the structure of the decision Tom Busey and John Palmer Indiana University, Bloomington and The University of Washington

Transcription

1 In visual search, the difference between "where" and "what" is mediated by the structure of the decision Tom Busey and John Palmer Indiana University, Bloomington and The University of Washington Please send correspondence to: Tom Busey or John Palmer Department of Psychology University of Washington Indiana University Psychology, Box Bloomington, IN Seattle WA

2 Abstract The relation between location and identity information was addressed by measuring set-size effects in several different localization and identification tasks in five experiments. Subjects identified the presence or location of a left-leaning grating in displays that contained 2 or 8 items, including right-leaning distractor gratings. Threshold angles were found for the two set sizes, and several independent channel models based on signal detection theory were developed to incorporate the different decision rules required by each task. In the models, the specific details of the tasks determined whether there was a smaller set-size effect for the localization task or the identification task. The independent channel model predictions accounted for the ordering of set-size effects across task with no free parameters. An extension to the independent channel model was introduced to account for an observed pattern of mislocalizations. This overlapping channels model results in good quantitative predictions for the set-size effects in all tasks. These results support the hypothesis that there is a close binding between location and identity information.

3 Recent research has suggested that when processing visual information, "what" and "where" information may be treated diffeently by the visual system. Within the domain of visual search, different hypotheses about the role of attention have been proposed for identification and localization. In one proposal, Treisman and Gelade (1980) suggested preattentive processing of identity information but not location information, which suggests a privaledged access for identity information. In an opposing proposal, Sagi and Julesz (1985) suggested preattentive processing of location information but not identity, reversing the access order. More recent work discussed below suggests a close binding between the two sources of information, which raises the question of how location and identity information is related in search tasks. In this paper, we test predictions of these general hypotheses, as well as the alternative of a close binding between identity and location information (e.g. Johnston and Pashler, 1990). Our emphasis is that the specific details of different tasks, often not made specific when testing hypotheses in the literature, determine the magnitude of set-size effects in visual search. To demonstrate this, we use models based on signal detection theory (Green & Swets, 1966; Creelman & Macmillan, 1979) to address the relation between location and identity information in several accuracy search tasks. Three Hypotheses Visual search paradigms have produced somewhat conflicting accounts of the relation between location and identity information, and below we describe three general hypothesis that bear on this relation. Identification without Localization By this hypothesis, identification is possible without localization. For example, in feature integration theory (Treisman & Gelade, 1980; Treisman & Gormican, 1988), there is a parallel preattentive stage that processes simple feature information and a serial attentive stage that processes relations among features. It is further assumed that discrimination of Page 1

4 simple features is mediated by a pooled response from the first stage (Treisman & Gormican, 1988). This pooled response loses location information but preserves identity. Localization requires processing by the serial attention stage. Thus set-size effects are predicted to be larger for localization than identification (Saarinen, 1996; Bennett & Jaye, 1995) Early suggestions of this hypothesis can be found in the ideas of feature and item perturbations (Wolford, 1975; Wolford & Shum, 1980; Lee & Estes, 1981) and in the differential loss of identity and location information in very short term visual memory (Irwin & Brown, 1987; Mewhort, Campbell, Marchetti & Campbell, 1981; Mewhort, Huntley & Duff-Fraser, 1993). Localization without Identification By this hypothesis, localization is possible without identification. For example, Sagi and Julesz (1985) (see also Müeller & Rabbitt, 1989; Donk & Meinecke, 2001) proposed a parallel preattentive stage that allows detection and localization of targets followed by a serial attentive stage that is necessary to identify targets. The first stage localizes targets by detecting feature gradients that are made by an "odd" target. In contrast, identification always requires the serial attention stage. Thus, set-size effects are predicted to be larger for identification than localization. This texture discrimination hypothesis has been investigated in more detail by Nothdurft (1985, 1991, 1992, 1993). He has demonstrated a variety of conditions where visual search depends on the local feature contrast rather than the specific values of the features. Closely bound Identification and Localization By this hypothesis, there are common processes that mediate identification and localization. For example, Johnston and Pashler (1990; see also Atkinson & Braddick, 1989; Bloem & van der Heijden, 1995) suggested that a two-stage search theory such as described by feature integration theory (Treisman & Gelade, 1980; Hoffman, 1979) be Page 2

5 modified so that a detection by the first stage calls attention to the relevant information for that stimulus, which includes both identity and location information (JP- check last part). Thus, set-size effects are predicted to be similar for identification and localization, all else equal (M. Green, 1992). This hypothesis is implicit in much work before the proposed distinctions between identification and localization. what does this last sentence mean? One of the first studies to compare set-size effects in accuracy search for identification and localization was Swensson and Judy (1981). Subjects viewed simulated medical images with pixel noise. The image contained a large ring structure with superimposed small disk targets. The targets could appear in 1 to 8 pre-specified locations. Performance was measured in several tasks including Yes/No Identification and n-alternative Localization. For each task, an appropriate d' measure was estimated for each set size. Using the d' measure tailored for each task, they showed that the set-size effects were similar for identification and localization. In summary, Swensson and Judy concluded that a simple model based upon signal detection theory and a common perceptual representation can account for both identification and localization performance. Theoretical Analysis and Relevant Literature Johnston and Pashler's Critique Most of the initial tests of these three hypotheses depended upon simple, atheoretic comparisons between identification and localization performance. Johnston and Pashler (1990) raised three problems with any casual comparison of different tasks, which are summarized below. Different properties problem Some tasks may offer more information for one kind of judgment relative to another. Consider an extreme example: Suppose one was to judge orientation of a line stimulus as either leaning left or right of vertical. In addition, one must judge if the line was to was to the Page 3

6 right or left of fixation. Only a single line is presented at one or the other side of fixation. The problem arises because different information is sufficient to make the two judgments. The orientation of the line is necessary for the orientation judgment. In contrast, any detectable attribute of the line may be sufficient to judge its position to the left or the right of fixation. To make the tasks comparable, the target can only be distinguished from the distractors by an attribute sufficient for identification. Negative information problem The next issue is whether guessing can differentially affect identification and localization. Suppose one was performing a disjunctive search with a color and a shape target (e.g. Treisman & Gelade, 1990). Further, suppose the color target was easier to detect than the shape target. Then on trials in which there was no obvious color target, one could improve performance by guessing there was a shape target and never guessing there was a color target. More generally, guessing raises the more general issue of evaluating whether the decision process is comparable in identification and localization. This is pursued in detail in the following sections. Location-reporting problem The three hypotheses refer to the internal representation of location, but one can only measure the reported location. Depending on the details of the experiment, errors may arise in translating the internal representation to a response. Indeed, if the stimulus locations are close together and confusible, such mislocalization errors are inevitable. Consequently, the more careful studies of identification and localization have used widely separated locations and/or liberal scoring criteria. More generally, one needs a measure of such mislocalizations to determine if this is a problem in any particular experiment. Page 4

7 To address these issues, Johnston and Pashler (1990) used a 2x2 paradigm... need to say more about their paradigm, what they found, and what are the limitations of their approach. Decision Analysis of Identification and Localization An alternative approach to that adopted by Johnston and Pashler (1990) is to introduce a formal model of the decision process for each task based on signal detection theory (Green & Swets, 1966). By doing so, the differences between the tasks, each with its own decision structure, are accounted for. This was the approach of Swensson and Judy (1981) and can also be seen in Burgess and Ghandeharian (1984) and Eckstein and Whiting (1996). Consider the simplest common identification and localization tasks: Yes/No Identification and 2AFC localization. The Yes/No task is to respond whether a single stimulus is the target or a distractor; The 2AFC localization task is to pick which of two simultaneous stimuli is the target. An analysis based on signal detection theory shows that there is more information available in the 2AFC task than in the Yes/No task. The relation between Yes/No and forced choice tasks is based on what is often called the "area theorem" (Green, Weber & Duncan, 1977): The percent correct in a forced choice task is equal to the area under the Yes/No ROC curve. If one further assumes Gaussian distributions for the noise and homogenous variances for each stimulus representation, then the predicted d' in the 2AFC task is larger than in the Yes/No task by a factor of a square root of 2. Thus, the structure of the task predicts different performance even though the underlying perceptual representations are identical. Further generalizations of this analysis can be found in Creelman and Macmillan (1979). Set-size effects for different tasks Decision analysis also predicts differences in the set-size effects for different tasks. From the beginning of signal detection theory, Tanner (1961) described how stimulus Page 5

8 uncertainty decreases performance even when one assumes the stimulus representations are independent. These ideas were brought into visual search by Shaw and others (e.g. Shaw, 1980; for review see Palmer, Verghese and Pavel, 2000). For example, the following calculations are also based on homogenous Gaussian distributions. For Yes/No search, increasing set size from 2 to 8 will decrease unbiased percent correct from 75% to 66%. In contrast, for n-alternative Localization, increasing set size from 2 to 8 will decrease percent correct from 75% to 37%. Thus, for this comparison of tasks, the set-size effects are predicted to be larger for localization than for identification even if based on identical stimulus representations. Set-Size Effects and Attention A central issue in visual search is whether set-size effects are due to divided attention affecting perception or affecting later processing, such as the decision. On one side of the debate are those that have interpreted any set-size or cueing effects as due to attention acting on perception (e.g. Posner, Snyder & Davidson, 1980). On the other side of the debate are those that assume no effect on perception and estimate the magnitude of the effect due to decision processing (e.g. Shaw, 1980; 1982; 1984). By this second hypothesis, perception has unlimited capacity. Versions of this hypothesis goes by several names including the noisy integration hypothesis (Palmer, Verghese & Pavel, 2000) and it is closely related to an earlier version of independent channels theory (e.g. Gardner, 1973). Recent research has found that the noisy integration hypothesis can predict the setsize effects for a variety of simple search tasks (Palmer, Ames & Lindsey, 1993; Palmer, 1994; for a short review see Palmer 1995). These results have been generalized to a variety of stimulus materials (e.g. Dobkins & Bosworth, 2001; Monnier & Nagy, 2001; Verghese & Stone, 1995) and limitations to these results have also been found (e.g. Morgan, Ward, Castet, 1998; Poder, 1999). Here we pursue this hypothesis as a baseline for comparison. If perceiving both identity and location have unlimited capacity, then the noisy integration Page 6

