Visual search with negative slopes: The statistical power ofnumerosity guides attention

Igor S. Utochkin # $National Research University Higher School of

Economics, Russian Federation

Four experiments were performed to examine thehypothesis that abstract, nonspatial, statisticalrepresentations of object numerosity can be used forattentional guidance in a feature search task.Participants searched for an odd-colored target amongdistractors of one, two, or three other colors. Anenduring advantage of large over small sets (i.e.,negative slopes of search functions) was found, and thisadvantage grew with the number of colored subsetsamong distractors. The results of Experiments 1 and 2showed that the negative slopes cannot be ascribed tothe spatial grouping between distractors but can bepartially explained by the spatial density of the visualsets. Hence, it appears that observers relied onnumerosity of subsets to guide attention. Experiments3a and 3b tested the processes within and between colorsubsets of distractors more precisely. It was found thatthe visual system collects numerosity statistics that canbe used for guidance within each subset independently.However, each subset representation should be seriallyselected by attention. As attention shifts from onesubset to another, the ‘‘statistical power’’ effects fromevery single subset are accumulated to provide a morepronounced negative slope.

Introduction

When using the visual search task in studies of visualattention, researchers often focus on the slopes of theresulting search functions that relate the reaction time(RT), or sometimes accuracy rates, to the number ofitems in a display (set size). The slopes of the searchfunctions are important for making distinctions be-tween two basic modes of search behavior: efficient andinefficient searches. The search for a target item amongmultiple distractors is typically considered efficient ifthe RT does not increase substantially with the numberof items in a display. This concept is supported by theshallow slopes of the search functions that are observedunder these conditions. In contrast, a visual search is

considered inefficient when the RT increases with thenumber of items in a display. In the latter case,substantial positive slopes of the search functions areobserved.

A more unusual visual search pattern was reportedin the early 1990s by Bacon and Egeth (1991) andBravo and Nakayama (1992). In both studies, observ-ers had to detect an odd-colored target among avariable number of distractors of another color. Inaddition, in some conditions of their experiments,Bravo and Nakayama required observers to reporteither the color or the shape of the odd element.Finally, Bravo and Nakayama compared variable-mapping tasks (when the color of an odd elementchanged unpredictably from trial to trial) and consis-tent-mapping tasks (when the color of an odd elementremained constant from trial to trial). Both studiesfound similar results, with RTs that decreased as thenumber of distractors increased. Remarkably, the mostdramatic decline of the function was found withinrelatively small set sizes. In large sets, the RTs remainedfar more constant. This trend, therefore, can becharacterized by search functions with deceleratingnegative slopes. It is noteworthy that in Bravo andNakayama’s study, a negative slope was found only ina shape-discrimination task under variable-mappingconditions while in the experiment by Bacon andEgeth, it was documented for a color-detection task.

The principal explanation for the negative slopes,which was supported in both studies, relies on thespatial organization of the displays, which can betreated as a factor guiding attention to an odd feature.However, the specific mechanisms behind this guidancesuggested by Bacon and Egeth (1991) differ from thosesuggested by Bravo and Nakayama (1992). Accordingto a hypothesis advocated by Bravo and Nakayama,this guidance is provided by local retinotopic interac-tions between all neighboring features. Two versions ofthis interaction are discussed by Bravo and Nakayama.One version suggests that the neighboring features arethe subjects of local comparison and contrast compu-

Citation: Utochkin, I. S. (2013). Visual search with negative slopes: The statistical power of numerosity guides attention. Journalof Vision, 13(3):18, 1–14, http://www.journalofvision.org/content/13/3/18, doi:10.1167/13.3.18.

Journal of Vision (2013) 13(3):18, 1–14 1http://www.journalofvision.org/content/13/3/18

doi: 10 .1167 /13 .3 .18 ISSN 1534-7362 � 2013 ARVOReceived December 6, 2012; published July 23, 2013

tations (Sagi & Julesz, 1987). The highest local contrastbetween a singleton and distractors immediately guidesattention to an odd feature. This version of a bottom-up guidance mechanism was later supported by Wolfe(1994) in his influential guided-search model ofattention. The other version suggests that there ismutual inhibition between neighboring similar itemsbecause they are represented within the same preat-tentive feature map. As soon as a singleton isrepresented in a different feature map and stands apartfrom the inhibitory interactions between distractors, iteasily wins the competition for prior attentionalselection (Koch & Ullman, 1985). In both versions, thelocal interactions are supposed to increase as the spatialseparation between neighbors decreases. The spatialdensity of elements, therefore, appears to be critical forsalience. Because visual search displays are typicallypresented within a constant visual angle, increasing thenumber of items causes the density to increase as well.This is most likely the simplest explanation for thenegative slopes observed by Bacon and Egeth andBravo and Nakayama.

However, Bacon and Egeth (1991) provided argu-ments against the explanation for negative slopes basedon target-distractor local contrasts. They manipulatedthe target-distractor spatial separation independentlyfrom the set size and observed no effect on the slopes.However, when Bacon and Egeth arranged thedistractors in a manner that maintained a constantdistractor-distractor separation for any set size, theyobserved the elimination of the negative slopes. Similarresults were obtained by the authors for the detectionof orientation singletons. Bacon and Egeth concludedthat the guidance principle is spatial by nature, but themechanism is based on the proximity grouping betweendistractors rather than the target-nontarget spatialseparation. The significance of the spatial grouping ofdistractors for search efficiency was demonstrated innumerous other experiments with feature and con-junction targets (Humphreys, Quinlan, & Riddoch,1989; Poisson & Wilkinson, 1992; Treisman, 1982).

