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Ensemble perception of multiple spatially intermixed sets Luo, Xiao 2016

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ENSEMBLE PERCEPTION OF MULTIPLE SPATIALLY INTERMIXED SETS by  Xiao Luo  B.Soc.Sc., The University of Hong Kong, 2014  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF SCIENCE in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  June 2016  Β© Xiao Luo, 2016 ii  Abstract The visual system is remarkably efficient at extracting summary statistics from the environment. Yet at any given time, the environment consists of many groups of objects distributed over space. Thus, the challenge for the visual system is to summarize over multiple sets distributed across space. My thesis work investigates the capacity constraints and computational efficiency of ensemble perception, in the context of perceiving multiple spatially intermixed groups of objects. First, in three experiments, participants viewed an array of 1 to 8 intermixed sets of circles. Each set contained four circles in the same colors but with different sizes. Participants estimated the mean size of a probed set. Which set would be probed was either known before onset of the array (pre-cue), or after that (post-cue). Fitting a uniform-normal mixture model to the error distribution, I found participants could reliably estimate mean sizes for maximally four sets (Experiment 1). Importantly, their performance was unlikely to be driven by a subsampling strategy (Experiment 2). Allowing longer exposure to the stimulus array did not increase the capacity, suggesting ensemble perception was limited by an internal resource constraint, rather than an information encoding rate (Experiment 3). Second, in two experiments, I showed that the visual system could hold up to four ensemble representations, or up to four individual items (Experiment 4), and an ensemble representation had an information uncertainty (entropy) level similar to that of an individual representation (Experiment 5). Taken together, ensemble perception provides a compact and efficient way of information processing.   iii  Preface The five experiments in this thesis work were designed by Dr. Jiaying Zhao and I with helpful input and comments from Dr. James Enns. Research assistants in Dr. Zhao’s lab assisted in data collection. Data analysis and statistical simulations were performed by me.   These five experiments are part of a collaborative project between Dr. Zhao and I conducted in the Behavioral Sustainability Lab. All experimental procedures in the project were approved by the UBC Behavioral Research Ethics Board (No. H13-02684).  Experiments 1, 2, 3 and 5 are included in a conference abstract and poster that have been published. Luo, X., and Zhao, J. (2016). The capacity limit of ensemble perception. Vision Sciences Society 16th Annual Meeting (Poster). I wrote most of the abstract and made the poster. This work was presented by Dr. Zhao at the 16th Vision Sciences Society Annual Meeting in May, 2016.   All five experiments are included in a manuscript under review. Contributions to the manuscript are: Dr. Zhao and I designed the experiments; Dr. Zhao and her research assistants collected data; I analyzed the data and simulated computational models; Dr. Zhao and I wrote the manuscript. iv  Table of Contents  Abstract .......................................................................................................................................... ii Preface ........................................................................................................................................... iii Table of Contents ......................................................................................................................... iv List of Tables ............................................................................................................................... vii List of Figures ............................................................................................................................. viii List of Symbols ............................................................................................................................. ix Acknowledgements ........................................................................................................................x Dedication ..................................................................................................................................... xi Chapter 1: Introduction ................................................................................................................1 1.1 Background ..................................................................................................................... 1 1.1.1 Capacity limit of ensemble perception ....................................................................... 2 1.1.2 Efficiency of ensemble perception .............................................................................. 3 1.2 Overview of the thesis .................................................................................................... 4 Chapter 2: General Methods ........................................................................................................5 2.1 Measuring capacity limits ............................................................................................... 5 2.1.1 Experimental paradigm ............................................................................................... 5 2.1.2 Participants .................................................................................................................. 5 2.1.3 Statistical power .......................................................................................................... 6 2.1.4 Apparatus .................................................................................................................... 6 2.1.5 Stimuli ......................................................................................................................... 6 2.2 Data analysis and modeling ............................................................................................ 7 Chapter 3: Capacity Limit of Ensemble Perception ...................................................................9 3.1 Experiment 1 ................................................................................................................. 10 v  3.1.1 Participants and procedure ........................................................................................ 10 3.1.2 Results ....................................................................................................................... 11 3.2 Experiment 2 ................................................................................................................. 13 3.2.1 Participants and procedure ........................................................................................ 13 3.2.2 Results ....................................................................................................................... 14 3.3 Experiment 3 ................................................................................................................. 17 3.3.1 Participants and procedure ........................................................................................ 17 3.3.2 Results ....................................................................................................................... 17 3.4 Chapter 3 discussion ..................................................................................................... 20 3.4.1 Common capacity limits ........................................................................................... 20 3.4.2 Discrete-slots hypothesis .......................................................................................... 22 3.4.3 Limitations ................................................................................................................ 23 Chapter 4: Efficiency of Ensemble Perception .........................................................................24 4.1 Experiment 4 ................................................................................................................. 25 4.1.1 Participants and procedure ........................................................................................ 25 4.1.2 Results ....................................................................................................................... 25 4.2 Experiment 5 ................................................................................................................. 27 4.2.1 Participants and procedure ........................................................................................ 27 4.2.2 Results ....................................................................................................................... 29 4.3 Chapter 4 discussion ..................................................................................................... 31 4.3.1 Prevalence of efficient representations ..................................................................... 32 Chapter 5: Conclusion .................................................................................................................34 Bibliography .................................................................................................................................36 Appendices ....................................................................................................................................42 Appendix A Discard rates and response time ........................................................................... 42 Appendix B Experiment 2 partial correlation ........................................................................... 44 vi  Appendix C Additional analysis ............................................................................................... 45 C.1 Experiment 1 additional analysis .............................................................................. 45 C.2 Experiment 2 additional analysis .............................................................................. 48 C.3 Experiment 3 additional analysis .............................................................................. 50 C.4 Experiment 5 additional analysis .............................................................................. 59  vii  List of Tables  Table 4.1  Entropy of individual and ensemble representations at different set sizes, and their KL divergence. .................................................................................................................................... 31 Table A.1 Discard rate and mean response time of participants in five experiments. .................. 43 Table B.1 Partial correlation in Experiment 2. ............................................................................. 44 Table C.1 Partial correlations in Experiment 1. ............................................................................ 47  viii  List of Figures  Figure 2.1 Examples of stimulus array for the ensemble tasks....................................................... 6 Figure 3.1 Experiment 1 procedure. ............................................................................................. 10 Figure 3.2 Experiment 1 results. ................................................................................................... 12 Figure 3.3 Experiment 2 procedure. ............................................................................................. 14 Figure 3.4 Experiment 2 results. ................................................................................................... 15 Figure 3.5 RMSE in Experiments 1 and 3. ................................................................................... 18 Figure 3.6 Guess probability Pg in Experiments 1 and 3. ............................................................ 19 Figure 3.7 Estimation precision (sd) in Experiments 1 and 3. ...................................................... 20 Figure 4.1 Results of Experiment 4. ............................................................................................. 26 Figure 4.2 Experimental procedure of Experiment 5.................................................................... 27 Figure 4.3 Results of Experiment 5. ............................................................................................. 29 Figure C.1 Post-cue π‘·π’ˆ (A) and 𝒔𝒅 (B) in Experiment 1. ........................................................... 46 Figure C.2 Post-cue π‘·π’ˆ (A) and 𝒔𝒅 (B) in Experiment 2 ............................................................ 49 Figure C.3 Post-cue π‘·π’ˆ (A) and 𝒔𝒅 (B) in Experiment 3 with 500ms exposure duration. .......... 51 Figure C.4 Post-cue π‘·π’ˆ (A) and 𝒔𝒅 (B) in Experiment 3 with 1500ms exposure duration. ........ 