9 hypothesis can predict performance in both tasks. If instead one task has limited capacity, then its set-size effects should be larger than predicted by this baseline. Another feature of the search studies just reviewed is that there are special provisions to isolate and control attention effects. Sensory interactions such as crowding are likely for displays with multiple stimuli and are easily confounded with set-size effects (e.g. Verghese & Nakayama, 1994). Therefore, we have limited our review to studies that use widely separated stimuli. In addition, recent studies have found that brief displays limited by masks show larger set-size effects than predicted by the simple version of the noisy integration hypothesis (Morgan, et al 1998). Therefore, we have limited ourselves to experiments without masks. Finally, the magnitude of the set-size effect interacts with the difficulty of the target-distractor discrimination and the heterogeneity of distractors (e.g. Duncan & Humphreys, 1989). Therefore, we have focused on studies that measure thresholds at a fixed performance level and use homogeneous distractors. Previous reviews of the search literature have concluded that under these conditions, the magnitude of set-size effects is quite predictable and consistent (Palmer, 1994; Palmer, et al., 2000). Recent Studies The advantage of models based on signal detection theory is that they provide a benchmark against which set size effects can be measured and the relation between localization and identification can be inferred. If a single class of models can account for performance in both tasks then this would suggest a common perceptual representation from which identity and location information is derived. Several recent papers have persued this logic, as reviewed below. Baldassi and Burr (2000) studied accuracy search for orientation discrimination. They used a two-target paradigm in which targets were either left-leaning or right-leaning in the presence of vertical distractors. Subjects made either a choice between left and rightleaning (2-Target Identification) or a choice among n-alternative locations. They found Page 7

10 larger set-size effects for identification than for localization. They argued that the results were consistent with different combination rules for the different tasks. Thus, they suggest common perceptual representations but with less optimal processing for identification than for localization. More recently, Baldassi and Verghese (2001) have updated this analysis with a treatment compatible with similar processing of identification and localization. Solomon and Morgan (2001) have also examined orientation discrimination using the same two-target search paradigm. They obtained both 2-alternative identification and n- Alternative Localization responses on each trial. The focus of their study was on duration effects and mislocalization errors and we return to mislocalization errors later in our article. Relevant here is a secondary analysis in their article that compared identification and localization thresholds. They found larger set-size effects for identification than for localization. They constructed explicit models for these tasks and found that their model predicted the observed difference in set-size effects. Thus, they replicated the results of Baldassi and Burr (2000) but interpreted it as consistent with similar processing of identification and localization. In summary, studies by Swensson and Judy (1981), Baldassi and Verghese (2001) and Solomon and Morgan (2001) all argue that models based on signal detection theory can account for identification and localization differences (at least under ideal conditions). Here we pursue this argument for a larger range of tasks. need to say more about why we are different and what our goals are. Experiment 1 To introduce our methods, consider Figure 1 that shows an example display. Subjects fixate a central cross, and either 2 or 8 tilted grating patches appear on an imaginary circle around a cross. A target is a grating patch tiled up and to the left, and depending on the task, the subject identifies the presence or absence of the target or reports its location. We vary the angle between the targets and distractors to create conditions of equal difficulty across Page 8

11 tasks, and then use the resulting psychometric function to find a performance threshold. The amount this threshold increases as more items are added to the display is our measure of the set-size effect. Within the visual search literature, the modal task involves identifying whether or not a target item is present (Yes/No search). For comparison with this literature, our first experiments contrast this task (which we will call Yes/No Identification) with a typical localization task: the target is always present and the subject indicates its location. We term this the n-localization task. because the target can be in any one of n locations. Later experiments consider a 2-Target Identification task, in which one of two targets is always presented and the subject indicates the identity of the target. We contrast this with the 2- Target Localization task in which the subject reports the location of the single target drawn from two possible targets (left or right leaning). Within the Yes/No Identification and n-localization tasks, we measure how performance degrades as additional items are added to the display. If one task displays more capacity limitations that the other, this is reflected in a larger increase in the threshold measured for that task with the larger set size. There are several factors related to the nature of the tasks that make a direct comparison difficult, but comparisons across tasks are possible within a modeling approach that takes into account the demands of each task. If we find capacity lmitations on one tasks but not another, this would suggest that the two sources of information differ in important ways, and would be consistent with a model that proposes a privaledged access or separate processing for once type of information. However, if we find a good account of performance across different tasks within a single modeling framework, we would suggest no need for a separate processing account. We have chosen to rely on accuracy search paradigms for several reasons, but perhaps the best resons is that we are able to derive parameter-free predictions from the models for the setsize effects in the different tasks. While these predictions rely on assumptions that may Page 9

12 have to be revised, they represent parameter-free property tests of the models rather than parameter estimation to produce fits to data. Methods Subjects The subjects were 6 young adults with normal or corrected-to-normal vision who received monetary compensation for their assistance. Each subject completed the experiment in 5-6 settings, each lasting approximately one hour. Stimuli An example display is found in Figure 1. The stimuli were 1.5 cycle/deg cosine gratings presented in a Gaussian window measuring 1 degree at half height. These were presented 6.8 degrees in the periphery around a central fixation cross. They were separated by 5.2 degrees. There were eight locations spaced evenly around an invisible circle, and the entire set of locations was rotated 15 clockwise to avoid displaying stimuli at the top, bottom or sides of the circle. To avoid specific alignments between the stimuli, the locations of the gratings was perturbed randomly up to 0.8 degrees horizontally and vertically. The phase of the grating was also randomized across items. The gratings were displayed on a 21" Macintosh grayscale monitor using luminance control and gamma correction provided by a Video Attenuator and the VideoToolbox software library (Pelli & Zhang, 1991). The monitor's background luminance was 40 cd/m 2. The contrast of the gratings was determined through a preliminary experiment for each subject in which contrast sensitivity was determined for a single grating displayed to the left or right of fixation. The subject indicated the location of the grating, and the contrast was adjusted to find the 82% contrast threshold using QUEST (Watson & Pelli, 1983). The final contrast was then tripled and the resulting value was used for all trials for that subject. The goal of this procedure was to make all gratings equally salient across subjects, conditions and tasks. The mean contrast Page 10

13 value used across subjects was 16%, and was the same for localization and identification tasks for each subject. Procedure Performance was measured at set size 2 and set size 8 for all tasks. For set size 2, the two items were always opposite each other to minimize the effects of eyemovements. Subjects were instructed to fixate centrally prior to the commencement of a trial. A block of 384 trials served as practice for each subject, and allowed the experimenters to watch for eyemovements on the part of the subjects and provide verbal feedback when eyemovements were made. This practice session also established a rough angle threshold for each condition using the Quest procedure. This procedure varied the angle between the target and the distractors so that on average the subject was 82% accurate at identifying the presence or absence of the target item. Because chance was 12.5% in the localization task for set size 8, threshold was fixed at 56.25% (halfway between chance and perfect). The actual experiment measured performance at 6 logarithmically-spaced angles centered around the preliminary threshold found via QUEST, and subjects completed 96 trials per angle. Each angle was tested in a separate block in order to allow subjects to adjust their yes/no criterion to maintain an equal bias. Feedback was given after each trial as well as at the end of the block. If subjects deviated substantially from the equal-bias condition they were given feedback and a warning to this effect. Subjects completed one task and then the other, but in different orders. The combination of 2 set sizes measured in two different tasks produces 4 psychometric functions. Each psychometric function was based on 96*6=576 trials. At the end of each block of trials subjects were informed of any biases in the yes/no task and cautioned if their percentage of one response type exceeded 60%. Responses were made on a numeric keypad and were not speeded. Page 11

14 Performance in the Yes/No task was averaged across the target-present and target-absent trials, under the assumption of equal bias. Subjects were given ample feedback about any biases, and the different angles were blocked to allow the subject to adapt a criterion to a specific angle differences. In addition, equivalent results obtain if target-present and targetabsent trials are analyzed separately. Results To illustrate our analysis, the top panel of Figure 2 shows an example of the two psychometric functions for the Yes/No Identification task for one subject. Performance in the Set Size 2 condition is better than the Set Size 8 condition. This implies that it takes a greater difference in angle between targets and distractors to achieve the 75% performance criterion. This decrease in performance can be quantified by fitting a Weibull function to each curve and finding the threshold angle that produces 75% correct performance for both set sizes. Thus subject MR has a threshold angle of 3.7 for set size 2 and 5.3 for set size 8. The middle panel of Figure 2 shows performance for the same subject in the n- Localization task. The analysis is the same as for the Yes/No Identification task, with the exception that the chance level for set size 8 is 12.5%, and thus the performance criterion half way between chance and perfect is 56.25%. Thus the two thresholds are measured at different performance criterion 1. However, the increase in threshold between set size 2 and set size 8 is still computed in the same manner, and it includes the fact that the performance criterion changed (one of many differences between identification and localization tasks). 1 Measuring all tasks at the same performance criterion is also possible, and would not change the substance of our modeling or predictions. However, there are some nice properties that emerge from our summary measure if we use the same criterion for all conditions (i.e. the performance criterion is always midway between chance and perfection). This is discussed in Experiment 5. Page 12