Certainly, spatial arrangements of items and localinteractions within these arrangements are very im-portant and biologically significant determinants ofrepresenting textures, surfaces, and objects that canguide attention through the environment. However, itappears that the visual system has a powerful tool torepresent large groups of objects in an abstract,nonspatial way. Recent studies within an explosivelyevolving ensemble-representation paradigm have es-tablished that observers are rather good at judging thesummary statistics for large sets of objects seen in amomentary glance (Ariely, 2001; Chong & Treisman,2003). At least two types of basic statistics are availableto the observer, namely the average values acrossdifferent dimensions (Ariely, 2001; Bauer, 2009; Dakin

& Watt, 1997; Watamaniuk & Duchon, 1992) and thenumerosity (Burr & Ross, 2008; Chong & Evans, 2011;Whalen, Gallistel, & Gelman, 1999). Perhaps the mostbeneficial effect of summary statistics is that they allowthe observers to overcome attention and workingmemory limitations (Cowan, 2001; Luck & Vogel,1997; Pylyshyn & Storm, 1988) and rapidly judge theproperties of multiple objects without paying attentionto individuals. Moreover, it appears that this type ofjudgment may be more precise and reliable than onebased on few sample objects (Alvarez, 2011; Chong,Joo, Emmanouil, & Treisman, 2008).

The main point of the present paper is that usingabstract ensemble statistics can be an effective strategyfor guiding attention through a visual search, especiallywhen spatial organization does not help. There are atleast three basic properties of ensemble summarystatistics that can make them useful for an efficientvisual search. The properties are applicable to bothaverage and numerosity estimations. First, the sum-mary statistics are extracted rapidly (Chong & Treis-man, 2003, but see Whiting and Oriet, 2011) and mostlyin parallel with equal efficiency for all set sizes (Ariely,2001). In fact, statistical judgments may even benefitslightly from larger sets (Chong et al., 2008; Robitaille& Harris, 2011). Second, both the average (Alvarez,2011; Chong & Treisman, 2005a; de Fockert &Marchant, 2008) and the numerosity (Chong & Evans,2011) can be better represented with broadly distrib-uted, rather than focused, attention. This is consistentwith the finding that an efficient visual search alsorequires distributed attention (Joseph, Chun, & Na-kayama, 1997). Third (and perhaps most important),the visual system is able to compute the averages(Chong & Treisman, 2005b) and numerosities (Hal-berda, Sires, & Feigenson, 2006; Treisman, 2006) fordifferent feature-marked subsets. This ability is quitegood for both spatially grouped and spatially over-lapped subsets (Chong & Treisman, 2005b) although itappears to be limited to approximately three subsets atone time (Halberda et al., 2006). In other words,differently featured items can be represented as separateand arrangement-independent statistical entities ordistributions. This notion can be used for thinkingabout visual search. If the visual system representsdifferently featured items as different distributions,then the search for a feature singleton can be performedas a direct comparison among these distributions.

So how can this statistically based strategy explainthe negative slopes of search functions? The answermost likely lies in the statistical concept of numerosity.In statistical terms, the visual search for a singleton isakin to testing a hypothesis in which only one feature-marked subset consists of a single member. Becausemultiple-subset enumeration can be processed inparallel, although with some limitations (Halberda et

Journal of Vision (2013) 13(3):18, 1–14 Utochkin 2

al., 2006), the visual system can then directly comparenumerosities. When discriminating the numerosities oftwo distinct subsets, their ratio is a powerful determi-nant of discrimination speed and accuracy (Barth,Kanwisher, & Spelke, 2003). If there is a largedifference in the number of items between subsets, thenit is easier to decide that the smallest one is perhaps asingleton. Certainly, I do not assume that a visualsearch is the same thing as an enumeration task, but Ihypothesize that the global contrast between thenumerosities of distracting and target subsets can beused to guide a visual search along with, or sometimesinstead of, local feature relationships.

This hypothesis was tested in the series of experi-ments described below. The main point of theseexperiments was to demonstrate that numerositycontributes to the negative slopes of search functionswhen both the local contrast (Bravo & Nakayama,1992; Sagi & Julesz, 1987; Wolfe, 1994) and thebetween-distractors spatial grouping (Bacon & Egeth,1991) are less useful. Two factors were manipulatedacross and within the experiments to test this hypoth-esis. First, I manipulated the density of sets across twopairs of experiments. In Experiments 1 and 3a, thedensity varied with the set size, like in standard visualsearch displays. In Experiments 2 and 3b, the densityremained the same for all set sizes, so local contrastsremained constant. The other factor manipulated in allof the experiments was the number of colors among thedistractors. Colors were intermixed so that they formedoverlapping subsets. When a singleton target issurrounded by homogeneous one-colored distractors,such as in experiments by Bacon and Egeth (1991) andBravo and Nakayama (1992), both the local contrastaround the target and the between-distractor groupingare high. Therefore, the typical search strategiesdescribed by those authors seem plausible under theseconditions. However, when the target is surrounded byheterogeneous, multicolored distractors, these guidancestrategies appear to be less useful for two reasons. First,there are many more regions with high local contraststhan in homogeneous displays. Second, the between-distractor grouping is weakened under multicolorconditions. Both of these circumstances predict thereduction of guidance tendencies based on spatialarrangement. Consequently, the negative slopes shouldalso decrease. However, in accordance with thenumerosity-based hypothesis, the negative slopesshould remain intact or even increase with the numberof colors because the visual system is supposed to beable to compare an abstract number of representationsof color subsets, regardless of their spatial arrangement(Halberda et al., 2006). It is also necessary to prove thatmulticolor displays are indeed treated as separate,overlapping subsets rather than a unitary, heteroge-neous superset. Otherwise, the numerosity-based hy-

pothesis makes no sense. Clear evidence in favor ofseparating dissimilar colors into distinct subsets and theserial attendance toward each subset will be presentedin all experiments. In addition, Experiments 3a and 3bclarify the nature of within- and between-subsetinteractions that eventually contribute to attentionalguidance.

Experiment 1

Experiment 1 was aimed at two main goals. The firstgoal was the replication of the negative slope patternreported by Bacon and Egeth (1991) and Bravo andNakayama (1992). A variable-mapping color searchtask was used as in the mentioned studies. Thereference condition was to include distractors of onlyone color. The second goal was the investigation ofgrouping processes within displays and their role inattentional guidance. The manipulation of the groupingfactors was achieved by adding colors to distractingsets. As was mentioned above, the spatial mixture ofheterogeneous elements increases the local contrasts atmultiple locations and diminishes the similaritygrouping between distractors. The theoretical consid-erations of Bacon and Egeth (1991) and Bravo andNakayama (1992) imply that both grouping reductionand increasing contrasts should eliminate negativeslopes. The current experiment tests this prediction.