53 Figure C.5 Post-cue π‘·π’ˆ (A) and 𝒔𝒅 (B) in Experiment 3 with 2000ms exposure duration. ........ 55 Figure C.6 Guess probabilities in the four exposure duration conditions in the post-cue condition (A), and the pre-cue condition (B). ............................................................................................... 56 Figure C.7 Post-cue π‘·π’ˆ (A) and 𝒔𝒅 (B) in Experiment 4. ........................................................... 58 Figure C.8 Post-cue π‘·π’ˆ (A) and 𝒔𝒅 (B) in Experiment 5. ........................................................... 60  ix  List of Symbols  Symbol Unit Property 𝒂 𝑝𝑖π‘₯𝑒𝑙2 Lower-bound of uniform distribution 𝒃 𝑝𝑖π‘₯𝑒𝑙2 Upper-bound of uniform distribution 𝑫𝑲𝑳 Nat Kullback-Leibler divergence 𝑯(𝑿) Nat Entropy of discrete variable 𝑋 𝒉(𝑿) Nat Entropy of continuous variable 𝑋 π’Ž - Number of sets 𝒏𝒄 - Number of trials in condition 𝑐 𝓝 - Probability density function of a normal distribution  π‘·π’ˆ - Guess probability π‘·π’ˆπ’„  - Estimate of guess probability in condition 𝑐 π‘·π’Ž - Probability that an item is held in working memory (in Zhang and Luck (2008)) π‘Ήπ’Š 𝑝𝑖π‘₯𝑒𝑙2 True set mean in trial 𝑖 π’“π’Š 𝑝𝑖π‘₯𝑒𝑙2 True circle size in trial 𝑖 π‘Ήπ’Šβ€² 𝑝𝑖π‘₯𝑒𝑙2 Size of flashed circle in the probe size RMSE 𝑝𝑖π‘₯𝑒𝑙2 Root mean squared error RMSEF 𝑝𝑖π‘₯𝑒𝑙2 Flashed circle-based RMSE RMSES 𝑝𝑖π‘₯𝑒𝑙2 Set mean-based RMSE π‘Ίβˆ— - Capacity limit (number of sets) 𝒔𝒅 𝑝𝑖π‘₯𝑒𝑙2 Standard deviation of normal distribution 𝒔𝒅𝒄 𝑝𝑖π‘₯𝑒𝑙2 Estimate of standard deviation in condition 𝑐 𝑺𝑻𝑫 π‘‘π‘’π‘”π‘Ÿπ‘’π‘’ Standard deviation of normal distribution (in Zhang and Luck (2008)) 𝓀 - Probability density function of a uniform distribution π’–π’Š 𝑝𝑖π‘₯𝑒𝑙2 Reported set mean in trial 𝑖 π’™π’Š 𝑝𝑖π‘₯𝑒𝑙2 Reported circle size in trial 𝑖 πœ½π’Š 𝑝𝑖π‘₯𝑒𝑙2 Signed error in trial 𝑖 𝝁 𝑝𝑖π‘₯𝑒𝑙2 Mean of normal distribution 𝝈 𝑝𝑖π‘₯𝑒𝑙2 Standard deviation of normal distribution 𝝈𝟐 (𝑝𝑖π‘₯𝑒𝑙2)2 Variance of normal distribution π“π’Š 𝑝𝑖π‘₯𝑒𝑙2 Difference between response and size of flashed circle in the probed set   x  Acknowledgements  I would first like to thank my thesis supervisors Dr. Jiaying Zhao and Prof. James Enns of the Department of Psychology at the University of British Columbia (UBC). They consistently encouraged independent thinking and exploration of novel ideas in the course of conducting my thesis work, and steered me in the right direction when I needed it.   My gratitude also goes to Prof. Alan Kingston (Department of Psychology, UBC), who was a member of my supervisory committee, and Dr. Miriam Spering (Department of Ophthalmology and Visual Sciences, UBC), who served as the external examiner in my thesis defense. Their thoughtful comments tremendously helped me improve this thesis work.   I would also like to thank Ru Qi Yu and Yu Luo, students in Dr. Zhao’s lab, for assistance in data collection.   Finally, special thanks are owed to my parents, who have always been supportive throughout my years of education.   xi  Dedication  To my parents. 1  Chapter 1: Introduction 1.1 Background Faced with the complexity of information in the environment, our visual system is found capable of extracting statistical summary representations of features and objects, including their mean, variance, or global correlations (Alvarez, 2011). Such process is termed ensemble perception. One form of ensemble perception, mean computation, has been shown to operate accurately and fast with many feature dimensions, such as size (Ariely, 2001, 2008; Chong & Treisman, 2003, 2005a), orientation (Parkes, Lund, Angelucci, Solomon, & Morgan, 2001), location (Alvarez & Oliva, 2008), moving direction (Watamaniuk & McKee, 1998), and facial expression (Haberman & Whitney, 2009).   Our understanding of the nature of ensemble perception has been significantly advanced in the past two decades. Ensemble perception is suggested to happen in an automatic manner, with little requirement of focal attention, as it can be computed from crowded objects presented in the peripheral vision (Parkes et al., 2001), or when attention is occupied by a dual task (Alvarez & Oliva, 2008). Extracting ensemble representations can happen rapidly. For instance, mean size of two arrays of 12 circles each could be accurately computed with 50ms exposure (Chong & Treisman, 2003), and mean emotion of 16 faces could be accurately perceived when the exposure time was 500ms (Haberman & Whitney, 2009). Ensemble perception could be influenced by attention allocation: in a mean size computation task, de Fockert and Marchant (2008) deliberately directed participants’ attention to a set member, and found mean estimation biased toward the set member. By contrast, encouraging global attention facilitates mean size computation (Chong & Treisman, 2005b). Moreover, recent studies show that mean size can be computed for sequentially presented circles, suggesting ensemble perception is a dynamic process actively involving working memory (Albrecht & Scholl, 2010).  Ensemble perception provides a useful β€œgist” of an otherwise complex scene. Yet within many natural scenes, there can be multiple spatially intermixed object groups, defined by similarity or proximity in one or more feature dimensions. Can the visual system group objects according to a common feature, and extract summary statistics for each group? Significance of this question is 2  two-fold: first, answers to this question can help us understand the interplay of selective attention, perceptual organization, and working memory in determining perceptual experience within a context of limited cognitive resources (Norman & Bobrow, 1975); second, understanding mechanisms of ensemble perception, along with other high-level, efficient perceptual capabilities of human vision may inspire development of artificial intelligent systems that use similar heuristics to reduce computational complexity and improve accuracy (Lowe, 1985).   This questions leads to two more specific inquiries: first, what is the capacity limit of ensemble perception, in terms of the number of ensemble representations that can be formed from a given display? Second, in what way can ensemble perception contribute to efficient representation of the complex visual world? Focusing on computation of mean size of sets of circles, this thesis work examines ensemble perception of multiple groups, with respect to its capacity and efficiency.  1.1.1 Capacity limit of ensemble perception Chong and Treisman (2005a) were among the first to explore ensemble perception of a selective subset of items. In their study, circles in two different colors were spatially intermixed, and participants were asked to report the mean size of one of the two color-defined sets. Performance did not differ whether participants knew which set would be probed before seeing the two sets or not. This suggests that participants were able to compute the mean size for both sets, and that that ensemble perception could operate over two sets. However, the maximum number of sets from which the visual system can extract an ensemble representation remains unknown.  Extracting summary representations can be a multi-stage operation, involving attentional selection of items (Kinchla, 1992), representing items in visual working memory (Marois & Ivanoff, 2005), as well as task-specific processing (i.e., deriving the statistical summary from visual inputs). Information processing bottleneck of these processes and their interactions could impose a capacity constraint on ensemble perception. Thus, ensemble perception is likely to be constrained by internal resource limits, as is most cognitive processes.  3  How is the pool of resources allocated? One hypothesis is that the pool of internal resources can be flexibly allocated to form ensemble representations of arbitrarily many sets, with a trade-off between the number of representations and their precision. This β€œflexible resources” hypothesis can be supported by the fact that statistical summary representation seems to require minimal attentional and processing effort (Alvarez & Oliva, 2008, 2009). An alternative hypothesis is that ensemble perception is subject to a fixed discrete capacity limit (β€œdiscrete slots” hypothesis). A limit of three or four items has been shown in visual working memory (Zhang & Luck, 2008; Luck & Vogel, 1997), attentional blink (Raymond, Shapiro, & Arnell, 1985), multiple object tracking (Cavanagh & Alvarez, 2005), and enumeration (Halberda, Sires, & Feigenson, 2006). In particular, visual working memory and attention may influence the capacity of ensemble perception by limiting the number of items or sets that can be simultaneously encoded, stored, and manipulated.   1.1.2 Efficiency of ensemble perception Previous findings suggest statistical summary representation may be an efficient resolution to the bottleneck of information processing, as accurate ensemble representations can be formed from noisy representations of individual items (Whitney, Haberman, & Sweeny, 2014). For instance, Parkes and colleagues (2001) presented an array of oriented Gabor patches in the periphery of a central Gabor patch. Orientations of the peripheral Gabors could not be accurately reported due to crowding, but their average orientation biased participants’ judgment of orientation of the central Gabor. Thus, encoding of summary statistics bypassed the bottleneck of processing individual items. Similarly, Sweeny, Haroz, and Whitney (2013) showed that estimation of the average walking direction of a crowd of people was more accurate than the estimation of the walking direction of an individual person in the crowd, or a foveally presented person.   These studies have examined efficiency of ensemble perception by comparing the precision of ensemble representations and that of individual item representations. They have not, however, compared the capacities of these two types of representations. As Ma, Husain, and Bays (2014) notes, in terms of visual working memory, capacity and precision characterize different aspects of the limited memory resource, and may not show a direct, linear trade-off. Other factors, such 4  as memory load and stimulus salience, may mediate their relationships and affect the total amount of information processed. Similarly, for ensemble perception, a strong argument for its being an efficient encoding strategy should show evidence from both the precision perspective, and the capacity perspective, as well as in connection with measures of the total information.   1.2 Overview of the thesis To summarize, the objective of this thesis is to investigate the capacity constraints on ensemble perception, and the extent to which it contributes to efficient information processing. Chapter 2 describes the general experimental paradigm that requires mean size computation from multiple spatially intermixed sets of colored circles.   Chapter 3 characterizes the capacity of ensemble perception by exploring whether a fixed discrete upper limit exists. Experiment 1 established that the maximum number of sets the visual system can summarize over was four. To examine whether the performance was driven by a subsampling strategy, Experiment 2 deliberately biased people’s attention to one circle in each set, while asking them to compute the set means. Results ruled out the possibility of subsampling one circle per set in mean computation. In Experiment 3, I manipulated exposure durations of the stimulus array to demonstrate that the ensemble capacity limit found in Experiment 1 was due to an internal resource constraint, rather than an encoding constraint.   Chapter 4 investigated the efficiency of ensemble perception by comparing the capacity of representing individual stimulus, and that of ensemble representation. In Experiment 4, participants were presented with the same stimulus arrays as in Experiment 1, but instructed to report the size of an individual circle. No more than four items could be reliably represented. Experiment 5 eliminated any requirement of mean computation and implicit set individuation, and still found a four-item capacity limit. Comparing results of Experiments 5 (item representation) and 1 (ensemble representation), I argued for the efficiency of ensemble perception from an information theoretic perspective.   Finally, Chapter 5 provides an overall summary of the thesis work.  5  Chapter 2: General Methods 2.1 Measuring capacity limits 2.1.1 Experimental paradigm To determine capacity limit of ensemble perception, I used the pre-cue and post-cue paradigm, which has been widely used in visual perception and working memory studies. In a typical pre-cue condition, the visual cue is presented prior to the onset of the stimulus array, such that attention is directed to the task-relevant information in the array (Griffin & Nobre, 2003; Woodman, Vecera, & Luck, 2003). In a typical post-cue condition, the cue is presented only after the offset of the stimulus array, such that attention is distributed among all items in the array during its presence (Griffin & Nobre, 2003; Hollingworth & Maxcey-Rchard, 2013).  