Optimal observer model of single-fixation oddity search predicts a shallow set-size function

Abstract

A common finding in oddity search, a search in which the target is unknown but defined to be different from the distractors, is that human performance remains insensitive or even improves with number of distractors (set size). A number of explanations based on perceptual and attentional mechanisms have been proposed to explain the anomalous set-size effect. Here, we consider whether the shallower set-size function for oddity search could be explained by stimulus information and task demands. We developed an ideal-observer and a difference-coding (standard-deviation) model for single-fixation oddity search and compared it to the ideal observer in the standard target-known search as well as to human performance in both search tasks. Performance for the ideal and difference-coding model in the oddity search resulted in a shallower set-size function than the target-known ideal observer and was a good predictor of human search accuracy. However, the ideal-observer model was a better predictor than the standard-deviation model for 10 of the 12 data sets. The results highlight the importance of using ideal-observer analysis to separate contributions to human performance arising from perceptual/attentional mechanisms inherent to the human brain from those contributions arising from differences in stimulus information associated with the tasks.

Introduction

In our daily lives, we must often search for various objects: our car in a parking lot, a soda in the refrigerator, friends in a crowd, and so forth. Along with the commonality of search, it is often quite important in the real world that whatever is being searched for is found, for example, a tumor in a mammogram. This makes it important to understand the processes underlying visual search and what factors influence performance. As such, visual search has become a ubiquitous focus of study in cognitive psychology (Cave & Wolfe, 1990; Eckstein, Thomas, Palmer, & Shimozaki, 2000; Palmer, 1995; Pashler, 1998; Treisman & Gelade, 1980). In the laboratory, researchers typically require observers to detect or localize a particular target item among a field of distractors. A common manipulation in many of these studies is to vary the number of items in the search display (set size). Typically, performance, as measured by decreased search accuracy or increased reaction time (Cave & Wolfe, 1990; Eckstein et al., 2000; Palmer, 1995; Treisman & Gelade, 1980), deteriorates as the number of distractors increases. This set-size effect has been the basis of several highly influential models of visual attention (feature integration theory: Treisman & Gelade, 1980; guided search model: Cave & Wolfe, 1990; signal detection models: Palmer, Verghese, & Pavel, 2000). 

Many of the studies examining the effects of set size on performance have observers search for a particular target they know in advance (target known exactly; Bochud, Abbey, & Eckstein, 2004; Burgess & Ghandeharian, 1984; Swensson & Judy, 1981). However, some studies have observers perform a search for a target that is randomly sampled from a group of targets (targets known statistically; Judy, Kijewski, & Swensson, 1997; Zhang, Pham, & Eckstein, 2004). Finally, in some instances, observers are given no knowledge about the target, other than that it is different from the distractors (Bacon & Egeth, 1991; Blough, 1989; Bravo & Nakayama, 1992; Santhi & Reeves, 2004). This is known as an oddity search (a special case of target known statistically/distractor known statistically where the target cannot be the same as the distractor). For oddity searches, some studies have indicated that that set-size effects flatten or even reverse, such that performance improves as set size increases (Bacon & Egeth, 1991; Bravo & Nakayama, 1992; Santhi & Reeves, 2004). A common explanation for this inverted set-size effect is distractor grouping. This explanation proposes that as similar items get physically closer together, they tend to group together, thus reducing the effective set size. In a previous study (e.g., Bacon & Egeth, 1991), grouping was varied and seemed to account for the reverse set-size effect. This effect is similar to that of texture segmentation, where many closely spaced items are sometimes perceived as a texture rather than individual items. 

In another study (Santhi & Reeves, 2004), it was suggested that improved performance in oddity search arose due to changes in effective target contrast. In their model, the additional attention allocated to the distractors in the oddity search boosts the effective contrast of the target by an amount proportional to the number of distractors. This increased target contrast improves overall performance in the task as set size increases. Santhi and Reeves suggest that this increase in effective target contrast in their study could be due to long-range color contrast mechanisms (Walraven, 1973). 

Another possible cause of this performance change as a function of set size that has not been considered is a difference in the amount of stimulus information available to the observer to perform the task. It is possible that the observer's lack of knowledge of the target and distractors, in the oddity search, which requires the observer to infer target identity, might inherently lead to different predictions of set-size effects. 

One may employ ideal-observer analysis as a quantitative method to assess the effect of differences in available information about target and distractor on set-size effects. Ideal-observer analysis predicts the best possible performance for a task, assuming all available information is used optimally. As such, it is an absolute standard for task performance and serves as an objective measure of the demands of a given task, in terms of how information should be used to make decisions. 

Ideal-observer analysis has been used to account for human performance in a variety of tasks including simple detection and discrimination (Barlow, 1978; Burgess & Ghandeharian, 1984; Eckstein, Ahumada, & Watson, 1997; Green & Swets, 1966; Kersten, 1984), object recognition (Braje, Tjan, & Legge, 1995; Liu, Knill, & Kersten, 1995; Tjan, Braje, Legge, & Kersten, 1995), and perceptual learning (Eckstein, Abbey, Pham, & Shimozaki, 2004; Gold, Bennett, & Sekuler,1999). It is important to note that some of these models are quasi-ideal in that they use simplified decision rules that closely approximate ideal. Ideal-observer analysis has also been applied to target-known search and accounts for typical human observer set-size effects (decreasing performance with increasing set size) quite well in many cases (Eckstein, 1998; Eckstein et al., 2000; Palmer, 1994, 1995; Verghese, 2001). 

The goal of this study was to assess whether differences in available stimulus information as quantified using ideal-observer analysis can account for the differences in set-size effects for target known and oddity single-fixation visual searches. Figures 1 and 2 illustrate examples of our search tasks. We note that in the present study, as in many previous studies (Baldassi & Verghese, 2002; Eckstein, 1998; Eckstein et al., 2000; Palmer, 1994, 1995), observers hold fixation on a central cross during the task and thus allows us to develop models that do not require explicit modeling of eye movements and the retinal eccentricity effects. Thus, we refer to the task as single-fixation search to differentiate it from multiple-fixation search (Najemnik & Geisler, 2005). 