Methods

Observers

Seventeen undergraduate students of the HigherSchool of Economics participated in this experimentfor extra credits in their general psychology lab classes.All students reported having normal or corrected-to-normal color vision and no neurological problems.

Apparatus and stimuli

The stimulation was developed and presentedthrough the StimMake for Windows (authors A.N.Gusev and A.E. Kremlev, UMK ‘‘Psychology,’’ Ltd.,Moscow, Russia, 1999–2012). The stimuli were pre-sented on a standard VGA-monitor with a refreshfrequency of 85 Hz and a spatial resolution of 800 ·600 pixels. To register the responses, a parallel port–compatible control pad for RT experiments was used.The construction of the control panel provided a shortdistance between the top (unpressed) and the bottom(pressed) key positions. This provided an RT mea-surement with a precision of no less than 62 ms.

Journal of Vision (2013) 13(3):18, 1–14 Utochkin 3

The stimuli were presented on a homogenous 158 ·158 white field. It was divided into 3 · 3¼ 9 imaginarysquares. These imaginary squares were considered to bepossible target locations. A target could appear at arandom location within a square, but its appearancefrequency was strictly uniform across squares. Thisprovided a globally uniform spatial distribution of agiven target over the field during the entire experiment.

The displayed items were colored dots, .68 indiameter. Four colors—red, green, blue, and yellow—were used for coloring the items. One to three colorscould be presented as distractors in negative trials(where no target was presented). In positive trials, colorsingletons were added to sets as targets. All colors wereequally likely to be presented as targets in each of thenine spatial positions. All colors were also equallylikely to be presented as distractors in the one-, two-,and three-color conditions. All possible color combi-nations were used with equal frequencies. Two set sizes,nine and 27 dots, were used to test the general RTtrend. The minimum set size was larger than those usedby Bacon and Egeth (1991) and Bravo and Nakayama(1992), but it was chosen intentionally to provide forthe further division of sets into smaller subsets. Thefailure to test small set sizes was compensated for inExperiments 3a and 3b. The dots were distributed overthe field so that small sets were sparser than large sets.The average between-dot distance was approximately4.48 (ranging from 3.18 to 5.28) for small sets andapproximately 3.18 (ranging from 1.38 to 4.48) for largesets. Examples of small and large sets are presented inFigure 1a and 1c, respectively.

Observers were seated approximately 65 cm from themonitor. A typical trial began with a 500-ms fixation ona small black cross at the center of the white field.Immediately following the fixation period, a stimulusarray appeared. Observers were asked to rapidly reportthe presence or absence of a singleton target in the

array by pressing one of two specially assigned keys onthe control panel. Half of the trials were positive, andthe other half of the trials were negative. The arrayremained on the screen until a response was made ordisappeared if a response had not been made within3000 ms. The intervals between the trials varied inlength between 1000 and 1500 ms, and a blank whitefield was presented during these intervals. Three 1-minbreaks were given during the experiment. In total, theexperimental design included three numbers of subsets· two set sizes · nine target positions (imaginarysquares) · four target colors · two types of trials(positive or negative)¼ 432 trials per observer. Twenty-four randomly chosen trials were used during a trainingsession that preceded the main block of trials. Allconditions were arranged in a random sequence withthe restriction that there could be no more than threepositive or negative trials in a row.

Results and discussion

The data from two observers were excluded from theanalysis due to their large error percentages (more than10%). For the rest of the data, only the trials with acorrect response were analyzed for RT.

The experimental effects of the number of subsetsand the set size on the RT were tested with a within-subjects ANOVA. In the ANOVA model, the set sizeand the number of subsets were defined as fixed factors.To handle individual differences in performance amongobservers, the model also included the observer’sidentity as a random factor.

The results of Experiment 1 are summarized inFigure 2 in the left panel. The effect of the number ofsubsets on the RT was found to be significant, F(2, 28)¼ 114.88, p , 0.001, g2p ¼ .89. As seen from the plot,the RT tended to increase with the number of subsets in

Figure 1. Examples of stimulus displays used in Experiments 1 and 2. (a) Set size¼ nine, sparse arrangement (Experiment 1); (b) set

size ¼ nine, dense arrangement (Experiment 2); (c) set size ¼ 27, dense arrangement (both experiments).

Journal of Vision (2013) 13(3):18, 1–14 Utochkin 4

a display. The set size also had a significant effect onthe RT, F(1, 14) ¼ 67.37, p , 0.001, g2p ¼ .83,producing negative slope functions similar to thosedescribed earlier for variable-mapping feature detectiontasks (Bacon & Egeth, 1991; Bravo & Nakayama,1992). The series of post-hoc t tests revealed that thisset size effect is maintained in all numbers of subsets (t¼ 2.19, p , 0.05 for the one-color condition; t¼ 7.29, p, 0.001 for the two-color condition; and t¼ 12.72, p ,0.001 for the three-color condition). Finally, theinteraction between the set size and the number ofsubsets also had a significant effect on the RT, F(2, 28)¼ 63.13, p , 0.001, g2p ¼ .81, indicating that theabsolute value of the slope increases with the number ofsubsets.

The experiment has succeeded in replicating thenegative slope effect under standard conditions, whichrequired the search for an odd target among one-colored distractors, that were the most similar to thetask used by Bacon and Egeth (1991). The finding thatthe overall performance tended to decrease with thenumber of distracting colors is in agreement with theresults obtained by Bundesen and Pedersen (1983) andSanthi and Reeves (2004).

However, the most remarkable result of Experiment1 is the character of interaction between the number ofsubsets and the set size. Paradoxically, it appears thatvisual search becomes slower in general but guidancebecomes stronger as the number of subsets increases. Ishall further refer to this pattern as the crescent slopeeffect. At first glance, this finding seems discouraging.As the number of overlapping colors increases, both thelocal contrast and the grouping tendencies betweendistractors are reduced, which is predicted to result in areduction in target salience. This reduction in targetsalience appears to be the case in terms of overallperformance. However, these conditions are alsopredicted to result in an impairment in the bottom-upguidance while assuming that guidance relies solely on

the parallel preattentive discrimination between groupsand textures (Bacon & Egeth, 1991; Bravo & Nakaya-ma, 1992; Sagi & Julesz, 1987; Wolfe, 1994).

