capacity, parallel models and all models that have a limited capacity that is either more or less limiting than is specified by a fixed capacity. Thus, this new comparison allows one to distinguish between fixed-capacity and other limited-capacity models. The unlimited-capacity, parallel model predicts superior performance in the repeated condition. The standard serial model and the fixed-capacity parallel model both predict equivalent performance in the sequential and repeated conditions condition. Figure 5 shows the predictions of each model for this comparison. The unlimited-capacity, parallel model posits that the extent of processing is determined solely by the stimulus duration. In the repeated condition, the stimulus duration (two frames) is twice the stimulus duration in the sequential condition (one frame). Thus, each stimulus is processed to a greater extent in the repeated condition than the sequential condition. This is illustrated in Panels A and B of Figure 6. The unlimitedcapacity, parallel model predicts superior performance in the repeated condition. The standard serial model posits that stimuli are processed one-at-a-time. As illustrated in Panels C and D of Figure 5, the repeated presentation yields no advantage over the sequential presentation when the observer can only attend one stimulus at a time. The switching of processes need not occur in the manner shown in the figure, the observer might linger longer on one stimulus than another, or attempt to rapidly attend to all four stimuli in each frame of the repeated condition. Regardless, the average degree of 16

17 processing for each stimulus is equivalent between the two conditions. Thus, the standard serial model predicts equivalent performance in the repeated and sequential conditions. The fixed-capacity, parallel model posits that processing capacity is divided between available stimuli. Panels E and F of Figure 5 illustrate how each stimulus is processed more slowly in the repeated condition than in the sequential condition. Counteracting this effect, each stimulus is displayed twice in the repeated display, compared to just once in the sequential display. Under the fixed-capacity, parallel model, the disadvantage of dividing capacity between multiple stimuli cancels out the advantage of multiple presentations. Thus, the fixed-capacity, parallel model predicts equivalent performance in each condition. This cancellation effect is not predicted by other limited-capacity, parallel models. For example, under some limited-capacity, parallel models, the rate of processing for each stimulus is less than r but greater than r / n. Such models do not predict that the effect of dividing attention and the effect of repeated presentations cancel out. Thus, these models do not predict equivalent performance in the repeated and sequential conditions. Memory assumptions for sequential-repeated predictions. In Appendix A, we describe the memory assumptions necessary to make the above predictions. As stated thus far, all of the predictions are consistent with an ideal memory integration model, in which information about stimuli are combined across the two repeated displays. We also consider a target-only memory assumption, in which observers only remember the most likely target from each display. This model is consistent with findings that observers 17

18 maintain memory for previously scanned locations, but do not integrate information across displays in an ideal fashion (Horowitz & Wolfe, 1998; Shore & Klein, 2000; Beck, Peterson, & Vomela, 2006; Beck et al, 2006). Under this model, the fixed-capacity, parallel model predicts an advantage for the sequential condition over the repeated, rather than equivalency. The predictions of the unlimited-capacity, parallel model and the standard serial model are unchanged. In summary, all but one of prediction is robust across the two memory assumptions. Methods Observers. Six observers participated in each experiment. They were either volunteers from within the laboratory or were paid $15 per hour. All observers had normal or corrected-to-normal acuity and had previous experience with psychophysical tasks. One of the authors (AS) participated in the experiments. Apparatus. The stimuli were displayed on a flat-screen cathode ray tube monitor (19-in View Sonic PF790) controlled by a Macintosh G4 (733 MHz, Mac OS 9.2) with an NVIDEA GeForce2 graphics card (832 by 624 pixels, viewing distance of 60 cm, subtending 32 by 24 with 25.5 pixel/deg at screen center; refresh rate of 74.5 Hz). The monitor had a peak luminance of 119 cd/m2, and a black level of 4.1 cd/m 2, most of which was due to room illumination. Stimuli were generated using the Psychophysics Toolbox Version 2.44 (Brainard, 1997; Pelli, 1997) for MATLAB (Version 5.2.1, Mathworks, MA). Observers were seated in an adjustable height chair in front of the display and used a chin rest to maintain a constant the distance from the monitor. 18

19 Procedure. Stimuli were displayed in three primary conditions: a simultaneous condition, a sequential condition, and a repeated condition. Each of the three conditions is shown schematically in Figure 1. In the simultaneous condition, four stimuli appeared in a single 100 ms frame. In the sequential condition, four stimuli were shown two-at-atime, in separate frames separated by an interval of 1000 ms. In the repeated condition, four stimuli appear twice, in frames separated by an interval of 1000 ms. On all trials, observers searched for a single target among three distractors. Stimuli appeared at four corners of an imaginary square surrounding a small fixation cross. After viewing each presentation sequence, the observer indicated the location of the target by pressing one of four keys corresponding to the four stimulus locations: 4, 5, 1, or 2 on the numeric keypad. In the sequential condition, two stimuli shown in the same frame always appeared at opposite corners of the square. The three primary conditions and target locations were counterbalanced. In the sequential condition, the pair of locations shown in the first frame was randomized. In the word categorization task, both target words and word categories were counterbalanced with conditions so that all categories and all target words occurred equally often in each condition. Sessions for the contrast increment and word categorization tasks were conducted separately. The display conditions were randomly ordered within a session. For each experimental task, observers trained by completing at least two practice sessions. After practicing, observers completed 10 sessions in each task, for a total of 960 trials in each display condition for each task. 19

20 Stimuli. The contrast increments were discs that were 0.5 in diameter. They all had a higher luminance than the surround. Distractors had a contrast of 20% and the target contrast was adjusted for each observer in order to achieve between 75% and 85% correct responses. The lowest target contrast used was 28% and the highest was 35%. Stimuli appeared an average of 6 away from fixation, and were independently jittered up to 1.5 horizontally and vertically. The word stimuli were short, high frequency words. Preceding each trial, a target category appeared on screen for 0.5 s. The target was a word chosen from the target categories, while distractors were random words from three distinct non-target categories. The words used in the experiment are listed in Appendix B. There were six word categories, each containing eight words. Each category had 2 three-letter words, 3 fourletter words, and 3 fice-letter words. Categories were roughly equated for average word frequency (Kucera & Francis, 1967). Three reviewers independently checked the word lists to ensure that each word fit its category unambiguously and did not fit any other categories. The word stimuli appeared 2 away from fixation and were not jittered. Words appeared in monospaced Courier 18-point font. The contrast of the stimuli was between 40% and 50%, adjusted for each observer. Noise. In both tasks, each stimulus was centered in a circular dynamic noise field. The noise field subtended 2.75 of visual angle. Each pixel of the noise varied in luminance according to a Gaussian function, with a mean luminance equal to the grey background and a standard deviation of 35% contrast. The noise changed every frame (75 20

21 Hz). We suggest that these noise fields have a similar purpose to using post-masks in making performance more sensitive to the duration of the stimulus. Results Simultaneous-sequential comparison. The results of the simultaneous-sequential comparison are plotted in Figure 7. The percentage of correct responses is plotted on the ordinate and the simultaneous and sequential conditions are plotted on the abscissa. For the contrast increment task, performance was indistinguishable between the simultaneous and sequential conditions. On average, the difference between the sequential condition and the simultaneous condition was -2% ± 1% (t(5) = 1.48, p >.1). Individually, only one of the six observers deviated in showing a reliable advantage in the simultaneous condition (SY: p <.01, all other observers: p >.1). Equivalent performance between the simultaneous and sequential conditions is consistent with the unlimitedcapacity, parallel model. It is not consistent with either the standard serial model or the fixed-capacity, parallel model. In the word categorization task, the difference between the sequential condition and the simultaneous condition was 15% ± 1%. This result was reliable (t(5) = 23.47, p <.0001) and all six observers showed this reliable difference. This inequality between conditions is consistent with standard serial and fixed-capacity, parallel processing. It is not consistent with unlimited-capacity, parallel processing. 21

22 Sequential-repeated comparison. The results of the sequential-repeated comparison are plotted in Figure 8. The percentage of correct responses is plotted on the ordinate and the sequential and repeated conditions are plotted on the abscissa. In the contrast increment task, the difference in performance between repeated and sequential conditions was 10% ± 1%, an advantage for the repeated condition. This difference was reliable (t(5) = 6.86, p =.001) and all six observers showed performance consistent with this overall pattern. This result is consistent with the unlimited-capacity, parallel model. It is not consistent with either the standard serial model or the fixedcapacity, parallel model. In the word categorization task, the difference in performance between repeated and the sequential conditions was 3% ± 1%. While this difference is reliable (t(5) = 4.36, p <.01), it is smaller than observed for contrast increments (3% vs. 10%). Individually, only two of the six observers showed reliable differences between conditions (p <.01 for each). The four other observers showed no reliable difference between the conditions (p >.1) While this result is not strictly consistent with the standard serial model or the fixed-capacity, parallel model, it does come relatively close to fulfilling the predicted equivalence property. Compared to the magnitude of other reliable effects found in this study, the advantage of the repeated condition over the sequential condition is modest. Secondary comparison: number of presentations. We assessed the effect of stimulus duration with a comparison of the simultaneous and repeated simultaneous conditions for each experiment. Compared to the simultaneous condition, the repeated presentation doubles both the physical display duration and the presumed persistence in 22