In my experiments, the capacity limit is measured as the minimum number of items or sets of items where task performance is reliably different between the pre-cue and the post-cue conditions. Similar definition of capacity limit has been used in Halberda, Sires, and Feigenson (2006), where the authors examined how many spatially intermixed sets of circles (define by different colors) could be enumerated from a briefly shown array. Comparing enumeration error in the pre-cue and post-cue conditions, they found performance significantly differed when the array contained more than two sets, thus concluding on a three-set capacity limit. The major advantages of measuring capacity limit by comparing pre-cue and post-cue performance is that it maintains the same perceptual input between conditions, and allows examining ensemble perception with manipulation of attentional allocation.  2.1.2 Participants Participants were undergraduate students at the University of British Columbia (UBC), and were recruited through the Human Subjects Pool in the Department of Psychology. All participants had normal or corrected-to-normal vision, provided informed consent, and received course credits as compensation. All experiments were approved by the UBC Behavioral Research Ethics Board.  6  2.1.3 Statistical power The required sample size was calculated a priori assuming a Type I error level of Ξ± = .05 and a medium effect size (Ξ·2 = 0.1) for a two-way repeated-measures ANOVA. Based on the power analysis, our sample size should be at least 19. Thus, 22 to 24 participants were recruited for each of the five experiments.  2.1.4 Apparatus In all experiments, participants were seated 60cm from a computer monitor (refresh rate = 60Hz). Stimuli were presented using MATLAB (Mathworks) and the Psychophysics Toolbox (http://psychtoolbox.org).   2.1.5 Stimuli   Figure 2.1 Examples of stimulus array for the ensemble tasks.  (A) shows one set of stimuli, and (B) shows four sets of stimuli. In Experiments 1 to 4, each set contained four circles as the figures show here. In Experiment 5, each set contained only one circle.           Each stimulus array consisted of 1 to 8 sets of colored circles presented against a gray background. Each set contained four circles in the same color (in Experiments 1-4), or only one circle in one color (in Experiment 5). The color was unique for each set, and randomly selected from a pool of eight colors (red, green, blue, yellow, cyan, magenta, orange, and black). The circles were randomly assigned to cells in an invisible 5Γ—7 grid (subtending 15.6°×21.8Β°) on the screen. Within each cell, the circle was placed at the center with random jittering of up to 0.2Β°. To determine the circle diameter for experiments with four circles per set (Experiments 1-4), the grand mean (i.e., the mean of all circles in the array) was first determined for trials where the set 7  size was an even number (2m, m = 1, 2, 3, 4). Then the individual set means were evenly spaced around the grand mean by steps of 0.3Β°. For trials where the set size was an odd number (2m-1, m = 1, 2, 3, 4), the grand mean of 2m sets was first determined, and the individual set means were evenly spaced around the grand mean by steps of 0.3Β°. Then one of the two set means closest to the grand mean was removed to obtain 2m-1 sets. Within each set, the diameters of individual circles were evenly spaced around the set mean by steps of 0.3Β°. The range of circle diameters was between 0.4Β° and 2.9Β°. In Experiment 5 with only one circle per set, the set means themselves were used as individual circle sizes.  2.2 Data analysis and modeling In the five experiments in my thesis, the task was either to estimate the mean size for a set of circles, or to estimate the size of an individual circle. Trials with response times greater than three standard deviations from the participant’s mean were removed (See Appendix A  for discard rates and response time for each experiment). I used two measures of error: the root mean squared error (RMSE), and the signed error. RMSE is computed for experimental condition 𝑐 as:  𝑅𝑀𝑆𝐸 = βˆšβˆ‘ πœƒπ‘–2𝑛𝑐𝑖𝑛𝑐 where πœƒπ‘– = 𝑒𝑖 βˆ’ 𝑅𝑖 is signed error in trial 𝑖, with 𝑒𝑖 being the participant’s estimate of set mean, and 𝑅𝑖 the true set mean probed in that trial. 𝑛𝑐 is the number of trials in a given cue and set size condition 𝑐. Alternative RMSE computations were used in Experiment 2.   The source of signed error could be the capacity limit of mean computation, and/or the noise in estimation or reporting. To identify the capacity limit and to separate the two factors, I assume the following uniform-normal mixture model of signed error distribution for each trial 𝑖, adapting from Zhang and Luck (2008):   𝑝(πœƒπ‘–)  =  𝑃𝑔 βˆ— 𝒰(π‘Ž, 𝑏) + (1 βˆ’ 𝑃𝑔) βˆ— 𝒩(πœ‡, 𝜎2) In this model, 𝑃𝑔 is the probability of guessing at random. A random guess follows a uniform distribution 𝒰(π‘Ž, 𝑏) with a lower-bound π‘Ž and an upper-bound 𝑏. Since values of π‘Ž and 𝑏 differ from trial to trial depending on the true set mean, they are heuristically estimated using the (2) (1) 8  smallest and the largest signed errors respectively, for each participant. With a probability of 1 βˆ’π‘ƒπ‘”, the probed set mean can be successfully reported with noise, captured by a Gaussian distribution 𝒩(πœ‡, 𝜎2). Since the expected error in this case is 0, so πœ‡ is fixed at 0. Precision of response is indicated by 𝑠𝑑 = 𝜎. Therefore, for each participant and each experimental condition 𝑐, there are two parameters to be estimated (𝑃𝑔𝑐 and 𝑠𝑑𝑐) by minimizing the negative log likelihood:  argmin𝑃𝑔𝑐,π‘ π‘‘π‘βˆ‘ βˆ’ log 𝑝(πœƒπ‘–)𝑛𝑐𝑖=1, 𝑠. 𝑑. 𝑃𝑔𝑐 β‰₯ 0, 𝑠𝑑𝑐 β‰₯ 0  Where 𝑛𝑐 is the number of trials in experimental condition 𝑐, and 𝑝(πœƒπ‘–) is defined by Eqn. (2).  Optimization of (3) is done using the Nelder-Mead algorithm (Nelder & Mead, 1965) over a range of different initial values in search for the global minimum.   Importantly, β€œcapacity limit” in this thesis work is operationalized differently than Zhang and Luck (2008). In that study, they draw on the probability that an item is remembered (π‘ƒπ‘š) and the representation precision (𝑆𝑇𝐷) to distinguish the flexible-resource and the discrete-slots hypothesis: for the former to be true, π‘ƒπ‘š should relatively constant, while 𝑆𝑇𝐷 should increase monotonically as set size increases; for the latter to be true with a capacity limit π‘†βˆ—, π‘ƒπ‘š should drop significantly when the set size exceeds π‘†βˆ—, and 𝑆𝑇𝐷 plateau. By contrast, the current thesis, capacity limit of ensemble perception is inferred from the minimum set number π‘†βˆ— at which the guess probability 𝑃𝑔 in the pre-cue and post-cue conditions are reliably different under paired-sample t-test.   (3) 9  Chapter 3: Capacity Limit of Ensemble Perception This chapter comprises of three experiments that explored the capacity limit of ensemble perception. In Experiment 1, I compared how well participants could report set means in the pre-cue and post-cue conditions, and inferred the capacity limit based on guess probabilities between these two conditions. To see whether the flexible-resource or the discrete-slots mechanism could better characterize ensemble perception, I examined the precision measure 𝑠𝑑 at different set sizes in the post-cue condition. A flexible-resource hypothesis would predict monotonically increasing 𝑠𝑑, and a discrete-slots hypothesis would predict a plateau in 𝑠𝑑, after the capacity limit π‘†βˆ— is reached (Zhang & Luck, 2008).    A long-term debate is whether participants’ performance in any ensemble perception task can actually be explained by strategic subsampling. Myczek and Simons (2008) statistically simulated a series of subsampling strategies in a mean size computation task, and argues subsampling just one or two items in the array can yield response accuracy numerically close to human performance. However, their statistical simulations were noise-free in stimulus encoding, mean computation, and response. Thus, they might not be directly comparable to human results, except at the correlation level. Experiment 2 of my thesis tested the possibility of subsampling strategies experimentally; in particular, I examined whether participants would rely on subsampling one item per set (termed β€œone-item subsampling” hereafter) to derive a response to an ensemble task. I deliberately biased participants’ attention to one circle in each set while instructing them to compute the set means. If their performance would be closer to the true set mean than to the biasing circle, and uncorrelated with the biasing circle, then the hypothesis of one-item subsampling can be refuted.   Finally, in an ensemble perception task, performance level is likely to be determined by two independent factors: an encoding limit that describes the maximum rate at which visual inputs can be processed and ensemble representations can be formed; and a capacity limit, which determines the maximum amount of information that can be ultimately encoded. Such a dual source of limit is associated with other high-level cognitive functions, such as working memory storage (Bays, Gorgoraptis, Wee, Marshall, & Husain, 2012). In Experiment 3, to study the 10  capacity limit of ensemble perception independently of its encoding limit, exposure durations of stimulus arrays were varied. The hypothesis was that, compared to results in Experiment 1, if increased exposure durations could not further improve the capacity limit, then the capacity limit observed should be due to an internal capacity constraint, rather than an effect of the encoding limit.    3.1 Experiment 1 3.1.1 Participants and procedure Twenty-two undergraduate students (14 females, mean age = 20.1yrs, SD = 1.9) participated in the experiment.  Figure 3.1 Experiment 1 procedure.  An example trial in the pre-cue condition (A) and in the post-cue condition (B). In each trial, the array contained 1 to 8 sets of circles, each set containing 4 circles in the same color. In the pre-cue example trial, the cyan fixation cross indicated the mean size of the 4 cyan circles would be probed later; and in the post-cue example trial, the fixation cross was hollow and which set would be probed was not known in advance  Experimental procedures, stimuli, and apparatus generally followed Chapter 2. There were two within-subject conditions: pre-cue and post-cue (Figure 3.1). In each trial, participants first fixed at a central cross for 1000ms. In the pre-cue condition, the fixation cross was filled with a stimulus color, signaling that participants would be asked to report the mean size of the four circles in that particular color. In the post-cue condition, the cross was hollow with a white outline, so participants did not know which set would be later probed. After fixation, an array with 1 to 8 sets of circles were presented for 1000ms, followed by a white-noise mask for 500ms. Then a probe circle (diameter = 0.3Β°) in a stimulus color appeared at the center of the screen. Participants reported the mean size of the four circles in that particular color from the array, by (A) Pre-cue condition (B) Post-cue condition 11  using the mouse to adjust the size of the probe circle. The size of the probe circle was bounded between 0.1Β° and 3.4Β°. No feedback was provided. The inter-trial interval was 500ms. The experiment began with 10 practice trials, followed by a total of 320 experimental trials. There were 20 trials in each cue Γ— set size condition.   3.1.2 Results The RMSE results are shown in Figure 3.2A. A 2 (cue condition: pre-cue vs. post-cue) Γ— 8 (set size: 1 to 8) repeated-measures ANOVA revealed a significant main effect of cue condition [F(1, 21) = 48.40, p < .001, Ξ·p2 = .70], and RMSE was lower in the pre-cue condition [95% CI = [-1692.04, -913.24], p < .001]. A main effect of set size [F(3.78, 79.46) = 6.38, p < .001, Ξ·p2 = .23] 1, and an interaction [F(3.96, 83.14) = 5.12, p = .001, Ξ·p2 = .20] were also observed. Paired-sample t-tests revealed that post-cue RMSE was higher than pre-cue RMSE at all set sizes except set size 1 [t’s(21) β‰₯ 1.76, p’s ≀ .046, d’s β‰₯ 0.38 at set sizes β‰₯ 2; t(21) = 0.85, p = .203, d = 0.18 at set size 1]. Thus, beyond set size 1, mean size estimation was less accurate in the post-cue condition than in the pre-cue condition.  To determine whether the reduced accuracy was due to a capacity limit in encoding multiple set means or reduced precision of representation, we estimated the guess probability 𝑃𝑔 and the standard deviation 𝑠𝑑 for each participant in each cue Γ— set size condition, using the signed error. Figure 3.2B-C show the average 𝑃𝑔 and 𝑠𝑑, respectively.   