Limited Capacity for Memory Tasks with Multiple Features within a Single Object

Abstract

Memory for multiple features might be limited by the number of features, the number of objects, or both. To focus on the role of features, we tested memory for a variable number of features within a single object. Subjects studied a single ellipse that varied in four features: size, orientation, contrast, and position. We conducted two experiments that differed in how memory was tested. If performance is limited by only the number of objects to be remembered, there should be no effect of the number of relevant features within a single object. Instead, for both experiments, the proportion correct was lower when four features had to be remembered compared to one. The magnitude of these effects varied with the details of the two experiments. While similar results have been reported for experiments using multiple objects, these are some of the first experiments to demonstrate such an effect for a single object. This result is inconsistent with theories in which visual memory has a discrete limit on the number of stored objects and no limit on the stored features within an object. Instead, it seems likely that objects and features both play a role in limiting performance in memory tasks.

It has long been known that there are sharp limits on the amount of visual information that can be remembered for more than a fraction of a second (e.g. Sperling, 1960; Phillips, 1974). Luck and colleagues (Luck & Vogel, 1997; Vogel, Woodman & Luck, 2001) described conditions in which this limit was determined by the number of visual objects (about 4), with no apparent limit on the number of features within the objects. This result stands in contrast with other conditions that reveal that the precision of feature judgments declines for even two objects relative to one (Palmer, 1990; Zhang & Luck, 2008).

In light of these observations, two hypotheses can be used to frame the debate about the nature of storage in visual memory, and by implication the source of memory limitations. According to the first hypothesis—the discrete-object limit—one can store a fixed number of objects regardless of their component features (Luck & Vogel, 1997; Vogel, et al, 2001; Zhang & Luck, 2008). Memory is discrete because an object is stored along with all its features, or it is not stored at all. According to the second hypothesis—the feature-precision limit—one can store a fixed amount of total information about the collection of features regardless of their number and the objects to which they belong (Palmer, 1990; van den Berg, Shin, Chou, George & Ma, 2012; Wilken & Ma, 2004). This hypothesis is related to the more general idea of a resource limit (Alvarez & Cavanagh, 2004; Bays & Husain, 2008). By the feature-precision hypothesis, memory is not discrete because the precision of information about features declines gradually with an increase in the number of remembered features. These two hypotheses are but a start. For example, one can have a discrete slot model that represents features as well as objects or one can have a continuous precision model that is a function of object properties as well as feature properties. A variety of such hypotheses have been intensely debated (e.g. Anderson & Awh, 2012; Fougnie, Asplund & Marois, 2010, Keshvari, van den Berg & Ma, 2013; Rouder, Morey, Cowan, Zwilling, Morey & Pratte, 2008; Zhang & Luck, 2011; for general reviews see Brady, Konkle & Alvarez, 2011; and, van den Berg, Awh & Ma, 2014).

Memory for a Single Object

Here we focus on an unexplored, but potentially revealing corner of this debate. Specifically, what is the effect of remembering multiple features for a single object? Is memory limited by the number of features that can be encoded, stored, and retrieved? Or is it unlimited? The case of a single object is special because there is no constraint imposed by multiple objects. For example, it minimizes any concern that feature information is lost because it is not linked to the correct object (e.g. Wheeler & Treisman, 2002). Testing a single object, therefore, allows a more direct interpretation of any effect of the number of features.

According to the discrete-object hypothesis, performance should not be limited by the amount of to-be-remembered feature information beyond sensory limitations (e.g., acuity and crowding). In contrast according to the feature-precision hypothesis, performance should decrease with an increase in the number of relevant features. Most previous studies address this question only for multiple objects.

There is one study of which we are aware that has examined perception and memory for one versus two features of a single object (Vogels, Eeckhout & Orban, 1988, Experiment 1). A briefly presented grating was varied in both orientation and spatial frequency. After an interval of 0.25 s, a second grating appeared and subjects judged “same” or “different” on the basis of an indicated relevant feature. The relevant feature was indicated either by a precue presented 1.0 s before the first grating or a postcue presented 0.2 s after the second grating. Thus, with the precue, subjects had to remember a single feature while with the postcue, they had to remember both features. Performance was worse when one had to remember two features rather than one. This result is consistent with a feature-precision limit.