23 memory. This comparison provides a useful measurement of the potential sensitivity of the experiment: if doubling the display duration does not reliably improve performance, then the simultaneous-sequential comparison is not expected to yield reliable differences under any processing model. For both tasks, observers performed better in the repeated condition than in the simultaneous condition. For the contrast increment task, the difference in performance was 8% ± 2%, a reliable difference (t(5) = 4.26, p <.01). In the word categorization task, performance was better by 20% ± 0.5% in the repeated condition, also a reliable difference (t(5) = 40.30, p <.0001). For both tasks, this comparison demonstrates a reliable improvement in performance with an increase in the duration of the stimulus display. This control demonstrates that the experiment is sensitive to potential capacity limitations. Thus, one can rule out overall insensitivity for the comparisons with equivalent performance. Secondary comparison: first versus second frame. This comparison tested the assumption that processing is equivalent for stimuli in the first and second frames of the sequential condition. Performance was compared for trials in which the first frame contained the target and trials in which the second frame contained the target. A systematic difference in performance between the two frames indicates that sequential presentation introduces a change in the nature of processing other than the intended manipulation of concurrent processing load. In both tasks, there was no effect of the frame in which the target appeared. In the contrast increment task, the difference between the frames was 0.1% ± 1% (t(5) = 0.33, p >.7). In the word categorization task, the 23

24 difference between frames was 0% ± 2% (t(5) =.08, p >.8). Thus, there is no evidence of differential processing for stimuli in the 1 st and 2 nd frames. Secondary comparison: target location. We conducted an analysis of the effect of target location in the simultaneous condition. In the contrast increment task, targets to the left of fixation were correctly identified 6% ± 8% more often than targets to the right of fixation, which was not a reliable difference (t(11) = 0.75, p >.4). Targets above fixation were identified 2% ± 2% more often than targets below fixation. This difference was also not reliable (t(11) = 0.75, p >.4). In the word categorization task, targets appearing to the right of fixation were identified 12% ± 12% more often than targets left of fixation. This difference was not reliable (t(11) = 1.01, p >.3). There was a reliable benefit for targets appearing above fixation rather than below it: targets above fixation were correctly localized 15% ± 5% more often than targets below fixation, a reliable advantage (t(11) = 3.20, p <.01). It is interesting to consider whether location biases are consistent with various models of divided attention. If the unlimited-capacity, parallel processing model underlies processing, an ideal observer would not bias responses towards any location: equivalent information is available for all stimuli. This is because we have matched these locations on eccentricity and hence discriminability. If capacity is limited, preferentially allocating resources to one location or another might affect expected performance. For example, in the standard serial model, the observer must process the stimuli in a certain order and not complete processes for the stimuli with the lowest priority. Thus, location effects also suggest limited capacity processing. 24

25 Discussion of Experiment 1 Contrast increment task. For the simultaneous and sequential conditions, there was no difference in performance. For the sequential and repeated conditions, there was a reliable advantage for the repeated condition. Both results are consistent with the predictions of the unlimited-capacity, parallel and are inconsistent with the predictions of the other models we consider. Word categorization task. In the simultaneous and sequential conditions, performance was reliably better in the sequential condition. This result is inconsistent with the prediction of the unlimited-capacity, parallel model. It is consistent with both the standard serial model and the fixed-capacity, parallel model. In the repeated and sequential conditions, there was a reliable but relatively small advantage for the repeated condition. Strictly interpreted, this result is not compatible with fixed-capacity processing. However, the small size of the effect sets it apart from larger effects we found in the study. The effect does not approach the magnitude of the repeated-sequential effect found in the contrast increment task the advantage of the repeated condition in the word categorization task was only 3%, which is modest in comparison to the 10% advantage found in the contrast increment task. In addition, this effect was only reliable for two out of six observers, whereas all other effects were replicated by at least five observers. We consider the results of Experiment 1 for the word categorization task to be roughly 25

26 consistent with the predictions of the standard serial model and the fixed-capacity, parallel model. We leave finer considerations of limited-capacity models to the future 1. Advantage of the simultaneous-sequential-repeated paradigm. The addition of the repeated-displays condition to the standard simultaneous-sequential design yields an important advantage: it allows the experimenter to assess the sensitivity of the experiment. All theories predict an improvement in performance between the simultaneous and repeated conditions, because the repeated condition doubles the physical and memorial availability of all stimuli. If no reliable difference is observed between the simultaneous and repeated conditions, the experiment is insensitive to any potential simultaneous-sequential difference. If a reliable simultaneous-repeated difference is found, sequential performance is predicted to be equivalent to the simultaneous performance if processing is unlimited-parallel and parallel, or equivalent to the repeated performance if processing has fixed-capacity (including serial processing). Other limited-capacity models predict that sequential performance is intermediate between simultaneous and repeated performance. Experiment 1 successfully distinguishes unlimited-capacity, parallel models from other alternatives. However, it does not distinguish between the standard serial and the fixed-capacity, parallel model. In Experiment 2, our goal is to distinguish between these two models. 1 There are several possible explanations for this small effect. Perhaps there is some modest amount of parallelism in word processing. For example, information about word shape, acquired in parallel, might8 aid identification on some proportion of trials. 26

27 Experiment 2 In the two comparisons described above, the standard serial model and the fixedcapacity, parallel model make the same predictions. The fixed-capacity, parallel model is considered for precisely this reason: its predictions often mimic those of a serial model. In Experiment 2, the repeated-new comparison can distinguish between the standard serial model and the fixed-capacity, parallel model. The two critical conditions are both part of the partially repeated display shown in Panel D of Figure 1. In this display, two stimuli are shown in the first frame and four are shown in the second, so that two of the stimuli shown in the second frame are new and two are repeated from the first frame. The new and repeated conditions are distinguished by looking at the subsets of trials in which the target was one of the stimuli only shown in the second frame or one was one of the stimuli that repeated, respectively. Observers were instructed to attend to the new stimuli shown in the second frame. Predictions We derive the formal predictions of the models in Appendix A. The standard serial model predicts equivalent performance in the repeated and new conditions. The unlimited-capacity, parallel model and the fixed-capacity, parallel model both predict superior performance in the repeated condition. However, these predictions for the parallel models are sensitive to the memory assumption. Figure 6 illustrates the models for this comparison. The unlimited-capacity, parallel model posits the extent of processing is determined solely by the stimulus duration. In the repeated condition, the target appears 27

28 for twice the duration that the target appears in the new condition. Thus, the location containing the target is processed a greater extent in the repeated condition than the new condition. This is illustrated in Panel A of Figure 6. The unlimited-capacity, parallel model predicts superior performance in the repeated condition. The standard serial model posits that stimuli are processed one-at-a-time. Given this limitation, it is in the observer s best interest to follow the instruction to attend to the new stimuli in the second frame and ignore the repeated stimuli in that frame. If they are able to do so, they process all locations to an equivalent extent, on average. This is demonstrated in Panel B of Figure 6. Thus, the standard serial model predicts equivalent performance in the repeated and new conditions. The fixed-capacity, parallel model posits that processing capacity is divided between available stimuli. This model predicts that the repeated stimuli are processed alone in the first frame, and then all stimuli receive an equal degree processing in the second frame. Thus, the repeated stimuli, however, would be processed to a greater extent than the new stimuli. Panel C of Figure 6 demonstrates this prediction. The fixedcapacity, parallel model predicts superior performance in the repeated condition. Memory assumptions for repeated-new predictions. In Appendix A, we describe the memory assumptions necessary to make the above predictions. that the equivalence predictions are robust across the memory assumptions, but the ideal memory assumption, in which information about stimuli are combined across the two repeated displays, predicts deviations form equivalency described above. Or, more precisely, it predicts effects that are small relative to the sensitivity of our experiment. Instead, the predictions 28