Figure 1

 

Panels A and B are stimulus examples from Experiment 1, and Panels C and D are stimulus examples from Experiment 2. Panels A and C are examples of Set Size 4, and Panels B and D are examples of Set Size 8. These examples illustrate extreme differences between the target and the distractors. In the actual experiment, differences were often much more subtle. Contrasts of these stimuli have also been increased for publication purposes.

Figure 2

 

Panels A and B are stimulus examples from Experiment 3, and Panels C and D are stimulus examples from Experiment 4. Panels A and C are examples of Set Size 4, and Panels B and D are examples of Set Size 8.

Observers participated in two separate two-alternative forced choice (2-AFC) tasks where the target differed from distractors along contrast (2D Gaussians with differing contrast; Panels A and B in Figure 1) and two 2-AFC tasks in which the target differed from distractors along orientation (2D Gabors with differing orientation; Panels C and D in Figure 1). In each alternative, there were two to eight items (only set sizes of 4 and 8 are illustrated in the figure). One alternative contained all distractors, and the other contained distractors and one target. It was the observer's task to determine which of two alternatives contained the target. In one condition, they are told target identity (target-known search), and in another, observers were told that the target is simply an item that is different from the rest of the items (oddity search). 

In an actual oddity search, the target can differ from the distractors by a potentially infinite number of features and feature values (contrast, orientation, shape, blur, etc.). However, for computational simplicity in the modeling, we constrained the possible number of features and feature values to a finite number. Implications of this simplification are discussed at the end of the Theory section. Two methods of presenting the alternatives were also used. In one method (illustrated in Figure 1), both alternatives were presented simultaneously; in the other, they were presented sequentially (illustrated in Figure 2). We compared the human results from these studies to predictions made by ideal-observer analysis, as well as to predictions made by a suboptimal model that bases its decision on the standard deviation of the observed values in each of the alternatives. 

Theory

Ideal observer—Target-known search

For a 2-AFC, the ideal observer calculates the posterior probability of the target being at each interval given the observed set of internal responses in both alternatives (Green & Swets, 1966). The posterior probability is the product of the likelihood of the observed internal responses given the presence of target in an alternative and the prior probability of the target being in that alternative (see the 1 for derivation). Because the prior probabilities for each alternative are equal, the ideal observer reduces to a comparison of likelihoods. However, because there is uncertainty about which element among the N elements in the alternative will be the target, the ideal observer has to consider all possible scenarios and, thus, sum the likelihoods across all the mutually exclusive events of each element within an alternative being the target. To illustrate the strategy for the task considered, we outline the ideal-observer strategy for a target-known exactly search for an example with only two possible distractor distributions. 

The ideal-observer model first extracts orientation or luminance values from each element in the display ( v ₁ – v ₄, Figure 3). It then creates a vector of those values from each of the alternatives ( x ₁ and x ₂, Figure 3). It then uses Equation 1 (which is a more general equation than is necessary for this example, as it accommodates any number of M intervals and Ddistractor types; see the 1 for derivation) to compute the likelihood of observing those values given a particular interval ( l) contains the target. It compares the hypothesized distractor intervals to vectors of mean distractor values ( d _j) and the hypothesized target interval to vectors of mean target values ( t _ij). (T refers to the transpose matrix operation.) Assuming that the probability density functions are Gaussian, we obtain  

$\tilde{P}({\mathbf{x}}_1\dots {\mathbf{x}}_M|l,{\mathbf{t}}_{1\dots N,1\dots D},{\mathbf{d}}_{1\dots D})=\sum _{i=1}^N\sum _{j=1}^1{e}^{({\mathbf{t}}_{ij}^{\mathrm{T}}{\mathbf{x}}_l+\sum _{m\ne l}^M{\mathbf{d}}_j^{\mathrm{T}}{\mathbf{x}}_m)}.$

(1)

The model must sum across for all possible D distractor distributions ( j) as well as all N possible target locations ( i). The sums of these likelihoods (

${\tilde{P}}_1$

and

${\tilde{P}}_2$

, Figure 3) are compared to each other, and the alternative with the greatest likelihood is chosen. 

Figure 3

 

Representation of how an ideal observer performs a target-known search. It first extracts observed orientation values from each item in the display. It then sums the likelihoods of observing those values given all possible target locations and distractor types for each alternative. It then picks the alternative with the greatest summed likelihood as the one that contains the target.

Because the decision variable is the sum across all items, as the set size increases, the overall amount of noise in the decision variable increases, even if the individual items have the same amount of noise. In other words, the probability that one of the distractors might appear to look much like the target increases as the number of items increases (Nolte & Jaarsma, 1966; Palmer, 1995; Shaw, 1980, 1984). This decreased performance as a function of set size describes the typical set-size (shown in Figure 4 as circles) effect found in target-known searches. Note, that the ideal observer for this 2 AFC target known search can be approximated for high contrasts by a max-model which bases its decision in the maximum response among the responses of each alternative/interval (Nolte & Jaarsma, 1966). 

Figure 4

 

Model predictions for all three models (ideal observer, target known; ideal observer, oddity; and standard deviation) at three levels of target/distractor discriminability for 60,000 simulation trials.

Ideal observer—Oddity search

The ideal-observer model performs an oddity search in much the same way it performs a target-known search, except that likelihoods must also be summed over all possible target types given that the target identity is unknown. For simplicity, the following example depicts two target and two distractor types. First, this model extracts the orientation or luminance values from each element in the display ( v ₁– v ₄, Figure 5). In this case, the summed likelihoods, computed using Equation 2 (

${\tilde{P}}_1$

and

${\tilde{P}}_2$

, Figure 5), include sums across all possible target types ( V types), as well as all distractor types ( V−1 types) and locations (note the extra rows for T ₁ and T ₂ in Figure 5). After this step, the model is identical to a target-known search; it picks the alternative with the greater summed likelihood as the one that contains the target.  