A 2 (cue condition) Γ— 8 (set size) repeated-measures ANOVA was run on Pg. A main effect of cue condition was observed [F(1, 21) = 5.25, p = .032, Ξ·p2 = .20], which was due to a lower Pg in the pre-cue condition [95% CI = [-0.11, -0.01], p = .025]. A main effect of set size [F(2.87, 60.35) = 8.42, p < .001, Ξ·p2 = .29] and an interaction [F(2.43, 50.95) = 5.63, p = .004, Ξ·p2 = .21] were also observed. One-way ANOVAs indicated a significant effect of set size in the post-cue condition [F(2.99, 62.78) = 8.88, p < .001, Ξ·p2 = .30], but not in the pre-cue condition [F(2.33, 48.94) = 1.99, p = .141, Ξ·p2 = .086]. Paired-sample t-tests revealed significantly higher post-cue                                                  1 Greenhouse-Geisser adjustment was applied to the degrees of freedom to correct for violation of sphericity. The same was true for all F scores with non-integer degrees of freedom. 12  Pg at set sizes β‰₯ 5 [t’s(21) ≀ 1.44, p’s β‰₯ .165, d’s ≀ 0.31 for set sizes up to 4; t’s(21) β‰₯ 2.18, p’s ≀ .041, d’s β‰₯ 0.47 for set sizes 5 to 8]. Hence, there was a higher probability of random guess in the post-cue condition, when the stimulus array contained more than four sets. This result suggests a four-set encoding limit of ensemble perception.  Figure 3.2 Experiment 1 results. (A) Estimation accuracy measured in RMSE (in area, pixel2) at each set size in the pre-cue and the post-cue conditions. (B) The guess probability Pg of the model in the pre-cue and the post-cue conditions. (C) The estimation precision as measured by the standard deviation of the model (sd, in area, pixel2) in the pre-cue and the post-cue conditions. (*p < .05, †p < .10, error bars reflect ο‚±1 SEM)   For the estimation precision 𝑠𝑑, a 2 (cue condition) Γ— 8 (set size) repeated-measures ANOVA revealed a main effect of cue condition [F(1, 21) = 46.62, p < .001, Ξ·p2 = .69], which was driven by lower pre-cue 𝜎 [95% CI = [-1261.88, -672.67], p < .001]. A main effect of set size [F(3.92, 82.29) = 3.76, p = .008, Ξ·p2 = .15], and an interaction [F(3.81, 80.03) = 2.94, p = .028, Ξ·p2 = .12] were observed. There were significant differences in 𝑠𝑑 between the two cue conditions at all set 13  sizes [t’s(21) β‰₯ 2.37, p’s ≀ .027, d’s β‰₯ 0.41], except for set size 1 [t(21) = 1.38, p = .181, d = 0.29]. The analyses of 𝑃𝑔 and 𝜎 suggest that within the four-set limit, the reduced accuracy at set sizes 3 and 4 measured by RMSE was driven by lower encoding precision, rather than by a higher guess probability. Moreover, there was no difference in the post-cue 𝑠𝑑 from set size 5 to 8 [F(3, 63) = 1.15, p = .337, Ξ·p2 = .05], suggesting the estimation precision remained constant above the capacity limit.  3.2 Experiment 2 3.2.1 Participants and procedure Experiment 2 examined whether the capacity limit observed in Experiment 1 was due to a heuristic strategy, in which participants randomly subsampled one circle per set, and based their response on those samples.    Twenty-two new participants (16 females, mean age = 19.7yrs, SD = 1.5) from UBC took part in the experiment. The stimuli and procedure were identical to those in Experiment 1, except for one critical difference. In each trial, the array appeared on the screen for 300ms after fixation, and one randomly selected circle in each set briefly disappeared for 200ms. Afterwards, all circles would be presented for 700ms (Figure 3.3). The brief flash of the circles intended to bias attention to one circle in each set and was completely task irrelevant.  As in Experiment 1, I computed the RMSE based on the set mean (RMSES) and the RMSE based on the flashed circle in the probed set (RMSEF). The former was calculated with equation (1), and the latter using:  𝑅𝑀𝑆𝐸𝐹 = βˆšβˆ‘ πœ™π‘–2𝑛𝑐𝑖 = 1𝑛𝑐 where πœ™π‘– = 𝑒𝑖 βˆ’ 𝑅𝑖′ is the difference between actual response (𝑒𝑖) and size of the flashed circle in the probed set (𝑅𝑖′), for trial 𝑖. Variable 𝑛𝑐 is the number of trials in a given cue Γ— set size condition. Comparison between RMSES and RMSEF would indicate whether participants based their estimation on the true set mean or on the flashed circle. As before, the uniform-normal mixture model was fitted to the signed error πœƒπ‘– = 𝑒𝑖 βˆ’ 𝑅𝑖 to obtain estimates of 𝑃𝑔 and 𝑠𝑑. (4) 14   Figure 3.3 Experiment 2 procedure.  An example trial in the pre-cue condition (A) and in the post-cue condition (B). During the flash display in both conditions, one circle from each set disappeared for 200ms (the white circle frame indicated the position of the disappeared circle and was not shown in the actual array).  3.2.2 Results The RMSE results are shown in Figure 3.4A. A two-way repeated-measures ANOVA on RMSE revealed a significant main effect of cue condition [F(1, 21) = 36.31, p < .001, Ξ·p2 = .63], and the pre-cue RMSES was lower than the post-cue RMSES [95% CI = [-1508.05, -734.23], p < .001]. A main effect of set size was also observed [F(7, 147) = 4.26, p < .001, Ξ·p2 = .17]. There was a reliable interaction [F(7, 147) = 7.05, p < .001, Ξ·p2 = .25], which could be explained by a significant effect of set size on post-cue RMSES [F(4.25, 89.33) = 8.21, p < .001, Ξ·p2 = .28], but not on pre-cue RMSES [F(7, 147) = 1.14, p = .346, Ξ·p2 = .05]. Paired-sample t-tests indicated significant differences between the two cue conditions at set sizes β‰₯ 3 [t’s(21) ≀ 1.21, p’s β‰₯ .121, d’s ≀ 0.26, for set sizes 1 and 2; t’s(21) β‰₯ 3.95, p’s ≀ .001, d’s β‰₯ 0.84, for set sizes 3 to 8]. This showed a deterioration of estimation accuracy at set sizes β‰₯ 3. (A) Pre-cue condition (B) Post-cue condition 15   Figure 3.4 Experiment 2 results.  (A) RMSE based on probed set mean (RMSES) and the size of the flashed circle in the probed set (RMSEF) in the pre-cue and the post-cue conditions. (B) The guess probability Pg in the set-mean based estimates. (C) The estimation precision as measured by the standard deviation sd in the set-mean based estimates. (*p < .05, error bars reflect Β±1 SEM)   To examine whether mean size estimation was influenced by flashed circles in the array, I compared RMSES and RMSEF across conditions and set sizes using a three-way ANOVA with a 2 (RMSE condition: RMSES, RMSEF) Γ— 2 (cue condition) Γ— 8 (set size) design. There was a main effect of RMSE condition [F(1, 21) = 52.53, p < .001, Ξ·p2 = .71], with RMSES being significantly lower than RMSEF [95% CI = [-585.42, -324.36], p = .030]. This was true in both the pre-cue condition [F(1, 21) = 47.68, p < .001, Ξ·p2 = .69] and the post-cue condition [F(4.18, 87.81) = 7.90, p < .001, Ξ·p2 = .27]. Therefore, estimation was more likely to be based on the mean of all circles in the set, rather than the flashed circle. Yet, were the participants still biased toward the flashed circles? A partial correlation test was run on each participant between their 16  estimation and size of the flashed circle in the probed set, after controlling for the probed set mean. Among the 22 subjects, only five showed significant partial correlations (Appendix B  ). This observation was consistent with the ANOVA analysis, and suggested a minimal biasing effect.   As in Experiment 1, I modeled the set mean-based signed error with the uniform-normal mixture distribution, and examined the guess probability 𝑃𝑔 (Figure 3.4B) and precision 𝑠𝑑 (Figure 4C). Foe 𝑃𝑔, repeated-measures ANOVA revealed a main effect of cue condition [F(1, 21) = 5.45, p = .030, Ξ·p2 = .21], with the pre-cue 𝑃𝑔 being significantly lower than the post-cue 𝑃𝑔 [95% CI = [-0.09, -0.01], p = .030]. A main effect of set size [F(3.84, 80.60) = 17.93, p < .001, Ξ·p2 = .46] and an interaction [F(2.86, 60.02) = 5.61, p = .002, Ξ·p2 = .21] were also observed. Paired-sample t-tests showed greater post-cue 𝑃𝑔 at set sizes β‰₯ 5[t’s(21) β‰₯ 2.35, p’s ≀ .028, d’s β‰₯ 0.50], but not set sizes ≀ 4 [t’s(21) ≀ 1.04, p’s β‰₯ .311, d’s ≀ 0.22], suggesting a four-set limit in ensemble perception. These results were consistent with those in Experiment 1, despite the deliberate attention bias.  A two-way repeated-measures ANOVA on 𝑠𝑑 showed a main effect of cue condition [F(1, 21) = 49.94, p < .001, Ξ·p2 = .70], with pre-cue 𝑠𝑑 being lower than post-cue 𝑠𝑑 [95% CI = [-1196.01, -652.05], p < .001]. There was also a main effect of set size [F(7, 147) = 7.32, p < .001, Ξ·p2 = .26] and an interaction [F(4.56, 95.66) = 7.18, p < .001, Ξ·p2 = .26]. Paired-sample t-tests revealed significant differences in 𝑠𝑑 between the two cue conditions at set sizes β‰₯ 3 [t’s(21) ≀ 0.95, p’s β‰₯ .353, d’s ≀ 0.20 for set sizes 1 and 2; t’s(21) β‰₯ 3.54, p’s ≀ .001, d’s β‰₯ 0.78 for set sizes  β‰₯  3]. Consistent with Experiment 1, within the four-set limit the reduced accuracy at set sizes 3 and 4 seemed to be driven by lower encoding precision, rather than by a higher guess probability. Moreover, there was no difference in post-cue 𝑠𝑑 from set size 5 to 8 [F(3, 63) = 2.00, p = .123, Ξ·p2 = .09], suggesting the estimation precision remained constant above the capacity limit.  To summarize, Experiment 2 suggested the participants based their ensemble estimation on the true set mean rather than an individual circle within the set, thus ruling out the one-circle subsampling strategy during ensemble perception. 17  3.3 Experiment 3 3.3.1 Participants and procedure Both Experiments 1 and 2 revealed a four-set capacity limit of ensemble perception. Experiment 3 explored whether this capacity limit could be improved with longer exposure durations.  Sixty-six new participants (42 females, mean age = 20.7yrs, SD = 2.7) were randomly assigned to three exposure duration conditions (500ms, 1500ms, and 2000ms) (N = 22 in each condition). The procedure was identical to that in Experiment 1, except for changes in exposure duration. Data from Experiment 1 (1000ms) was reused for comparison.   3.3.2 Results The RMSE are presented in Figure 3.5 for the 500ms, 1000ms, 1500ms, and 2000ms conditions. For the 500ms, 1500ms, and 2000ms conditions, post-cue RMSE was significantly higher than pre-cue RMSE at set sizes β‰₯ 3 [F’s(1, 21) β‰₯ 40.29, p’s < .001, Ξ·p2 β‰₯ .66]. For the 1000ms condition in Experiment 1, a marginal difference occurred at set sizes β‰₯ 2.  Repeated-measures ANOVA was conducted on guess probabilities (𝑃𝑔’s). Main effects of cue condition and set size were observed for all exposure durations [cue condition: F’s(1, 21) β‰₯ 5.21, p’s ≀ .033, Ξ·p2 β‰₯ .20; set size: F(3.85, 80.75) = 16.24, p < .001, Ξ·p2 = .44 for 500ms; F(2.87, 60.35) = 8.42, p < .001, Ξ·p2 = .29 for 1000ms; F(2.44, 51.15) = 14.13, p < .001, Ξ·p2 = .40 for 1500ms; F(3.34, 70.06) = 11.66, p < .001, Ξ·p2 = .36 for 2000ms]. In Experiment 1, when set sizes increased to 5 and beyond, 𝑃𝑔 in post-cue trials reliably differed from that in post-cue trials (Figure 3.6B). In the current experiment, when exposure duration decreased to 500ms, post-cue 𝑃𝑔 was at least marginally different (𝛼 = .10) from pre-cue 𝑃𝑔 at set sizes β‰₯ 3 [t’s(21) β‰₯ 1.87, p’s ≀ .075, d’s β‰₯ 0.40], except at set size 7 [t(21) = 1.32, p = .200, d = 0.28]. Post-cue 𝑃𝑔 was not different from pre-cue 𝑃𝑔 at set sizes ≀ 2 [t’s(21) ≀ 0.96, p’s β‰₯ .346, d’s ≀ 0.20] (Figure 3.6A). Possibly, 500ms was not sufficient to reach the maximum ensemble representation capacity. This bottleneck could be driven by either the speed of set individuation, or that of mean computation, or a combination of both.   18   Figure 3.5 RMSE in Experiments 1 and 3. The four panels show RMSE in pre-cue (solid curves) and post-cue (broken curves) conditions of the ensemble perception task with exposure durations of (A) 500ms, (B) 1000ms in Experiment 1, (C) 1500ms, and (D) 2000ms. (*p < .05, †p < .10, error bars reflect Β±1 SEM)   On the other hand, when the exposure duration increased to 1500ms, post-cue 𝑃𝑔 significantly differed from pre-cue 𝑃𝑔 at set sizes β‰₯ 5 [t’s(21) β‰₯ 2.17, p’s ≀ .042, d’s β‰₯ .46], but not at set sizes ≀ 4 [t’s(21) ≀ 0.60, p’s β‰₯ .556, d’s ≀ 0.13] (Figure 3.6C). Increasing the exposure duration to 2000ms did not alter this trend, except that the difference at set size 5 now become marginally significant [𝑑(21) = 1.75, 𝑝 = .094, 𝑑 = 0.37]; the difference was significant at set sizes 6 to 8 [𝑑′𝑠(21) β‰₯ 3.14, 𝑝′𝑠 ≀ 0.005, 𝑑′𝑠 β‰₯ 0.67], but not at set sizes 1 to 4 [t’s(21) ≀ 1.02, p’s β‰₯ .321, d’s ≀ 0.22] (Figure 3.6D). In addition, for 1000ms and beyond, difference between pre-cue and post-cue 𝑃𝑔’s at set sizes 5 to 8 was accompanied by a relatively constant post-cue 𝑠𝑑 [F’s(3, 36) ≀ 2.02, p’s β‰₯ .121, Ξ·p2 ≀ .09] (Figure 3.7).   19   Figure 3.6 Guess probability π‘·π’ˆ in Experiments 1 and 3.  The four panels show π‘·π’ˆ in the pre-cue (dark-gray bars) and post-cue (light-gray bars) conditions of the ensemble perception task with exposure durations of (A) 500ms, (B) 1000ms, (C) 1500ms, and (D) 2000ms. (*p < .05, †p < .10, error bars reflect ο‚±1 SEM)   Taken together, the four-set capacity limit of ensemble perception was not improved by extending exposure duration beyond 1000ms, suggesting that 1000ms might be a saturating point where the encoding limit no longer imposed a constraint on ensemble perception. The four-set capacity limit consistently observed at or beyond 1000ms was likely to stem from internal resource constraint.  