15 To provide a single summary measure that quantifies the decrease in performance as the set size is increased, we plot the threshold against set size, as shown in the bottom panel of Figure 2. These Threshold-vs.-Set Size plots have both axes logarithmically scaled to present multipicative effects as additive. The slope of these lines characterizes the set-size effect, which we can compute computationally as log( θ8) log( θ2) log - log slope = log( 8) log( 2) where θ 2 and θ 8 are the thresholds for set size 2 and 8 respectively. This log-log slope is a measure of the performance decrease as additional items are added to the display. Larger numbers imply a steeper slope and therefore a larger set-size effect. Thus Subject MR has overall better performance in the n-localization task (lower thresholds) but a smaller setsize effect for identification. Figure 3 shows the performance averaged across the 6 subjects. Across subjects, performance is better overall for the n-localization task, which also has a larger set-size effect. In fact, the obtained set-size effect for the n-localization task (0.57±0.09) is almost twice that of the Yes/No Identification task (0.32±0.08). Figure 4 shows the threshold-vsset-size graphs for the six subjects, and demonstrates the consistency of the effects across subjects: All have better overall performance and larger set-size effects for the n- Localization task. Models of Yes/No and n-localization Tasks To compare the log-log slopes from the two tasks, we turn to models based on signal detection theory. In particular, we consider the signal-detection theory analysis of the identification and localization tasks with the independent channels theory of set-size effects in visual search. There are three kinds of assumptions. First, we assume that the relevant percepts are noisy. We assume they correspond to single, real-valued random variables. Furthermore, for most specific predictions we assume equal-variance Gaussian Page 13

16 distributions. Second, we assume that the percepts for each stimulus are independent. Two aspects of independence are relevant: Unlimited capacity and statistical independence. For unlimited capacity, the quality of the representation of any one stimulus is independent of the number of the stimuli. For statistical independence, the value of the representation of any one stimulus is uncorrelated with another. Third, the two tasks require different decision rules. For Yes/No Identification, the subject is assumed to set an internal criterion, and say 'yes' if at least one of the items on the screen is tilted more than the criterion amount. For n- Localization, the subject is assumed to pick the location with the largest perceived tilt to the left. Thus to compare these two tasks we need a framework that builds in these different task demands. Predicted Set-Size Effects The top panel of Figure 5 shows the representations for the two stimuli under target absent (left half) and target present (right half) trials. The abscissa specifies the stimulus representation, which is an internal transformation of angle, with target items appearing more tilted to the left (which is further to the right on the Figure 5 graphs). The distractors shown in the Figure 5 are Gaussian and the numerical predictions also assume Gaussian distractors. However, the theory can make predictions from any reasonable distribution. For the Yes/No Identification task, we adopt a decision rule in which the subject makes a 'target present' (or 'yes') response if at least one stimulus item in the display exceeds an internal criterion. This rule is not quite ideal, although it is very close to ideal. Thus on a target absent trial the model samples twice from the target absent distribution, and reports a 'yes' response if either exceed the decision criterion. The assumptions of unlimited capacity and statistical independence imply that as additional items are added to the display, processing of a single item doesn't change. However, noisy integration in the decision process can still produce an increase in threshold for larger set sizes. Recall that within signal detection models we represent targets and Page 14

17 distractors as distributions centered around different means. For Yes/No Identification, this implies that on a target absent trial for set size 2, the subject has two chances to make a false alarm. However, with set size 8, the subject has eight chances to make a false alarm, as shown in the lower panel of Figure 5. To compensate and keep equal bias, the subject must increase the criterion for saying 'present', at the cost of reducing the hit rate. Thus the independent channels theory predicts non-zero set-size effects even with unlimited capacity. The model for the n-localization task is similar. Begin with the left panel of Figure 6 for set size 2. The n-localization task differs in that a target is always present and the subject reports the location that is most likely to contain a target. Thus there is always one target and one distractor distribution for the signal detection model of the set size 2 condition for localization. In our version, the subject chooses the maximum stimulus location and reports that as the target. This decision rule is ideal. Thus on a trial with set size 2, one samples from the two distributions and picks the location with the largest value. The subject makes an error if the distractor value exceeds the target value. With set size 8 there are 7 distractors, each of which might exceed the target item and lead to an error. This can be seen in the right panel of Figure 6, which plots the distribution of the maximum of the 7 distractors along with the target distribution. There is now much greater overlap between the two distributions, leading to reduced performance and higher thresholds for set size 8. Thus, this model also produces non-zero set-size effects even with independent channels. Complete descriptions of the models are found in the Appendix. For the present purposes we are concerned with comparing the predictions for the Yes/No Identification and n-localization tasks. Figure 7 shows the predicted values for the two tasks derived from the relevant signal detection models, plotted against the obtained means. Note that predictions for the log-log-slope depend on the exact values for the two set sizes, and are parameter free. For the set size 2 vs 8 comparison, the models predict a log-log slope of 0.22 for Yes/No Identification and 0.34 for n-localization. The models predict a lower set-size effect for Yes/No Identification than n-localization, which is what is found in Experiment 1. Page 15

18 However, the predicted values are substantially lower than those observed, which can be seen by the points lying above the oblique line in Figure 7. We will return to this quantitative discrepancy, but first we examine two new tasks and several control experiments. Experiment 2 The primary conclusion from Experiment 1 is that the n-alternative Localization task has a larger set-size effect than the Yes/No Identification task. On the face of it, this is consistent with the identification-without-localization hypothesis. This effect is also qualitatively captured by the signal detection models, suggesting that at least some of this difference is inherent in the task demands. To distinguish these interpretations, one might ask whether localization always has a larger set-size effect than identification. To test this possibility, we turn to two tasks that have been used in similar experiments by (Baldassi and Burr, 2000; Solomon and Morgan, 2001). In these variants, the distractor items are always oriented vertically, while the target can be tilted to the left or right of fixation. All trials contain exactly one target. In the 2-Target Identification task, the subject must indicate whether the target is tilted to the left or the right. In the 2-Target Localization task, the subject must indicate the location of the target, regardless of its orientation. Note that the identification task is fairly different than the Yes/No Identification task, since the subject must decide which item is tilted the most from vertical as opposed to setting an internal criterion for the amount of evidence they require for a 'yes' response in Yes/No Identification. However, the n-alternative and 2-Target Localization tasks are very similar. The only difference is that in the 2-Target Localization task there are two possible targets, Page 16

19 while in the n-alternative case there is only one possible target 2. Moreover, the identity of the target in the two-target case is irrelevant for the localization task. Experiment 2 measures set-size effects in both the 2-Target Identification and Localization tasks, using similar procedures as Experiment 1. We will again develop versions of the signal detection models appropriate to each task and compare the observed set-size effects to those predicted by the models. Methods Subjects The subjects were four naive subjects and the first author. Each subject received laboratory credit or payment for their participation. Stimuli The stimuli were identical to those of Experiment 1, with the exception that all distractors were vertically oriented and targets were tilted to the left or right of vertical. As before, the stimuli were set to a maximum contrast value that was three times their detection threshold. The mean contrast value used across subjects was 15%, and was the same for localization and identification tasks within a subject. 2 A second difference in our implementation is that the distractors are also tilted in the n-alternative case and vertical in the two-target case. However, this difference is irrelevant for modeling, and plays a role only in human data to the degree to which vertical is treated as a special case by the perceptual system (e.g. Gentaz & Tschopp, 2002). Page 17

20 Procedure Each subject participated in a contrast threshold procedure, followed by a practice session to establish the approximate angle threshold in each task and set size. The subject then completed 1154 trials per task over the course of two days to establish performance at 6 different target-distractor angle differences. The order of the two tasks was counterbalanced across subjects. Results Figure 8 shows the data from a typical subject in Experiment 2. The top and middle panels illustrate the psychometric functions for the 2-Target Identification and 2-Target Localization tasks. Comparing the thresholds for the two tasks (the width between the vertical lines), we see that the 2-Target Identification task has a larger set-size effect than the 2-Target Localization task for this subject. This is reflected in a steeper slope for the identification task in the bottom panel of Figure 8. This subject has better performance for the identification task at set size 2 despite the larger set-size effect. This pattern of data is repeated in the average data, as shown in Figure 9. At set size 2, the threshold is better in the 2-Target Identification task (4.7 ± 0.4 vs. 6.2 ± 0.8 for 2- Target Localization), but the 2-Target Identification task has a larger set-size effect than the 2-Target Localization task (0.57±0.10 for 2-Target Identification vs. 0.29±0.03 for 2- Target Localization). The pattern of results in Figure 9 is replicated by four of the five subjects, as illustrated by Figure 10. The exception is Subject TB, an author, who has extensive practice in these tasks. The finding that 2-Target Identification has a larger set-size effect than the associated localization task may be surprising when compared to Experiment 1. Recall that Experiment 1 found the opposite effect: larger set-size effects for localization than for Yes/No Identification. This reversal of set-size effects for the localization and identification tasks demonstrates that localization does not always have a larger set-size effect than Page 18