It is natural to ask if this result is due to a limit on perception or memory. Much of the Vogels et al. paper pursues conditions that minimize the role of memory. Indeed, they termed their effect one of feature uncertainty. In contrast, we concentrate on their Experiment 1 which was an explicit memory task. Our focus is on performance in explicit memory tasks that are intended to reveal memory phenomena. The question of the relative contribution of perception and memory is deferred to the discussion.

Two Models

We addressed the question of memory for multiple features in the context of two contrasting models of how processing is limited: unlimited capacity and fixed capacity. These two models serve as landmark alternatives within which to ask the question.

The unlimited-capacity model needs little introduction. By it, there should be no effect of the number of relevant features for the paradigms that probe memory for a particular feature. In contrast, fixed-capacity models predict an effect of the number of relevant features (Palmer, 1990). Fixed-capacity processing for features means that the total amount of information that is processed about the features within a given amount of time is limited. As applied to memory, it implies that one can remember a large amount of information about a single feature or remember less information about multiple features. If, for example, processing is allocated equally across the features, then for n features, one would remember 1/n of the amount of information about each feature relative to the case with only a single relevant feature. Of course, more complicated versions of fixed-capacity processing models are possible in which processing is allocated to different features in a weighted manner. Our specific fixed-capacity predictions are based on Shaw’s sample size model and assumptions of equal weighting and Gaussian distributions for the feature representations (Palmer, 1990; Shaw, 1980). Details of the model and its predictions are provided in the Appendix.

The concept of capacity is used in multiple ways. Here we are emphasizing processing capacity, which is rooted in early theories of attention (Broadbent, 1958). This is distinct from the concept of storage capacity that is at the heart of some memory models (Miller, 1956). A fixed-capacity storage model is limited by something like the number of ‘slots’ available for storage. A fixed-capacity processing model is limited by encoding, consolidation and/or retrieval processes that act on the relevant information. Such a processing model can predict effects of increasing the number of features from 1 to 2, whereas a typical storage model with a fixed number of slots, does not predict limitations until storage capacity is reached.

Experiment 1

In this first experiment, we used a procedure similar to Vogels, et al. (1988) and Palmer (1990). The stimuli were single objects that varied along four different features. A postcue indicated which feature of a previously presented object was to be reported. The number of to-be-reported features – or memory set size – was either 1 or 4. This was achieved by manipulating whether the postcue was constant within a given block of trials (memory set size 1) or varied across trials (memory set size 4). When the postcue was constant, observers could commit only the relevant feature to memory, whereas when the postcue was unknown, all four features had to be committed to memory until the postcue appeared. The discrete-object hypothesis predicts no effect of memory set size in this experiment because there is always only a single object.

Method

Subjects.

The four subjects were 20- to 40-year old adults with normal or corrected-to-normal vision. All subjects were volunteers, and some received a small payment in compensation for their time. Subject BB was an author. The others were naïve with regard to the purpose of the study.

Apparatus.

Luminance calibrated displays were presented on a video monitor with a 67 Hz refresh and 640-by-480 resolution. Viewed from a distance of 61 cm, the center-to-center spacing of pixels in the center of the screen was two arc min. Stimuli were presented against a background that appeared “white” with a luminance of 200 cd/m².

Stimuli.

The stimuli were based on an ellipse with a 2:1 aspect ratio. Four features were varied: size, orientation, contrast and horizontal position (relative to fixation). The standard stimulus had a size of 120 arc min (major axis), an orientation of 45°, a contrast of 70%, and a position at fixation. The variable stimulus varied in all four features across trials. For a given trial, each feature took on one of two possible values that either increased or decreased the value in the standard stimulus. This yielded 16 possible stimuli (see Figure 1). The frame around each ellipse is to illustrate the change in horizontal position and was not present in the experiment. The differences among the stimuli in this figure are exaggerated relative to the differences in the experiment. The particular values were adjusted for each subject based on pilot data so that discrimination performance for each feature was in the range 65 to 85% correct. For example, Subject BB used size changes of ±2 arc min, orientation changes of ±1°, contrast changes of ±5%, and position changes of ±10 arc min.

Figure 1.

A schematic illustration of the 16 possible stimuli in Experiment 1. The ellipses varied in size, orientation, contrast and horizontal position. The frame is used here to show the change in horizontal position and was not present in the experiment.

Procedure.