20   Figure 3.7 Estimation precision (sd) in Experiments 1 and 3.  The four panels show sd in the pre-cue (dark-gray bars) and post-cue (light-gray bars) conditions with exposure durations of (A) 500ms, (B) 1000ms, (C) 1500ms, and (D) 2000ms. (*p < .05, †p < .10, error bars reflect ο‚±1 SEM)   3.4 Chapter 3 discussion  Experiments in this chapter determined capacity limit of ensemble perception of spatially intermixed sets. Experiment 1 found that observers could accurately estimate the mean size of circles from maximally four sets, and Experiment 2 suggested their performance was unlikely to be driven by a one-item subsampling strategy. With varying exposure durations, Experiment 3 found that the four-set capacity limit was better accounted for by internal resource limits, rather than an encoding limit.  3.4.1 Common capacity limits The capacity limit found in the current study is consistent with previously observed limits on visual working memory (Zhang & Luck, 2008; Luck & Vogel, 1997), attentional tracking (Pylyshyn & Storm, 1988; Cavanagh & Alvarez, 2005), and enumeration (Halberta et al., 2006). 21  David Marr (1982) distinguishes three levels of analysis for a cognitive function: the computational level, which defines the objectives of the function, and the constraints that it should satisfy; the algorithmic/representational level, which investigates how input and output of the cognitive function are represented and manipulated; and the implementation level, which studies physical basis for the algorithms and representations. The convergence of capacity among various resource-limited cognitive functions suggest they may be underlain by similar algorithmic/representational principles, or share common neural basis (the implementation level).   For instance, at the algorithmic/representational level, selective attention may exert a β€œgatekeeping” effect on both visual working memory and ensemble perception, directing the encoding effort to items that are goal-relevant in early perceptual processing (Awh, Vogel, & Oh, 2006; Gazzaley, 2012). Rutman, Clapp, Chadick, and Gazzaley (2010) presented superimposed face and scene stimuli to participants, and asked them to selectively attend to one of them. They found early modulation of visual cortical activity (P1 component of ERPs), which signaled selectively attention to goal-relevant stimuli. More importantly, P1 magnitude predicted working memory performance in a subsequent face or scene recognition task. This study and others (Zanto & Gazzaley, 2009) provide evidence that attention enhances working memory representation by selectively filtering irrelevant distractors at the early perceptual stage. In forming ensemble representations of multiple object groups, it is crucial to individuate each group, which creates a great demand for selective attention. As attention is a capacity-limited resource (Xu & Chun, 2011; Alvarez & Franconeri, 2007), it could thus constrain the number of sets for which an ensemble representation can be formed. On the other hand, attention-based rehearsal is necessary for working memory maintenance (Awh & Jonides, 2001; Awh, Jonides, & Reuter-Lorenz, 1998). In the ensemble perception tasks of this thesis work, if mean sizes are computed sequentially, then attention-based rehearsal of the set means that are already encoded is needed. Thus, attention could also impose a limiting factor on ensemble perception during the maintenance stage.   22  At the physical level, brain areas and neural computations underlying top-down modulations may provide a substrate for selective attention, which in turn serves as a capacity-limiting factor on visual working memory and ensemble perception. A recent study applied functional magnetic resonance imaging (fMRI)-guided, repetitive transcranial magnetic stimulation (rTMS) to disturb the right inferior frontal junction (rIFJ) in the prefrontal cortical (PFC) region, which had been found underlying early top-down modulation (Zanto, Rubens, Bollinger, & Gazzaley, 2010). The study observed impaired performance in a selective-attention, delayed color recognition task (Zanto, Rubens, Thangavel, & Gazzaley, 2011). Thus, a causal role of PFC-mediated top-down modulation in information selection and distractor filtering during the early stage of working memory encoding was established, confirming previous correlational studies (Gazzaley et al., 2007; Mayer, Bittner, Nikolic, Blendowski, Geobel, & Linden, 2007). Since constructing ensemble representations for multiple sets requires target selection and distractor filtering, it is reasonable to postulate its dependence on PFC-mediated top-down modulation, the limitation of which may undermine the signal-to-noise ratio in ensemble perception, and ultimately leads to failure in encoding beyond the capacity limit.    3.4.2 Discrete-slots hypothesis In the discrete-slots model proposed by Zhang and Luck (2008), working memory resources are quantized into a fixed number of slots. They further distinguish two subclasses of discrete-slots models: the β€œslots + resources model”, and the β€œslot + averaging model”. The slot + resources model assumes a one-to-one correspondence between objects and slots, and that working memory resources are allocated in a graded manner to each slot. It is thus tantamount to a flexible-resources models with a fixed upper limit on the number of items to be stored. In the β€œslots + averaging model”, working memory resources can only be allotted to each slot in an all-or-none manner, yet, each object can be represented with a flexible number of slots. That is, multiple slots can be allocated to encode one item, thus boosting the recall precision.   Results from the three experiments of this chapter seem to be consistent with the slot + resources model. First, I consistently observed a decrease in post-cue representation precision (higher 𝑠𝑑), which gradually plateaued as set size increased. Such a plateau in precision rules out a pure 23  flexible-resources model, which would have predicted monotonic decrease in precision without a lower bound. Second, post-cue precision plateaued before the capacity limit was reached (see additional analysis in Appendix C  . Possibly, when there is one set mean to represent, all resources are allocated to that slot, conferring it a high precision. As set size increases, the amount of resources allocated to each slots reduces, hence a decrease in precision. Finally, when set sizes further increases beyond the capacity limit, there are no available slots to represent additional set means. As now resources remain constant in each slot, so does the representation precision. This would fit the slots + resources model.   Further tests for a slots + resources model for ensemble perception can include manipulating attention to set means, and see if resources can be flexibly directed to set means that receive more attention (Zhang & Luck, 2008; Bays & Husain, 2008). Previous studies also suggest stimulus complexity (information load) mediates the maximum number of items that can be held in working memory (Alvarez & Cavanagh, 2004), so it needs to be tested whether the four-set capacity limit found in this thesis work would hold for different feature dimensions and ensemble statistics.   3.4.3 Limitations Experiment 3 consists of four sub-experiments, each examining capacity limit of ensemble perception with a different exposure duration. Different participants were assigned to each sub-experiment. Although I derived the capacity limits from within-subject difference between the pre-cue and post-cue conditions in each sub-experiment, which minimized random effects across the four exposure conditions, having the same participants going through all four conditions would lend Experiment 3 additional power.    24  Chapter 4: Efficiency of Ensemble Perception Previous studies demonstrate efficiency of ensemble perception in terms of its speed, little requirement of overt attention, and being able to be formed from noisy representations of individual objects (Ariely, 2001; Parkes et al., 2001; Alvarez & Oliva, 2008). However, it remains unclear whether an ensemble representation is able to carry comparable amount of information to the individual representations it is abstracting, which is another important aspect of efficiency. In this chapter, I compared information of individual representation and corresponding ensemble representation from two perspectives: their capacity limits, and their information difference.    In Experiment 4, the pre-cue/post-cue paradigm outlined in Chapter 2 was applied to a task of individual circle perception. Capacity limit was found at set size 1, suggesting no more than four individual items could be reliably represented. In previous experiments, I identified a four-set capacity limit for ensemble representation, which is an abstraction of 16 individual items. Thus, ensemble perception might be able to encode more visual information than individual representation.   In Experiment 5, within an information theoretic framework, I compared the amounts of information carried by individual representation and ensemble representation. β€œInformation theory” is first developed by Shannon (1948) to characterize the dynamics in signal processing; central to it is the concept of β€œentropy”, which quantifies the amount of uncertainty associated with a random variable. Given two random variables, their difference can be measured by the Kullback-Leibler (KL) divergence. A number of studies have applied information theory to analyze cognitive phenomena, such as dynamical evolution of decision-making (Stephen, Boncoddo, Magnuson, & Dixon, 2009), and how new concepts, categories, and memories are created (Stephen, Dixon, & Isenhower, 2009). Since information theory facilitates quantitative comparisons of complex information systems, it provides a powerful analytic tool for characterizing ensemble representation in relation to individual representation. Results from Experiment 5 suggest an individual representation and an ensemble representation could be encoded at similar uncertainty levels.  25  4.1 Experiment 4 4.1.1 Participants and procedure Twenty-two new participants (18 females, mean age = 20.6yrs, SD = 2.4yrs) were recruited. As this experiment set out to compare the capacities of ensemble representation and individual representation, participants viewed the same stimulus arrays as in Experiment 1, but were only asked to report the size of a probed circle, rather than the set mean. The fixation cross, either colored (pre-cue condition) or hallow (post-cue condition), was always placed at the center of the screen. This encouraged participants to encode all four circles in a set as accurately as possible. The probe, which was a colored circle, was placed at the center of a randomly chosen circle. Participants were instructed to report the size of the probed circle by adjusting the probe. There were 20 trials in each cue Γ— set size condition, totaling 320 trials. The mixture model outlined in Chapter 2 was applied to the signed error πœƒπ‘– = π‘₯𝑖 βˆ’ π‘Ÿπ‘–, where π‘₯𝑖 is the reported circle size, and π‘Ÿπ‘– is the target circle size, in trial 𝑖. RMSE is as defined by Eqn. (1).   4.1.2 Results For RMSE (Figure 4.1A), a 2 (cue) Γ— 8 (set size) ANOVA showed significant main effects of cue condition [𝐹(1, 21) = 274.51, 𝑝 < .001, πœ‚π‘2 = .93], and set size [𝐹(7,147) = 15.37, 𝑝 <.001, πœ‚π‘2 = .42]. Interaction between these two factors was also observed [𝐹(4.73, 99.40) =11.04, 𝑝 < .001, πœ‚π‘2 = .35], which was driven by a significant effect of set size on post-cue RMSE [𝐹(4.87, 102.36) = 20.75, 𝑝 < .001, πœ‚π‘2 = .50], but not on pre-cue RMSE [𝐹(4.85, 101.79) = 0.49, 𝑝 = .778, πœ‚π‘2 = .02]. Paired-sample t-tests revealed difference between pre-cue and post-cue RMSE’s at set sizes β‰₯ 3 [𝑑′𝑠(21) β‰₯ 3.67, 𝑝′𝑠 β‰₯ .001, 𝑑′𝑠 β‰₯0.78].   In terms of guess probability 𝑃𝑔 (Figure 4.1B), two-way ANOVA identified main effects of cue condition [𝐹(1, 21) = 9.47, 𝑝 = .006, πœ‚π‘2 = .31], and set size [𝐹(3.04, 63.77) = 19.76, 𝑝 <.001, πœ‚π‘2 = .49]. The interaction was not significant [𝐹(3.48, 73.15) = 0.68, 𝑝 = .591, πœ‚π‘2 =.03]. Difference between pre-cue and post-cue 𝑃𝑔’s did not show a clear trend as set size increased: a marginal difference was observed at set size 1 [𝑑(21) = 1.80, 𝑝 = .087, 𝑑 = 0.38]; 26  at set sizes 2 and 3, the difference was not significant [𝑑′𝑠(21) ≀ 1.55, 𝑝′𝑠 β‰₯ .136, 𝑑′𝑠 < 0.33]; at others, the difference was significant [𝑑′𝑠(21) β‰₯ 2.00, 𝑝′𝑠 ≀ .059, 𝑑′𝑠 β‰₯ 0.43]. It thus seemed participants could not reliably represent more than one set (four individual items)   Figure 4.1C shows the estimation precision 𝑠𝑑. At set size 1, pre-cue 𝑠𝑑 and post-cue 𝑠𝑑 did not differ [𝑑(21) = 0.05, 𝑝 = .096, 𝑑 = 0.01]. Their difference was marginally significant at set size 2 [𝑑(21) = 1.75, 𝑝 = .095, 𝑑 = 0.37], and reliably significant at larger set sizes [𝑑′𝑠(21) β‰₯3.03, 𝑝 ≀ .006, 𝑑 β‰₯ .0.65].  