21 identification. In Experiment 1, we saw how task demands could account (at least qualitatively) for differences in set-size effects across tasks. Might the same be true for the two-target case? Below we apply signal detection theory to the two-target paradigm (c.f. Baldassi & Verghese, 2001; Solomon & Morgan, 2001). Signal Detection Models of the Two-Target Paradigms To develop these models, we begin from the same assumptions as before. We assume noisy representations that satisfy the independence properties of unlimited capacity and statistical independence. In addition, for specific illustrations and predictions, we assume equal-variance Gaussian distributions. For the two-target case, we assume that the distractors have a mean of zero and the target distributions have positive and negative means (corresponding to left-leaning and right-leaning respectively). See the Appendix for a more formal treatment. The two tasks must have different decision rules. For the 2-Target Identification task, the subject must indicate whether the target is left- or right-leaning. We assume that the subject picks the median of the distractor distribution (i.e. zero) as the measuring point and then finds the stimulus item with the maximum absolute deviation from this zero. The sign of this item determines the response. For the localization task, the subject also determines the maximum absolute deviation from zero, but then reports the location. Despite the independence assumptions, the models predict that both tasks will have nonzero set-size effects as the number of items increases from 2 to 8. The appendix contains the formal development of the models while here we provide an intuitive explanation for the set-size effects. For both tasks, an error is likely to be made if the maximum absolute deviation from zero is greater for one of the distractors than that of the target. For localization, if this occurs then an error will always be made. However, for identification the maximum absolute Page 19

22 deviation can come from a distractor, but if it is at the correct orientation the subject will make the correct response for the wrong reason. The predicted log-log slope for the 2-Target Identification task is 0.42, and for 2-Target Localization is Recall that the Yes/No Identification task has a predicted value of 0.22, and the n-localization task has a predicted value of Thus the 2-Target Localization task has a set-size effect that is approximately (but not exactly) half of the n-localization task, while the 2-Target Identification task has a significantly larger predicted value than the Yes/No Identification task. Figure 11 summarizes the predictions from the models and results from Experiments 1 and 2, averaged across subjects. These predictions are parameter-free. The signal detection theory-based models capture the qualitative ordering of the tasks: for example, the 2-Target Localization task has the lowest observed set-size effect (0.29) and is predicted to have the lowest value (0.17). In addition, the models capture the reversal of set-size effect magnitude seen in Experiments 1 and 2. However, despite this qualitative agreement, all four points lie above the identity line, and thus the current models are not consistent with the data at a quantitative level. In Experiments 3-5 we explore possibilities for this relatively poor quantitative fit. In Experiment 3 we examine possible spatial-temporal interactions with the two tasks that are suggested by some neurophysiological models. In Experiment 4 we address the role of sensory interactions in set size 2 with a cueing experiment, and in Experiment 5 we address the role of the number of responses in the localization tasks. Experiment 3: Sensory Interactions in Space and Time The signal detection models of Experiments 1 and 2 placed all the differences between the various tasks at the decision processes, and assumed a common perceptual representation across tasks. One test of this assumption is to vary the nature of the perceptual representation to see whether this affects the localization and identification tasks Page 20

23 in different ways. If decision processing accounts for the task differences, then the results observed for one pair of spatial and temporal frequencies will generalize to other frequencies. In contrast, if perceptual processing differs for what vs. where, then spatial and temporal frequency manipulations allow one to vary the mediating mechanism. In particular, our choice of stimulus parameters was motivated by research on parallel pathways in the visual system (Merigan & Maunsell, 1993). The idea is that one can selectively stimulate one or another visual pathway using different combinations of spatial and temporal frequencies. While this may not isolate a particular pathway completely, it is likely to vary the contributions of the different pathways. To test the spatial and temporal frequency dependencies of the two tasks and determine whether these affect the capacity limitations, in Experiment 3 we measured the set-size effects at various combinations of spatial and temporal frequencies in the Yes/No Identification and n-alternative Localization tasks of Experiment 1. In addition, to assess the dependence on contrast level, we included a higher-contrast condition for some combinations of spatial and temporal frequencies. Method The methods were similar to Experiment 1, with the exception that the gratings were either 1.5 or 3 cycles per degree, and were modulated at either 2 or Hz. Subjects completed n-localization and Yes/No Identification, at six different combinations of spatial frequencies (1.5 or 3 cpd), temporal frequencies (2 or Hz) and contrast (3x contrast threshold or 5x contrast threshold). The mean contrast value varied across flicker rates and spatial frequency. For the 2 Hz flicker rate, the mean contrasts across subjects were 14%, 27% and 16% for the 1.5 cpd, 1.5 cpd higher contrast and the 3 cpd conditions respectively. For the Hz flicker rate, the values were 35%, 60% and 76% respectively. These values were identical for paired localization and identification tasks. Page 21

24 Subjects Three naive subjects and the first author served as subjects. All were experienced with the general paradigm. Subjects completed the identification task first and then the localization task. Procedure The procedures were similar to Experiment 1, except that the temporal waveform of the stimulus was modulated either at 2 Hz or 18 Hz. This modulation included the reversal of the grating (rather than modulating to gray). Results Figure 12 shows the results of the various combinations of the three variables. To assess whether the three types of manipulations (spatial frequency, temporal frequency and contrast) had any effect at all on the set-size effects for the two tasks, we performed a within-subjects one-way ANOVA for each task. For both the Yes/No Identification and n- Alternative Localization tasks, no significant differences were found between any of the six conditions (Identification: F(5,15)<1; Localization: F(5,15) = 1.65, p>0.05). Based on these results, we conclude that the identification and localization tasks do not show strong spatial and temporal frequency dependencies in our measure of capacity. This is consistent with an account in which both identification and localization tasks rely on a representation that has similar spatial and temporal properties, such as the signal detection theory models that assume that both tasks access a common perceptual representation. Without differences across spatial and temporal frequencies, we combined across conditions to produce an overall summary measure for the two tasks for comparison with Experiment 1, as shown in Figure 13. For identification, the mean log-log slope is 0.27±.04, and for localization it is 0.46±0.03. These are consistent with the Experiment 1 results, and Page 22

25 are both above the predictions from the unlimited capacity signal detection models. Thus, we can generalize our results to a range of spatial and temporal frequencies. Experiment 4- Cueing Control One possible explanation for our poor quantitative fits may be that sensory interactions between items play a role in our displays. We deliberately tried to avoid sensory interactions by using widely-spaced items: the nearest neighbor is a distance three times the width of a single stimulus item. In addition, we attempted to disrupt sensory interactions by randomly jiggling each item around its location, and randomizing the phase of the 1.5 cpd cosine gratings. However, it is possible that despite our efforts some kind of sensory interactions were present in our tasks. To test this we had subjects complete two conditions. The first was a control condition identical to the n-alternative Localization task from Experiment 1. The second condition measured set-size effects by cueing a subset of items in otherwise identical displays. For the set-size 2 case, the display had two pre-cues prior to the onset of the trial that indicated the relevant locations on this trial. This was always followed by 8 stimulus items, and the target appeared in one of the two pre-cued locations. For set size 8, all 8 locations were pre-cued. The essential question is whether the set-size effect differs systematically between the original localization task and the cued version of the localization task. If the set-size effects using cues are smaller than when simply varying set size, we will have evidence for sensory interactions or ineffective cues. If the set-size effects using cues are larger, we can attribute this to a mechanism such as crowding (Bouma, 1970; Pelli, Palomares & Majaj, 2003). Methods Subjects The subjects were five new naive subjects. Each subject received laboratory credit or payment for their participation. Page 23

26 Stimuli The stimuli were identical to those of Experiment 1. The mean contrast value used across subjects was 17%, and was identical for cued and no-cued conditions for each subject. Procedure There were two different localization tasks. One was identical to the n-alternative task in Experiment 1. The other always presented 8 items on each trial, but for set-size 2 pre-cued two locations with small plus signs prior to the start of the trial. The stimulus onset asynchrony (SOA) between the cue and the stimuli was 800 ms. The target then appeared in one of the two pre-cued locations, with the other location containing a distractor. The set size 8 condition used 8 precues in this task, with one target and 7 distractors. Each subject participated in a contrast threshold procedure, followed by a practice session to establish the approximate angle threshold in each task and set size. The subject then completed 1154 trials per task over the course of two days to establish performance at 6 different target-distractor angle differences. The order of the two tasks was counterbalanced across subjects. Results Table 1 contains the thresholds for set size 2 and 8 for the two localization tasks. Averaged across subjects we find no differences between the two localization tasks. The obtained set-size effect for the original version of the n-alternative Localization task is 0.47±0.07, and for the cueing version is 0.43±0.10. We find no evidence for crowding, since the set-size effect is smaller (though non-significantly so) than the original localization task. In addition, the obtained log-log slopes are similar to those found in Experiments 1 and 3. Page 24

27 Experiment 4 demonstrates that one can obtain the same set-size effect without changing the stimulus. Thus, subtle sensory interactions such as crowding do not account for the setsize effects observed here. Experiment 5- Coarse Localization All of the localization tasks discussed so far require a relatively complex decision rule for set size 8, since the subject must choose between one of 8 locations. However, the associated identification tasks require selecting from one of two choices. This difference may inflate the threshold at set size 8 for the localization tasks. To address this possible effect, in Experiment 5 we reduced the number of possible responses from 8 to 2 by slightly separating the 8 locations into two identifiable groups, one on the left and one on the right side of the screen. If the target appeared in one of the four locations on the left side of the screen, the subject indicated 'left' with a button press, and if the target appeared in one of the four locations on the right side of the screen the subject indicated 'right' with a different button press. Subjects also completed the Yes/No Identification task, which was identical to Experiment 1 with the exception that subjects used the same spatial layout as the new Coarse Localization task. Methods Subjects The subjects were three new naive subjects and the first author. Each subject received laboratory credit or subject payment for their participation. Stimuli The stimuli were identical to those of Experiment 1, with the exception that the eight locations were separated into two groups by separating the positions into two groups and shifting the middle items horizontally away from the midline. The horizontal shift was done Page 25