29 follow from a target-only memory assumption, in which observers remember only the most likely target from each display. This memory assumption is consistent with findings that observers maintain memory for previously scanned locations, but do not integrate information across displays in an ideal fashion (Horowitz & Wolfe, 1998; Shore & Klein, 2000; Beck, Peterson, & Vomela, 2006; Beck et al, 2006). In summary, the usefulness of the repeated-new comparison depends on accepting the target-only memory assumption. Methods Apparatus and stimuli were as in Experiment 1. Two conditions were used: a sequential condition, as in Experiment 1, and a partially-repeated condition. The partially-repeated condition is shown in Panel D of Figure 1. In the first frame, two stimuli appeared. In the second frame, four stimuli are presented: two new stimuli were alongside the two previously presented stimuli. The two stimuli that repeated appeared in opposite quadrants of the display. Partially-repeated trials were randomly intermixed with sequential trials so observers did not know whether they would have another chance to view the stimuli in the first frame. Additionally, observers were instructed to attend to the new stimuli in the second frame. The pair of locations shown in the first frame was randomized. For each of six observers, there were 1440 trials in each condition for each task, completed in 10 sessions. Sessions for the contrast increment and word categorization tasks were conducted separately. Each observer trained on each task by completing at least two practice sessions. 29

30 Results Data from the partially-repeated condition were split into two subsets for analysis: performance in trials in the target appeared in one of the locations that repeated and performance in trials in which target appeared in a location that was seen only once. These are referred to as the repeated-location condition and new-location condition, respectively. Results of the repeated-new comparison are plotted in Figure 7. The percentage of correct responses is plotted on the ordinate and the conditions are plotted on the abscissa. In the contrast increment task, performance was better in the repeated-locations condition by 12% ± 1%. This difference was reliable (t(5) = 8.32, p <.001). All six observers showed performance consistent with this overall pattern (all observers: p <.01). This advantage from multiple presentations is comparable to the advantage from multiple presentations of contrast increments in the simultaneous-repeated comparison from Experiment 1 (12% in Experiment 2 compared to 8% in Experiment 1). This result is consistent with the unlimited-capacity, parallel model. It is not consistent with either the standard serial model or the fixed-capacity, parallel model. In the word categorization task, there was no reliable difference between the repeated-locations condition and the new-locations condition. Accuracy was 4% ± 3% higher in the repeated-locations condition, but not reliably (t(5) = 1.43, p >.2) Of the six observers, three performed reliably better in the repeated-locations condition (Observers AS, MK, & MT: <.01), one performed reliably better in the new locations condition (VN: p <.01), and two had no reliable difference in performance between the conditions (JL: p >.4, SY: p >.05). There is a striking contrast between this 4% ± 3% effect for 30

31 multiple presentations of words here (repeated-locations) and the 18% effect for multiple presentations of words in the simultaneous-repeated comparison from Experiment 1. Equivalent performance between the repeated and new locations is consistent with the standard serial model. It is not consistent with either the unlimited-capacity parallel model or the fixed-capacity, parallel model. Secondary comparison: first versus second frame. This comparison was carried out on the sequential condition data, as in Experiment 1. In the contrast increment task, observers performed somewhat better when stimuli appeared in the second frame. The average difference between the intervals was 4% ± 1%, a reliable difference (t(5) = 3.67, p <.05). Individually, two observers showed performance consistent with this effect (observers JL and MT, both p <.01), the rest showed no reliable effect. This is the only time in four tests that we observed this effect. In Experiment 1, the same task and condition showed no difference between first and second frame. It is small (4%) compared to the repeated-new effect (12%). In the word categorization task, there was no reliable difference between cases where the target appeared in the first and second frame. On average, performance was 2% ± 1% better when the target appeared in the second interval, a non-reliable difference (t(5) =.85, p >.6.). Overall, 3 of 4 tests showed no effect of the frame in which the target appeared. If there is an effect of frame, then it is likely to be small (< 5%) relative to the comparisons of interest ( 10%). Nevertheless, this comparison is something to check in future studies with this paradigm. 31

32 Secondary comparison: Sequential and partially-repeated conditions. We made two comparisons between related conditions within the sequential and partially-repeated displays. First, we compared performance from the first-frame of the sequential condition with the repeated-locations sub-condition of the partially-repeated condition. In both of these situations, the target appears in the first frame of the display. Second, we compared performance from the second-frame of the sequential condition with the new-locations of the partially-repeated condition. In both of these conditions, targets appear only in the second frame. In the contrast increment task, performance was reliably better for the repeatedlocation condition than in the first frame of the sequential condition, with an average difference of 12% ± 1% (t(5) = 8.32, p <.001). Performance did not differ reliably between the new-location condition and the second frame of the sequential display with a difference of 4% ± 3% (t(5) = 1.55, p >.1). These results are consistent with the predictions of the unlimited-capacity, parallel model. They are inconsistent with the predictions of the standard serial model and the fixed-capacity, parallel model. In the word categorization task, neither comparison showed reliable differences. For the repeated-locations condition and the first frame of the sequential condition, the difference was 2% ± 1% (t(5) = 1.51, p >.19). For the second frame of the sequential condition and the new-locations condition, the difference was 3% ± 2% (t(5) = 1.42, p >.2) These results are consistent with the standard serial model and inconsistent with the unlimited-capacity, parallel and the fixed-capacity, parallel models. 32

33 Discussion of Experiment 2 Contrast increment task. In the contrast increment task, there was advantage in performance for the repeated condition over the new condition. This result is consistent with both the unlimited-capacity and fixed-capacity, parallel models. It is not consistent with the standard serial model. Combined evidence from the three main comparisons in this study indicates that the contrast increment task is consistent with the unlimitedcapacity, parallel model. Word categorization task. In the word categorization task, there was no reliable difference in performance between the repeated and the new condition. This result is consistent with the standard serial model, and not consistent the unlimited-capacity or fixed-capacity, parallel models. Of the three models we consider, the standard serial model offers the best account of the word categorization task. However, one prediction of the standard serial model was not met by the data: performance in the repeated and sequential conditions was not equivalent. This inconsistency might mean that processing of words was not strictly serial in this experiment. For example, perhaps word-shape information, gathered in parallel, helped observers localize the targets on some proportion of trials. An alternative model: controlled parallel processing. There is a plausible alternative model that can mimic the predictions of the standard serial model in Experiment 2. If observers were able to selectively stop processing of the repeated stimuli in the second frame, as instructed, then any parallel model would predict equivalent 33

34 performance in the new-locations and repeated-locations conditions. Pashler (1998) called this kind of model a controlled parallel model. A controlled, fixed-capacity, parallel model can account for the results of the three word categorization task comparisons. This raises the question of whether a such a model can ever be distinguished from a standard serial model it is essentially a parallel model that can act like a serial model under many circumstances. For any single experiment designed to identify serial processes, there may be a plausible controlled parallel model that can mimic the predictions of the standard serial model. This does not mean that serial and controlled parallel processes cannot be distinguished, but it does suggest that converging evidence from multiple experiments is necessary to distinguish them. It seems likely that if models differ in meaningful ways, it is possible to craft an experiment that can reveal those differences and differentiate the models. For example, controlled parallel models suggest that instructions might be enough to switch from one kind of processing to the other. General Discussion Summary of Results In this study, we tested three models of visual processing under conditions of divided attention with two different tasks. In the three primary comparisons, all results for the contrast increment task were consistent with an unlimited-capacity, parallel model of divided attention. The standard serial model and all kinds of limited-capacity, parallel models were rejected for this task. For the word categorization task, results were generally consistent with a standard serial model. The unlimited-capacity, parallel model 34

35 was rejected. The results of the repeated-sequential comparison for word categorization deviated slightly from the predictions of the standard serial model. This smaller degree of limited capacity in the word categorization task might be explained by occasional parallel identifications from word shape information. Other Paradigms Distinguishing Serial and Parallel Processes Here we consider alternative experimental approaches to distinguishing serial and parallel processes. For a previous, quite different review of such approaches, see Townsend (1990) and Townsend & Wenger (2004). Each approach has its advantages, and each one requires its own set of assumptions about the nature of processing. Here we emphasize the simultaneous-sequential approach because it requires few assumptions (Pashler, 1998). The major disadvantage of this approach is possibility of undesirable consequences of using sequential displays. For example, sequential displays, and in particular the additional memory demands of sequential displays compared to simultaneous displays, require assumptions about the nature of memory. In Appendix A, we explore alternative memory assumptions for the paradigm. Set-size paradigms. As discussed in the introduction, paradigms that rely on the interpretation of set-size effects must contend with issue of error: performance degrades as set-size increases, irrespective of the presence of capacity limitations. Consider next recent studies distinguished parallel and serial models by fitting quantitative models to set-size effects and redundant target effects (Dosher, et al, 2004; Thornton & Gilden, 35