The procedure for a single trial is illustrated in Figure 2. An initial fixation was followed by a first stimulus for 0.1 s, a blank interval of 1 s, a second stimulus for 0.1 s, a blank interval of 1 s and a postcue until response. The postcue indicated the relevant feature with a label such as “size”. For the cued feature, subjects used a key press to indicate the relative magnitude of the second stimulus compared to the first. For example, was the size of the second ellipse larger or smaller than the first? Subjects were instructed that for orientation increasing “magnitude” corresponded to clockwise changes in orientation and for location increasing “magnitude” corresponded to leftward shifts. As illustrated in Figure 2, the two ellipses always varied in all four features. The second stimulus was always the standard. Despite the second stimulus never changing, subjects often reported basing their decision upon a comparison of the two stimuli because comparing to the second stimulus yielded better performance than comparing to a long-term memory standard. For discussions of the pros and cons of this design see Dyjas, Bausenhart and Ulrich (2012) and Thiele, Pratte and Rouder (2011).

Figure 2.

A schematic illustration of the display sequence in Experiment 1. A first stimulus display was followed by a second stimulus display and then a postcue indicating the relevant feature. For this illustration, the relevant feature was size and the correct answer was that the second display was smaller than the first. For conditions with a single relevant feature (memory set size 1), only a single feature was tested for a block of trials. For conditions with four relevant features (memory set size 4), different features were tested within a block. Thus, one had to remember only one feature in memory set size 1 and all four features in memory set size 4.

Design.

The number of relevant features (memory set size) was either 1 or 4. For memory set size 1, the to-be-reported feature was the same for a block of trials and indicated by an instruction at the beginning of the block (e.g. “size” or “orientation”). For this condition, the subject knew to remember only that one feature. For memory set size 4, the to-be-reported feature varied from trial-to-trial. Thus, the subject had to remember 4 features. The displays were drawn from an identical set of possible displays regardless of the number of relevant features. After practice, 8 sessions were run with 512 trials per session resulting in 2048 trials per set size per subject.

Results and Discussion

The results are shown in Figure 3, which plots the proportion correct against the number of relevant features (memory set size). Performance was worse when more features had to be perceived and remembered. Mean proportion correct was 0.84±0.01 with one feature and 0.69±0.01 with four features for a reliable difference of 0.14±0.01 (t(3)= 11.9, p < .001). This difference was found for all features (size, 0.15±0.02; orientation, 0.12±0.02; contrast, 0.17±0.02; position 0.14±0.03).

Figure 3.

The results of Experiment 1. The proportion correct is shown as a function of memory set size (the number of relevant features). Performance declined for the larger memory set size and the effect matched that predicted by fixed-capacity models.

Consider theories of visual working memory in which the discrete-object limit is the only limit on memory. When there is only one object, the processing of multiple features should be independent and thus have unlimited capacity. As can be seen in the Figure 3, the data from this experiment fall closer to what is predicted for fixed-capacity processing than for unlimited-capacity processing. Thus, this result is inconsistent with the discrete-object limit being the only limit on memory.

Experiment 2

In this second experiment, we pursued the same question with several changes in procedure to achieve three goals. First, in order to reduce the potential for condition-specific strategies, we mixed set size within blocks of trials, rather than having it blocked. Second, we made the experiment more similar to other recent visual memory experiments (e.g., Luck & Vogel, 1997; Vogel, et al, 2001) by increasing the duration of the study and test display, and by presenting the postcue with the test display. Third, rather than a single memory standard throughout the experiment, we used multiple standards – often called a roving standard (e.g. Bull & Cuddy, 1972; Harris, 1952). The purpose of the roving standard is to minimize the use of categorization strategies. For example in Experiment 1, one might remember the single standard and judge the study display stimuli against that standard. In Experiment 2, subjects might still remember the center of the range and encode stimuli relevant to that point but it would be less effective. In addition, one can examine performance as a function of the standard to see if stimuli at the ends of the range result in better performance than stimuli in the center of the range. To foreshadow the result, there was no sign of such categorization effects.

Method

Subjects.

Six subjects were 20- to 60-year old adults with normal or corrected-to-normal vision. All were volunteers, and some received a small payment in compensation for their time. Subject JP was an author. The others were naïve with regard to the purpose of the experiment.

Apparatus.

Luminance calibrated displays were presented on a video monitor with a 75 Hz refresh and 624-by-832 resolution. Stimuli were presented against a background that appeared “white” with a luminance of 108 cd/m².