The results, however, become clearer if we presumethat a limited-capacity process, typically referred to asattention, is involved in this type of visual search. Onemight argue that limited-capacity attention shouldproduce positively, rather than negatively, slopedsearch functions because attention serially switchesfrom one location occupied by an object or a group toanother. However, as was previously mentioned,attention can also select spatially overlapping, feature-marked subsets (e.g., Halberda et al., 2006). This typeof selection implies that the RT should increase withthe number of subsets rather than number of items. Theresults of the current experiment suggest that such asubset-oriented type of attentional selection may beinvolved in this task. Given the serial character ofattending to subsets, a further presumption can bemade regarding the crescent slopes observed. Accord-ing to this presumption, separate guiding processes arebeing launched within each subset. Taken together,these processes provide a more efficient overallguidance effect or cumulative guidance. The idea of theaccumulation of several guidance processes, providedby several separable features, was earlier suggested byWolfe, Cave, and Franzel (1989) for conjunction-searchtasks. I hypothesize that, in my feature-search task, thiscumulative guidance effect may be caused by anumerosity comparison between subsets. However, thepresent experiment does not provide sufficient data totest this hypothesis because subset numerosities wereincomparable for one-, two-, and three-color condi-tions. This hypothesis will be properly addressed belowin Experiments 3a and 3b.

Experiment 2

As was mentioned in the Introduction, the reasoningregarding bottom-up attentional guidance is typicallybased on the concepts of density and proximity. Inother words, a prevailing significance is given to thespatial factors of display organization. Increasing thenumber of items is thought to produce a negative slopeonly as far as it is positively related to density andproximity. Alternatively, numerosity can be consideredto be an independent factor affecting guidance.Experiment 2 tests whether the negative slopes ob-served in Experiment 1 were the result of spatial densityor numerosity. In this experiment, the spatial densityfor small and large sets was identical. A density-oriented hypothesis predicts a shallow-slope pattern forsingleton detection because guidance tendencies areequated in small and large sets. In contrast, a

Figure 2. Reaction times for Experiments 1 and 2. Error bars

denote 61 SEM.

Journal of Vision (2013) 13(3):18, 1–14 Utochkin 5

numerosity-oriented hypothesis predicts the mainte-nance of a negative slope despite the density manipu-lations.

Methods

Observers

Sixteen undergraduate students of the Higher Schoolof Economics participated in this experiment for extracredits in their general psychology lab classes. Allstudents reported having normal or corrected-to-normal color vision and no neurological problems.None of the observers that participated in thisexperiment participated in Experiment 1.

Stimuli and procedure

The stimulation and procedures used were identicalto those used in Experiment 1. The only exceptionconcerned the spatial arrangement of small sets. Thedensity of these sets was equated with that of the largesets, and both were the same as the large set densitiesfrom Experiment 1. In this experiment, the small setswere organized as compact groups occupying approx-imately 20% of the ‘‘working’’ field. To provide auniform distribution of the target locations during thisexperiment, these small sets were centered along each ofthe nine imaginary squares of the visual field with equalfrequencies. An example of a dense small set used inthis experiment is provided in Figure 1b.

Results and discussion

The data from one observer were excluded from theanalysis due to the large error percentage (more than10%). The method of analyzing the RT and thestatistics used were the same as those used inExperiment 1.

The number of subsets had a significant effect on theRT, F(2, 28) ¼ 66.99, p , 0.001, g2p ¼ .83, indicatingthat the search efficiency decreased with the number ofpresented colors. The effect of the set size was alsosignificant, F(1, 14) ¼ 18.24, p¼ 0.001, g2p ¼ .57),indicating the global maintenance of negative slopes.The series of post-hoc t tests indicated that the slopeswere negative in all conditions (t¼ 2.69, p , 0.01 forthe one-color condition; t¼ 2.66, p , 0.01 for the two-color condition; and t¼ 6.88, p , 0.001 for the three-color condition). Finally, the interaction between theset size and the number of subsets also had a significanteffect on the RT, F(2, 28)¼ 10.46, p , 0.001, g2p¼ .43,demonstrating a crescent slope trend similar to the oneobserved in Experiment 1. As seen from the effect sizeestimates, the effect of the number of subsets remained

largely identical to that observed in Experiment 1.However, the effect sizes were reduced for both the setsize and the set size · number of subsets interactionwhen compared with Experiment 1. Indeed, as seenfrom the plot (Figure 2, right panel) the negative slopefunctions are shallower than those observed in Exper-iment 1 at least for the two- and three-subsetconditions.

As predicted by traditional accounts regardingbottom-up guidance (Bacon & Egeth, 1991; Bravo &Nakayama, 1992; Sagi & Julesz, 1987; Wolfe, 1994), thespatial densification of visual elements in small setsmade the visual search for a feature singleton moreefficient. However, this manipulation failed to removethe negative slopes and the crescent slope trendcompletely. This result is also in agreement with theprevious finding that the effect of spatial density on afeature search is limited (Cohen & Ivry, 1991).Theoretically, there could be an alternative interpreta-tion for this pattern. This interpretation is based on thenotion that perceiving small but dense sets requires theadditional operation of narrowing attention. Indeed, inExperiment 2, the set size varied randomly from trial totrial, so observers did not know which region would beoccupied until the dots appeared. This could result inthe adoption of a strategy of using a wide attentionalwindow with subsequent narrowing as needed. Thisnarrowing operation could yield an additional cost inthe RT that resulted in negative slopes. This narrowingof attention is a plausible strategy but is still unable toexplain the crescent slope effect. If the narrowingstrategy acts alone, then its effect on the slopes shouldbe independent of the number of subsets because theregion occupied by small sets was always the same.Hence, the remaining factor that could contribute tothe crescent slope pattern is numerosity.