36 2007). These studies distinguish serial and parallel processes on the basis of predicting the magnitude of the set-size effect. Dosher, Han and Lu (2004) developed models of unlimited-capacity, parallel and serial processing in search tasks in which the observer indicated whether or not a target was present. They used these models to fit data from asymmetric search tasks, in which swapping the set of targets and distractors alters the difficulty of the search. For example, searching for a C among Os is easy, while searching for an O among Cs is difficult (Treisman & Souther, 1985; Rosenholz, 2001). Dosher and colleagues found that their unlimited-capacity, parallel model fit the speed-accuracy data for both the C-in-O and the O-in-C searches, while their serial model did not fit to either set of data. Dosher and colleagues did not consider limited-capacity, parallel models. Thornton and Gilden (2007) developed simulation-based models of parallel and serial search processes in multiple-target search tasks, in which both the number of stimuli and the number of targets varied. Observers indicated whether or not at least one target was present in each display. Thornton and Gilden compared the fits of their serial and parallel models to results of 29 different search tasks, each using different kinds of stimuli. A few tasks were best fit by their serial model, including searches for specific direction of rotating motion (e.g., clockwise among counterclockwise) and some tasks reliant on spatial relationships (e.g., phase of a gabor-like stimulus). Most of their tasks were best by parallel models. Capacity limits were assessed by fitting a free parameter which placed each task on a continuum from unlimited-capacity, parallel to fixedcapacity. On the unlimited-capacity side of this continuum were simple feature tasks such as size color, and orientation discriminations, as well as a some letter searches like 36

37 looking for an A among Bs or a T among Ls. On the fixed-capacity side of the spectrum were spatial configuration tasks, such as a discrimination of whether a left-facing C-like figure was present among rightward facing C-like figures. Thornton and Gilden did not test any word stimuli. These studies were successful in distinguishing between serial and parallel models. The limitation of this approach is that quantitative models of parallel and serial search always requires making specific assumptions about the nature of the search. For example, the serial model tested by Thornton and Gilden is not the same as the serial model tested by Dosher and colleagues. Redundant target paradigms. Redundant targets experiments depend upon search tasks with multiple targets. The targets are redundant in the sense that identifying any target is sufficient for responding correctly (e.g., van der Heijden, 1975; Snodgrass & Townsend, 1980; Egeth, Folk & Mullin, 1988; Mullin & Egeth, 1989; Townsend & Wenger, 2004). The critical trials are those in which all stimuli are targets. In these alltarget trials, the unlimited-capacity, parallel model predicts that response time decreases as set size increases. This is because the fastest of n processes becomes faster as n increases. This effect is termed redundancy gain. Serial models predict that response time is constant for all set sizes. This is because a response is always made following the identification of first of n stimuli (Snodgrass & Townsend, 1980). One treatment of redundant targets was part of the larger analysis by Thornton and Gilden (2007), described above. For a description of accuracy-based, redundant-targets paradigms, see Palmer, Verghese, and Pavel (2000). 37

38 The advantage of the redundant targets paradigm is that it provides general, straightforward predictions for both the unlimited-capacity, parallel and serial model. Unfortunately, there is no general prediction can be made for fixed-capacity, parallel models (at least without making further assumptions). Thus, fixed-capacity parallel models might be devised that mimic the predictions of either unlimited-capacity, parallel models or serial models. A second disadvantage is that redundancy gains may be very small, going to zero as the response time variance approaches zero. When response time variance is small, redundancy gain effects may be unreliable. Lastly, there is a possible complication in that a redundancy gain can be predicted by a serial model if the greater number of targets improves the probability that a target appears at a favored position (e.g., Mordkoff & Miller, 1993). Independent stimulus manipulation paradigm. Another approach involves separately manipulating the processing rate of stimuli by degrading visual quality. This can be achieved by reducing contrast or adding visual noise to some or all stimuli (Pashler & Badgio, 1991, Egeth & Dagenbach, 1991; Townsend & Wenger, 2004). If one assumes that this manipulation selectively influences the processing rate for the affected stimuli, unique response time predictions can be made for parallel and serial models. For example, in Egeth and Dagenbach s (1991) study, observers searched 2-item displays. The critical analysis was on response times in target-absent trials in which the observer must complete processing of both stimuli before responding. A typical serial model predicts an additive effect on response time. In contrast, parallel models predict underadditivity: response time is determined by the completion time of the slowest-completing 38

39 stimulus process, regardless of the time it takes to process other stimuli. Townsend and colleagues (e.g., Wenger & Townsend, 2006; Fific, Townsend & Eidels, 2008) have combined this technique with the analysis of response time distributions and redundant targets manipulations. To the best of our knowledge, this method has not been applied to search among simple features or words. A current limitation of this work is that it has not yet been generalized to account for errors. Until this issue is addressed, this method has limited appeal. Dual task paradigms. One can also distinguish between unlimited-capacity and limited-capacity processing using the dual-task paradigm (Sperling & Melcher, 1978; Bonnel, Stein & Bertucci, 1992; Bonnel & Hafter, 1998). In a dual task, two unrelated tasks are performed at the same time or seperately in a control. Performance on each task in the dual-task condition is compared with performance in the corresponding single-task control. If performance suffers in the dual-task condition, it is assumed that the two tasks rely on a common limited-capacity mechanism. If the trial-to-trial performance between the two tasks is inversely correlated, this suggests serial processing. In this review we emphasize unspeeded dual tasks rather than speeded dual tasks (PRP paradigm, Pashler, 1994). Both are relevant to the general topic but the unspeeded paradigm is easier to apply to perception. The dual-task method makes distinct predictions for unlimited-capacity, parallel models, fixed-capacity parallel models, and serial models. The disadvantage of this method is that it requires two independent responses, which potentially allows intrusions response-related capacity limitations (Duncan, 1980; Hafter, et al, 1998). 39

40 Summary. Our review emphasizes four paradigms in addition to the simultaneoussequential paradigm. Three of these depend on search: the set-size paradigm, the redundant target paradigm, and independent stimulus manipulation paradigm. These search paradigms have several variations in how to address accuracy and response time measures. In addition, unspeeded dual tasks also show promise. Clearly, none of these paradigms is perfect. We suggest the best path ahead is to build consensus for using some subset of these paradigms by seeking agreement in the interpretation of chosen test cases such as simple features and word categorization. Prior Studies of Divided Attention and Simple Features The evidence that contrast increment detection is an unlimited-capacity process is consistent with a growing body of evidence for unlimited-capacity, parallel processing of simple features. Here we briefly review relevant results from several paradigms. Simultaneous-sequential experiments. Using the simultaneous-sequential comparison method, Huang and Pashler (2005) found evidence for unlimited-capacity processing in searches for size singletons and for size-orientation conjunctions. Some kinds of searches for single alphanumeric characters also yield no differences between simultaneous and sequential conditions (e.g, Shiffrin & Gardner, 1972; Pashler & Badgio, 1987). In a study of bimodal perception, Shiffrin and Grantham (1974) observed no sequential advantage in a task that required processing of simple auditory, tactile and 40

41 visual signals, indicating that simple stimuli from these three modalities can be processed in parallel and without capacity limitations. Accuracy set-size experiments. Set-size effects in searches for many simple features, including contrast increments, size differences, and orientation, can explained by an unlimited-capacity, parallel model (Palmer, 1995; Monnier & Nagy, 2001, Dobkins & Bosworth, 2001, Baldassi & Verghese, 2002). These results contradict the notion that all perceptual processes are subject to capacity limitations (e.g., Posner, 1980) and support the view that many low-level perceptual processes are parallel and unlimited capacity (e.g., Gardner, 1972). Redundant targets experiments. Egeth, Folk, and Mullin (1989: Experiment 6) conducted a redundant targets study of line-orientation judgments. The targets were horizontal or vertical line segments against a background of diagonal line segments. Egeth and colleagues observed a steady decrease in response time as the number of targets increased, a result consistent with unlimited-capacity, parallel processing. Verghese and Stone (1995) conducted a 2-alternative, forced-choice speed discrimination experiment and found that speed difference thresholds improved as they varied the number of moving grating stimuli from 1 to 6. This result was consistent with unlimitedcapacity, parallel processing. Dual-task experiments. Bonnel, Stein & Bertucci (1992) found no dual-task performance decrement in a dual-task that involved simultaneously detecting two 41