$\tilde{P}({\mathbf{x}}_1\dots {\mathbf{x}}_M|l,{\mathbf{t}}_{1\dots N,1\dots V,1\dots V},{\mathbf{d}}_{1\dots V,1\dots V})=\sum _{i=1}^N\sum _{j=1}^V\sum _{k\ne j}^V{e}^{({\mathbf{t}}_{ijk}^{\mathrm{T}}{\mathbf{x}}_l+\sum _{m\ne l}^M{\mathbf{d}}_j^{\mathrm{T}}{\mathbf{x}}_m)}$

(2)

A web-based implementation of both the ideal-observer target-known and the ideal-observer oddity searches that allows fitting of these models to observed data can be found online at http://www.psych.ucsb.edu/research/viu/cyberfits.htm and is available to interested users. 

Figure 5

 

Representation of how an ideal observer performs an oddity search. It first extracts observed orientation values from each item in the display. It then sums the likelihoods of observing those values given all possible target locations, distractor types, and target types for each alternative. It then picks the alternative with the greatest summed likelihood as the one that contains the target.

Why is the set-size effect flattened for the ideal-observer oddity search?

As in the target-known search, increasing the number of items increases the overall amount of noise in the decision variable; however, the model does not predict the typical decrease in performance as a function of set size. The effect is counteracted by the fact that there is increased certainty as to what the distractor is as the number of distractors increases. Imagine an oddity search with a set size of 1. Clearly, this task would be impossible even for an optimal observer as no item can be unambiguously defined as the odd target. The odd target can only be defined with a set size of 2 or more. However, assuming that the representations of the distractors are noisy, as the set size increases, the estimates of what the actual distractor is become more accurate. As shown in Figure 4 as squares, the combination of these two effects (increasing decision variable noise and decreasing distractor uncertainty as set size increases) almost eliminates for the ideal observer the set-size effect found in the target-known search, producing a nearly flat line as a function of set size. 

Standard-deviation/difference-coding model—Oddity search

The second model we tested was a standard-deviation model for the oddity search ( Figure 6). This model assumes that when the human observers are uncertain about the target and the distractor, they resort to a suboptimal strategy of looking at the variability in contrast/orientation within alternatives. This model falls into a class that has been referred to as relative coding models (Palmer et al., 2000). The decision is based on differences between the observed stimuli. The first step of this model is the extraction of luminance or orientation values from each of the items in the display (v₁ – v₄, Figure 6). It next computes the standard deviation of these values from each of the alternatives (

${\tilde{P}}_1$

and

${\tilde{P}}_2$

, Figure 6). The alternative containing all distractors should be more homogenous than the one containing a target. The model then picks the alternative with a greater standard-deviation value as the one containing the target. As shown in Figure 4 as diamonds, this model also predicts that performance is nearly unchanged as a function of set size. Again, the noise in the display increases as set size increases, but the typical set-size effect is counteracted by the fact that the model obtains better estimates of the standard deviation of each alternative as set size increases. 

Figure 6

 

Representation of how the standard-deviation model performs an oddity search. It first extracts observed orientation values from each item in the display. It then calculates the standard deviation of those values for each alternative. It then picks the alternative with the greatest standard deviation as the one that contains the target.

Figure 4 illustrates the performance of all models at three levels of task difficulty (difficulty corresponding to the only free parameter in these models). The ideal-observer target-known model has the best performance of the three models and shows the typical set-size effect. The fact that this model has the best performance should come as no surprise considering that this model has the most information about the environment and is using it optimally. The ideal-observer oddity model has the second best performance and shows a slight increase in performance from Set Size 2 to Set Size 4 but is relatively flat. Overall, the drop in performance from the target-known model is a direct quantification of the performance drop that should be expected in this task by simply removing information about the target; all information still available is being used optimally. Although the standard-deviation model has lower performance than the ideal-observer model, it still preserves the relatively shallow set-size predictions. The drop in performance for the standard-deviation model compared to the ideal-observer oddity-search model reflects that the standard-deviation model is using the available information suboptimally. Note that the difference between the models diminishes as the index of detectability ( d′) increases. At high values of d′, the standard-deviation model approximates the ideal observer for the oddity-search task. 

Effect of the number of target distributions

In our version of the oddity-search task, the target and the distractor are each chosen from one of Ndistributions without replacement. In a typical oddity search, the number of distributions could potentially be infinite; in our task, N = 6. It is important to understand how the number of possible target distributions affects the predictions made by these models. The effect of adding more distributions is illustrated in Figure 7. 

Figure 7

 

The effect of the number of possible distributions on the predictions made by the three models: ideal observer, target known; ideal observer, oddity search; and standard-deviation model for the oddity search. Performance drops for all three models as the number of distributions increases. Performance for the ideal-observer oddity search drops the most and can be approximated by the standard-deviation model for higher numbers of target distributions.

In our task, we use index of detectability, d′, to refer to the distance from one potential target distribution to the adjacent distribution ( Figure 8). To investigate the effect of number of possible target/distractor distributions while controlling for the average pairwise index of detectability, we equated the average d′ (i.e., expected distance between a target and a distractor across trials). If we were to have simply equated d′ across adjacent distributions, performance would increase due to an increasing average d′. ¹ Thus, we generated model simulations for increasing number of distributions but equated the average d′. 

Figure 8

 

d′ was the only free parameter in this model. The target/distractor distributions were always equally spaced, but when fitting the models, the distance between them was free to vary to match human performance.

Increasing the number of target distributions lowers the performance of all models. This is due to increased uncertainty as to which distribution the target and the distractor came from. The ideal-observer oddity search drops the most quickly, whereas the standard-deviation model drops the least quickly. Interestingly, as the number of distributions ( N) increases, the performance of the ideal-observer oddity model is increasingly better approximated by the standard-deviation model. It is important to note that the shallowness of the set-size function for the oddity task (for both ideal-observer and standard-deviation models) and the steepness of the function for the target-known ideal observer remain relatively invariable to the number of distributions. 

Methods

Participants

An author (W.S.) and five naive observers participated in this series of studies. Two observers were female (B.P. and J.C.), ages 20–45, and the other four were male, ages 20–35. Observers had normal or corrected-to-normal vision. 

Apparatus

Stimuli were displayed on an Image Systems M17LMAX monochrome monitor at a resolution of 1,024 × 768 pixels (Image Systems, Minnetonka, MN) with a refresh rate of 70 Hz. The luminance versus gray-level relationship was set to a linear response function (calibrated using Dome software by Planar Systems, Inc.). The monitor was calibrated with the “black” luminance set to 0.00 cd/m ² and the “white” luminance set to 50.00 cd/m ². The study was conducted in a dark room. Observers were at an approximate distance of 55 cm from the display, resulting in a subtended angle of 0.034° per pixel. 