To label this link between numerosity and searchefficiency, I use the term ‘‘statistical power effect.’’ Thisterm is prompted by a statistical rule that states that anincreasing sample size (numerosity) typically lowers thethreshold value of a statistical test sufficient forconfirming or rejecting an H1 hypothesis. In visualsearch for a singleton, the visual system tests thehypothesis that the size of any of subset is equal to one.In a general statistical sense, the task is akin to testing aproportion hypotheses (chi-square hypotheses). I foundthat the visual system makes decisions regarding searchoutcomes faster with larger sets. Larger differencesbetween numerosities make the differences betweensubsets more evident; therefore, hypotheses are beingconfirmed or rejected more readily. However, thenotion of statistical power does not limit the possiblemechanisms of guidance by direct numerosity com-parisons. The other mechanism that can mediate searchfacilitation is variance reduction. In statistics, with anincreasing number of items, a population of similar

Journal of Vision (2013) 13(3):18, 1–14 Utochkin 6

items tends to its mean, and the observed variancetends to be reduced. A recent finding by Robitaille andHarris (2011) supports the notion that this statisticalrule acts in judging ensemble summary statistics inhuman observers. In a visual search, a variancereduction can be directed at diminishing the noisyinteractions between subset representations and en-hancing discrimination. Both direct numerosity com-parisons and variance reduction are plausiblemechanisms of the statistical power effect. Futureexperiments will be needed to clarify their possiblecontributions to negative slopes. In the present paper, Imainly highlight the apparent similarity between thestatistical power rule and the observed RT patterns tojustify the use of the term ‘‘statistical power effect.’’ Inmy opinion, the statistical power effect is useful whenthinking about visual sets as statistically representedentities.

Experiments 3a and 3b

In the discussion of Experiment 1, a hypothesis waspresented that separate guidance processes are beinggenerated by each color subset and accumulated in avisual search once attention is paid to a subsequentsubset. This hypothesis could likely explain the crescentslope trends observed in Experiments 1 and 2.However, the above experiments do not provideenough data to conclude whether this hypothesis iscorrect due to the inability to separate the set size andthe subset size. In standard set-size manipulations, thetotal number of items is an endpoint of an experimentalcondition. However, if the set is then divided into equalor nearly equal subsets, the number of items per subsetis reduced. For example, in a positive trial of a nine-item set, we have eight items in one distracting subset;four and four items in two distracting subsets; or three,three, and two items in three distracting subsets. Theguidance tendencies within subsets cannot be comparedunder these conditions. As was clearly demonstrated inthe experiments conducted by Bacon and Egeth (1991)and Bravo and Nakayama (1992), the negative slope isa decelerating function of the number of items. Thisdeceleration can provide an alternative explanation forthe crescent slope trend: As the number of subsetsincreases, the subset numerosity is reduced, and thisresults in the sharpening of guidance processes withineach subset. In summary, two alternative hypothesesshould be considered regarding the guidance processesunderlying the crescent slope trend. In accordance withthe first hypothesis, independent guidance processesthat evolve within each subset are being seriallyaccumulated as soon as attention is directed to othersubsets in a serial manner. This hypothesis supports the

concept of cumulative guidance as suggested above. Thesecond hypothesis asserts that there is no serialaccumulation of numerical information. Rather, itappears that information is being accumulated inparallel within all subsets but is only accessible whenattention is paid to a particular subset. In this case, theRT slopes should not be dependent on the number ofsubsets. Note that both hypotheses assume thatattention inspects the subsets serially as was demon-strated in Experiments 1 and 2.

Experiments 3a and 3b address these hypotheses andprovide advanced investigations into the within- andbetween-subset processes that result in bottom-upguidance. The subset numerosities were controlled inthese experiments rather than the set sizes. Theconditions of the subset numerosity factor were termed‘‘iso-numerosity points.’’ An iso-numerosity point is avisual set in which the number of items per subsetremains constant regardless of the number of subsets.To assert that spatial proximity is not an exclusivefactor affecting bottom-up guidance, the densitymanipulations from Experiments 1 and 2 were repli-cated with iso-numerosity stimuli. Thus, the itemdensity was manipulated across Experiments 3a and 3b.

Given the serial character of attending to subsets (asExperiments 1 and 2 revealed), two predictions can bemade regarding the results when considering the abovehypotheses. If the guidance processes are beingaccumulated from subsets with serial shifts of attention,then the absolute slope should increase with thenumber of subsets (cumulative guidance hypothesis). Ifthe guidance processes are being generated in parallelprior to attending to a particular set, then the slopesshould remain the same, regardless of the number ofsubsets (noncumulative guidance hypothesis). These twopredictions are summarized in Figure 3a and 3b,respectively. Moreover, Experiments 3a and 3b test theshape of the numerosity–RT function in greater detailthan in Experiments 1 and 2, in which only two pointswere used. Based on the results reported by Bacon andEgeth (1991) and Bravo and Nakayama (1992) for one-color–condition experiments, I predict a rapidly decel-erating shape for the numerosity–RT function as the

Figure 3. Two predictions regarding between-subset interactions

in providing attentional guidance by subset numerosity: (a)

cumulative guidance or (b) noncumulative guidance.

Journal of Vision (2013) 13(3):18, 1–14 Utochkin 7

number of items per subset increases. Figure 3 takesthis prediction into account.

Methods

Observers

Nineteen and 18 undergraduate students of theHigher School of Economics participated in Experi-ments 3a and 3b, respectively, for extra credits in theirpsychology courses. All students reported havingnormal or corrected-to-normal color vision and noneurological problems. Observers did not overlapbetween the two experiments, and none of themparticipated in Experiments 1 or 2.

Stimuli and procedure

The stimulation and procedures for both experi-ments were similar to those used in Experiments 1 and2. However, there were several important differences.

The ‘‘working’’ visual field was 158 · 9.78. The fieldwas divided into six imaginary squares, serving asspatial positions for the targets, and the sizes of thesquares were the same as in Experiments 1 and 2. It canbe seen that the total size of the field and the number ofimaginary squares were reduced as compared to theabove experiments. This change was caused by changesin set size range brought by iso-numerosity manipula-tions. In Experiment 1, only two set sizes, nine and 27items, were used that provided two items per square onaverage across conditions. In Experiments 3a and 3b,the set size ranged from three to 37 items, depending onthe number of subsets and subset numerosity (seedetailed set size calculations below). Using a six-squarefield provided an average density of about 2.2 items persquare that was closer to Experiment 1 than using anine-square field (1.5 items per square). As in Exper-iment 1, the targets appeared with equal frequencies in

all imaginary squares during the experiments. The samefour colors and rules for coloring targets and dis-tractors were implemented as were used in the previousexperiments.