Stimuli.

Experiment 2 used a roving standard. This change from Experiment 1 is illustrated in Figure 4 which compares this aspect of the design of Experiments 1 and 2 in Panels A and B. In Experiment 1, each feature of the study display could have two possible values while the test display always had a single standard value. This is illustrated on the left side of the figure by a graph that connects possible values of a feature in the study and text displays: red lines for increasing values and black lines for decreasing values. The possible stimulus values (e.g. size) are on the ordinate and the different displays are distinguished by when they occur in time on the abscissa. An alternative representation is shown on the right in which the possible study and test value features are shown in a array: a single test value is paired with two possible study values.

Figure 4.

The designs of Experiments 1 and 2 are shown in Panels A and B, respectively. On the left side is a graph the possible stimulus values in the study display linked to the possible values used in the following test display. On the right is the same information in a matrix format. In brief, Experiment 1 used a fixed standard for the test display while Experiment 2 used a roving standard with many possible values.

Panel B shows the design of Experiment 2. Now there are seven possible values of both study and test. The graph on the left shows how the feature value of the study is always followed by a test value that is one step “more” or “less” than the study value. Of course there is a value on the end of the range that is followed by only one possible test. These end-of-range cases are marked by dashed lines. The same design is shown in the array on the right. Now the end-of-range cases are shown by circles while the other cases are shown by x-characters. This design has 12 possible stimulus pairs which if one ignores order reduces to six possible stimulus pairs (1,2), (2,3), …, (6,7). In the analysis of standards, these six pairs are referred to as standards 1 through 6. These pairs were shown with equal probability.

As before, the stimuli were based on an ellipse with a 2:1 aspect ratio and four features were varied: size, orientation, contrast and horizontal position. To allow the use of roving standards, ellipses could have seven different values on the four features: The center of these values was a size of 120 minutes of visual angle (major axis), an orientation of 45°, a contrast of 55%, and a position at fixation. For three subjects, the steps between stimuli were 4 arc min for size, 3 for orientation, 10% for contrast, and 10 arc min for position. To roughly equate performance, the other three subjects had smaller steps.

Procedure.

Another change in this second experiment was that the number of relevant features was manipulated using a precue that varied on every trial (see Figure 5). The precue indicated the to-be-remembered feature(s) with a label. The label either indicated a single feature (e.g., “size”) or indicated that all four had to be remembered by “no cue”.

Figure 5.

A schematic illustration of the display sequence in Experiment 2. The two memory-set-size conditions are shown in the two columns. Trials began with a fixation and precue display. This was followed by a study and then a test display. The test display also contained a postcue and was present until response. For this illustration, the relevant feature was size and the correct answer was that the second display was smaller than the first.

The procedure for a single trial is illustrated in Figure 5. An initial fixation and cue display was presented for 1 s. After an interval of 1 s, the study stimulus was presented for 1 s. This was followed by a retention interval of 2 s and then a test display that remained present until response. The test display contained both a comparison stimulus and a postcue indicating the relevant feature. Observers used a key press to indicate the relative magnitude of the cured feature in the second display compared to the first display. For example, was the size of the second ellipse more or less than the first? As illustrated in the figure, the two ellipses always varied in all four features. After practice, 10 sessions were run with 192 trials per session resulting in 960 trials per set size per subject.

The details of this procedure include three more differences between this Experiment and Experiment 1. First, the study display was 1 s rather than 0.1 s. Second, the test display was presented until response rather than for 0.1 s. And third, the postcue indicating the to-be-reported feature was simultaneous with the test display rather than following it by a second. All of these changes made this experiment more similar to recent visual memory experiments, rather than to a sequential-comparison experiment on which the design of Experiment 1 was based. Specifically, in Experiment 2, the test stimulus is made more visible so that the role of perception and memory for the study stimulus is emphasized. In contrast, the design of the first experiment kept the roles of the two stimuli more symmetric.

Results

The results of Experiment 2 are shown in Figure 6. It plots proportion correct as a function of the number of to-be-remembered features (memory set size). Performance was worse when there were more relevant features. Mean proportion correct was 0.772±0.019 with one relevant feature and 0.750±0.014 with four relevant features for a reliable difference of 0.022±0.006 (t(5)= 3.6, p < .01). Considering each feature separately, this difference was present but not reliable (size, 0.01±0.01; orientation, 0.02±0.02; contrast, 0.03±0.03; position 0.04±0.02).