42 luminance increments. Similarly, Bonnel & Hafter (1998) found no performance decrement in a study that required simultaneously detecting a visual luminance increment and an auditory intensity increment. However, in both studies Bonnel and colleagues found that dual-task effects emerged when the task was to identify whether the signal change was a decrement or increment. They initially concluded that the task difference identification rather than detection was responsible for difference. Later, they found that memory was the real culprit: apparent capacity limitations emerged because they identification task required that each stimulus be compared to a standard in long-term memory (Hafter, et al, 1998). When they altered the tasks so that they instead relied on a comparison to a standard that was persisting in sensory memory, the dual-task effects disappeared. Overall, the results of these dual-task studies are consistent with simple feature judgments based on unlimited-capacity, parallel processing. The memory issue serves as a cautionary tale: if the task is constructed such that it relies on memory, capacity limitations may emerge due to these memory demands, even when the perceptual processing is unlimited-capacity and parallel. Response-time, set-size experiments. The apparent consensus regarding simple features is not found for traditional interpretations of response time experiments that ignore errors. For highly discriminable simple features, set-size effects are small, consistent with the traditional interpretation of the unlimited-capacity, parallel model. But for difficult-to-discriminate simple features, set-size effects are large, which is not consistent with the traditional interpretation of the unlimited-capacity, parallel model. 42

43 Our interpretation of this result is that response time models without error are not appropriate for visual search and must be replaced with models that consider error (Palmer & McLean, 1995; Thornton & Gilden, 2007). Summary. For simple features, our reading of the literature is that several paradigms converge on the same interpretation. Under ideal conditions, the processing of the individual stimuli occurs in parallel and with unlimited capacity. However, there remain exceptions to this generalization that require further attention. Prior Studies of Divided Attention and Words The serial result in the word categorization task supports the more general claim that reading is a strictly serial process. This idea is contested in the literature. Simultaneous-sequential experiments. Two prior studies used the simultaneoussequential comparison method to assess word identification tasks. In two simultaneoussequential experiments using word stimuli, Harris, Coburn and Pashler (2004) found an advantage for sequential presentations, just as we found in our simultaneous-sequential comparison. Patterson (2006) also found an advantage for the sequential condition in task in which observers indicated which number-word in a display had the highest value. Neither of these studies distinguished between fixed-capacity, parallel and serial models. Set-size, redundant targets, and dual-task experiments. Karlin and Bower (1976) conducted a response time, set-size experiment in which observers searched for specified 43

44 words among distractor words in displays with up to 6 words. The set-size effect in target-present displays was half the effect in target-absent displays, a result consistent with serial search. This analysis depends on models that do not account for error. Mullin and Egeth (1989) conducted a redundant targets experiment in which observers responded to words drawn from a specific semantic category. The observed no redundancy-gain effect, a result consistent with serial processing and inconsistent with unlimited-capacity, parallel process. To our knowledge, the dual-task paradigm has not been used to examine the serial or parallel nature of reading. Reading-like experiments. Rayner and colleagues have studied the serial-parallel question in the context of sentence reading. They concluded that evidence favors serial extraction of semantic information from words (see Starr & Rayner, 2001 for a review). For example, a study by Rayner, Balota and Pollatsek (1986) demonstrated that physical information about words that have not yet been fixated can affect reading, but semantic information from these words has no effect. The experimenters used a paradigm in which a critical word changed as the observer was fixating on the preceding word. Relative to the target, the preview word was either the same (song-song), orthographically similar (sorp-song), semantically related (song-tune), or semantically unrelated (door-song). Relative to the same-word control, the orthographically similar preview word had no effect on reading time. Both semantically related and unrelated preview words slowed reading, and each was equally disruptive. These preview effects suggest that low-level physical information is gathered prior to fixation, but not semantic information. 44

45 Rayner s findings suggest a model in which physical word information can be extracted in parallel, but extracting semantic word information requires focal attention. The preview effects described above suggests that feature judgment processing, done in parallel, can improve the efficiency of subsequent semantic processing. This theory could account for the modest advantage in the repeated condition over the sequential condition in Experiment 1. Compared to the sequential condition, the repeated condition allows more time to pre-process all stimuli. Color-word interference experiments. The color-word interference effect first described by Stroop (1935; reviewed in MacLeod, 1991) can occur even when distractor color words are spatially separated from target color patches (Dyer, 1973). The fact that irrelevant words can interfere with processing can be construed as evidence pre-attentive, parallel processing of words. But this is not the only possible interpretation. Indeed, a narrower interpretation is to favor parallel processing of color information and a single word. The heart of any explanation is as a failure of selection: a word demands the observer s attention despite its irrelevance to the task. Kahneman & Chajczyk (1983) found adding an additional irrelevant words to the display reduces the magnitude of the interference effect and concluded that word processing in Stroop tasks is involuntary but strictly serial. This account was supported in a study by Risko, Stolz, and Besner (2005) in which the Stroop interference effect was diminished when the target color patch appears in an unpredictable location in each trial. If words were processed in parallel, one expectd the Stroop effect to persist under such conditions. Consequently, we argue that 45

46 evidence from the Stroop interference effect is more relevant to the failure of selective attention than to the nature of divided attention across words. Summary. Several experiments are consistent with serial identification of words. On the other hand, a variety of Stroop-like experiments suggest to some that words are processed in parallel. We argue that these experiments are best explained by a failure of selective attention by words. The range of paradigms applied to simple features have not yet been applied to word tasks. Thus, this area seems ripe for further study. Conclusions The simultaneous-sequential paradigm distinguishes unlimited-capacity models from all others. In the present study, we developed variations on that paradigm that can distinguish other models of divided attention. Collectively, the experiments described here constitute a method for distinguishing between serial and parallel processes. For each of several plausible models, there is an explicit pattern of predictions across the three experimental comparisons. The contrasting results of our two test cases establish that this method can distinguish different effects of divided attention for different perceptual tasks. The results of Experiment 1 reject any kind of limited-capacity processing in the contrast increment task, thus we conclude that perceptual processing of contrast increment discs is unaffected by divided attention. This result is consistent with the longstanding notion that the perception of simple visual features has unlimited-capacity, parallel processing. On the other hand, some kinds of perception are subject to capacity limitations. The results of 46

47 Experiment 1 rule out the possibility of unlimited-capacity, parallel processing for semantic categorization of words and suggest capacity limitations as severe as is expected by serial processing. Experiment 2 gives further support for serial processing under these conditions. We conclude that reading words is likely to be a serial process. Author Notes The authors thank Cathleen Moore for her support and helpful suggestions throughout this project. We also thank Serap Yigit for many helpful comments and discussion. This research was support in part by a grant from the National Institute of Health and the National Institute of Mental Health (MH067793) to Cathleen Moore. 47

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53 Appendix A Signal-Detection Models of the Simultaneous-Sequential Paradigm In this appendix, we derive the predictions of signal detection theory-based models of the localization tasks used in this article. Predictions are presented for each of the three comparisons in the body of the paper simultaneous-sequential, repeatedsequential, and repeated-new, and each of the three models described in this article: the standard serial model; unlimited-capacity, parallel model; and fixed-capacity, parallel model. These models reinforce the informal descriptions of the predictions in the body of the article. Notation and assumptions We follow signal detection theory (Green & Swets, 1966) in assuming the internal representation of each stimulus corresponds to a real-valued random variable representing the evidence that a particular stimulus is a target rather than a distractor. For targets and distractors, these random variables are designated T and D, respectively. Random variables for n distractors are subscripted D 1, D n. We assume statistical independence between the random variables: on a particular trial, the value of any one random variable, say D 1, is independent of any other representation on the same trial, say D 2. We further assume that all distractor random variables D i are identically distributed. The distractor density function is designated f D (x) and the target density is function is designated f T (x). The corresponding cumulative distributions are F D (x) and F T (x), respectively. 53

54 Parallel Processing Model We follow the analysis of the n-alternative localization task described by Shaw (1980). On a given trial, the observer chooses the stimulus with the largest value in the relevant dimension (e.g., the brightest disc). This is equivalent to a maximum rule on an appropriately standardized representation. For n stimuli, the probability of choosing the correct location is p(correct) = p[ T > max(d 1, D n-1 )], [1] which is equal to # p(correct) = $ f T (x)f D (x) n"1 dx. "# [2] This equation integrates the product of a shifted density function representing the target, multiplied by the cumulative distributions representing the distractors. Thus far, these equations describe parallel model describes both unlimited- and fixed-capacity, parallel models. For the fixed-capacity, parallel model, we assume that the variance of each random variable is proportional to the number of stimuli being processed at one time. For the unlimited-capacity, parallel model, we assume the variance of each random variable is invariant of the number of stimuli being processed. Standard Serial Model To describe serial processing, we assume that in a given display, some stimuli are identified, while others are not. The number of stimuli that can be identified in each 54