To investigate the role of subset numerosity inattentional guidance, four iso-numerosity points wereused. The iso-numerosity points were two, four, eight,and 12 items per subset. The total set size included thenumber of subsets multiplied by the iso-numerositypoint value, and one item was reserved to be a target inpositive trials (Figure 4). In negative trials, anadditional member of a randomly chosen distractorsubset was presented instead of the target. The total setsizes, consequently, were 2þ 1¼ 3, 4þ 1¼ 5, 8þ 1¼ 9,and 12þ 1¼ 13 for the one-distractor subsets; 2 · 2þ1¼ 5, 2 · 4þ 1¼ 9, 2 · 8þ 1¼ 17, and 2 · 12þ 1¼ 25for the two-distractor subsets; and 3 · 2þ 1¼ 7, 3 · 4þ 1 ¼ 13, 3 · 8þ 1 ¼ 25, and 3 · 12 þ 1 ¼ 37 for thethree-distractor subsets. In Experiment 3a, the itemswere uniformly distributed across the field in all setsizes as described in Experiment 1. Consequently, theaverage density of all items varied with the set size.However, the average within-subset density remainedconstant for any given iso-numerosity point. InExperiment 3b, the densities were equated for all setsizes as described in Experiment 2.

In total, the designs of Experiments 3a and 3bincluded three numbers of subsets · four subsetnumerosities (iso-numerosity points) · six targetpositions (conditional squares) · four target colors ·two types of trials (positive or negative)¼ 576 trials perobserver. Trials of different types were intermixed in arandom order with the restriction that there could beno more than three positive or negative trials in a row.Five breaks were provided during the main session ofthe experiments. Twenty-four randomly chosen trialswere used during a training block that preceded themain session. All of the other conditions were identicalto those used in the previous experiments.

Figure 4. Examples of stimulus displays used in Experiment 3a demonstrating iso-numerosity manipulation. All panels depict the same

iso-numerosity condition, namely the two-item condition; that is, there are always two items of the same color regardless of the total

number of colors. This causes the overall set size to increase but keeps subset numerosity constant: (a) one-subset condition: two

blue distractorsþ red target¼ three items; (b) two-subset condition: two green distractorsþ two yellow distractorsþ blue target¼five items; (c) three-subset condition: two red distractors þ two blue distractors þ two yellow distractors þ green target ¼ seven

items.

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Results and discussion

The data from one observer who participated inExperiment 3b was excluded from analysis due to thelarge error percentage (more than 10%). The methodsfor analyzing the RTs and the statistics used wereidentical to those described in previous experiments.

The results of Experiment 3a are plotted in Figure5a, in the left panel. The effect of the number of subsetswas significant, F(2, 36)¼ 117.61, p , 0.001, g2p¼ .87,indicating that the addition of subsets yielded anincrease in the RT as observed in the previousexperiments. The second principal result was that theRT function gradually declined as the subset numer-osity increased from two to 12 items, F(3, 78)¼ 69.17, p, 0.001, g2p ¼ .79, which replicated the negative slopefunction observed in the previous experiments. Aregression analysis was applied to determine whichshape of the function fit the experimental data better.Five function shapes were tested, including linear,inverse, logarithmic, quadratic, power, and exponen-tial. The best fit was provided by the inverse function,R2¼ .25, F(1)¼ 268.01, p , 0.001, and the powerfunction with the absolute exponent less than one, R2¼.25, F(1) ¼ 260.79, p , 0.001. Both functions aredecelerating. The worst fit was provided by the linearfunction, R2¼ .18, F(1)¼ 185.54, p , 0.001. Therefore,it was more likely that the general trend was slightlydecelerated, which is consistent with that reported byBacon and Egeth (1991) and Bravo and Nakayama(1992). The most dramatic difference was obtained inthe transition from the iso-numerosity points of two to

four items. Remarkably, in the one-subset condition,this transition was also accompanied by the slightfacilitation of performance, followed by a flat RTfunction at all other numerosities. This inhibition ofresponse to the target in the presence of only two one-colored distractors could be due to a strong competi-tion between the three items. Both local contrast andbetween-distractor grouping appear to be insufficientfor guiding attention under this condition.

The absolute slope of the function tended to increasewith the number of subsets (subset numerosity ·number of subsets interaction, F(6, 108) ¼ 20.84, p ,0.001, g2p ¼ .54), replicating the crescent slope trendobserved in the previous experiments. Yet the effect sizeestimate was reduced when compared with Experiment1 because the uncertainty regarding the influence of thenumber of subsets and the subset numerosity, describedabove, was removed.

The results of Experiment 3b are plotted in Figure 5ain the right panel. As seen from the plot, the results donot differ substantially from those observed in Exper-iment 3a. Again, the effect of the number of subsets wassignificant, F(2, 32) ¼ 86.41, p , 0.001, g2p ¼ .85. Theeffect of subset numerosity was also significant, F(3, 48)¼ 67.31, p , 0.001, g2p ¼ .81, demonstrating overallnegative slope. The results of the regression analysis ofthe search function general shape yielded substantiallythe same results as for Experiment 3a. The best fit wasprovided by the power function with the exponent lessthan one, R2¼ .26, F¼256.60, p , 0.001, and the worstfit was provided by the linear function, R2 ¼ .16, F ¼160.84, p , 0.001, indicating that the slope tends todecelerate as subset numerosity increases. Finally, the

Figure 5. RTs for Experiments 3a and 3b (a) as a function of subset numerosity (iso-numerosity points). The functions are

characterized by negative slopes with deceleration. The absolute RT tends to increase with the number of subsets as depicted by

separate lines, but the absolute slopes are showed to increase as the number of subsets increases, demonstrating the crescent slope

trend. (b) The same results averaged across experiments and replotted using alternative coordinates (x-axis depicts the number of

subsets) to emphasize an effect of subset numerosity (separate lines) on serial search among subsets. As the numerosity increases,

the slope tends to be reduced, indicating attentional guidance. Error bars denote 61 SEM.

Journal of Vision (2013) 13(3):18, 1–14 Utochkin 9

crescent slope trend was retained in the presentexperiment, subset numerosity · number of subsetsinteraction, F(6, 96)¼ 31.75, p , 0.001, g2p ¼ .66.