Figure 6.

The results of Experiment 2. The proportion correct is shown as a function of memory set size (the number of relevant features). Performance declined for the larger memory set size and the difference was reliable.

To provide context, we show two predictions for the memory set size 4 condition based on the performance observed in memory set size 1. For unlimited capacity, performance with memory set size 4 should be identical to memory set size 1. For fixed capacity, performance is predicted to be considerably worse for memory set size 4. The observed difference, while reliable, was much less than that predicted by fixed capacity. Thus, unlike Experiment 1, this result is inconsistent with a fixed-capacity model of processing multiple features.

In order to assess the possibility that observers adopted a categorical-judgment strategy as described in the introduction to this experiment, we looked at performance separately for each of the 6 standards (see Figure 7). The data are shown for each of the four different feature types in separate panels with a consistent order from “less” to “more”. This ordering is clear for size and contrast. For orientation, the order is with respect to which is more clockwise; and for position, the value is with respect to which is more leftward. For example, size has standards that are 108, 112, 116, 120, 124, and 128 arc min.

Figure 7.

The effect of the standard is illustrated for Experiment 2 with a separate panel for each feature. In each panel, the proportion correct is shown as a function of the standard’s ordinal value. For comparison, a flat line is shown to represent no effect of the standard. In general there was no effect although there is a hint of a decline for larger values of size and contrast. Of primary relevance here, there was no sign of improved performance for the standards at both ends of the range as expected from a categorization strategy.

There is no evidence of categorical judgments. There is no sign of a U-shaped function such as would be expected from a categorical judgment. There is no improvement for the standards at the end of the range. The hint of an improvement with size and contrast for larger standards is consistent with the commonly observed Weber’s Law in which sensitivity declines for larger magnitudes.

Discussion of Experiment 2

Up until now, we have focused on the issue of whether there is any effect of the number of features. In addition, it is desirable to measure the magnitude of this effect. The magnitude of the effect is much smaller in Experiment 2 than in Experiment 1. In the first experiment the magnitude of the effect was close to the that predicted by fixed-capacity models and in the second experiment it was a small fraction of that prediction. Thus, the second experiment allows one to reject the fixed-capacity model. Why the difference in Experiment 2?

Because several things were changed between experiments, we don’t know which was responsible. Perhaps the roving standard was important. Perhaps the improved viewing conditions for the test was important. However, our primary concern is that the blocked design in Experiment 1 might have allowed different strategies in the two set-size conditions. When only a single feature was relevant, observers might judge that feature of a single object without waiting for the comparison object. In contrast, when multiple features were relevant, observers might do as expected and compare the postcued feature present in the first and second objects. For Experiment 1, this strategy confounds the number of relevant objects (study vs. study and test) with the number of relevant features (1 vs. 4). This possibility undermines the feature interpretation of Experiment 1. In contrast, the mixed design combined with the roving standard makes such confounding strategies impossible for Experiment 2. It is the stronger experiment for distinguishing effects of features and objects.

In summary, Experiment 2 provides an improved test of the effect of the number of features given a single object. It’s design makes it unlikely that subjects used different strategies in the different conditions. This is our best evidence for an effect of the number of features.

General Discussion

We found that varying the number of to-be-remembered features from a single object affected performance on a memory task. The key innovation of this study is that memory for multiple features was measured within a single object, rather than memory for features across multiple objects (see also Vogels et al., 1988). The special case of a single object is important because it eliminates any possibility of effects being mediated by the presence of multiple objects. For example, there is no opportunity for effects to arise from interference between multiple objects (e.g. McConnell & Quinn, 2000) or from confusion regarding which features go with which objects (e.g. Wheeler & Treisman, 2002). In general, using a single object is a way to study the effects of having to remember multiple features with no possibility of confounding effects due to multiple objects. That the number of to-be-remembered features affected memory performance with a single object indicates that feature memory does not have unlimited capacity and thus a discrete-object limit on visual memory cannot be the only limitation in visual memory. In addition, the relatively small effects found in Experiment 2 allow one to reject the fixed-capacity model as well. The effect of the number of features falls between the two extreme models.