Figure 4.1 Results of Experiment 4.  (A) RMSE in the pre-cue and the post-cue conditions. (B) The guess probability Pg. (C) The estimation precision as measured by the standard deviation sd. (*p < .05, †p < .10, error bars reflect ο‚±1 SEM) 27  4.2 Experiment 5 4.2.1 Participants and procedure This experiment aimed to determine whether internal representations of ensemble perception resembled those of individual object perception. Specifically, I focused on comparing the error distributions of these two processes using the pre-cue/post-cue paradigm. For their resemblance to be true, their error distributions should share similar capacity limits and statistical properties.  Figure 4.2 Experimental procedure of Experiment 5.  An example trial in the pre-cue condition (A) and in the post-cue condition (B) are shown. In each trial, the array contained 1 to 8 sets of circles, each set containing one circle. In the pre-cue example trial, the cyan fixation cross indicated the mean size of the cyan circle would be probed later; and in the post-cue example trial, the fixation cross was hollow and which set would be probed was not known in advance.  Twenty-two new participants (17 females, mean age = 20.0yrs, SD = 2.1) completed the experiment. The stimuli were similar to those in Experiment 1, except that each set now contained only one circle, so that each array contained between 1 and 8 circles. The circles were randomly assigned to the invisible 5Γ—7 grid. The procedure was identical to that in Experiment 1: each trial started with a central fixation cross for 1000ms, which might be filled with a specific color that indicated which circle would be probed (pre-cue condition), or hallow with a white outline and the probed circle would not be known until after the array disappeared (post-cue condition). After the cue, an array with 1 to 8 circles, each in a unique color, appeared for 1000ms, and participants were instructed to remember the size of a specific circle (pre-cue condition) or every circle (post-cue condition). The array was then replaced by a 500ms white-noise mask, followed by a probe circle in a specific color. Participants reported the size of the circle in that color in the stimulus array.  28  Analyses of RMSE and model parameters followed the same procedure as in Experiment 1. Additionally, I drew on Information Theory (Shannon, 1948) to measure how well ensemble representation resembled individual representation. The information of a discrete random variable 𝑋 at 𝑋 = π‘₯ is quantified by the logarithm of the inversed probability:  log1𝑃(𝑋=π‘₯)= βˆ’ log 𝑃(𝑋 = π‘₯). The entropy of 𝑋 is defined as the average information we gain from observing 𝑋, thus:  𝐻(𝑋) = 𝔼[βˆ’ log 𝑃(𝑋)] = βˆ‘ βˆ’π‘ƒ(𝑋) log 𝑃(𝑋)𝑋  The Kullback-Leibler (KL) divergence 𝐷𝐾𝐿(𝑃||𝑄) measures the difference between two distributions 𝑃(𝑋) and 𝑄(𝑋) about random variable 𝑋, and is defined as:  𝐷𝐾𝐿(𝑃||𝑄) = βˆ‘ 𝑃(π‘₯𝑖) log𝑃(π‘₯𝑖)𝑄(π‘₯𝑖)𝑖 In other words, 𝐷𝐾𝐿(𝑃||𝑄) quantifies how much information loss would be incurred if one switches form 𝑃(𝑋) to 𝑄(𝑋).   The intuitive extension of entropy and KL divergence to a continuous variable 𝑋 is:  β„Ž(𝑋) = 𝔼[βˆ’ log 𝑃(𝑋)] = ∫ βˆ’π‘“(π‘₯) log 𝑓(π‘₯) 𝑑π‘₯ And:  𝐷𝐾𝐿(𝑃||𝑄) = ∫ 𝑓(π‘₯) log𝑓(π‘₯)𝑔(π‘₯)𝑑π‘₯ Where 𝑓(π‘₯) and 𝑔(π‘₯) are two continuous probability densities for 𝑋. 𝐷𝐾𝐿(𝑃||𝑄) is guaranteed to be non-negative, but β„Ž(𝑋) is not (El Gamal & Kim, 2011).   In the context of ensemble perception, let 𝑓(π‘₯) be the error distribution of an individual representation; let π‘Œ be the error distribution of ensemble representation. Suppose the distributions 𝑓(π‘₯) and 𝑔(π‘₯) both follow the Uniform-Gaussian mixture model defined by Eqn. (2). Parameters of 𝑓(π‘₯), namely,  𝑃𝑋, 𝑠𝑑𝑋, π‘Žπ‘‹ and 𝑏𝑋, are approximated using the average parameter values from participants in the current experiment; π‘ƒπ‘Œ, π‘ π‘‘π‘Œ, π‘Žπ‘Œ and π‘π‘Œ for 𝑔(π‘₯) are (5) (6) (7) (8) 29  approximated using average parameter values from participants in Experiment 1. The lower- and upper-bounds for integration are π‘Žπ‘‹ and π‘Žπ‘Œ respectively.   4.2.2 Results As before, ANOVA on RMSE (Figure 4.3A) showed significant main effects of set size [F(3.99, 83.68) = 14.89, p < .001, Ξ·p2 = .42], and cue condition [F(1, 21) = 145.40, p < .001, Ξ·p2 = .87], as well as an interaction [F(4.01, 84.21) = 16.26, p < .001, Ξ·p2 = .44]. Paired-sample t-tests showed that post-cue RMSE was reliably greater than pre-cue RMSE at all set sizes [t’s(21) β‰₯ 3.00, p’s ≀ .004, d’s β‰₯ 0.64], except for set size 1 [t(21) = 0.34, p = .370, d = 0.07].  Figure 4.3 Results of Experiment 5.  (A) RMSE in the pre-cue and the post-cue conditions. (B) The guess probability Pg. (C) The estimation precision as measured by the standard deviation sd. (*p < .05, †p < .10, error bars reflect ο‚±1 SEM)  30  Analyses of the guess probability 𝑃𝑔 (Figure 4.3B) showed that pre-cue and post-cue 𝑃𝑔’s were significantly different at set sizes β‰₯ 6 [t’s(21) β‰₯ 2.37, p’s ≀ .028, d’s β‰₯ 0.50] and marginally different at set size 5 [𝑑(21) = 1.76, 𝑝 = .094, 𝑑 = 0.37], but not at set sizes 2 to 4 [t’s(21) ≀ 0.38, p’s β‰₯ .706, d’s ≀ 0.08]. Curiously, pre-cue/post-cue difference at set size 1 was significant [𝑑(21) = 2.17, 𝑝 = .041, 𝑑 = 0.46]. This was however driven by a low post-cue 𝑃𝑔, which did not suggest a capacity limit was reached. Despite this, since a higher probability of random guess in the post-cue condition was still consistently observed at set sizes β‰₯ 5, I concluded that individual representation had a four-item capacity limit. This limit coincided with those in Experiments 1 and 2, and was consistent with previous studies that suggested a capacity of three or four items for visual working memory (Luck & Vogel, 1997). I also examined the standard deviation 𝑠𝑑 (Figure 4.3C). Similar to the previous experiments, post-cue 𝑠𝑑 at set sizes 5 to 8 did not differ [F(3, 63) = 0.99, p = .401, Ξ·p2 = .05], suggesting the estimation precision remained constant above the capacity limit.  Next, I computed the entropy and KL divergence of the mixture model of post-cue trials at each set size. As Table 4.1 shows, information uncertainty associated with individual representations in the current experiment was generally lower than that associated with ensemble representations in Experiment 1. This is intuitive, since ensemble representations are likely to be noisier than individual representations within their respective capacity limits, due to computation error and increased task difficult. KL divergence between individual and ensemble representation at each set size was small relative to the entropy. Thus, our visual system seemed to be able to hold an ensemble representation as well as it holds an individual representation. These results provide further evidence that a common representational scheme may underlie perception of individual objects and ensemble perception.       31  Set size  Entropy (nats): individual representation, 𝒇(𝒙) Entropy (nats): ensemble representation, π’ˆ(𝒙) KL divergence (nats) 1  8.8441  9.4218  0.016760 2  8.9779  9.4697  0.0086696 3  9.1661  9.6202  0.014023 4  9.2408  9.6667  0.017891 5  9.4161  9.7608  0.024548 6  9.4750  9.7731  0.033219 7  9.5327  9.8140  0.033686 8  9.6120  9.8227  0.032339 Table 4.1  Entropy of individual and ensemble representations at different set sizes, and their KL divergence.  4.3 Chapter 4 discussion In this chapter, Experiment 4 showed the visual system could reliably represent up to four individual items. By contrast, ensemble perception allows us to represent maximally four summary statistics, which is abstraction of 16 individual items. In this sense, ensemble perception enables us to hold information for a greater number of objects. Recent studies in Bayesian inferences in cognition suggests perceptual organizations embedded in multiple objects can be incorporated into a β€œcognitive prior” that benefits working memory recall (Brady & Tenenbaum, 2013; Brady, Konkle, & Alvarez, 2009). Similarly, when encoding an array with multiple sets, such as those in Experiment 4, the visual system may encode the set means as the prior, and the probabilistic relationship between a mean and its member items as the likelihood. Combination of these two statistics can then be used to derive information about an individual item. A recent study showed psychophysical data consistent with the Bayesian interference model in spatial working memory recall, where the prior was the center of the items, and the likelihood was computed based on the distance between the center and the items (Lew & Vul, 2015). However, it did not offer data to prove their likelihood function, so it is unclear whether alternative likelihoods, such as one that based on relative distances between items (rather than between a center and items), would be more relevant. Thus, to show how our visual system utilizes ensemble perception is our next challenge.    Experiment 5 confirmed the four-item capacity of individual representation. By computing the entropy, I also showed the four individual items encoded in Experiment 5 and the four set means encoded in Experiment 1 may carry similar amount of information. That is, the four items and 32  the four set means could be represented with similar levels of uncertainty. It should be noted that data from Experiment 1 seem to have greater standard deviations of the Gaussian, in general. How could this still result in small information difference between the two experiments? One possibility is that the uniform distribution component, together with the guess probability, offset the difference in Gaussian standard deviation.   4.3.1 Prevalence of efficient representations Shannon and Weaver (1949) defines β€œredundancy” as inputs that carry very little information and waste channel capacity, if encoded. β€œRedundancy reduction” is the step that achieves economy without losing any information. Barlow (1961) is among the first to extend information theory and redundancy reduction to neurophysiological properties of sensory neurons. In particular, he stipulates the β€œefficient coding” hypothesis, which states neuronal populations should encode information as compactly as possible, so as to maximize the information they convey about the environment (Barlow 1961, 2001). It is important to distinguish the definition of β€œefficiency” here from one that refers to a rapid encoding rate, or minimal demand of focused attention. From Barlow’s efficient coding hypothesis, two implications can be made about neural code: first, it minimizes the number of spikes needed to transmit a signal; second, it should match the statistics of the signals it represents (Barlow, 1961). Thus, efficient coding should be able to make use of statistical regularities in the environment (Shepard, 1994; Simoncelli & Olshausen, 2001).   Ensemble perception may be a cognitive manifestation of the efficient coding principle, as it provides a way to achieve information compression and utilizing statistical regularities. Also related are other forms of cognitive processes that capitalize on environmental statistical regularities, such as the β€œgist”, which summarizes the semantic categories and global perceptual structures of a complex scene (Oliva, 2004; Oliva & Torralba, 2001); perceptual organizations, or Gestalt perception, which extracts structures and groupings from visual inputs (Wertheimer, 1955; Rubin, 1921; Palmer, 1992); and statistical learning, which derives the probabilistic co-occurrence and other temporal or spatial patterns between items (Baker, Olson, & Behrmann, 2004). Similar to ensemble perception, these cognitive processes are found to operate rapidly, 33  without requirement of focused attention. For example, global gist, such as the degree of naturalness, openness, and mean depth, can be extracted with a 75% accuracy rate within 19 to 67ms’s exposure (Greene & Oliva, 2009a), and from images of very low resolutions (Oliva & Schyns, 2000). Statistical learning can occur automatically without overt attention (Fiser & Aslin, 2001), and be adopted by infants as young as 2 months old (Kirkham, Slemmer, & Johnson, 2002). Objects organized by Gestalt laws are more rapidly detected in visual research than unstructured items (Palmer, 1977; Wagemans, et al., 2012). Moreover, it has been established that these abstraction processes facilitate performance in a variety of cognitive tasks. For instance, perception of gist enables rapid scene categorization (Greene & Oliva, 200b); our visual system can also cultivate perceptual organization and statistical learning to boost the number of objects that can be held in working memory (Brady & Tenenbaum, 2013).   