Experiment 1: 2AFC—Contrast

Stimuli

The stimuli in Experiment 1 were 2D Gaussian dots with a spatial full width at half height of 0.74° presented on a background with a luminance of 25 cd/m ². To reduce location uncertainty, we placed four small squares adjacent to each dot; these squares subtended 0.1° × 0.1° of visual angle, had a luminance of 7.8 cd/m ², and were located 0.79° up, down, left, and right from the center of the dot (see Figures 1A and 1B). There were 4, 8, 12, or 16 dots in each display. They were placed at locations chosen from 20 possible equidistant locations along the circumference of an imaginary circle at an eccentricity of 9.08° from a central fixation point. Each dot was placed adjacent to another dot, except that half the dots were displaced from the other half by skipping one of the possible locations (see Figure 1). The skipped location created a gap that was used to distinguish between the two alternatives. No observers reported difficulty in perceptually segregating the two groups of dots forming the two alternatives. The dots were placed next to each other to ensure that each of the alternatives had equal item density regardless of set size. On each trial, the 20 possible locations were randomly rotated along the imaginary circle by a random overall rotation (the rotation amount was chosen from a uniform distribution of 1–360°). Each Gaussian's contrast was one of six possible contrasts (−0.548, −0.352, −0.116, 0.116, 0.352, 0.548) perturbed by independent contrast noise with a standard deviation of 0.18. The noise standard deviation was chosen so that the signal-to-noise ratio for the stimuli resulted in behavioral performance in the range of 70–90% correct. Contrast values both above and below the background luminance were chosen to allow for observers to perceptually discriminate the six different contrast values. For each trial, two contrasts were chosen: one was the target contrast and the other was the distractor contrast. Only one of the Gaussians displayed had the target contrast; the rest had the distractor contrast. Each set size, target contrast, distractor contrast, and target location were equally likely to be used on every trial. It is important to note that means of the contrast distributions were not equally spaced (i.e., −0.3, −0.2, −0.1, 0.1, 0.2, 0.3). The ideal-observer model calculations for this search task assumed that these distributions are equally spaced; however, we spaced the distributions unevenly for humans to control for the effect described by Weber's law. This effect posits that the just-noticeable difference between two stimuli increases with intensity of the stimulus. Thus, to achieve equivalent perceptual discrimination across neighboring contrasts, we increased the contrast differences as the element contrasts increased. If we had used equally spaced contrasts, the perceptual discriminability for human observers would have been unequal. 

Procedure

On each trial, the observers' task was to determine whether the set of dots clockwise or counterclockwise from the gap contained the target dot ( Figure 1). There were two conditions: target-known search and oddity search. The procedure for target-known trials was as follows. During the first 750 ms of the trial, observers were shown the reference target Gaussian in the center of the screen. This reference Gaussian had the luminance of the mean of the target distribution for that trial, not the exact luminance of the target dot for that trial. After the first 500 ms, a fixation cross appeared. The Gaussian elements were then displayed for 150 ms followed by a mask of white noise for 500 ms. Next, observers responded by key press if they thought that the target was clockwise from the gap or if they thought that it was counterclockwise. Feedback was given immediately upon response, and the next trial began when the space bar was pressed. Each session consisted of 200 trials (50 trials of each set size), and the target type remained constant throughout the session. 

The procedure for the oddity-search condition was identical to the procedure used in the target-known search except that instead of displaying the target dot for 500 ms at the beginning of the trial, a fixation cross was presented (this resulted in the fixation cross being displayed for 1,250 ms at the beginning of each trial). Unlike for the target-known search, the target type was randomly chosen from one of the six possible distributions of contrast on each trial. Observers completed 30 sessions alternating between target-known and target-unknown conditions. 

Experiment 2: 2AFC—Orientation

Experiment 2 was identical to Experiment 1 except that instead of using Gaussian dots of varying contrast, Gabor patches were used with varying orientations. The Gabor patches had the same spatial full width at half height as the Gaussian dots and a spatial frequency of 2.12 cycles/deg. Instead of six contrast distributions, the target and distractors were chosen from six orientation distributions with means ranging from 45° counterclockwise from vertical to 45° clockwise from vertical in 18° steps. The standard deviation of the distributions was 10°. As with the contrast discrimination task, the standard deviation was chosen so that the signal-to-noise ratio for the stimuli resulted in human performance in the range of 70–90% correct. There were no squares placed around the Gabor patches because these were highly visible regardless of the orientation. 

Experiments 3 and 4: 2IFC—Contrast and 2IFC—Orientation

Experiments 3 and 4 were identical to Experiments 1 and 2 except that they differed in the method of presentation of the two alternatives. In Experiments 1 and 2, both alternatives were displayed on the screen simultaneously and a small gap was used to distinguish between them. In Experiments 3 and 4, the intervals were presented sequentially, each for 150 ms with a mask (500 ms) and fixation cross (500 ms) presented after the first interval. There was also no adjacency rule; all of the items in each interval were rotated independently of the other interval such that items in the second interval could fall on the same location as items in the first. The observers' task was to determine whether it was the first or the second interval that contained the target. 

Results

Figures 9, 10, 11, and 12 present human observers' proportion correct as a function of set size for all experiments ( Figure 9: Experiment 1, 2-AFC contrast; Figure 10: Experiment 2, 2-AFC orientation; Figure 11: Experiment 3, 2-IFC contrast; Figure 12: Experiment 4, 2-IFC orientation). Target-known searches are represented by dark blue circles, and oddity searches are represented by light squares (light blue squares for observers whose data were best fit by the ideal-observer oddity-search model and light green for observers whose data were best fit by the standard-deviation model). Overall performance was significantly higher for the target-known search than for the oddity search for all conditions and observers with the exception of B.P. in Experiment 2 and T.S. in Experiment 4 where the difference did not reach statistical significance ( p < .05). Observer performance for the target-known condition followed the typical set-size effect (Baldassi & Verghese, 2002; Cave & Wolfe, 1990; Eckstein et al., 2000; Monnier & Nagy, 2001; Palmer, 1995; Treisman & Gelade, 1980) with percentage correct decreasing by about 5% on average as the set size increases from 2 to 8. Comparatively, the set-size effects for the oddity-search condition are much shallower with percentage correct from Set Size 2 to Set Size 8 decreasing on average by 2%. 