28 proportionately, such that the items at the edges of the display did not move at all, and items near the midline of the monitor move the most. This slight movement clearly identifies the locations for the subjects as belonging to the left or right group without enlarging the overall circle by any substantial degree. The mean contrast value used across subjects was 17%, and was identical for localization and identification tasks for each subject. Procedure Each subject participated in a contrast threshold procedure, followed by a practice session to establish the approximate angle threshold in each task and set size. The subject then completed 1154 trials per task over the course of two days to establish performance at 6 different target-distractor angle differences. The order of the two tasks was counterbalanced across subjects. The yes-no task was identical to that in Experiment 1. The Coarse Localization task differed from the n-alternative Localization task in that subjects made one of only two possible responses. If the target appeared in any of the four locations on the left of the screen, the subject pressed the '5' key, and if it appeared on the right side of the screen they pressed the '6' key. Results With the previous n-alternative Localization task we used a threshold criterion of 56.25% correct since chance was 1/8. For this Coarse Localization task, chance is 1/2, and so the Yes/No Identification and Coarse Localization tasks have the same threshold criterion of 75% correct. One benefit of keeping the decision criterion at midway between chance and perfection for all task is that the predictions of the signal detection models for localization do not change: the predicted log-log slope (set size 2 and 8) for Coarse Localization is 0.34, which is the same as for the n-alternative Localization task (see Appendix). Page 26

29 Figure 14 shows the data from Experiment 5, averaged across the 4 subjects. The obtained set-size effects for the two tasks, averaged across subjects, was 0.44±0.08 for the Coarse Localization task and 0.28±0.08 for the Yes/No Identification task. These are in agreement with those from Experiment 1 and confirm the finding (as predicted by the models) that n-alternative Localization has a larger set-size effect than Yes/No Identification. The obtained values are higher than the predicted values from the signal detection models, as was observed in Experiments 1-4. The obtained value for the Coarse Localization task is somewhat smaller than that found in Experiment 1, but it is in good agreement with other n-alternative Localization results reported in Experiments 3 and 4. We conclude that the Coarse Localization manipulation does not have a major effect on the magnitude of the set-size effect or the degree to which the models can (or cannot) provide a quantitative account of the data. Summary Of Results The results of Experiments 1-5 are summarized in Figure 15. The models based on signal detection theory capture the qualitative ordering of the tasks, but fail to capture the quantitative fits of the data. The dashed line in Figure 15 is the best-fitting linear regression line, and emphasizes both the good qualitative ordering and the poor quantitative fit of the predictions. In addition, the model accurately predicts the reversal seen in Experiments 1 and 2 with regard to whether localization or identification has a larger set-size effect. As we have seen, the answer depends on the nature of the tasks, and the signal detection models accurately capture this reversal by building in different decision rules appropriate for each task. Note that the predictions in Figure 15 are parameter free. Mislocalization Analysis One possible explanation for the poor quantitative fit of the data in Figure 15 is a failure of independent processing of the individual stimuli, which can lead to, among other things, Page 27

30 mislocalizations. Such mislocalizations were found by Solomon & Morgan (2001) in a very similar paradigm. To measure the degree of mislocalization with set size 8, we computed the error rates for each position conditioned on how far away each position was from the target. The data for Experiments 1-4 are shown in Figure 16. To combine across subjects, we divided the six angles that were used to test each subject by that subject's threshold angle. This produces a relative angle, where 1.0 is the angle that produces approximately threshold performance (56% correct for set size 8), and smaller angles represent more difficult conditions. The angles used for each subject were then assigned to one of 8 relative angles based on the angle's relation to the threshold angle for that subject in that condition. Performance at each relative angle was then averaged across subjects to produce the graphs shown in Figure 16. As shown in Figure 16, positions adjacent to the target have systematically more errors than those locations 2 and 3 positions away from the target. The simple signal detection model presented previously does not predict this effect. One might propose that these mislocalizations come from confusions by subjects when translating their chosen location into a response on the keypad. However, two lines of evidence make this explanation implausible: In all of the set size 2 conditions, subjects can make any of 8 possible responses, but they made virtually no responses that do not specify one of the two stimuli. In addition, in Experiment 4 relevant set size 2 condition, subjects almost never made responses other than at one of the two pre-cued locations, despite the fact that 8 stimuli were presented. Thus, the basis for the mislocalizations is probably not response confusion on the part of the subjects, at least in terms of how they translate their internal representation into a keypress. Below we consider an alternative explanation that can not only account for our mislocalizations, but also better account for our set-size effects. Page 28

31 Extending the Signal Detection Models: Overlapping Channels To account for mislocalizations in a similar task, Solomon & Morgan (2001) proposed a model in which subjects pick the incorrect nearby location on a certain percentage of the trials. An alternative suggestion in Parkes, Lund, Angelnui, Solomon and Morgan (2001) is that subjects pool information (average) from nearby stimuli. We use this pooling idea as the basis of our generalized model. In it, we relax the assumption of stochastic independence and replace it with a weighted pooling among neighboring stimuli. This is also similar to models of crowding (e.g. Pelli et al., 2003) and the older interactive channel models (Estes, 1982). Our approach is to develop the model with one free parameter to account for the mislocalization data, and then with this parameter fixed see if the amended models also account for the set-size effects. The overlapping Channels Model assumes that on a given trial the representations of the 2 or 8 stimuli are obtained with unlimited capacity processing. In other words, increasing the number of items does not change the quality of the information at each location. However, the representation is dependent on its neighbors in another way. The interaction is by assimilation where the values of the representations associated with each stimulus blend together. We call this an Overlapping Channel model to distinguish it from the Independent Channels and the more general interactive channel models. The Overlapping Channels model is formulated as follows. Assume that stimulus location i has value s i, and that the two nearby locations have values s i-1 and s i+1. Assimilation produces a new value for location i, denoted s i ' ' s i si + ωsi + ωsi = 1+ 2ω by the weighted average: where ω represents a free parameter that controls the amount of assimilation from nearby items. If ω is zero, no information is combined from nearby items and we have the independent channels model. If ω is positive then nearby items are blended with item i. A ω of 1 indicates the stimuli are averaged with equal weights. This process occurs in parallel for Page 29

32 all 8 locations. This has no effect for set size 2 because the two items are always wellseparated. To find the appropriate value for ω, we computed the predictions for the model with various values of ω. To directly compare with our data from Figure 16, we fit the predicted percent correct psychometric function and found the value of d' (i.e. difference between target and distractor angles) that produced the criterion performance of percent correct. We then computed the relative angles as in Figure 16 and re-fit the model on these angles. This produces comparable values on the abscissa and allows for direct comparisons between theory and data. Although the Overlapping Channels model is simple in principle, it is difficult to solve analytically. Thus we resorted to Monte Carlo simulation with 1 million simulated trials per point. This produces predicted error probabilities that have standard errors of less than 1%. Threshold values for particular set sizes were found via linear interpolation, and hierarchical simulations were used to avoid biases in the linear interpolation procedures. The best-fitting parameters for ω differ depending on the task. The best ω values for the n-alternative Localization tasks were 0.42 and 0.38 for Experiments 1 and 3, and the cued and uncued versions of Experiment 4 were 0.36 and 0.38 respectively. Based on these four estimates, we picked a value of 0.38 for all n-alternative Localization predictions. The bestfitting value for the 2-Target Localization task was This implies greater assimilation in this task. Thus we used different values of ω for the one-target and two-target tasks 3. 3 The larger value of w for the 2-Target tasks is not a consequence of the fact that the 2-Target tasks had vertical distractors while the n-localization and Yes/No Identification tasks had tilted distractors. The 2- Target Localization data seem to require more assimilation. Page 30

33 Consequences of Mislocalizations for Set-Size Effects The lack of independence implied by the assimilation process changes the expected setsize effects. Because the assimilation process is only relevant to the set size 8 condition, the assimilation model predicts an increase of the set-size effect relative to independent processing (see Appendix for details). These predictions are closer to the observed values for the localization tasks. The results of the spatial and temporal manipulations of Experiment 3 suggest that the identification and localization tasks rely on a common perceptual representation, since there were no interactions between the tasks and the perceptual variables for the set-size effect. If this is indeed the case, then we should also apply the assimilation process to the identification task. The mislocalizations do not affect the identification decision directly, but the assimilation process may increase the false alarm rate (and decrease the hit rate) selectively in the set size 8 condition. Thus the overlapping channels model also increases the predicted set-size effects (see Appendix for details). Figure 17 shows the fits of the Overlapping Channels models, using weights estimated from the mislocalization data. The assimilation models provide a better account of the setsize effects (and do so with no parameters fit directly to the set size data). The data points are distributed much more around the line of best prediction, and the only outlier is the 2- Target Localization task 4, which we discuss below in conjunction with a discussion of other models. 4 The set-size prediction is extremely senstive to the value of ω. A value of 0.55 for ω gives essentially a perfect fit in to the set size data while producing a somewhat worse fit of the mislocalization data. Page 31