It is possible that further understanding of the efficient coding principle would offer insights into ensemble perception, gist perception, perceptual organization and statistical learning; and knowledge about these cognitive processes could advance our understanding of efficient coding. To date, efforts have been to identify neuronal and circuit properties that could support efficient coding. For instance, Baddeley et al. (1997) recorded from V1 of cats and inferior temporal (IT) areas of monkeys for neuronal responses to natural scene video. They found that neurons maximized their information transmission for a fixed long-term average firing rate. Neurons in V2 and V4 in nonhuman primates are found sensitive to texture statistics of natural images (Movshon & Simoncelli, 2014; Freeman & Simoncelli, 2011). Further experimental and theoretical efforts are needed to bridge efficient coding principle and high-level information compression in cognition. Moreover, other neural coding principles, such as redundant coding (Puchalla, Schneidman, Harris, & Berry, 2005), may be also considered in formulating a good sensory processing mechanism for ensemble perception.      34  Chapter 5: Conclusion In summary, my thesis work demonstrated a four-set capacity limit of ensemble perception of multiple spatially intermixed sets. This result could not be explained by a heuristic strategy of subsampling one item from each set. Allowing longer exposure to the stimulus array did not increase the capacity, suggesting ensemble perception was limited by an internal resource constraint, rather than an information encoding rate limit. Moreover, our visual system could hold up to four ensemble representations, or up to four individual items. As the four ensemble representations abstract information about more than four individual items, they provide a compact and efficient way of information processing. Finally, I showed that, in my experimental paradigm, an ensemble representation may have an information uncertainty (entropy) level similar to that of an individual representation. Thus, we can represent a summary statistic as well as we can represent an individual item. Taken together, my results suggest ensemble perception is a resource-limited cognitive process that enables efficient processing of visual information. Understanding properties of ensemble perception helps reveal how the visual system extracts and abstracts goal-related information from a complex environment.  For future studies, it would be important to understand whether the capacity limit of ensemble perception observed in the current study is generalizable to other feature dimensions, probing methods (e.g., discrimination tasks), and experimental paradigms. Moreover, it would be interesting to see how spatial organization of items or groups interacts with the capacity limit and efficiency of ensemble perception. For one thing, ensemble perception could be improved by a more salient spatial grouping cue, as a recent study by Im and Chong (2014) suggests enhancing spatial clustering of items in a set can increase the accuracy of mean size computation. Another way to manipulate spatial organization is to place items in a set in the same or different visual fields (left versus right, or foveal versus peripheral). Previous studies on attentional tracking found that participants can track twice as many targets when they are divided between the left and right hemifields as when they are confined to only one hemiefield (Alvarez & Cavanagh, 2005). Possibly, independent resources are linked to the two visual fields, or to different spatial cues.   35  Finally, in the experiments of my thesis, participants were explicitly instructed to compute the mean of a set. 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Nature, 453, 233-235.    42  Appendices Appendix A  Discard rates and response time Experiment 1  Experiment 2  Experiment 3 (500 ms)  Experiment 3 (1500 ms)  Experiment 3 (2000 ms) ID Discard rate (%) Mean RT (sec) ID Discard rate (%) Mean RT (sec) ID Discard rate (%) Mean RT (sec)  ID Discard rate (%) Mean RT (sec)  ID Discard rate (%) Mean RT (sec) 1 1.88 1.25 1 1.88 1.88 1 0.94 1.01 1 1.88 2.21 1 2.19 1.48 2 2.50 1.78 2 1.88 1.78 2 1.88 2.17 2 2.19 1.79 2 0.94 2.15 3 1.56 1.73 3 1.88 1.21 3 0.62 1.74 3 0.94 2.05 3 0.94 2.08 4 1.56 1.55 4 1.25 1.70 4 1.25 1.81 4 0.94 1.89 4 1.88 1.50 5 0.62 1.89 5 1.56 1.63 5 0.94 1.74 5 2.81 1.19 5 1.25 1.46 6 0.62 3.14 6 1.25 1.61 6 1.56 1.28 6 1.88 2.34 6 0.62 1.32 7 1.88 1.95 7 1.25 2.84 7 1.56 1.59 7 1.25 2.17 7 1.56 1.21 8 1.88 1.87 8 1.88 1.56 8 1.88 1.54 8 2.19 1.67 8 1.56 1.26 9 1.56 2.25 9 1.88 1.96 9 2.19 1.59 9 2.19 3.13 9 0.62 0.96 10 0.94 2.17 10 1.88 1.58 10 1.25 3.73 10 2.50 3.13 10 1.25 1.60 11 0.62 1.54 11 2.19 2.30 11 2.19 1.10 11 1.25 2.20 11 0.31 1.33 12 3.13 1.09 12 2.19 1.64 12 0.62 1.26 12 1.88 3.79 12 1.25 2.02 13 1.88 1.06 13 2.50 1.34 13 1.88 1.72 13 3.44 1.28 13 1.88 1.49 14 0.31 1.53 14 1.88 1.07 14 2.50 3.02 14 0.94 1.96 14 1.56 1.28 15 1.88 1.01 15 1.56 1.06 15 1.25 1.30 15 2.19 1.34 15 2.19 1.88 16 0.31 0.97 16 1.56 1.60 16 1.88 1.51 16 1.88 1.00 16 2.19 1.40 17 2.19 1.48 17 0.94 1.47 17 1.88 1.26 17 1.25 1.43 17 1.88 1.26 18 3.13 1.63 18 1.88 1.51 18 1.25 1.52 18 2.50 1.33 18 1.88 1.11 19 1.88 1.53 19 1.88 1.75 19 1.88 0.96 19 0.94 1.50 19 1.25 1.87 20 0.31 1.57 20 1.56 1.97 20 0.31 1.76 20 1.56 1.51 20 1.88 2.02 21 1.56 1.53 21 1.56 1.35 21 0.94 1.48 21 2.50 1.34 21 1.56 1.42 22 1.56 1.25 22 1.25 1.38 22 0.62 1.26 22 0.62 2.21 22 1.56 4.03  M 1.53 1.63  M 1.70 1.64  M 1.42 1.65  M 1.80 1.93  M 1.46 1.64 SE 0.18 0.10 SE 0.27 0.29 SE 0.13 0.14 SE 0.16 0.15 SE 0.11 0.13 43  (cont’d) Experiment 4  Experiment 5 ID Discard rate (%) Mean RT (sec) ID Discard rate (%) Mean RT (sec) 1 1.56 1.53 1 1.88 1.33 2 2.81 2.61 2 2.19 1.71 3 1.25 1.27 3 1.25 1.56 4 1.56 1.61 4 1.56 2.13 5 0.94 1.65 5 1.88 2.39 6 0.62 1.92 6 1.25 2.31 7 1.56 1.63 7 2.81 1.92 8 1.88 2.98 8 0.94 2.09 9 1.25 1.21 9 1.25 2.03 10 0.94 1.38 10 1.88 1.69 11 0.62 2.49 11 2.81 1.38 12 1.88 1.56 12 1.56 1.48 13 1.25 1.16 13 1.88 1.74 14 1.25 1.41 14 0.62 1.67 15 0.94 1.29 15 1.56 2.12 16 2.50 1.75 16 1.56 1.22 17 0.62 1.45 17 1.25 1.34 18 1.25 1.28 18 1.56 1.13 19 2.19 1.71 19 0.62 1.70 20 0.31 2.13 20 3.44 1.85 21 1.56 1.91 21 1.25 2.06 22 1.25 1.51 22 0.62 1.09  M 1.36 1.70  M 1.62 1.73 SE 0.13 0.10 SE 0.15 0.08 Table A.1 Discard rate and mean response time of participants in five experiments.  β€œDiscard rate” is calculated by dropping the trials with a response time (RT) greater or less than three standard deviations from the participant’s mean RT of all trials. The remaining trials are the β€œvalid trials”. β€œMean RT” is the average RT of the valid trials. β€œM” and β€œSE” are the mean and standard error for all participants in the experiment.   44  Appendix B  Experiment 2 partial correlation  Subject Partial correlation 𝒑 Significance 1 -0.03 0.614  2 0.05 0.358  3 0.30 <.001 * 4 -0.02 0.679  5 0.00 0.950  6 0.04 0.449  7 -0.09 0.127  8 0.01 0.929  9 0.04 0.526  10 0.10 0.069 † 11 -0.03 0.647  12 -0.01 0.925  13 -0.02 0.766  14 -0.03 0.604  15 0.03 0.592  16 0.02 0.711  17 0.12 0.034 * 18 0.15 0.008 * 19 -0.07 0.208  20 -0.07 0.213  21 0.46 <.001 * 22 0.01 0.920  Table B.1 Partial correlation in Experiment 2.  The second column shows the partial correlation of estimation and size of the flashed circle in the probed set, after controlling for size of the probed set mean. The third column shows significance of the partial correlation (*p < .05, †p < .10).   45  Appendix C  Additional analysis C.1 Experiment 1 additional analysis C.1.1 RMSE: p-vales of multiple comparisons with Bonferroni correction Pre-cue RMSE:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 0.577 0.205      S4 1.000 1.000 1.000     S5 1.000 1.000 1.000 1.000    S6 1.000 1.000 1.000 1.000 1.000   S7 1.000 1.000 1.000 1.000 1.000 1.000  S8 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Post-cue RMSE:  S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 1.000 1.000      S4 0.705 1.000 1.000     S5 0.107 0.263 1.000 1.000    S6 0.362 0.196 1.000 1.000 1.000   S7 0.005 0.011 0.046 0.009 0.183 0.457  S8 0.040 0.059 0.494 0.184 1.000 1.000 1.000  C.1.2 Guess probability 𝑃𝑔: p-values of multiple comparisons with Bonferroni correction Pre-cue 𝑃𝑔:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 1.000 1.000      S4 1.000 1.000 1.000     S5 1.000 1.000 1.000 1.000    S6 1.000 1.000 1.000 1.000 1.000   S7 0.887 1.000 1.000 1.000 1.000 1.000  S8 0.615 1.000 1.000 1.000 1.000 1.000 1.000 Post-cue 𝑃𝑔:  S1 S2  S3 S4 S5 S6 S7 S2 0.684       S3 1.000 1.000      S4 0.293 0.801 1.000     S5 0.024 0.098 1.000 1.000    S6 0.002 0.013 0.418 1.000 1.000   S7 0.003 0.019 0.497 1.000 1.000 1.000  S8 0.007 0.036 0.552 1.000 1.000 1.000 1.000  46  C.1.3 Estimation precision 𝑠𝑑: p-values of multiple comparisons with Bonferroni correction Pre-cue 𝑠𝑑:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 1.000 1.000      S4 1.000 1.000 1.000     S5 1.000 1.000 0.649 1.000    S6 1.000 1.000 1.000 1.000 1.000   S7 1.000 1.000 0.392 1.000 1.000 0.496  S8 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Post-cue 𝑠𝑑:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 1.000 1.000      S4 1.000 1.000 1.000     S5 0.076 0.342 1.000 1.000    S6 1.000 0.521 1.000 1.000 1.000   S7 0.056 0.151 0.161 0.599 1.000 1.000  S8 0.141 0.145 0.274 0.539 1.000 1.000 1.000  Comparison with Zhang and Luck (2008): Figure C.1 Post-cue π‘·π’ˆ (A) and 𝒔𝒅 (B) in Experiment 1.  Here, pot-cue π‘·π’ˆβ€™s at set sizes 5 to 8 were significantly different from that at set size 1, suggesting a significantly impaired encoding probability; 𝒔𝒅 at set sizes 5 to 8 were not different. This lends support to the discrete-slots hypothesis for cognitive resource allocation. However, as the tables above indicated, 𝒔𝒅 at set size 8 had n.s. difference from those in all other set sizes if tested with Bonferroni correction. The p-values were sufficiently large (p > .50) at set sizes β‰₯ 4.     47  C.1.4 The effect of grand mean:  Partial correlation tests indicated that, for about half of the participants, their estimated set mean was significantly correlated with the grand mean, after controlling for the true set mean. However, all participants had significant partial correlations between their estimated set mean and the true set mean, after controlling for the grand mean. Thus, it seemed the true mean and the grand mean both had influenced what participants reported.  Subject Partial correlation of estimated set mean and true set mean, given the grand mean p-value  Partial correlation of estimated set mean and grand mean, given true set mean p-value 1 0.385 <.001 0.017 0.763 2 0.546 <.001 0.107 0.061 3 0.526 <.001 0.138 0.015 4 0.645 <.001 0.087 0.123 5 0.623 <.001 0.177 0.002 6 0.363 <.001 0.297 <.001 7 0.600 <.001 -0.058 0.305 8 0.660 <.001 0.014 0.806 9 0.478 <.001 0.083 0.142 10 0.567 <.001 0.241 <.001 11 0.489 <.001 0.103 0.066 12 0.226 <.001 0.071 0.213 13 0.156 0.006 0.241 <.001 14 0.464 <.001 0.167 0.003 15 0.249 <.001 0.176 0.002 16 0.493 <.001 0.137 0.014 17 0.529 <.001 0.119 0.036 18 0.517 <.001 0.095 0.095 19 0.451 <.001 -0.033 0.566 20 0.524 <.001 0.033 0.553 21 0.616 <.001 0.072 0.206 22 0.425 <.001 0.046 0.412 Table C.1 Partial correlations in Experiment 1.    