Figure 9

 

Observer data from Experiment 1, the 2-AFC contrast task, with best fitting ideal-observer model predictions. The best fitting model was the ideal-observer oddity search for W.S. and C.L. and the standard-deviation model for D.K.

Figure 10

 

Observer data from Experiment 2, the 2-AFC orientation task, with best fitting ideal-observer model predictions. The best fitting model was the ideal-observer oddity search for all three observers.

Figure 11

 

Observer data from Experiment 3, the 2-IFC contrast task, with best fitting ideal-observer model predictions. The best fitting model was the ideal-observer oddity search for all three observers in this experiment.

Figure 12

 

Observer data from Experiment 4, the 2-IFC orientation task, with best fitting ideal-observer model predictions. The best fitting model was the ideal-observer oddity search for J.C. and T.S. and the standard-deviation model for W.S.

Observer performance was compared to predictions made by the ideal-observer and standard-deviation models. Two fits were performed. In the first, the ideal-observer target-known and oddity model predictions were simultaneously fit to the target-known and oddity-search data, respectively. Thus, this fit assumed that humans behaved like an ideal observer for both search conditions. Continuous light blue lines in Figures 9, 10, 11, and 12 show the best fitting model predictions for the ideal observer. Reduced chi-square goodness-of-fit calculations are presented in Table 1. In the second fit, the ideal-observer target-known and standard-deviation model predictions were simultaneously fit to the target-known and oddity-search data, respectively. Thus, this second fit assumed that observers behaved like an ideal observer for the target-known search but followed a suboptimal standard-deviation model for the oddity search. Continuous light green lines in Figures 9 and 12 show the best fitting model predictions for the standard-deviation model. Table 2 presents the goodness-of-fit results. Only one free parameter corresponding to the separation in standard-deviation units between the target–distractor and distractor–distractor distributions, d′ (illustrated in Figure 10), was used to fit all models to the human data. 

Table 1

 

Reduced chi-square goodness-of-fit statistics for ideal-observer models.

	Contrast		Orientation
	Observer	Ideal target known/ideal oddity	Observer	Ideal target known/ideal oddity
AFC	W.S.	0.3175	W.S.	0.0412
	D.K.	1.8203	D.K.	0.3959
	C.L.	1.1074	B.P.	9.5204*
IFC	W.S.	0.5413	W.S.	3.8059*
	J.C.	0.8438	J.C.	1.2454
	T.S.	1.2399	T.S.	3.1382*

 

Note: *Probability of observing the data given the model <.01%.

Table 2

 

Reduced chi-square goodness-of-fit statistics for ideal-observer target-known model and standard-deviation model.

	Contrast		Orientation
	Observer	Ideal target known/standard deviation oddity	Observer	Ideal target known/standard deviation oddity
AFC	W.S.	1.2624	W.S.	0.5228
	D.K.	1.3513	D.K.	1.4618
	C.L.	1.6676	B.P.	13.0028*
IFC	W.S.	0.7389	W.S.	2.9424*
	J.C.	2.0394	J.C.	1.6453
	T.S.	2.5502	T.S.	5.6136*

 

Note: *Probability of observing the data given the model <.01%. Values in boldface highlight the conditions and observers for which the standard-deviation model fit better than the ideal-observer model.

The ideal-observer model fit to target-known and oddity data could not be rejected for 9 of the 12 observers. The average chi-square value for these models was 2.001. The ideal-observer target-known and standard-deviation model predictions were also not rejected for 9 of the 12 observers (see Table 2). The average chi-square value for these models was 2.90. Fits involving the standard-deviation model fit better for 2 of the 12 observers ( Figures 9B and 12B). On average, fits involving the ideal-observer oddity-search model were significantly better than fits involving the standard-deviation model ( p < .01). 

The efficiency of the ideal observer can be computed for the experiments involving orientation. For Experiment 2, average observer efficiency was 24%. For Experiment 4, average observer efficiency was 36%. 

Discussion

Here, we have presented an ideal-observer model for single-fixation oddity-search task. The ideal-observer model quantifies inherent performance differences between target-known and oddity-search tasks assuming all available information is used optimally. These performance differences are predicted independent of special perceptual processes that are activated specifically for oddity search. The ideal-observer models also account for observer data quite well but could be rejected for 3 of 12 data sets. 

Previous models have predicted differences between these target-known and oddity-search tasks (Santhi & Reeves, 2004) but use an additional parameter where the effective contrast of the target increases with number of distractors. The ideal-observer oddity-search model predicts a shallower set-size effect for oddity search than for the target-known search but attributes this difference to information available to perform the task and without the need for an additional parameter that is a function of set size. 

Both the ideal-observer oddity-search model and the standard-deviation model predict flatter set-size effects, but the performance drop predicted by the ideal-observer oddity model is more than that predicted by the standard-deviation model. While a flatter set-size effect for the oddity search is not unique to the ideal-observer model, when the models were fit to the data, the actual degradation in performance demonstrated by our observers was overall better predicted by the ideal-observer oddity-search model. 

Oddity search can be thought of as a specific type of search in a more general framework of search tasks. This framework is presented in Table 3. This framework describes search in terms of the type of information available to the observer. At one extreme, the observers might know exactly what they are looking for and know exactly what the distractors are going to be. This would be a target-known exactly/distractor known exactly search. At the other extreme is our oddity-search task where the observer only knows that the target will be one of T targets and the distractors could be one of D distractors (with the added constraint that the target will never be the same as the distractor). Previous ideal-observer models for single-fixation search have been proposed to predict ideal performance in three of the four boxes in Table 3. Here, we extended the modeling efforts to an ideal-observer model for the target-known statistically/distractor known statistically case. 

Table 3

 

Visual search tasks can be categorized based on the knowledge about the targets and distractors to create a general framework of visual search.