The following discussion has four parts. The first part relates our study of a single object to the larger literature on multiple objects. The next two parts address the theoretical interpretation of the effect of the number of relevant features. And the fourth part discusses the possible reasons for the differences in the magnitude of effects found in Experiments 1 and 2.

The Effect of the Number of Features with Multiple Objects

An influential study on visual working memory (Luck and Vogel, 1997) came to a contrasting conclusion that visual working memory is not limited by the number of to-be-remembered features, but by only the number of objects in which those features are instantiated. In those experiments, subjects were presented with brief study displays of 2 to 6 colored lines followed 0.9 s later by a test display that was either unchanged or different for one object. Subjects responded either “same” or “different” to the test display. In different blocks of trials, subjects were instructed to detect changes in the color, in the orientation, or in both. Performance decreased as the number of objects increased, but there was little if any effect of the number of relevant features on performance. This result with multiple objects is in apparent conflict with the current study.

Later studies, however, have indicated that features do play a role (Cowan, Blume & Saults, 2013; Davis & Holmes, 2005; Olsen & Jiang, 2002; Marshall & Bays 2013; Wheeler & Treisman, 2002; and Xu, 2002). A particularly simple demonstration of an effect of the number of relevant features was reported in Fougnie et al. (2010). They presented three stimuli that varied in color and orientation and probed the memory of one feature of one object. Subjects had to report the relevant feature using the adjustment method of Wilken and Ma (2004; see also Zhang & Luck, 2008, 2011) which allows an estimate of both the precision of memories and the number of items remembered. In different blocks of trials, subjects had to remember the color, the orientation, or both. Increasing the number of relevant features resulted in a decrease in precision but no effect on the number of remembered objects. Fougnie and colleagues (Fougnie et al., 2010; see also Fougnie, Cormiea & Alvarez, 2013) suggested a hybrid model in which visual memory is limited by both the number of to-be-remembered objects and, separately, by the number of to-be-remembered features. The number of objects influences both storage capacity and precision while the number of features influences only precision. The current results are also consistent with such a hybrid model in that we found that processing was limited by the number of to-be-remembered features. Our single-object experiments do not speak to whether or not there is a further limit imposed by the number of to-be-remembered objects.

The Nature of the Capacity Limits: Storage capacity versus processing capacity

The capacity limitations observed in the current study might concern processing capacity in the sense of the amount of information processed per display (Broadbent, 1958), rather than storage capacity defined by the number of objects per display (Miller, 1956). Discussions regarding capacity limitations of visual memory have often focused on limitations in storage of visual information. This is perhaps a natural focus when thinking about memory. But, it is important to also consider possible processing limitations on encoding, maintenance and retrieval of visual information (see Cowan & Morey, 2007; Fougnie & Marois, 2009).

For an intuition regarding a precision limit due to limited processing capacity, consider Shaw’s (1980) sample size model, which was the basis for our fixed-capacity predictions. According to this model, one can make n samples of a stimulus representation per unit time. For a single relevant feature, these samples can all be on one feature. For four relevant features, the samples must be divided which results in a less precise estimate for the individual features. For the case of evenly divided samples, the variance of the estimate is inversely related to the number of samples and thus the standard deviation doubles when the memory set size increases by a factor of 4. Thus the capacity limits observed in this study could reflect a processing capacity limitation like that captured in this model and would manifest as a limit on the precision of the representation of the visual information. This sampling intuition is a start, but what it does not speak to what specific aspect of processing is limited. Is it encoding, maintenance, retrieval, or all of the above?

A specific processing-capacity hypothesis has been suggested for the effect of multiple features by Bayes, et al. (2011). They suggest a combination of a memory storage limit for objects and a memory encoding limit for features within an object. To test this idea, they varied both stimulus duration and the number of features and objects and found results consistent with this two component hypothesis. Recently this idea has been elaborated by Sewell, Lilburn and Smith (2014).

A discrete-object limit can sometimes mimic a precision limit. Zhang and Luck (2008) suggested that the effects on precision that they inferred through a cue estimation procedure could be accounted for under a model that assumed a discrete-object limit on storage. Specifically, they proposed a model in which when memory set size is below the storage capacity limit, multiple copies of some objects can be stored, resulting in a measured improvement in precision. This solution does not work for the current study because multiple copies of the (single) study object would improve the precision of all the features of the object. There would be no advantage for specifying a single feature as relevant rather than all four features.