34 Alternative Models of Mislocalizations An alternative approach is the 'response substitution' model of Solomon & Morgan (2001), in which on a certain percentage of trials the subject mislocalizes the target to a nearby location. Because it assumes errors on a certain percentage of trials, asymptotic performance does not reach perfection. While we do not have data at extremely large angles, some of our experiments offer some support for this prediction. Consider the data of Figure 16. For Experiments 1, 3 and the cueing version of Experiment 4, the error rates for the positions next to the target appear larger than that predicted by the assimilation process at the larger angles. This appears to be a limitation of assimilation and is consistent with the response substitution mechanism proposed by Solomon and Morgan. At this point, we can only say that the data are consistent with aggregation between neighbors by either assimilation or response substitution. General Discussion We asked how various tasks that fall under the loose 'what' vs. 'where' distinction are processed differently by subjects. Our primary conclusion is that, for simple feature search, identification and localization show different set-size effects, but virtually all of the differences between the tasks are due to the specific details of the decision mechanism required to make a response. The most typical task in accuracy search is a precence/absence judgement, and for these tasks (Experiments 1, 3 and 4) we found larger set size effects for localization than for identification. This held even when the localization task involved just 2 choices. The order of this effect reversed when two targets were used in either a 2-alternative forced choice identification task or the analagous localization task (Experiment 2). All of these orderings were qualitatively accounted for by models based on signal detection theory, which include the specific details of each task when making parameter-free predictions for the set-size effects. Adding an additional assimilation mechanism to the perceptual representation allowed the models to make better quantitative predictions for the tasks. Page 32

35 These results are consistent with identification and localization of simple features being based on a common perceptual representation. These assumptions, while sufficient for simple feature search, may break down for more complex search where relational information is required to identify targets (e.g.. Exp 5 Palmer, 1994). These tasks probably depend on processes that have some kind of limited capacity. But even here the details of the tasks probably still influence the magnitude of the set-size effects. relate to Johnston and Pasher? Limitations of their approach that we overcome? Converging evidence? Intuitions about Task Differences While the signal detection-based models are well defined, more is needed to develop some intuitions about the various predictions. In particular, why does 2-Target Identification have such a large set-size effect in Experiment 2, compared to 2-Target Localization? To address this question, we next develop some intuitions starting from what is known as the two-by-two paradigm. The two-by-two paradigm was designed to match performance in identification and detection experiments (Graham, 1989; Klein, 1985; Thomas, 1985) and can be adapted to compare identification and localization. It requires two responses, one for each of the two judgments of interest. In a typical application, two displays are presented sequentially. One display has one of two possible targets and the other display has a blank stimulus. One must make a 2IFC response as to which display contained a target (detection) and a forced choice response as to which of the two targets was present (identification). For a variety of assumptions about the decision rule used in each task, there is predicted to be equal performance between the detection and identification tasks. One can apply this paradigm to identification and localization in accuracy search by presenting two search displays. One display contains a target and the other display contains only distractors. Subjects make two responses: a two-choice identification response Page 33

36 indicating which display had the target and a two-choice coarse localization response indicating on which side of the display was the target. This design makes the same set-size predictions for localization and identification. For the set sizes 2 vs. 8 comparison, the single-target version (e.g. the target is right leaning and distractors are vertical) has a predicted log-log slope of 0.278, while the 2-Target version (where the target could be either right- or left-leaning) has a predicted log-log slope of Thus, all else equal, the 2-target case produces a smaller set-size effect for both localization and identification tasks. This 2IFC paradigm provides a standard against which to compare the other tasks (see Table 2). Consider the n-alternative Localization task of Experiments 1, 3 and 4, and the course localization task of Experiment 5. The predicted log-log slope is 0.34 for both n- Localization and Coarse Localization tasks, but the 2IFC has two intervals and thus twice as many stimuli. If we double the number of stimuli for the n-alternative and Coarse Localization tasks, both tasks have predicted log-log slopes of 0.278, which matches that of the 2IFC tasks. Thus Coarse Localization, n-alternative Localization and 2IFC predictions match if the number of stimuli is appropriately defined. Changing the number of potential targets makes the above relation no longer true, which explains in part why the 2-target case is different. For the 2-Target case, the 2IFC predicted set size is 0.176, and the 2-target Coarse Localization matches that if the set size is doubled. However, the n-alternative Localization task with 2 targets and twice the number of stimuli as used in Experiment 2 (doubled for comparisons with the 2IFC task) does not match. The predicted set-size effect for 2-target, n-alternative Localization is 0.17, which is close but not the same as the for 2IFC and 2-Target Coarse Localization. The predictions diverge much more for the identification tasks. For the 1-target case, the 2IFC prediction is For Yes/No with double the set size, the prediction is Thus Page 34

37 there is a large difference in the set-size effects for the 2IFC and Yes/No Identification tasks. The 2-Target Identification case also has a difference from 2IFC. The 2-target version for 2IFC has a log-log slope prediction of The 2-Target Identification version with the set size doubled (for direct comparison with 2-Target 2IFC), has a predicted set-size effect of 0.305, which is quite different. The Yes/No 2-Target Identification task (which we did not run) has a doubled set-size effect of Thus, the difference between Yes/No and the 2AFC identification cases in the present work produce the major portion of the interaction seen in Experiments 1 and 2 with the reversal of the identification/localization relation. To summarize, a signal detection theory analysis using the 2IFC baseline reveals the following effects: 1) The two-target task reduces the set-size effect relative to the one-target task, all else being equal. It doesn't affect the identification/localization comparison per se. 2) Predicted localization set-size effects are in line with the predictions from the 2IFC task as long as the number of stimuli are doubled to make comparisons with the two-interval task. The 2-target, n-localization task differs only very slightly from the Course Localization and 2IFC 2-target predictions. 3) The yes/no and 2-Target Identification tasks both significantly differ from the 2IFC task, and the main reason for these differences arises from the Yes/No and 2AFC identification tasks being very different for the 2-target case. Thus, the major source of interactions seen in Experiments 1 and 2 arises from the difference between yes/no and two-choice identification in the 2-target case. The above discussion highlights the need to consider the details of the tasks if meaningful comparisons are to be made. This can be accomplished either through a modelbased approach, as we have done, or through close matching of tasks such as with the two- Page 35

38 by-two paradigm. Regardless, before establishing a difference between identification and localization tasks one must address the different decision rules required by each task. Relation to Prior Studies Thus far, we have focused on using set-size effects to compare localization, and identification tasks. In the next two sections, we consider two other methods of comparing these tasks. Absolute Performance Perhaps the simplest comparison of localization and identification is a comparison of absolute performance. One of the first studies to directly address this issue in this way was Atkinson and Braddick (1989). In the conditions of interest, they displayed stimuli varying in orientation and required both two-target Coarse Localization and 2-Target Identification. They adjusted the time between the display and a mask to estimate a threshold and found the localization/identification threshold ratio to be 0.8. Thus, localization performance was better than identification performance. Such a direct comparison was criticized by M. Green (1992) as not taking into account the decision structure of the task (as also argued here). Green used a two-by-two task with two display intervals with varying orientation. One response was a 2IFC identification judgment(called detection in the article) and the other was a coarse left-right localization judgment. Using this combination of tasks, performance was essentially identical for the two tasks. Green argued on the basis of signal detection theory that this particular pair of tasks should result in identical performance while other parings such as used by Atkinson and Braddick (1989) need not be identical in performance. More recently, Solomon and Morgan (2001; see also Baldassi & Burr, 2000) measured threshold values for a 2-Target Identification and an n-localization task (similar to those here). In their low set-size condition, they found a localization/identification threshold ratio Page 36

39 of around 1.2. They determined that the signal detection theory predication for their combination of tasks was a ratio of 1.5. Thus, they interpret the deviation of the observed ratio as roughly consistent with a close binding of location and identity. In our own experiments, we can also measure the localization/identification threshold ratio. Consider the set size 2 conditions to make analysis of the most distinct from the setsize analysis. For the 1-target tasks (Yes/No and n-localization), the observed ratio was 0.5 and the prediction was 0.5. For the 2-target tasks (2-Alternative Identification and n- Localization), the observed ratio was 1.3 and the prediction was 1.6. Thus, in both cases the observed ratios were near the predications. Moreover, the ratios reversed between the two experiments just as did the relative magnitude of the set-size effects. This evidence is consistent with the decision structure of the tasks determining performance rather than identity verses localization, per se. Dependency Between Identity and Localization Another important approach to the relation between identity and localization are direct estimates of the dependency between these judgments when they are made within a single trial. Specifically, one estimates the probability of identification given no location information or the complement (the probability of localization given no identity information). Close binding predicts both of these conditional probabilities are low. The localization-without-identification and identification-without-localization hypotheses predict one or the other probability is above chance. Early experiments found support for both identification without localization (e.g. Treisman & Gelade, 1980) and localization without identification (e.g. Müller & Rabbitt, 1989). A second generation of studies on this question began with Johnston & Pashler (1990). As reviewed in the introduction, they identified several ways the conditional probabilities can be influenced by poor choice of tasks. They attempted to control for these effects and were able to account for much if not all of the observed conditional probability. Since then a Page 37

40 variety of studies have pursued these analyses further (e.g. Bloem and Van der Heijder, 1995; Bronwer and Van der Heijder, 1997; and Donk and Meineike, 2001). While room for debate remains, most of these studies support a close binding between identity and location. Conclusions Prior studies have shown large differences in set-size effects for localization and identification tasks. We show here that these effects are specific to the details of the tasks and not specific to localization verses identification. Thus, these effects do not support hypotheses specifying differential processing of location and identity information. Instead, we find that the observed results can all be accounted for by simple models based on signal detection theory and independent channels theory that are tailored to the decision structure of each task and allow overlap between channels. Page 38