Distractor knowledge	Target knowledge
	Exact	Statistical
Exact	Bochud et al., 2004; Burgess & Ghandeharian, 1984; Palmer, 1995; Swenson & Judy, 1981	Baldassi & Burr, 2004; Cameron, Tai, Eckstein, & Carrasco, 2004; Judy et al., 1997; Solomon & Morgan, 2001; Zhang et al., 2004
Statistical	Rosenholtz, 2001	Bacon & Egeth, 1991; Blough, 1989; Monnier & Nagy, 2001; Santhi & Reeves, 2004

The ideal-observer models did not fit well for all observers in all conditions. For observer B.P. in Experiment 2 ( Figure 10C) and observer T.S. in Experiment 4 ( Figure 12C), performance in the oddity-search condition sometimes surpassed their performance in the target-known condition. This implies that knowledge of the target somehow interfered with their performance in the task. They would have performed better if they had simply ignored the target information and treated the task as an oddity search. A possible explanation for this is that the demand of keeping the target in working memory detracts from resources that could be helping to perform the task. 

The ideal-observer model also assumes that the observer has perfect memory. This may be a problem, however, particularly in the 2-IFC task, because it assumes that the observer is able to remember the observed contrast or orientation values at each location in the display across both intervals before attempting to make a decision. It is possible that the observer's representation of the first interval fades from memory as the representation of the second interval is acquired. This would obviously degrade performance below that of the ideal-observer models. This is possibly why the ideal-observer model was also rejected for W.S. (author) in Experiment 4. This explanation, however, would also predict poorer fits for Experiment 3. It could be that the brain is better able to maintain the representation of contrast values than those of orientation values. 

It is important to note that our version of the oddity-search task is not the typical oddity search. In a typical oddity search in the literature, the odd item could be any of an infinite number of items. The oddity-search task in the present work is a simpler version of the typical oddity-search task. In our task, the observer knows that the target will be chosen from one of six possible target distributions. In a typical oddity search, the observer is not informed about which categories from which the odd item will be sampled, and the number of categories is larger than that in the present study. Thus, one might wonder whether the present results with simplified stimuli are applicable to the typical oddity search. We contend that the simulations showing the relative constancy of the shape of the set-size function for the oddity search as a function of the number of distributions ( Figure 7) indeed show that the principles reducing the steepness of the set-size effect should apply to the more complex and typical oddity search. 

Conclusions

For the tasks investigated in this study, differences in the set-size function between target-known and oddity search can be attributed to differences in task demands as quantified by the ideal observer, leaving no need for additional perceptual factors specific to oddity search to affect performance in the task. Finally, we suggest that ideal-observer analysis should be used when comparing oddity search to more standard searches to appropriately partition human performance differences due to task demands from human perceptual/attentional effects specific to oddity search. 

Appendix A

The following are the derivations of the ideal-observer oddity-search and the ideal-observer target-known models. In choosing which alternative contains the target, the ideal observer computes the posterior probability of the target being present in each of the alternatives given the data observed in all alternatives ( x ₁… x _M) and chooses the alternatives with the highest posterior probability. Observed data are the contrast or orientation values of each of the items in the display for this experiment. The posterior probability at the lth alternative can be related to the likelihood of the data in all alternatives given target presence in the lth alternative, through Bayes' rule (Green & Swets, 1966; Peterson, Birdsall, & Fox, 1954). The posterior probability of a given alternative, l, containing the target is given as a normalized product of the likelihood of the observed data, P(x₁…x_M∣l), and the prior probability of that alternative P(l) 

$P(l|{\mathbf{x}}_1\dots {\mathbf{x}}_M)=\frac{P\left(l\right)P({\mathbf{x}}_1\dots {\mathbf{x}}_M|l)}{P({\mathbf{x}}_1\dots {\mathbf{x}}_M)}.$

(A1)

P(x₁…x_M) is the probability of the data, which acts as a normalization factor for the posterior probability. Because it is independent of l, it can be neglected without affecting the outcome of the decisions. In this work, each alternative is equally likely to contain the target, and hence, P(l) is the same for all l, and can also be neglected, leaving the decision completely determined by the likelihood term. Thus, the alternative chosen by the ideal observer,

${\widehat{l}}_{\mathrm{I}\mathrm{O}}$

, is given by 

${\widehat{l}}_{\mathrm{I}\mathrm{O}}=\underset{l}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}\left(P\right(l|{\mathbf{x}}_1\dots {\mathbf{x}}_m\left)\right)=\underset{l}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}\left(P\right({\mathbf{x}}_1\dots {\mathbf{x}}_m|l\left)\right).$

(A2)

Implementing the ideal observer consists of computing the likelihood term for each of the possible alternatives and choosing the alternative with the highest value as the one that contains the target. 

The target and distractors are defined in terms of features such as contrast or orientation features. Let us define F ₁… F _V as the set of mean feature values of the possible target and distractor distributions. The mean feature values were uniformly spaced over a range of contrast or orientation, with spacing Δ f (therefore, F _i = F ₁ + ( i − 1) Δ f). Each observed feature value is normally distributed about a mean feature value (the particular mean feature value is described below) with a common variance, σ ². Observed feature values were controlled by the spacing of the mean feature values and the standard deviation of the noise. For the human observer experiments, the spacing and standard deviation were set to give a reasonable level of task difficulty and perceived variability in the stimuli (values are described in the Methods section), while avoiding any issues of saturation (contrast) or wraparound (orientation). For convenience, the ideal observer was computed with the variance fixed at 1, and difficulty was controlled solely through the distance between the feature values. Performance of the ideal observer is equivalent for equal values of the spacing divided by the noise standard deviation. This ratio is d′ in Figure 8. 