55 display is designated m. In this serial model, the observer has information about m stimuli and has no information about n - m stimuli. This approach is similar to that of Davis, Shikano, Peterson, and Michel (2003). If all stimuli are identified (m = n), the observer chooses the stimulus with the largest value in the relevant dimension. In this case, the capacity of the standard serial model has not been exceeded and the model is equivalent to the unlimited-capacity, parallel model. If all not all stimuli are identified (m < n), the standard serial model can be distinguished from the unlimited-capacity, parallel model. In this case, the observer chooses the identified stimulus with the largest value, unless that value falls short of criterion c. For simplicity, we fix the criterion at a point halfway between the mean target and distractor values. If the highest of the m identified stimuli does not exceed c, the observer guess the target from the n m unidentified stimuli. The probability of choosing the correct location is p(correct) = [ (m/n)p(t > max(d 1, D m-1 ) > c)] [3] +[(1-m/n)p(c > max(d 1, D m-1 )g] The first line of Equation 3 counts cases in which the target is among the m identified stimuli. These cases occur with probability m/n. The observer first checks whether any of the identified stimuli have a value that exceeds the criterion c. If any identified stimuli have values that exceed c, the observer chooses the stimulus with the largest value. If no identified stimulus has a value that exceeds c, the observer randomly chooses among the n m remaining unidentified stimuli. In this case, the guess will always be incorrect, because the target was among the m identified stimuli. Thus, for the 55

56 observer to respond correctly, T must be greater than all values of D and must also be greater than the criterion. The second line of Equation 3 counts the cases in which the target is not among the m identified stimuli. These cases occur with probability 1/(n m). If any of the m identified distractors exceed the criterion c, the observer will incorrectly choose that the distractor as the target. If the observer properly rejects all the distractors as below the criterion, she can then guess the target from the remaining n m unidentified distractors, guessing correctly with probability g = 1 (n / m). Equation 3 describing the standard serial model is equivalent to p(correct) = m n + (1 # c ( f T (x)f D (x) m 1 )dxk c m n )( f 1 D (x)dx)m ( n m ) # [4] The first line of Equation 4 counts cases in which the target is identified. The probability of correct response in this case is the integral of the product of the density function of the target and the cumulative function of all identified distractors from c to infinity. The second line of the equation counts cases in which the target is not identified. The integral term yields the probability that none of the identified distractors exceed the criterion and are falsely selected as the target. The integral of the density function of the distractor from - to c gives the probability that one identified distractor exceeds c. The product of m such integrals gives the joint probability that none of m identified distractors exceed c. If the observer correctly rejects all of the identified distractors, she has a chance to correctly guess among the remaining identified stimuli, and does so with probability g = 1/(n m). 56

57 Modeling the simultaneous-sequential comparison In order to model performance of the various models in the simultaneous and sequential conditions, we assume that all random variables are Gaussian. The density function of each distractors has a mean of zero and a variance of one, except where otherwise specified. The density function of the target has a variance of one, and a mean that is a parameter that ranges from zero to six in our examples. Because the variance of both the target and distractor distributions is one, and the mean of the distractor distribution is zero, the mean of the target distribution can be thought of as the d measure of discrimination difficulty. For all predictions given below, n is set equal to 4. Unlimited-capacity, parallel model. For the unlimited-capacity, parallel model, we assume that the quality of information about each stimulus is equivalent for all stimuli that are presented for an equivalent duration. Thus, quality of information about each stimulus is equivalent in the simultaneous and sequential conditions. Both simultaneous and sequential conditions are modeled according to the parallel search equation, Equation 2. Panel A of Figure A1 shows the predictions of the unlimited-capacity, parallel model in the simultaneous and sequential conditions. Difficulty is represented on the abscissa as the difference in means of the target and distractor random variables, units of the standard deviation (d ). The percent correct in each condition is represented on the ordinate. For the unlimited-capacity, parallel model, the curves representing performance in the simultaneous and sequential conditions were generated from identical equations, 57

58 signifying that the model predicts that at any level of difficulty, performance is equivalent in the two conditions. Fixed-capacity, parallel model. For the fixed-capacity, parallel model, we adopt the sample-size model of Shaw (1980). The number of samples that are obtained from each stimulus is inversely proportional to the number of simultaneously presented stimuli. Thus, the variance of all random variables is proportional to the number of stimuli being processed at once. In our model, this assumption has the effect of doubling the variance of the random variables representing each stimulus in the simultaneous condition, relative to the variance of the variables in the sequential condition. The fixedcapacity, parallel model is otherwise the same as the unlimited-capacity, parallel model. Panel B of Figure A1 shows the predictions of the fixed-capacity model for the simultaneous and sequential conditions. The model predicts that performance between the two conditions is similar when the task is very easy or very difficult, but performance is better in the sequential condition at intermediate difficulties. Standard serial model. To model the predictions of the standard serial model in the simultaneous and sequential conditions, we assume that in the sequential condition, the variable m is double that of the same variable in the simultaneous condition. For example, if conditions allow the observer to identify only one stimulus in the simultaneous display, they can identify two stimuli in the two sequential displays. The variable m is capped at n, so that number of identified stimuli does not exceed the number of stimuli in the display. From this point on, we refer to the value of m used in the 58

59 simultaneous condition. In sum, there are two separate determinants of performance in this model: the parameter m and the target-distractor discriminability. Panel C of Figure A1 shows predictions of the standard serial model. This figure plots the average of the predictions of the m = 2 and m = 3 variants of the standard serial model. We show this mixture because it accounts for the data collected in our word experiment. In this prediction, sequential performance is better than simultaneous performance for all difficulty levels above chance. Discussion of the simultaneous-sequential comparisons. We can compare the quantitative theory with our data by using an operating characteristic plot, shown in Figure A2. This figure plots percent-correct performance in the sequential condition against percent-correct performance in the simultaneous condition. In this figure, difficulty sweeps out a curve such that the bottom left corner represents the most difficult conditions (d = 0) and the upper right corner represents the easiest conditions (d = 6). The vertical distance from the diagonal to a model prediction is the predicted magnitude of the simultaneous-sequential effect. The results of the two simultaneous-sequential comparisons in this article are plotted as points with error bars representing standard error of the mean performance in each of the two conditions. The predictions of two standard serial models are plotted: one that assumes that 2 stimuli per display are identified (m = 2) and one that assumes that 3 stimuli per display are identified (m = 3). The results of the word-categorization experiment fall between these predictions. Thus, these results experiment are consistent with observers identifying serially two or three words in each display. The curve for m = 4, not shown in the figure, runs along the diagonal. Thus, the 59

60 results from the contrast increment experiment cannot reject a serial model that operates by quickly identifying all four stimuli in the simultaneous display. The figure supports the informal arguments in the body of this article: the results of the word categorization experiment are consistent with either a fixed-capacity model or a serial model, but reject the unlimited-capacity model. The results of the contrast increment task are consistent with the unlimited-capacity, parallel model, but reject serial models that process fewer than all four stimuli. Two memory models for repeated displays The next two comparisons include conditions in which stimuli are repeated in separate displays. In order to model these conditions, it is necessary to make assumptions about the role of memory in the experiment: how do observers integrate information across two displays? We describe two memory assumptions for the experiment. For each condition with repeated displays, we model the outcome using both models. These two memory assumptions yield identical predictions in the simultaneous-sequential comparison just described. For the ideal memory assumption, we assume that the observer perfectly integrates information across the two displays. In signal-detection terms, the effect of integrating two views of a stimulus is to halve the variance of the random variable representation of that stimulus. For the target-only memory assumption, we assume that the observer retains only the most likely target from one display to the next. Rather than integrating evidence about 60

61 each stimulus across displays, the observer determines the most likely target from each display separately, then chooses which of those two possible targets is more likely. To illustrate the memory assumptions, consider the performance of the unlimitedcapacity, parallel model in the repeated-display condition. Under the ideal memory assumption, the decision is based on the maximum of the four stimulus representations. The variance of each random variable representation is halved, because each stimulus is viewed twice. If the target stimulus yields the highest evidence value, the observer responds correctly. In contrast, the target-only assumption pits two target representations (one from each frame) against 6 distractor representations (three from each frame). The variance of each representation is constant. If either of the target representations yields the highest evidence value, the observer responds correctly. Modeling the sequential-repeated comparison To model the sequential-repeated comparison, we use the same conventions as described above. The parallel models depend on the memory assumption, so we generate separate predictions for each memory assumptions. Figure A3 shows the operating characteristics for this comparison. Panel A shows predictions made with the ideal memory assumption. Panel B shows predictions made with the target-only memory assumption. In each panel, percent correct in the sequential condition is plotted against percent correct in the simultaneous condition. Difficulty sweeps out a curve such that the bottom left corner represents the most difficult conditions (d = 0) and the upper right corner represents the easiest conditions (d = 6). The vertical distance from the diagonal to a model prediction is the predicted size of the sequential-repeated effect. The results of 61