As in Experiments 1 and 2, the strongest statisticaleffect was observed for the number of subsets. Notethat in Experiments 3a and 3b the set sizes increasedproportionally to the number of subsets while inExperiments 1 and 2 the set sizes remained constantand the subset numerosities were variable. Yet thesubset-related increments in the RT were nearlyidentical for all of the experiments, which leads to animportant conclusion that this effect is only marginallydependent on the manipulations of the set size. Rather,changes in the RT appear to be dependent on the subsetnumerosities. These results provide evidence that, in mytask, attention operates serially with subsets ratherthan with single items or with the whole set at once(otherwise the RT should not be dependent on thenumber of subsets).

Strikingly, Experiments 3a and 3b showed nosubstantial effects of the item density on the slopeswhile substantial effects of item density were observedin Experiments 1 and 2. There is unlikely to be astimulus variable that could interfere with densitybetween the two pairs of experiments because they allused principally the same stimuli. However, there couldbe a procedural difference responsible for this discrep-ancy. In Experiments 1 and 2, observers were exposedto only two set sizes while in Experiments 3a and 3bthere were four iso-numerosity points, which, in turn,produced an even greater variety of set sizes. Therandom mixture of all these set sizes in a trial sequenceresulted in a greater level of unpredictability, whichmay have complicated the spatial adaptation to smallersets. In other words, it became quite difficult to predictwhich visual angle would be occupied by informativeitems in any given trial and, therefore, to adjust theproper size and resolution of an attentional window.This hypothesis should definitely be tested properly infuture experiments. The most important result, in thecontext of the present paper, is the finding that densitymanipulations appear to have a more volatile effect onsearch performance than subset numerosities do.

The main goals of Experiments 3a and 3b were toexamine the roles of within- and between-subsetprocesses in attentional guidance. First, I obtained amore precise numerosity–RT function due to theincreased number of iso-numerosity points. Moreover,my numerosity manipulations concerned the mostguidance-sensitive span that did not exceed 12 items persubset. In general, the shapes of the functions(decelerating and negatively sloped lines) were similarto those reported by Bacon and Egeth (1991) andBravo and Nakayama (1992). Given the difficulties inderiving a purely spatial explanation for the guidanceeffect, the latter similarity appears to be rather

important. Presumably, both types of functions canreflect the statistical power effect, which is thefacilitation of reaching a decision regarding the entireensemble as the number of items increases. A moredetailed view of the statistical power effect and its rolein guidance will be given in the General discussionsection.

I also sought to understand whether the guidanceeffects caused by each subset were being accumulatedduring a visual search or not. As seen from the results,the crescent slope trend was maintained with iso-numerosity manipulations. Consequently, the results ofthese experiments correspond with the predictioninferred from the ‘‘cumulative’’ hypothesis. It appearsthat independent guidance processes generated byindividual subset statistics are then accumulated asattention switches from one subset to another.

In Figure 5b, results of Experiments 3a and 3b arereplotted in an alternative coordinate system to makethe guidance effects of statistical power more evident. Itcan be seen from the plot that additional subsets resultin a linear increase in the RT as was predicted by thestandard, serial, item-by-item attentional deploymenthypothesis (Treisman & Gelade, 1980). A similar lineareffect of additional color subsets on search perfor-mance was reported by Bundesen and Pedersen (1983).The statistical power of subsets has little effect on thelinearity but reduces the slopes of the functions, whichcan serve as a criterion for attentional guidance(Friedman-Hill & Wolfe, 1995).

General discussion

Attentional guidance by statistical power

Attentional guidance is an adaptive mechanism thatfacilitates the visual search for various targets. Itappears that this mechanism is served by numerousprocesses. In the present study, I tested one suchmechanism, namely using the statistical power ofnumerosity for detecting feature singletons. It isimportant to emphasize that I consider this numerosity-based method of guidance to be an additional strategythat the visual system can utilize to control detection.Both local contrast comparisons and spatial groupingare the major tools used for guiding visual search(Bacon & Egeth, 1991; Bravo & Nakayama, 1992; Sagi& Julesz, 1987; Treisman, 1982; Poisson & Wilkinson,1992; Wolfe, 1994), especially in highly regular orhomogeneous displays. However, when different ele-ments are arranged in an irregular manner (which istypical for many natural textures, objects, and scenes),other tools can aid visual search. These tools, however,

Journal of Vision (2013) 13(3):18, 1–14 Utochkin 10

should be less dependent on spatial arrangement toprovide effective guidance.

My experimental manipulations were based on thenotion that arrangement-based and numerosity-basedstrategies should yield different effects on the slopes ofthe search functions when grouping and density aremanipulated. Increasing the number of colors amongdistractors made arrangement-based strategies lessuseful for guiding attention because there were toomany concurrent discontinuity regions in the visualfield. These conditions should abolish the guidancepatterns that are manifested as negative slopes whenonly arrangement-based strategies are used. However,observers demonstrated an opposite pattern, thecrescent slope effect, which demonstrated that thepower of guidance tended to increase with the numberof heterogeneous colors. At the same time, densitymanipulations had only a limited effect on the slopesobserved in Experiment 2 and no substantial effect wasobserved in Experiments 3a and 3b. This led me toconclude that observers rely on the numerosities ofspatially overlapping color subsets to judge whether asingleton is present or absent. This statistical powereffect was found to have the properties of a guidingfactor.

The notion of statistical power implies that there canbe an analogy between a visual search and a statisticaldecision. This analogy is a natural extension of theprevious ideas regarding the global ensemble repre-sentation as summary statistics (Alvarez, 2011; Ariely,2001; Chong & Treisman, 2003; Treisman, 2006). If thevisual system is indeed able to rapidly extract rathercompact statistical descriptions from an image insteadof representing every feature and every item, why can’tit use such descriptions to distinguish between differentensembles? As summary statistics for feature-markedsubsets are collected independently (Chong & Treis-man, 2005b; Halberda et al., 2006) and in parallel fromthe entire visual field (Chong et al., 2008), then directcomparisons among statistical properties (numerosities,averages, etc.) can be a more economic strategy thanlocal feature comparisons among all neighboring items.