In the present experiments, the number of feature means, V, was fixed at six. On each trial, two different mean feature values were selected: one as the mean value of the target and one as the mean value of the distractors. For the alternative containing the target stimuli, t, the observed feature value of one item is generated by adding noise to the mean target feature value and the remaining items are generated using the distractor mean with the addition of noise. Nontarget alternatives, d, are generated using the distractor feature value alone with noise added. The following example represents the mean values of the target and distractor alternatives when the target is at the first position, the set size is 4, the target feature is F ₆, and the distractor feature is F ₁. The expected value of the target alternative would be t = [ F ₆, F ₁, F ₁, F ₁], and the expected values of the distractor alternative(s) would be d = [ F ₁, F ₁, F ₁, F ₁]. The probability distribution describing the observed values in a particular alternative ( x) is the multivariate normal density where μ is a vector of expected values (i.e., [ F ₆, F ₁, F ₁, F ₁])  

$P({\mathbf{x}}_l|\mathbf{\mu })=\frac{1}{(2\pi {)}^{N/2}}{e}^{(-\frac{1}{2}({\mathbf{x}}_l-\mathbf{u}{)}^{\mathrm{T}}({\mathbf{x}}_l-\mathbf{u}))}.$

(A3)

Note that we have incorporated into this density the constraints that observed values are independent with a standard deviation of 1. A more general formulation without these constrains would use a covariance matrix Σ to model statistical dependencies in the observed values. 

The model decides which alternative contains the target by evaluating the likelihood that each alternative is the target and all other alternatives are distractors. It compares the presumed target alternative, x _l, to a vector that contains expected values for the target alternative ( t). The remaining alternatives are compared to a vector that contains the expected values for the distractor ( d). For the moment, let us consider the t and d vectors to be fixed (this requirement is relaxed below to accommodate uncertainty within an alternative). Because each alternative is independent, we can write the conditional distribution of all alternative data as the product of likelihood of alternative l given the target vector and the likelihood of the remaining alternatives given the distractor vector  

$P({\mathbf{x}}_1\dots {\mathbf{x}}_M|l,\mathbf{t},\mathbf{d})=P({\mathbf{x}}_l|\mathbf{t})\prod _{m\ne l}^{M}P({\mathbf{x}}_m|\mathbf{d}).$

(A4)

Equation A4 considers both the target and distractor features as known, as well as the target's position within the alternative; however, in the actual experiment, the target alternative contains multiple locations that could potentially be the target ( N, the set size within each alternative). We now address this uncertainty in the target and distractor vectors. 

We begin by considering uncertainty due to the fact that the target can appear in any one of N locations within an alternative. This results in N target vectors that contain the target at different locations. We therefore add a subscript to the target vector, t _i ( i = 1, …, N), where i indicates the location of the target within the alternative. The likelihood that a particular alternative contains the target is the sum of the likelihoods of the target across all locations in the alternative. This generalizes Equation A4 to  

$P({\mathbf{x}}_1\dots {\mathbf{x}}_M|l,{\mathbf{t}}_{1\dots N},\mathbf{d})=\sum _{i=1}^{N}P\left({\mathbf{x}}_l|{\mathbf{t}}_i\right)\prod _{m\ne l}^{M}P({\mathbf{x}}_m|\mathbf{d}).$

(A5)

Because there are also multiple potential distractor features for a given alternative ( D), the model needs to sum across all possible distractor features in addition to the possible target locations. This adds another subscript to both the target vector variable, t _ij, and to our distractor vector variable, d _j. For example, for a given target feature value of F _Targ, the target vector t ₁₂ = [ F _Targ, F ₂, F ₂, F ₂], and distractor vectors are represented by d ₂ = [ F ₂, F ₂, F ₂, F ₂]. When distractor uncertainty is accounted for, the likelihood in Equation A5 is generalized to  

$P({\mathbf{x}}_1\dots {\mathbf{x}}_M|l,{\mathbf{t}}_{1\dots N,1\dots D},{\mathbf{d}}_{1\dots D})=\sum _{i=1}^N\sum _{j=1}^{D}P\left({\mathbf{x}}_l|{\mathbf{t}}_{ij}\right)\prod _{m\ne l}^{M}P\left({\mathbf{x}}_m|{\mathbf{d}}_j\right).$

(A6)

This equation is representative of the likelihood in the target-known search where F _Targ is known to the observer. For an oddity search, there is also uncertainty as to the target feature value. We incorporate this additional uncertainty by adding an additional subscript, k, to the target vector, which indicates the feature value of the target. For an alternative with four possible locations, a feature vector t ₂₁₃ = [ F ₁, F ₃, F ₁, F ₁]. Note that the feature value of the target cannot be the same as the distractor, and hence, k ≠ j. The target and distractor feature values are otherwise independent, and hence, any pairing of feature values as target/distractor is equally likely although it is straightforward to generalize this as well. The model must now sum across all target feature values ( V values) in addition to the other components shown in Equation A6,  

$P({\mathbf{x}}_1\dots {\mathbf{x}}_M|l,{\mathbf{t}}_{1\dots N,1\dots V,1\dots V},{\mathbf{d}}_{1\dots V,1\dots V})=\sum _{i=1}^N\sum _{j=1}^V\sum _{k\ne 1}^{V}P\left({\mathbf{x}}_l|{\mathbf{t}}_{ijk}\right)\prod _{m\ne l}^{M}P\left({\mathbf{x}}_m|{\mathbf{d}}_j\right).$

(A7)

We now describe the implementation of this model with Gaussian densities. 

Target-known ideal-observer search model

Using the assumption that the vectors representing the mean values of responses from a particular alternative have a multivariate Gaussian distribution, we can compute the probability of observing a particular set of responses by substituting the multivariate Gaussian equation, Equation A3, into Equation A6, yielding  

$P({\mathbf{x}}_1\dots {\mathbf{x}}_M|l,{\mathbf{t}}_{1\dots N,1\dots D},{\mathbf{d}}_{1\dots D})=\sum _{i=1}^N\sum _{j=1}^D\frac{1}{(2\pi {)}^{N/2}}{e}^{(-\frac{1}{2}({\mathbf{x}}_l-{\mathbf{t}}_{ij}{)}^{\mathrm{T}}({\mathbf{x}}_l-{\mathbf{t}}_{ij}))}\times \prod _{m\ne l}^M\frac{1}{(2\pi {)}^{N/2}}{e}^{(-\frac{1}{2}({\mathbf{x}}_m-{\mathbf{d}}_j{)}^{\mathrm{T}}({\mathbf{x}}_m-{\mathrm{d}}_j))}$

(A8)

as the target-known search likelihood. This expression can be simplified in a way that will not affect the maximum likelihood alternative chosen by the ideal observer. Let us denote by

$\tilde{P}$

quantities that are equivalent to likelihood terms not affecting the decision of the ideal observer. 