62 the two sequential-repeated comparisons in this article are plotted as points with error bars representing standard error of the mean performance in each of the two conditions. The predictions of each model are plotted as lines. Standard serial model. The standard serial model predictions assume that the effect of the repeating the display is equivalent to doubling the value of m, as m stimuli are identified in the first display and an additional m stimuli are processed in the second display. This effect is equivalent to the effect of the sequential presentation. Thus, serial models with m of two or less predict equivalent performance in the sequential and repeated displays, falling along the diagonal in both panels of Figure A3. Parallel models with ideal memory. With ideal memory, we assume that observers ideally integrate information from two frames, resulting in a more accurate representation of each stimulus. Predictions of the two parallel models with ideal memory are shown in Panel A of Figure A3. These curves were generated using Equation 2, with some changes to the variance of stimulus representations. For the unlimited-capacity, parallel model, viewing each stimulus twice in the repeated display has the effect of halving the variance of each representation in the repeated display, resulting in superior performance in that condition. For the fixed-capacity, parallel model, variance is halved because the display is repeated, but doubled because twice as many stimuli are being sampled in each display so there is no overall change in variance. Thus, performance in sequential and repeated conditions are equivalent for the for the fixed-capacity, parallel 62

63 model and the predictions of this model fall along the diagonal axis in Panel A of Figure A3. Parallel models with target-only memory. With target-only memory, we assume that observers remember only the most likely target from each frame. With target-only memory in the repeated condition, observers compare the two candidate targets from each of the two frames and choose the one with the highest value. Observers respond successfully when either view of the target yields a value that is greater than the maximum of all the distractors values. The model with the target-only memory assumption requires one to consider each frame separately. To do so, use T 1 and T 2 to designate the random variables corresponding to the target s evidence in the first and second frame, respectively. Similarly, the random variables for the distractors are designated D 1,n and D 2,n for the nth distractor in the first or second frame, respectively. For target-only memory, we assume that the observer responds correctly when the value of either target representation exceeds the all of the other representations: p(correct) = [ p(t 1 )p(t > max(t 2,D 1,1, D 1,n-1, D 2,1, D 2,n-1 ) + [5] which is equal to p(t 2 )p(t > max(t 1,D 1,1, D 1,n-1, D 2,1, D 2,n-1 )] # p(correct) = 2 ( ( f T (x)f T (x)f D (x) # 2(n 1) )dx) [6] The integral in Equation 6 counts all cases in which the value of one of the targets exceeds the value of all other stimulus representations (both the distractors and the other 63

64 target). To count all cases in which one or the other target yields the highest value, the integral is multiplied by two. Panel B of Figure A3 shows the predicted performance in the repeated-sequential comparison for with target-only memory. For the unlimited-capacity, parallel model, target-only memory results in a smaller advantage for the repeated condition relative to the same model with ideal memory. For the fixed-capacity, parallel model, target-only memory predicts an advantage for the sequential condition, whereas an ideal memory condition predicts equivalence between the two conditions. The predictions of the standard serial model do not depend upon the memory assumptions, so it remains at the diagonal. Discussion of the sequential-repeated comparison. These models echo the informal predictions in the body of this article: the unlimited-capacity, parallel processing predicts the repeated display have better performance than the sequential display. This holds true for both the ideal and target-only memory assumptions. The equivalence prediction for fixed-capacity, parallel processing holds for the ideal memory but not the target-only memory assumption. Under the target-only memory assumption, performance is predicted to be better in the sequential condition. Comparing the data from the two sequential-repeated comparisons in this article with these predictions, the contrast increment task data is most consistent with the unlimited-capacity, parallel model with both memory assumptions. The word task data is consistent with both the fixed-capacity, parallel model and serial model under the ideal memory model, but with target-only memory it is consistent with only the standard serial model. 64

65 Modeling the repeated-new comparison The parallel models of the repeated-new comparison depend on the memory assumption, so we again generate separate predictions for each memory assumptions. Figure A4 shows the operating characteristics for the repeated-new comparison. Standard serial model. The standard serial model predictions assume that observers identify m stimuli in the first display and then identify m of the new stimuli in the second display, ignoring the repeated stimuli. If m is two or less, the total number of identified stimuli in the display is equal to two times m. Stimuli in the new and repeated conditions are treated identically by the model. Thus, serial models with m of two or less predict equivalent performance in the sequential and repeated displays. The predictions of the standard serial model fall along the diagonal in both panels of Figure A4. Parallel models with ideal memory. For ideal memory, we assume that observers integrate information from two frames, resulting in a more accurate representation of the stimuli at repeated locations than the stimuli at new locations. For the unlimited-capacity, parallel model, integrating two views of each of the repeated-locations stimuli has the effect of halving the variance of each representation, the consequence of doubling the number of samples. The unlimited-capacity, parallel model with an ideal memory assumption calls for a maximum rule on four stimuli, in which the representations of new-locations stimuli have twice the variance of the representations of repeated-locations stimuli. 65

66 For the fixed-capacity, parallel model, information from two views of the repeated-locations stimuli is integrated in the same way. Additionally, the effect on variance that each view of a stimulus is inversely proportional to the number of stimuli present in that view, so that a view of each stimulus in the second frame contributes half as many samples to the stimulus representations as a view of stimuli in the first frame.. Combing these two effects, we find that representations of the new-locations have three times the variance of the representations of the repeated-locations stimuli. Under this memory assumption, the gain in the precision of the representations of repeated-locations stimuli is counteracted by a loss in precision of the representations of new-locations stimuli. The loss in precision makes these new-locations stimuli more likely to be chosen as targets. Predictions of the two parallel models assuming the ideal memory model are shown in Panel A of Figure A4. For the ideal memory assumption, the predictions depend on difficulty: superior performance in the repeated-locations condition is predicted for more difficult conditions and but not for easier conditions. This comparison is ineffective at distinguishing among the three models: the predicted effects that distinguish any two models are never larger than about 3%. The results of are contrast increment experiment are not consistent with any of the three models under the ideal memory condition. Parallel models with target-only memory. Consider next the target-only memory assumption. In the repeated-locations condition, observers compare the two candidate targets from each of the two frames and choose the one with the highest value. Note that this model introduces a bias for the repeated locations, because they each have two 66

67 chances to be considered the most likely target, as compared to just one chance for the new locations. For target-only memory, we assume that the observer responds correctly when either of the value of any target representation exceeds the all other representations. For the repeated locations, there are two target representations (one from each frame) and four distractor representations (one from the first frame and three from the second). The probability of correct response is p(correct) = [ p(t 1 )p(t > max(t 2,D 1,1, D 2,1, D 2,n-1 ) + [7] which is equal to p(t 2 )p(t > max(t 1,D 1,, D 2,1, D 2,n-1 )] # p(correct) = ( $ ( f T (x)f T (x)f D (x) n )dx) 2. "# [8] The integral in Equation 7 counts all cases in which the value of one of the targets exceeds the value of all other stimulus representations (both the distractors and the other target). To count all cases in which one or the other target yields the highest value, the integral is multiplied by two. For new locations, there is one target representation (from the second frame) and five distractor representations (two from the first frame and three from the second). The probability of correct response is p(correct) = [ p(t 1 )p(t > max(d 1,1, D 1,n-2, D 2,1, D 2,n-1 ) [9] which is equal to # p(correct) = ( $ ( f T (x)f D (x) n +1 )dx). "# [10] 67

68 The integral in Equation 10 counts all cases in which the value the target representation exceeds the value of all distractor representations. Panel B of Figure A4 shows the predicted performance in the repeated-new comparison for the fixed-capacity and unlimited-capacity parallel models with the targetonly memory assumption. Both parallel models predict a sizable advantage for the repeated locations. With target-only memory, this analysis is consistent with the informal analysis presented in the body of this article: both parallel models yield superior performance for repeated locations. Discussion of the repeated-new comparison. The predictions in this comparison are strongly dependent on the memory assumptions. With ideal memory, the predicted effects are small compared to the precision of our experiments. With target-only memory, the predicted effects are substantial for both of the parallel models, consistent with the informal arguments in the body of this article. In comparison with our data, the ideal memory assumption cannot account for the large repeated-advantage found in the contrast increment task. For the target-only memory the models fare better. Under this assumption, both parallel models are consistent with the contrast increment result, and only the standard serial model is consistent with our word categorization data. In summary, the assumptions of the various models are presented in Tabler A1. The first set of these assumptions are shared by all models. The second set of assumptions define the three primary models. The third set of assumptions describe 68