41 Appendix In this appendix we derive the predictions for the various independent channels models. These are developed for arbitrary set sizes, but for examples we will focus on the set size 2 and set size 8 conditions used throughout the present work. We also discuss the Overlapping Channels models. Common Notation and Assumptions We follow the notation commonly used in signal detection theory (Green & Swets, 1966; for more detail see Palmer et al, 2000). The representation of each stimulus corresponds to a real-valued random variable. For targets and distractors, these random variables are designated T and D, respectively (in our task corresponding to left- and righttilting gratings). We assume two independence properties. First, for all of the models, we assume unlimited capacity in the processing that results in the relevant representations. Hence, the mean and variance for any T or D random variable are independent of the set size. Second, for all but the Overlapping Channels models, we assume statistical independence between the random variables. As a result, on a particular trial, the value of any random variable, say D1, is independent of the value of any other representation on that same trial, say D2. We further assume that all the distractor random variables D i are identically distributed with a density function of f i and a cumulative distribution function of F i. In addition, we assume that targets are represented by a shift family of the same distributions. Specifically, if the distractor density is f(x) the target density is f(x-ws) where s is the physical description of the target distractor difference and w is a free parameter relating physical units to the units of the internal representation (e.g. d'). Finally, for all specific numerical calculations, we make the more specific assumption that all individual target and distractor random variables are Gaussian. However, similar calculations can be done for another distribution. Page 39

42 Yes/No Identification The analysis of set-size effects for Yes/No tasks dates to Tanner (1961) and a derivation of the effect on threshold can be found in Palmer et al Performance on target present and target absent trials is characterized by the hit and false alarm rates, respectively. Both measures summarize the probability of saying 'target present'. In the model, we assume that each stimulus engenders an internal response which can be compared with a criterion c. We further assume that the subject indicates 'target present' if at least one stimulus response has a value above the criterion c. First, consider the distributions associated with a single stimulus item. For a single distractor, the probability of a false alarm is p(fa) = p(d>c), which becomes = 1-F(c). A1 Similarly, the probability of a Hit is: p(hit) = p(t>c), which becomes = 1-F(c-ws). A2 For n items where n is greater than 1, we have that p(fa) = p(d 1, D 2,..., or D n > c) = 1-F(c) n. A3 With 1 target and n-1 distractors, we have: p(hit 1 target, n-1 distractors) = p("at least one above criterion"), which becomes = 1 - F(c-ws) F(c) n-1. A4 This computes the probability of a miss of the target times the probability of correct rejections for all n-1 distractors. One minus this product is the probability of a hit. This implies that we can have a correct response on a target-present trial because one of the distractors exceeded threshold. In fact, this happens more and more as the set size n Page 40

43 increases. Of course, this also occurs on target-absent trials, and the subject must adjust their criterion accordingly. We assume that subjects adjust their criterion in the set size 2 and set size 8 blocks so that they have equal bias, which implies P(Hit) = P(CR). To solve these equations to find an angle threshold, we assume that subjects have equal bias, and solve for the unknowns c, w and s. The goal is to predict the change in threshold sensitivity as a function of set size. Thus, although we can't measure s directly, we can measure the threshold sensitivity as a function of set size, which is s t (n). The next step is to solve for s t (n). To do this, note that the criterion false alarm probability is k =.25 and the criterion hit probability for threshold is 1-k =.75. Setting Equations A3 and A4 equal to these numbers and substituting s t (n) for s in Equation A4, we get: k = 1 F( c) n A5 and k = F( c ws ( n)) F( c) t n 1. A6 The parameter w is a free parameter that indicates the scaling function for perceptual sensitivity for each subject. The next step is to solve for c from equation A5: 1 Fc () = ( 1 k) / n [ ] 1 1 n c = F ( 1 k) / A7 and then use this to solve for s t (n) by substitution with equation A6 (see Palmer, 1998): / n n / n st ( n) = ( / w) F 1 1 [( k) ] F K/( 1 K) ( 1 ) A8 { [ ]} where F -1 represents the inverse normal cumulative function. Note that we still have the free parameter w, which represents perceptual sensitivity. Rather than fit this parameter, we instead consider the ratio of two set sizes, which causes both w values to cancel. By convention, we express this by the log-log slope, which equals: log( st( n2)) log( st( n1)) α = log( n ) log( n ) 2 1 A9 Page 41

44 where α is the set-size effect expressed as a ratio of log differences (actually a ratio of ratios), n 1 and n 2 are the first and second set-size effects used (2 and 8 in the present study) and s t (n) is the angle threshold for set-size n. Assuming a criterion of k =.25 and 1-k =.75 for false alarm and hit rates, respectively, and w=1, the predicted threshold angle for set size 2 is 1.67 degrees and for set size 8 is 2.27 degrees. This gives a predicted log-log slope of 0.22 for these set sizes. n-localization Here we follow the analysis of n-localization of Shaw (1980). As with identification, the stimulus is uni-dimensional, varying only on angle. For localization, the model assumes that the subject simply chooses the location of the stimulus that appears to lean the left the most. This is equivalent to a max rule on an appropriately standardized representation. For set size n, the probability of choosing the correct location is: p[t>max( D 1,.. D n-1 )], which becomes n 1 p( correct) = f ( x d) F( x) dx A10 This equation integrates the product of a shifted density function representing the target distribution, multiplied by the cumulative distribution representing the distractor distribution (for n = 2) or distributions (for n greater than 2). To find a threshold angle, we assume w=1 and use a numerical procedure to find the s value that produces 75% correct for n= 2 and 56.25% correct for n=8. This produces a threshold of 0.95 for n= 2 and 1.53 for n=8. Using the log-log slope formula (Equation A9), the log-log slope is 0.34, which is slightly higher than the 0.22 for Yes/No. Coarse localization In Coarse Localization, the target item is grouped with n/2-1 distractors on one side, and there are n/2 distractors on the other side. Subjects indicate which side of the display Page 42

45 contained the target. Subjects can get the right answer accidentally by erroneously believing that one of the distractors on the target side was the target. If they choose one of the distractors on the side opposite the target, they get the wrong answer. The probability of selecting the correct side is: p(c) = max (T, D 1,.. D n/2-1 )> max (D n/2,.. D n-1 ), which becomes n n 2 p( correct) = 1 f( xfx ) ( wsfx ) ( ) dx 2 A11 With ws set to 1.0, this function equals.76 with n = 2, and.65 with n=8. The lower percent correct for the n=8 case is due to the fact that with more distractors on the wrong side, the more likely that one of them exceeds the target value. As with Fine Localization, to find an angle threshold, a numerical procedure is used to find the value s t that produces 75% correct for both n= 2 and n=8. For w=1, this produces a threshold of 0.95 for n= 2 and 1.53 for n=8. Perhaps surprisingly, these are the same values computed for Fine Localization (although recall that for Fine Localization we fit an angle threshold of 56.25%). Indeed, using the log-log slope formula from above, we again compute a log-log slope of 0.34, identical to the Fine Localization value and slightly higher than the 0.22 value for Yes/No. 2AFC Identification (Discrimination) Here, we follow previous analyses of Solomon and Morgan (2001) and Baldassi & Verghese (2002). In our example of 2-Target Identification (also called discrimination), the target item can either be a left-leaning or right-leaning grating, and the distractors are all vertical. Two responses are possible on each trial, called the 'larger' response and the 'smaller' response. Define the target deviating from the distractors in a positive direction as T and the target deviating in a negative direction T'. The model includes the assumption that the subject chooses the response that corresponds to the percept that deviates the most from the median distractor percept, which we define as zero. Suppose that a larger target is Page 43

46 presented on a trial. Two events lead to a correct response of 'larger'. First, the target item may be perceived as deviating most, and deviating in the positive direction. Second, a distractor item may be perceived as deviating most, and deviating in a positive direction, which also leads to the correct answer (but for the wrong reason). Thus the probability of correctly saying 'larger' if T is presented on the trial equals [ 2 3 n 2 3 n ] A12 p(" larger " T) = P max( T, D, D,..., D ) > min( T, D, D,..., D ) which expresses the logic that the correct "larger" response will be given if the largest deviation from zero is in the positive direction, regardless of whether it comes from a target or one of the n-1 distractors. A similar formulation exists for the 'smaller' response: [ 2 3 n 2 3 n ] A13 p(" smaller " T' ) = P max( T', D, D,..., D ) min( T', D, D,..., D ) which says that the correct 'smaller' response will be given if the target T' or one of the distractors provides the largest deviation in the negative direction. The probability correct is given by: P( correct) = P(" l arg er" T) P( T) + which is P(" smaller" T) P( T' ) [ 2 n 2 n ] + P( correct) P max T, D,..., D min T, D,..., D P( T) = ( )> ( ) [ ( 2 n) ( 2 n) ] A14 P max T', D,..., D min T', D,..., D P( T' ) which computes the probability of a correct response for each type of target times the probability of each target appearing (set by the experimenter;.5 in our case). To derive equations for this expression, first consider the case in which the larger target, T, is presented: [ 2 n 2 n ]. P( correct T) P max T, D,..., D min T, D,..., D = ( )> ( ) A15 Page 44