The normalization factor in front of the exponentials can be removed as all likelihoods that will be computed have identical normalization factors and, thus, are essentially constant, which will not affect the outcome of the decision.  

$\tilde{P}({\mathbf{x}}_1\dots {\mathbf{x}}_M|l,{\mathbf{t}}_{1\dots N,1\dots D},{\mathbf{d}}_{1\dots D})=\sum _{i=1}^N\sum _{j=1}^D{e}^{(-\frac{1}{2}({\mathbf{x}}_l-{\mathbf{t}}_{ij}{)}^{\mathrm{T}}({\mathbf{x}}_l-{\mathbf{t}}_{ij}))}\times \prod _{m\ne l}^M{e}^{(-\frac{1}{2}({\mathbf{x}}_m-{\mathbf{d}}_j{)}^{\mathrm{T}}({\mathbf{x}}_m-{\mathrm{d}}_j))}$

(A9)

By summing the multiplied exponentials, the equation yields  

$\tilde{P}({\mathbf{x}}_1\dots {\mathbf{x}}_M|l,{\mathbf{t}}_{1\dots N,1\dots D},{\mathbf{d}}_{1\dots D})=\sum _{i=1}^N\sum _{j=1}^D{e}^{(-\frac{1}{2}({\mathbf{x}}_l-{\mathbf{t}}_{ij}{)}^{\mathrm{T}}){\mathbf{x}}_l-{\mathbf{t}}_{ij})+\sum _{m\ne l}^M-\frac{1}{2}){\mathbf{x}}_m-{\mathbf{d}}_j{)}^{\mathrm{T}}({\mathbf{x}}_m-{\mathbf{d}}_j\left)\right)},$

(A10)

which can be rewritten as  

$\tilde{P}({\mathbf{x}}_1\dots {\mathbf{x}}_M|l,{\mathbf{t}}_{1\dots N,1\dots D},{\mathbf{d}}_{1\dots D})=\sum _{i=1}^N\sum _{j=1}^D{e}^{(-\frac{1}{2}{\mathbf{x}}_l^{\mathrm{T}}{\mathbf{x}}_l+{\mathbf{t}}_{ij}^{\mathrm{T}}{\mathbf{x}}_l-\frac{1}{2}{\mathbf{t}}_{ij}^{\mathrm{T}}{\mathbf{t}}_{ij}+\sum _{m\ne l}^M-\frac{1}{2}{\mathbf{x}}_m^{\mathrm{T}}{\mathbf{x}}_m+{\mathbf{d}}_j^{\mathrm{T}}{\mathbf{x}}_m-\frac{1}{2}{\mathbf{d}}_j^{\mathrm{T}}{\mathbf{d}}_j)}.$

(A11)

Because the model will be making comparisons between these likelihoods, we are able to remove components of this equation that do not depend upon l (they are essentially constants). Removing these terms from the equation results in  

$\tilde{P}({\mathbf{x}}_1\dots {\mathbf{x}}_M|l,{\mathbf{t}}_{1\dots N,1\dots D},{\mathbf{d}}_{1\dots D})=\sum _{i=1}^N\sum _{j=1}^D{e}^{({\mathbf{t}}_{ij}^{\mathrm{T}}{\mathbf{x}}_l+\sum _{m\ne l}^M{\mathbf{d}}_j^{\mathrm{T}}{\mathbf{x}}_m)}.$

(A12)

As described above, the ideal observer chooses the alternative with the greatest value as

${\widehat{l}}_{\mathrm{I}\mathrm{O}}$

. 

Ideal-observer oddity-search model

The equation for computing the posterior probabilities for oddity search is essentially identical to that of the target-known search except that there is an extra variable (corresponding to the multiple target types that could potentially be present) that must be summated over. Again, we can substitute the multivariate normal equation into the equation and simplify it following the same steps used for the target-known model. The equation can be simplified to  

$\tilde{P}({\mathbf{x}}_1\dots {\mathbf{x}}_M|l,{\mathbf{t}}_{1\dots N,1\dots V,1\dots V},{\mathbf{d}}_{1\dots V,1\dots V})=\sum _{i=1}^N\sum _{j=1}^V\sum _{k\ne j}^V{e}^{({\mathbf{t}}_{ijk}^{\mathrm{T}}{\mathbf{x}}_l+\sum _{m\ne l}^M{\mathbf{d}}_j^{\mathrm{T}}{\mathbf{x}}_m)}.$

(A13)

This posterior probability is calculated for each of the Malternatives, and again, the model chooses the alternative with the greatest value. 

Standard-deviation oddity-search model

This model is similar to that introduced by Palmer et al. (2000) and can be referred to as a relative coding model. The stimuli are encoded relative to one another before contributing to the decision. Each element in the display elicits a scalar noisy response. The model computes the standard deviation of these responses and chooses the alternative with the greatest standard deviation as the one that contains the target. The decision variable can be expressed as 

${\widehat{l}}_{\mathrm{s}\mathrm{t}\mathrm{d}}=\underset{l}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}(\mathrm{S}\mathrm{t}\mathrm{d}\mathrm{D}\mathrm{e}\mathrm{v}({\mathbf{x}}_l\left)\right).$

(A14)

 

Acknowledgments

This research was supported by NSF-0135118. 

Special thanks to Craig Abbey for help with the Appendix. 

Commercial relationships: none. 

Corresponding author: Wade Schoonveld. 

Email: [e-mail]. 

Address: Department of Psychology, University of California, Santa Barbara, CA 93106, USA. 

Footnote

Footnotes

¹  If there are six equally spaced distributions with a distance of 1 d′ unit between adjacent distributions, then the target will most often differ from the distractor by 1 d′ unit; the maximum difference is 5 d′ units, and the average d′ will be 2.33 d′ units. If there are 32 distributions, each separated by a distance of 1 d′ unit, the target will still most often differ from the distractor by 1 d′ unit, but the maximum difference is now 31 d′ units, and the average d′ would be 11 d′ units. This would lead to higher ideal-observer performance in the 32-distribution case than in the 6-distribution case because there are more “easy” trials.