69 memory. And, the last set of assumptions add the details necessary for the quantitative predictions but not the general equality predictions. 69

70 Appendix B Words Used In Word Categorization Task Animals: Cat, dog, lion, bear, wolf, snake, mouse, tiger. Body parts: Lip, ear, chin, neck, face, elbow, thumb, wrist. Clothing: Tie, hat, belt, coat, vest, scarf, pants, shirt. Food: Ham, pie, soup, rice, meat, bread, fruit, salad. Professions: Nun, cop, chef, maid, poet, nurse, actor, judge. Transportation: Bus, jet, boat, ship, taxi, truck, train, plane. 70

71 Table 1. Predicted equivalencies and non-equivalences for three models in three comparisons. Models Comparisons Simultaneous- Repeated- Repeated-New Sequential Sequential Unlimited-capacity parallel p(sim) = p(seq) p(seq) < p(rep) p(new) < p(rep) Standard serial p(sim) < p(seq) p(seq) = p(rep) p(new) = p(rep) Fixed-capacity parallel p(sim) < p(seq) p(seq) = p(rep) p(new) < p(rep) Other limited-capacity p(sim) < p(seq) p(seq) < p(rep) p(new) < p(rep) parallel models 71

72 Table A1. Assumptions of signal detection-based models. Assumptions of all models Single random variables represent the evidence that given stimulus is a target. Representations of stimuli are statistically independent. Distractor random variables are identically distributed. Assumptions of specific models Variance of random variable representations: o Independent of the number of stimuli, except in the case of the fixedcapacity, parallel model, in which it is proportional to the number of simultaneous stimuli. o Inversely proportion to the number of stimulus repetitions Number of stimuli identified: o Parallel models: all identified processed. o Serial model: m stimuli identified in each display. Decision rule: o Parallel models: maximum rule for decision. o Serial model: maximum rule with guessing. Guessing: if no stimulus exceeds a decision criterion, the observer guesses randomly among the unidentified stimuli. Decision criterion for guessing in the standard serial model defined as midpoint between the means of the distractor and target representation. 72

73 Assumptions regarding memory Ideal memory: assume that information from separate frames is ideally integrated for each stimulus. Repeating a stimulus has the effect of halving the variance of the random variable representation for that stimulus. Target-only memory: assume that a tentative decision is made for each frame and only the potential target is remembered. This has the effect of causing each framepresentation of a stimulus to be considered as a separate random variable. Assumptions for numeric results All random variables are Gaussian distributions. Distribution of target stimulus has a mean value that varies with difficulty. Distribution of each distractor stimulus has a mean of zero. 73

74 Figure Captions Figure 1. Display conditions used in this study. (a) Simultaneous condition: All four stimuli are displayed in one 100 ms frame. (b) Sequential condition: Stimuli are divided between two 100 ms frames shown sequentially. Each frame displays two stimuli. (c) Repeated condition: All four stimuli are displayed twice in two 100 ms frames shown sequentially. (d) Partially-repeated condition: Two stimuli are shown in the first frame, then all four stimuli are shown in the second frame. In the second frame, two of the stimuli are repeated from the first frame. Figure 2. Predictions of three models for the simultaneous-sequential comparison. In each figure, a grey bar represents the presence of a stimulus. Stimuli appear in four locations, indicated by the numbers 1 through 4 arranged vertically. Depending on the condition, stimuli can occur in the first and/or the second frame. The black arrows represent the hypothesized course of processing for each model. A solid black line represents a stimulus being processed to the fullest extent possible. Dashed lines indicate less complete processing. (a,b) Predictions of the unlimited-capacity, parallel model for the simultaneous and sequential conditions, respectively. The unlimited-capacity, parallel model predicts that performance in two conditions is equivalent. (c,d) Predictions of the standard serial model for the simultaneous and sequential conditions, respectively. The standard serial model predicts superior performance in the sequential condition. (e,f) Predictions of the fixed-capacity, parallel model for the simultaneous and sequential 74

75 conditions, respectively. The fixed-capacity, parallel model predicts superior performance in the sequential condition. Figure 3. Sample display from the simultaneous condition of the contrast increment detection task. The three distractors are discs of lower contrast and the target is a single disc of higher contrast. In this illustration, the contrast values have been exaggerated. The stimuli are embedded in dynamic Gaussian noise patches. The screenshot has been cropped to better show the stimuli. Figure 4. Sample display from the simultaneous condition of the word categorization task. The stimuli are embedded in dynamic Gaussian noise patches. The screenshot has been cropped to better show the stimuli. Figure 5. Predictions of three models for the sequential-repeated comparison. See Figure 2 caption for key. (a,b) Predictions of the unlimited-capacity, parallel model for the repeated and sequential conditions, respectively. The unlimited-capacity, parallel model predicts that performance is superior in the repeated condition. (c,d) Predictions of the standard serial model for the repeated and sequential conditions, respectively. The standard serial model predicts equivalent performance between the repeated and sequential conditions. (e,f) Predictions of the fixed-capacity, parallel model for the simultaneous and sequential conditions, respectively. The fixed-capacity parallel model predicts equivalent performance between the repeated and sequential displays. 75

76 Figure 6. Predictions of three models for the repeated-new comparison. See Figure 2 caption for key. For repeated locations, target stimuli occur in Locations 1 or 2. For new locations, target stimuli occur in Locations 3 or 4 (a) The unlimited-capacity, parallel model predicts superior performance for repeated locations. (b) The standard serial model predicts equivalent performance for repeated locations and new locations. (c) The fixedcapacity, parallel model predicts superior performance for the repeated locations. Figure 7. Results of the simultaneous-sequential comparison. The percentage of correct responses is plotted as a function of the number of simultaneous stimuli. The solid line and closed circles represent the contrast increment task. The dashed line and open circles represent the word categorization task. Figure 8. Results of the sequential-repeated comparison. The percentage of correct responses is plotted as a function of the number of stimulus repetitions. The solid line and closed circles represent the contrast increment task. The dashed line and open circles represent the word categorization task. Figure 9. Results of the repeated-new comparison. The percentage of correct responses is plotted as a function of the number of stimulus repetitions. The solid line and closed circles represent the contrast increment task. The dashed line and open circles represent the word categorization task. 76

77 Figure A1. Quantitative predictions for three models in the simultaneous-sequential comparison. Predicted percent correct for each condition is plotted as a function of task difficulty. (a) Predictions of the unlimited-capacity, parallel model. (b) Predictions of the fixed-capacity, parallel model. (c) Predictions of the standard serial model. This figure was generated using an equal mixture of standard serial models with m values of 2 and 3. Figure A2. Comparison of the simultaneous-sequential models and data. Percent correct in the simultaneous condition is plotted against percent correct in the sequential condition. Results from Experiment 1 are shown for the contrast increment and word categorization tasks. Figure A3. Comparison of the sequential-repeated models and data. Percent correct in the sequential condition is plotted percent correct in the repeated condition. Results from Experiment 1 are shown for the contrast increment and word categorization tasks. (a) Predictions assuming ideal memory. (b) Predictions assuming target-only memory. Figure A4. Comparison of the repeated-new models and data. Percent correct at repeated locations is plotted against percent correct at new locations. Results from Experiment 2 are shown for the contrast increment and word categorization tasks. (a) Predictions assuming ideal memory. (b) Predictions assuming target-only memory. 77

78 Figure 1. Distinguishing Serial and Parallel Processes A) Simultaneous Condition Fixation 250 ms All locations 100 ms B) Sequential Condition Fixation 250 ms Locations 1 & ms C) Repeated Condition Fixation 1000 ms Locations 3 & ms Fixation 250 ms All locations 100 ms D) Partialy-repeated Condition Fixation 1000 ms All locations 100 ms Fixation 250 ms Locations 1 & ms Fixation 1000 ms All locations 100 ms 78

79 Figure 2. Distinguishing Serial and Parallel Processes A Models Unlimited-capacity, parallel model Standard serial model Fixed-capacity, parallel model 1st Frame 2nd Frame C 1st Frame 2nd Frame E 1st Frame 2nd Frame B 1st Frame 2nd Frame D 1st Frame 2nd Frame F 1st Frame 2nd Frame Sequential Conditions Simultaneous 79

80 Figure 3. Distinguishing Serial and Parallel Processes 80

81 Figure 4. Distinguishing Serial and Parallel Processes 81