47 To solve this equation, we need to consider a density function equal to the max of the target and n-1 distractors P max ( T, D,..., D )= x [ 2 n ] and determine how many percepts this [ ] quantity dominates that are in the opposite direction P min ( T, D2,..., Dn )< x because if one of these dominates our density function of correct responses, the subject responds incorrectly. Thus we can derive the integral: [ 2 n ] ( 2 n)< 0 [ ] P( correct T) = P max ( T, D,..., D )= x P min T, D,..., D x dx A16 where the first term includes all of the different values of our density function that lead to a correct response, and the second term includes all of the negative values. When the max of our positive values exceeds all the negative values ( min ( TD,,..., ) correct response. 2 D n ) then one makes the Equation A16 is equal to: [( 2 n ) ( ( 2 n)< )] = P(Correct T) = P max ( D,..., D )< x and min D,..., D x P( T x) dx 0 [( n ) ( ( n)< )] = ( n ) P max ( TD,,..., D)< xand min TD,,..., D x PD ( xdx ) 0 A17 The first term computes the probability that the target wins, and the second term computes the probability that one of the n-1 distractors wins. This reduces to: n [ ] ( 1 ) P(" larger " T) = f ( x ws) F( x) F( x) dx 0 n [ ][ ] ( 2 ) + ( n 1) f( x) F( x ws) F( ( x ws)) F( x) F( x) dx 0 A18 With w set to 1.0, the percent correct identification is 0.76 with set size 2 and 0.63 with set size 8. Again, we predict a set-size effect even though we assume unlimited capacity and stochastically independent processing. This set-size effects is due only to the fact that with Page 45

48 more noisy distractors there is a greater chance that one of the distractors exceeds the target percept in the opposite direction and forces the incorrect response. With set sizes 2 and 8, we predict a log-log slope of 0.42, which is larger than the 0.22 of yes/no and 0.34 of the localization tasks. 2AFC Localization Computing the probability of choosing the correct target location is chosen is a straightforward modification of the 2-Target Identification equation (Equation A18). [ ] ( n ) [ ] ( n 1) P( correct) = f ( x ws) F( x) F( x) dx f ( x ws) F( x) F( x) dx 0 which computes in the first term the probability that the target item dominates all distractors on its side. The second term computes the probability that the target exceeds all distractors on the other side. If we again assume P(T)=P(T')=0.5 and symmetric distributions, we do not need to include the probabilities for the smaller target T'. Setting w to 1, the percent correct is.635 for n=2 and is.270 for n=8. This gives a localization threshold of 1.49 for n= 2 and 1.89 for n=8. This results in a surprisingly low log-log slope of 0.17 for the 2-alternative localization task. A19 2IFC Identification For comparison to the other models, we also state the model for a two-interval identification task (cf. Palmer et al, 1993). Here n stimuli are presented in two successive intervals and the subjects chooses which interval contains the target. With the max rule, the probability correct is: [ 2 3 n n ] A21 pc ( ) pmax TD,, D,.., D max D, D,.., D = ( )> ( ) 2n 2 = 1 f( xfx ) ( wsfx ) ( ) dx For w=1, the 75% correct threshold for n = 2 is 0.95 and for n=8 is This yields a loglog slope of Page 46

49 Overlapping Channels Models The Overlapping Channels models differ from the previous models in that it explicitly breaks the independent processing assumption. Instead, stimulus items influence their neighboring items in such a way that the value of one item is not independent of the other items. The lack of independence proves formidable in terms of analytic predictions, and thus we resort to Monte Carlo simulations in order to produce quantitative predictions. Below we describe the procedures used to estimate the mislocalizations found in the n-localization and 2AFC Localization experiments, and then produce predictions for set-size effects for the various models once the magnitude of the assimilation effects has been estimated. In all cases the Overlapping Channels model is fit only to the set size 8 data, since the set size 2 data demonstrate no mislocalizations. That is, in the set size 2 condition two positions were chosen to receive stimuli, but the subject could still make one of 8 responses. However, errors were almost always limited to the position opposite the target location. This is also true for the relevant set size 2 condition in Experiment 4 where 8 locations received stimuli but only 2 were precued. Thus the source of mislocalizations does not appear to arise from difficulty translating a location on the screen to a position on the response keypad. n-localization To fit the mislocalization data, recall that performance (shown in Figure 16) was computed as a function of relative angle, or that angle that produces threshold performance. Localization performance (and therefore threshold performance) varies as a function of the weighting parameter in the assimilation models. Thus to predict the mislocalization data we first conducted Monte Carlo simulations of 100,000 trials per angle and fit a Weibull function to the data to find the threshold angle. Note that an angle in the context of the model is scaled in arbitrary units, because the relation between units of angle and physical angle is unknown. However, by converting to relative angle by finding a threshold angle Page 47

50 (both in the data and for a particular theory) we can compare theoretical predictions to human data. Within the Monte Carlo simulation, a single trial was conducted as follows. The values associated with the 7 distractor locations were sampled from a normal distribution with mean zero and unit variance. The target distribution was sampled from a distribution with unit variance and mean equal to one of 6 angles being evaluated. The assimilation procedure was then applied to the 8 numbers using the weighted sum: ' s i si + ωsi + ωsi = 1+ 2ω A22 where i indexes stimulus location, and si-1 and si+1 are the two stimulus locations immediately adjacent to location si. The position with the greatest value is then chosen as the response, and the trial is then characterized as either correct, a mislocalization on a nearby item, or an error. Performance at each of the angles is fit by a Weibull and the threshold angle is computed. Once the threshold angle has been determined, six angles at log-equal steps are chosen to produce a psychometric function that maps out the mislocalizations similar to those found in Figure 16. The threshold angle is set to be the fourth angle in this series (as was done in the experiments) and the lower and higher angles are log-equal steps above and below the threshold angle. The simulation is re-run with 1 million trials per angle, and this produces performance predictions that are directly comparable with the data in Figure 16. This procedure was repeated for different values of ω the weighting parameter in the assimilation process, and the value that minimized the root mean squared error with the Figure 16 data was chosen as the most accurate summary of the mislocalization process. Once the appropriate weight value is found, the angle that produces threshold performance for this weight value is used with the angle from the set size 2 condition to produce a predicted log-log slope that can be directly compared with those from the individual subjects. Page 48

51 Coarse Localization The procedures used to derive predictions for the Coarse Localization data were very similar to those of the n-localization data. The only difference is that the decision rule varies slightly. The location associated with the largest value was chosen after the assimilation procedure was applied, and this location was then re-interpreted as a left-right decision depending on the location of the greatest value. This produces set-size effects that differ slightly from the n-localization set-size effects for a fixed value of the weighting parameter ω. Yes/No Identification Yes/No Identification data do not have mislocalization data associated with them, and thus we used the weighting value associated with the mislocalizations in the n-localization task. Once a weighting value is known, the threshold angle can be derived by computing hits and false alarm rates for different values of angle and finding the angle associated with hit rates and correct rejection rates of 75%. One complication is that Yes/No data involve a decision criterion. We assume that subjects place this decision criterion to balance the hit and correct rejection rates. In the simulation we computed hit and correct rejection performance for a variety of angles and decision criterions in a grid-like fashion and fit a plane to the resulting surface for both the hit and correct rejection data. Using planar interpolation we were able to find the combination of angle and decision criterion that produced hit-rate and correct-rejection rates of 75%. This procedure was repeated twice more using increasingly-narrow step sizes to ensure that our surface approximation procedures did not underestimate performance at threshold angle. Once the threshold angle had been determined, this value was used in conjunction with that obtained from set size 2 in order to produce predictions for the log-log slopes. These are shown in Figure 17. Page 49

52 2-Target Localization Predictions for the 2-Target Localization are very similar to those of the n-localization task. The major difference is that the value of the target location was set either by sampling from a normal distribution with the mean equal to the angle or minus the angle for left or right-leaning targets respectively. This sets the distractors to vertical on average, and makes the target left- or right-leaning on average. On each trial the weighting procedures were the same as described above, and equation 22 was applied to the value associated with the 8 stimulus locations. To choose a response the model picked the location that had the greatest absolute deviation from zero. As with n- Localization, the threshold angle was computed and then a million trials per angle were used to produce predicted mislocalization and predicted set-size effects for the best-fitting weighting value. 2-Target Identification (Discrimination) The 2-Target Identification data do not have mislocalizations, and so the value for the weighting parameter found for the 2-Target localization data was used here as well. The simulation procedures were similar to those of the 2-Target Localization task, including the distribution of the target and distractor values and the weighting function applied to these values. The only difference resides in the decision rule. For 2-Target Identification the Monte Carlo procedures determine whether the maximum deviation from zero is greater than zero or less than zero. If it is greater than zero the model interprets this as a 'right' response, and if it is less than zero the model interprets this as a 'left' response and computes accuracy accordingly. The threshold angle is determined using the Monte Carlo procedures described above and this threshold angle is used in conjunction with that from set size 2 in order to produce a log-log slope prediction as shown in Figure 17. Page 50

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58 Table 1 Experiment 4: Set-Size Effects using cues or varying the number of stimuli. Set Size 2 8 No Cue 2.5± ±0.6 Cue 2.3± ±0.6 Note: Values are the orientation threshold in degrees

59 Table 2. Comparison of predicted set-size effects across different tasks. Set-Size Effects (2 vs. 8) 1-Target 2-Targets Two-By-Two Paradigm: -2IFC Identification Coarse Localization * n-alternative Localization * Yes/No Identification * Alternative Identification * n/a 0.31 Note: * indicates that the set-size has been doubled to match the number of stimuli in the 2IFC Identification task.

60 Figure 1. Example display from the set-size 8 condition.