The tendency to focus on storage limitations when considering memory might be a reason why the special case of a single object has been neglected. If one is focused on only possible limitations of memory storage, then it would be unlikely to consider a single-object case because it would be unlikely to reveal the limits of the system.

Memory or Perception?

The study-test paradigm used here is intended to measure visual memory, but might the observed effect of set size be due to perception? This is a difficult question and few studies have directly addressed it (e.g. Bays, Gorgoraptis, Wee, Marshall & Husain, 2011; Fougnie & Marois, 2009; Mazyar, van den Berg, & Ma, 2012; and Sewell et al., 2014). Consider two opposing arguments. An argument for a perceptual account comes from the experiment of Mazyar et al. (2012) that found similar effects of the number of objects/features using matched search and memory paradigms (but see McLean, 1999). Mazyar and colleagues interpret this result as consistent with perceptual limits and no memory limits.

An argument against the perceptual account are experiments that show no effect of the number of simultaneous objects/features using search tasks (e.g. Huang & Pashler, 2005; Scharff, Palmer & Moore, 2011). These authors interpret their results as showing there are no perceptual limits for these kinds of simple stimuli and tasks and thus memory must be the limit in study-test paradigms. Critically relevant to memory, this simultaneous-sequential procedure has been applied to the effect of multiple objects on memory performance by Sewell et al. (2014; see also Mance, Becker & Liu, 2012; Liu & Becker, 2013). Sewell and colleagues found equal performance for simultaneous and sequential displays consistent with no limit on perception despite an effect of the total number of items to be remembered. The next step is to use these methods to study the effect of the number of features with a single object. Sorting out this issue is important because one way to “save” a simple memory hypothesis is to attribute to perception any exception to the memory hypothesis.

Experiment 1 compared to Experiment 2

Experiments 1 and 2 yielded effects that differed markedly in magnitude. The effect in Experiment 1 was as large as that predicted by fixed-capacity models, whereas the effect in Experiment 2, though inconsistent with unlimited-capacity models, was much less than predicted by fixed-capacity models. There were several procedural differences between the two experiments, and therefore we cannot say with certainty what is the source (or sources) of the difference. We offered some speculations in the discussion of Experiment 2.

A broader view is that whatever the specific cause for the difference in effect size between the two experiments, it is an example of how the details of the procedure can change the effect of the number of relevant features. The task can change, for example, how the number of decisions varies with the number of features (see Busey & Palmer, 2008 for an example of this in visual search). The task can simplify or complicate the retrieval process (see Anderson & Bower, 1972; Kintsch, 1970 for examples of this in verbal memory). The experiments reported here, for example, used a cued recognition task in which subjects had to judge a particular cued aspect of the study display. Specifically, one of four features was cued. In prior studies, an entire object (or set of objects) was cued (e.g., Palmer, 1990). This cued recognition task was chosen because it minimizes potential for contributions from decision and retrieval processes to the magnitude of the observed memory-set-size effects. The more common task of probe recognition, in which subjects judge whether a given stimulus was present anywhere in the memorized display (e.g. Mazyar, et al., 2012) probably presents a more difficult retrieval and decision problem because all of the elements of the study display are relevant to the response. But that task, in turn, presents a simpler decision and retrieval problem compared to other change detection (e.g., Keshvari, et al., 2013; Luck & Vogel, 1997; Scott-Brown & Orbach, 1998). Change detection provides present the subject with many decisions since all of the elements of the study and test display are relevant. It remains to be worked out how the specific task influences the effect of the number of to-be-remembered features.

Conclusions

We measured performance in a memory task as a function of the number of relevant features in a single object. We found that increasing the number of relevant features decreased performance. This effect indicates that feature perception and memory does not have unlimited capacity and therefore rejects theories that assume only a discrete-object limit on performance. The current findings are consistent with hybrid models such as that offered by Fougnie et al. (2010) and Bays et al. (2011).

To address these questions further, we suggest that there are two special cases of visual working memory that deserve particular study. The familiar case is: How many objects can one remember when the visual discriminations are relatively coarse? This case has been studied extensively. The other case is the one considered here: For a single object, how does the precision of feature memory depend on the number of relevant features? This case has received little study and has the potential to demonstrate limits on feature memory that have nothing to do with objects.