Attentional selection of statistical subsets

The second important and very articulated result ofthe present study is the finding that the visual systemappears to be unable to process all subsets at a time,and processing is likely to require attentional selection.Furthermore, it appears that attentional switchingbetween subsets was being performed in a strictly serialmanner as the RT tends to increase linearly with thenumber of subsets (Experiments 3a and 3b). Thisfinding is in agreement with those reported byBundesen and Pedersen (1983). Yet Bundesen and

Pedersen understood that subsets are spatial groupsdetermined by the combination of proximity andsimilarity. They found that the search time positivelycorrelates with the number of subjectively perceivedgroups. A somewhat similar conclusion was made byTreisman (1982) for conjunction searches. My results,however, provide an alternative explanation for theirresults. When all of the elements of one color werespatially grouped, the average distance between themwas shorter than in a mixed arrangement. We can seefrom Experiment 2 that the densification of visual setsfacilitated the visual search performance to somedegree, but it appears that attention, nevertheless,operates with the whole subset rather than its spatiallyseparated fragments. Otherwise, no statistical powereffect could be observed.

The other reference for an increase in RT with anincreasing number of subsets is the framework pro-vided by Duncan and Humphreys (1989). According totheir theory, the efficiency of a visual search decreasesas the heterogeneity of distractors increases. However,this rule appears to apply only if the target is similar tothe distractors; otherwise, the distractor heterogeneityhas no effect on efficiency. As only basic colors wereused in the experiments, the targets were alwaysdissimilar from the distractors and, hence, could notproduce a substantial loss in efficiency; this is exactlywhat was found in other color detection tasks (Duncan,1989; Smallman & Boynton, 1990). However, theDuncan and Humphreys’ theory implies that a targettemplate in working memory controls the allocation ofattention in a visual search. In studies by both Duncanand Smallman and Boynton, observers had suchtemplates. In my variable-mapping experiments, therewas no definite template for a target color, andtherefore, attention selected subsets one by one.

The notion that attention is able to operate withsubsets of spatially distributed objects sharing acommon feature as single units is not novel. Theensemble representation paradigm provides examplesof the representation of global features of color subsetsindependently of spatially overlapping subsets (Chong& Treisman, 2005b; Halberda et al., 2006; Treisman,2006). More remarkable, in the context of my study, isan example from visual search. In conjunction-searchexperiments, Friedman-Hill and Wolfe (1995) demon-strated that observers can produce flat search functionsmore typical for feature searches. Once an observerselects a subset of all commonly featured items (forinstance, attends to all red items, ignoring all greenitems), the task transforms into a trivial feature search(a target is the only differently oriented item in thatsubset). An interesting observation, made by Fried-man-Hill and Wolfe, is that the serial or parallelcharacter of a conjunction search depends on the setsize: The more items that are presented in a display, the

Journal of Vision (2013) 13(3):18, 1–14 Utochkin 11

more the slope tends to zero. This is considered to be amanifestation of attentional guidance. This observationis important in light of the statistical power effectdiscussed in this paper. As proposed by Friedman-Hilland Wolfe and Michod, Wolfe, and Horowitz (2004),the failure of guidance for small set sizes can beascribed to the slow speed of subset selection. It takesapproximately 250–300 ms for this type of guidance todevelop, but visual searches with small sets typicallyfinish faster (Michod et al., 2004). This is the likelyexplanation for the failure of guidance with small sets,given recent estimates of the time required forrepresenting ensemble properties (Whiting & Oriet,2011). However, this notion does not explain theadvantage of large sets found in my color detectionexperiments. As the visual search in a large set isfinished faster than in a small set, the guidance effectdevelops before a singleton is found in a small set.Again, only the statistical power of numerosity appearsto provide a satisfactory explanation. What the presentexperiments add to the knowledge of subset-orientedattention is the notion that attention appears to haveaccess to the statistical representation of subsets to guidevisual search, not merely a simple feature map (otherwisenumerosity could not be taken into account).

Cumulative guidance

The final point of my analysis concerns theinteraction between the processes responsible for thestatistical representation within subsets and the allo-cation of attention between subsets. Taken together,these processes yield an unusual combination of totalsearch efficiency and attentional guidance. As attentionis required for the serial selection of subsets, the RTtended to increase with the number of subsets. At thesame time, the statistical power effect became strongerwith the number of subsets. I concluded, hence, that theguidance processes generated by every individual subsetare simply accumulated (or summated) as the searchprogresses.

The model of such accumulation is based on thediscrimination hypothesis, proposed above, for thelikely mechanism of statistical guidance. I suggestedthat discrimination between a target and distractorsbecomes easier in large sets (due to numerosity contrastor decreasing variance). My results demonstrated thatattention likely switches from one subset to another toprovide discrimination. As more subsets are attendedto in a visual search, more serial discriminationoperations are performed. It is obvious that thereshould be a more relative benefit from two, serial, easydiscriminations than from only one such discrimina-tion. As a result, the crescent slope trend is observed asthe number of subsets increases. In other words,

numerosity statistics are being collected within eachsubset independently, but only when attention is paid to asubset does it enact subset statistics to guide the visualsearch.

Conclusion

Recent studies have convincingly shown that statis-tical features are being rapidly encoded from largedisplays by a broad attentional window like many otherfeatures. The central idea of the present study was thatthe basic statistical properties of ensembles (such asnumerosity) can guide attention when searching for aunique target. In the present study, I found thatnumerosity statistics affect the bottom-up type ofguidance and yield negative slopes of search functions.I referred to this effect as the statistical power effectand demonstrated that it is more than simply spatialretinotopic interactions between targets and distrac-tors. Finally, I showed the critical role of attention inthe selection of statistical representations and theaccumulation of guidance effects during a visual search.The results of the present study can be considered to bea step in bridging the gap between the two aspects ofperceiving objects and scenes, the ensemble represen-tation, and the visual search for an individual memberof that ensemble.

Keywords: visual search, attention, feature search,ensemble representation, numerosity

Acknowledgments

The study was implemented in the framework of theBasic Research Program at the National ResearchUniversity Higher School of Economics in 2012. Theauthor thanks Yulia Stakina for her assistance incollecting experimental data.

Commercial relationships: none.Corresponding author: Igor S. Utochkin.Email: isutochkin@inbox.ru.Address: National Research University Higher Schoolof Economics, Russian Federation.

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