48  C.2 Experiment 2 additional analysis C.2.1 RMSE: p-vales of multiple comparisons with Bonferroni correction Pre-cue RMSE:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 1.000 1.000      S4 1.000 1.000 1.000     S5 1.000 1.000 1.000 1.000    S6 1.000 1.000 1.000 1.000 1.000   S7 1.000 1.000 1.000 1.000 1.000 1.000  S8 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Post-cue RMSE:  S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 0.050 1.000      S4 0.066 0.494 1.000     S5 0.012 0.348 1.000 1.000    S6 0.041 0.339 1.000 1.000 1.000   S7 0.001 0.051 0.304 1.000 1.000 1.000  S8 0.010 0.233 1.000 1.000 1.000 1.000 1.000  C.2.2 Guess probability 𝑃𝑔: p-values of multiple comparisons with Bonferroni correction Pre-cue 𝑃𝑔:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 1.000 1.000      S4 0.452 1.000 1.000     S5 0.130 1.000 1.000 1.000    S6 0.233 1.000 1.000 1.000 1.000   S7 0.064 0.884 1.000 1.000 1.000 1.000  S8 0.222 1.000 1.000 1.000 1.000 1.000 1.000 Post-cue 𝑃𝑔:  S1 S2  S3 S4 S5 S6 S7 S2 0.508       S3 0.096 1.000      S4 0.006 1.000 1.000     S5 <.001 0.146 0.339 1.000    S6 <.001 0.122 0.255 1.000 1.000   S7 <.001 0.041 0.089 0.524 1.000 1.000  S8 <.001 0.049 0.085 0.168 0.649 0.978 1.000 49  C.2.3 Estimation precision 𝑠𝑑: p-values of multiple comparisons with Bonferroni correction Pre-cue 𝑠𝑑:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 1.000 1.000      S4 1.000 1.000 1.000     S5 1.000 1.000 1.000 1.000    S6 1.000 1.000 1.000 1.000 1.000   S7 1.000 1.000 0.507 0.588 0.384 0.173  S8 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Post-cue 𝑠𝑑:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 0.010 0.337      S4 0.002 0.017 0.964     S5 0.001 0.053 1.000 1.000    S6 0.004 0.077 1.000 1.000 1.000   S7 0.000 0.009 0.390 1.000 1.000 1.000  S8 0.017 0.213 1.000 1.000 1.000 1.000 1.000  Comparison with Zhang and Luck (2008):  Figure C.2 Post-cue π‘·π’ˆ (A) and 𝒔𝒅 (B) in Experiment 2    50  C.3 Experiment 3 additional analysis C.3.1 Sub-experiment 1: 500 ms exposure duration C.3.1.1 RMSE: p-vales of multiple comparisons with Bonferroni correction Pre-cue RMSE:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 1.000 1.000      S4 0.568 0.270 0.156     S5 1.000 1.000 1.000 1.000    S6 1.000 1.000 1.000 1.000 1.000   S7 1.000 1.000 1.000 0.018 1.000 1.000  S8 1.000 1.000 1.000 0.425 1.000 1.000 1.000 Post-cue RMSE:  S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 0.315 0.013      S4 1.000 0.431 0.420     S5 1.000 0.063 1.000 1.000    S6 1.000 1.000 1.000 1.000 1.000   S7 0.253 0.021 1.000 0.338 1.000 0.099  S8 0.226 0.020 1.000 0.459 1.000 0.043 1.000  C.3.1.2 Guess probability 𝑃𝑔: p-values of multiple comparisons with Bonferroni correction Pre-cue 𝑃𝑔:   S1 S2  S3 S4 S5 S6 S7 S2 0.519       S3 0.163 1.000      S4 0.280 1.000 1.000     S5 0.291 1.000 1.000 1.000    S6 0.121 1.000 1.000 1.000 1.000   S7 0.019 0.169 0.122 0.039 0.724 0.701  S8 0.004 0.067 0.051 0.011 0.537 0.613 1.000 Post-cue 𝑃𝑔:  S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 0.081 0.285      S4 0.126 0.515 1.000     S5 0.024 0.089 1.000 1.000    S6 0.016 0.076 1.000 1.000 1.000   S7 <.000 0.003 0.813 1.000 1.000 1.000  S8 <.000 0.002 0.441 1.000 1.000 1.000 1.000  51  C.3.1.3 Estimation precision 𝑠𝑑: p-values of multiple comparisons with Bonferroni correction Pre-cue 𝑠𝑑:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 1.000 1.000      S4 0.803 0.448 0.774     S5 1.000 1.000 1.000 1.000    S6 1.000 1.000 1.000 0.497 1.000   S7 1.000 1.000 1.000 1.000 1.000 1.000  S8 1.000 1.000 1.000 0.523 1.000 1.000 1.000 Post-cue 𝑠𝑑:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 0.081 0.041      S4 1.000 0.179 1.000     S5 0.992 0.180 1.000 1.000    S6 0.644 0.088 1.000 1.000 1.000   S7 0.096 0.004 1.000 0.424 1.000 1.000  S8 0.480 0.100 1.000 1.000 1.000 1.000 1.000  Comparison with Zhang and Luck (2008):  Figure C.3 Post-cue π‘·π’ˆ (A) and 𝒔𝒅 (B) in Experiment 3 with 500ms exposure duration.     52  C.3.2 Sub-experiment 2: 1500 ms exposure duration C.3.2.1 RMSE: p-vales of multiple comparisons with Bonferroni correction Pre-cue RMSE:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 1.000 1.000      S4 1.000 0.090 0.394     S5 1.000 1.000 1.000 0.198    S6 1.000 0.334 1.000 1.000 0.180   S7 1.000 1.000 1.000 0.301 1.000 1.000  S8 1.000 1.000 1.000 1.000 1.000 0.349 1.000 Post-cue RMSE:  S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 1.000 0.392      S4 0.874 0.361 1.000     S5 0.026 0.040 1.000 1.000    S6 1.000 0.012 1.000 1.000 1.000   S7 0.079 0.000 0.591 1.000 1.000 0.059  S8 0.007 0.001 0.287 1.000 1.000 0.308 1.000  C.3.2.2 Guess probability 𝑃𝑔: p-values of multiple comparisons with Bonferroni correction Pre-cue 𝑃𝑔:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 1.000 1.000      S4 0.734 1.000 1.000     S5 0.299 1.000 1.000 1.000    S6 0.246 1.000 1.000 1.000 1.000   S7 0.143 1.000 1.000 1.000 1.000 1.000  S8 0.085 1.000 1.000 1.000 1.000 1.000 1.000 Post-cue 𝑃𝑔:  S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 0.482 1.000      S4 0.352 1.000 1.000     S5 0.026 0.371 1.000 1.000    S6 0.005 0.087 0.505 0.498 1.000   S7 0.004 0.034 0.134 0.150 1.000 1.000  S8 0.003 0.030 0.134 0.153 1.000 1.000 1.000   53  C.3.2.3 Estimation precision 𝑠𝑑: p-values of multiple comparisons with Bonferroni correction Pre-cue 𝑠𝑑:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 1.000 1.000      S4 0.829 1.000 1.000     S5 1.000 1.000 1.000 0.558    S6 1.000 1.000 1.000 1.000 1.000   S7 1.000 1.000 1.000 1.000 1.000 1.000  S8 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Post-cue 𝑠𝑑:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 1.000 0.619      S4 0.447 0.017 1.000     S5 0.457 0.077 1.000 1.000    S6 1.000 0.010 1.000 1.000 1.000   S7 0.363 0.008 1.000 1.000 1.000 1.000  S8 0.076 0.005 0.552 1.000 1.000 1.000 1.000  Comparison with Zhang and Luck (2008):  Figure C.4 Post-cue π‘·π’ˆ (A) and 𝒔𝒅 (B) in Experiment 3 with 1500ms exposure duration.    54  C.3.3 Sub-experiment 3: 2000 ms exposure duration C.3.3.1 RMSE: p-vales of multiple comparisons with Bonferroni correction Pre-cue RMSE:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 1.000 1.000      S4 1.000 1.000 1.000     S5 1.000 1.000 1.000 1.000    S6 1.000 1.000 1.000 1.000 1.000   S7 1.000 1.000 1.000 1.000 1.000 1.000  S8 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Post-cue RMSE:  S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 0.267 1.000      S4 1.000 1.000 1.000     S5 0.004 0.510 1.000 1.000    S6 0.424 1.000 1.000 1.000 1.000   S7 0.008 0.168 1.000 1.000 1.000 1.000  S8 0.005 0.209 1.000 1.000 1.000 0.561 1.000  C.3.3.2 Guess probability 𝑃𝑔: p-values of multiple comparisons with Bonferroni correction Pre-cue 𝑃𝑔:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 1.000 1.000      S4 0.566 1.000 1.000     S5 0.555 1.000 1.000 1.000    S6 1.000 1.000 1.000 1.000 0.066   S7 0.570 1.000 1.000 1.000 1.000 1.000  S8 0.475 1.000 1.000 1.000 1.000 1.000 1.000 Post-cue 𝑃𝑔:  S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 1.000 1.000      S4 0.135 0.168 0.659     S5 0.051 0.078 0.512 1.000    S6 0.006 0.011 0.159 1.000 1.000   S7 0.010 0.019 0.247 1.000 1.000 1.000  S8 0.013 0.027 0.303 1.000 1.000 1.000 1.000   55  C.3.3.3 Estimation precision 𝑠𝑑: p-values of multiple comparisons with Bonferroni correction Pre-cue 𝑠𝑑:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 1.000 1.000      S4 1.000 1.000 1.000     S5 1.000 1.000 1.000 1.000    S6 1.000 1.000 1.000 1.000 1.000   S7 1.000 1.000 1.000 1.000 1.000 1.000  S8 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Post-cue 𝑠𝑑:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 0.026 0.298      S4 1.000 1.000 1.000     S5 0.008 0.099 1.000 0.417    S6 0.238 0.811 1.000 1.000 1.000   S7 0.692 0.887 1.000 1.000 1.000 1.000  S8 0.007 0.004 1.000 0.548 1.000 0.895 1.000  Comparison with Zhang and Luck (2008):  Figure C.5 Post-cue π‘·π’ˆ (A) and 𝒔𝒅 (B) in Experiment 3 with 2000ms exposure duration.    56  C.3.4 Three-way ANOVA on 𝑃𝑔: A 4 (between-subjects factor: exposure duration) Γ— 2 (within-subject factor: cue) Γ— 8 (within-subject factor: set size) ANOVA was run on all 𝑃𝑔 data. Exposure duration did not have a significant main effect [𝐹(3, 84) = 1.032, 𝑝 = .383, πœ‚π‘2 = .04]. The main effects of cue [𝐹(1, 84) = 21.31, 𝑝 < .001, πœ‚π‘2 = .20], and set size [𝐹(3.59, 301.45) = 46.54, 𝑝 < .001, πœ‚π‘2 =.38], as well as their interaction [𝐹(2.94, 247.28) = 11.95, 𝑝 < .001, πœ‚π‘2 = .13] were all significant. Exposure duration did not have significant interaction with any of these factors [𝐹′𝑠 ≀ 1.00, 𝑝′𝑠 β‰₯ .617]. Post-hoc Tukey’s honest significant difference (HSD) test showed no pairwise difference in 𝑃𝑔 between the four exposure duration conditions [𝑝′𝑠 β‰₯  .430]. Next, I ran a two-way ANOVA on pre-cue 𝑃𝑔 and post-cue 𝑃𝑔 separately (Figure C.3). First, for the pre-cue 𝑃𝑔, the between-subjects effect of exposure duration was not significant, [𝐹(3, 84) =0.70, 𝑝 = .552, πœ‚π‘2 = .03]. Set size had a main effect, [𝐹(3.21, 269.60) = 14.49, 𝑝 < .001, πœ‚π‘2 =.15], which did not interact with exposure duration, [𝐹(9.63, 269.60) = 0.95, 𝑝 = .487, πœ‚π‘2 =.03]. Consistently with the n.s. effect of exposure duration, Tukey’s HSD test showed no pairwise difference between the four exposure duration conditions, [𝑝′𝑠 β‰₯ .582]. Second, for the post-cue 𝑃𝑔, exposure duration did not show a significant main effect, [𝐹(3, 84) = 0.64, 𝑝 =.589, πœ‚π‘2 = .02], or an interaction with set size, [𝐹(9.66, 270.40) = 0.40, 𝑝 = .942, πœ‚π‘2 = .01]. A significant main effect of set size was still observed, [𝐹(3.22, 270.40) = 36.09, 𝑝 < .001, πœ‚π‘2 =.30]. Tukey’s HSD also showed no pairwise between the four exposure durations, [𝑝′𝑠 β‰₯ 574].   Figure C.6 Guess probabilities in the four exposure duration conditions in the post-cue condition (A), and the pre-cue condition (B).  Analyzing the pre-cue and post-cue data separately, exposure duration did not seem to have altered the pattern of guess probabilities as a function of set size. 57  C.4.1 RMSE: p-vales of multiple comparisons with Bonferroni correction Pre-cue RMSE:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 1.000 1.000      S4 1.000 1.000 1.000     S5 1.000 0.455 1.000 1.000    S6 1.000 1.000 1.000 1.000 1.000   S7 1.000 1.000 1.000 1.000 1.000 1.000  S8 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Post-cue RMSE:  S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 0.117 0.288      S4 0.003 0.000 1.000     S5 0.000 0.000 0.046 0.723    S6 0.015 0.031 1.000 1.000 0.941   S7 0.000 0.000 0.040 0.910 1.000 0.334  S8 0.000 0.000 0.000 0.003 0.003 0.001 1.000  C.4.2 Guess probability 𝑃𝑔: p-values of multiple comparisons with Bonferroni correction Pre-cue 𝑃𝑔:   S1 S2  S3 S4 S5 S6 S7 S2 0.152       S3 0.032 1.000      S4 0.002 1.000 1.000     S5 <.000 0.337 0.411 1.000    S6 <.000 0.679 1.000 1.000 1.000   S7 <.000 0.093 0.260 1.000 1.000 1.000  S8 <.000 0.197 0.216 1.000 1.000 1.000 1.000 Post-cue 𝑃𝑔:  S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 0.225 1.000      S4 0.030 0.137 1.000     S5 0.007 0.042 0.604 1.000    S6 0.003 0.047 0.951 1.000 1.000   S7 0.001 0.014 0.301 1.000 1.000 1.000  S8 0.002 0.036 0.677 1.000 1.000 1.000 0.141  C.4.3 Estimation precision 𝑠𝑑: p-values of multiple comparisons with Bonferroni correction 58  Pre-cue 𝑠𝑑:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 1.000 1.000      S4 0.085 0.807 1.000     S5 1.000 1.000 1.000 1.000    S6 1.000 1.000 1.000 1.000 1.000   S7 1.000 1.000 1.000 1.000 1.000 1.000  S8 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Post-cue 𝑠𝑑:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 0.030 1.000      S4 1.000 1.000 1.000     S5 0.117 1.000 1.000 1.000    S6 0.723 1.000 1.000 1.000 1.000   S7 0.040 0.457 1.000 1.000 1.000 1.000  S8 <.000 0.002 0.089 0.013 0.437 0.054 1.000  Comparison with Zhang and Luck (2008):  Figure C.7 Post-cue π‘·π’ˆ (A) and 𝒔𝒅 (B) in Experiment 4.  Notably, 𝒔𝒅 at most set sizes were different from that at set size 8. Such as pattern was not observed in other experiments. Probably, the task in Experiment 4 was much more demanding than the other experiments, and the distractors impose increasing interference with perceptual processing as set size increased, which continuously reduced estimation precision.     59  C.4 Experiment 5 additional analysis C.5.1 RMSE: p-vales of multiple comparisons with Bonferroni correction Pre-cue RMSE:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 0.269 1.000      S4 1.000 1.000 1.000     S5 1.000 1.000 1.000 1.000    S6 1.000 1.000 1.000 1.000 1.000   S7 1.000 1.000 0.346 0.420 0.906 1.000  S8 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Post-cue RMSE:  S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 0.219 1.000      S4 0.018 1.000 1.000     S5 <.000 0.019 0.128 0.012    S6 <.000 <.000 0.121 0.007 1.000   S7 <.000 <.000 0.001 0.000 0.247 1.000  S8 <.000 <.000 0.002 <.000 0.119 0.596 1.000  C.5.2 Guess probability 𝑃𝑔: p-values of multiple comparisons with Bonferroni correction Pre-cue 𝑃𝑔:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 1.000 1.000      S4 1.000 1.000 1.000     S5 1.000 1.000 1.000 1.000    S6 1.000 1.000 1.000 1.000 1.000   S7 1.000 1.000 1.000 1.000 1.000 1.000  S8 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Post-cue 𝑃𝑔:  S1 S2  S3 S4 S5 S6 S7 S2 0.137       S3 0.021 1.000      S4 0.006 1.000 1.000     S5 0.002 0.388 0.863 0.248    S6 <.000 0.033 0.068 0.023 1.000   S7 <.000 0.006 0.016 0.006 0.935 1.000  S8 <.000 0.001 0.002 0.001 0.165 0.416 1.000 60  C.5.3 Estimation precision 𝑠𝑑: p-values of multiple comparisons with Bonferroni correction Pre-cue 𝑠𝑑:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 1.000 1.000      S4 1.000 1.000 1.000     S5 1.000 1.000 1.000 1.000    S6 1.000 1.000 1.000 1.000 1.000   S7 1.000 1.000 1.000 1.000 1.000 1.000  S8 1.000 1.000 1.000 1.000 1.000 1.000 1.000  Post-cue 𝑠𝑑:   S1 S2  S3 S4 S5 S6 S7 S2 1.000       S3 0.927 0.518      S4 0.032 0.046 1.000     S5 0.008 0.077 0.834 1.000    S6 0.000 0.000 0.165 1.000 1.000   S7 0.002 0.004 0.142 0.447 1.000 1.000  S8 0.006 0.012 0.064 0.101 1.000 1.000 1.000  Comparison with Zhang and Luck (2008):   Figure C.8 Post-cue π‘·π’ˆ (A) and 𝒔𝒅 (B) in Experiment 5.    

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