Appendix

In this appendix, we describe in detail the unlimited- and fixed-capacity models that are used in this article. They are based on the sample size model (Shaw, 1980) and have been discussed in Palmer (1990) with the key derivation given in the “perception hypothesis” of the appendix of Palmer, Ames and Lindsey (1993).

We use similar notation to that in signal detection theory (Green & Swets, 1996) and in related articles (e.g. Busey & Palmer, 2008). In these theories, the judgment is based on a representation of a feature that corresponds to a single random variable, U. For the case of n multiple features, there are corresponding random variables U_i, i = 1, …, n. We assume that these random variables are independent of one another and that they are identically distributed with a density function f_i, and a cumulative distribution function F_i. In addition, we assume features that vary in their value on a single dimension correspond to members of a shift family of the distribution function (i.e. f(x-a) where a is the shift in terms of the random variable). For our experiment with a standard and comparison values of a given feature, denote the distribution of the standard value as f(x) and the comparison value by f(x-wδ) where the amount of shift is given by wδ with δ the physical description of the feature and w the free parameter relating physical units to the signal/noise units of the representation. Finally, for numerical calculation of specific predictions, we make the further assumption that the random variables are Gaussian.

Consider the case in which a single stimulus of n features is presented but only one of the features is relevant to the task (memory set size 1). We will focus on the design of Experiment 1 in which each feature in the study display could take on either an increment δ₊ or a decrement δ₋ relative to the value in the study display. The subject’s task is the indicate whether the relevant feature in the second display was larger or smaller than the first display which is a variation on a two-interval, forced-choice task. Lets indicate the first and second displays by j = 1, 2 and the n features by i = 1, …, n, so that the random variables are U_ij. Given the feature had a value of δ in the first display and the standard value denoted by 0 in the second display, the corresponding two random variables are U_i1(wδ) and U_i2(0). By this theory, the judgment is determined by which of these two random variables has the larger value on a given trial. The probability of judging the second display as “larger” than the first is thus

$$P\left("\text{larger}"|\delta \right)=P\left[U_{i2}(0)>U_{i1}(w\delta )\right].$$ (1)

The probability correct for an increment is

$$P(\text{correct})=1-P\left("\text{larger}"|{\delta }_{+}\right).$$ (2)

Because the random variables are independent, one can solve this equation by taking the integral of the product of the density and cumulative of the two random variables

$$P(\text{correct})=1-{\int }_{-\infty }^{\infty }f(x)F(x-w\delta )dx.$$ (3)

The probability correct for a decrement is

$$P(\text{correct})=P\left("\text{larger}"|{\delta }_{-}\right).$$ (4)

Given independence, this equation can be similarly solved as

$$P(\text{correct})={\int }_{-\infty }^{\infty }f(x)F(x+w\delta )dx.$$ (5)

Since these cases occur with equal probability, the combined probability correct is

$$P(\text{correct})=0.5[1-P\left("\text{larger}"|{\delta }_{+}\right)]+0.5[P\left("\text{larger}"|{\delta }_{-}\right)].$$ (6)

We can rewrite this equation using Equations (3) and (5):

$$P(\text{correct})=0.5\left[1-{\int }_{-\infty }^{\infty }f(x)F(x-w\delta )dx\right]+0.5\left[{\int }_{-\infty }^{\infty }f(x)F(x+w\delta )dx\right].$$ (7)

From this result, we can calculate performance for any given difference in features δ.

Now consider the case in which one must perceive and remember multiple features. The key assumption of all postcue experiments is that the postcue allows one to retrieve just the relevant feature and to retrieve it independently of other features in memory. Thus, one can make a single decision based on the retrieved single value rather than multiple decisions for multiple memories as in probe recognition or change detection. Given unlimited capacity in both perception and memory, performance for multiple features is the same as for a single feature.

For the fixed-capacity model, we instead assume that a constant amount of information is remembered from the set of features (see Lindsay, Taylor & Forbes, 1968 for a discussion of information theory, capacity limits and signal detection). The concept of fixed capacity is perhaps more easy to understand in the context of Shaw’s sample size model. For this model, when only a single feature is relevant, all of the available samples can be directed toward that single feature. When multiple features are relevant, the samples are divided among the features. Given, equal sampling of n features, the variance of the random variable for a given feature is inversely proportional to the number of samples. Equivalently, the variance as a function of the number of relevant features U(n) is proportional to the number of features:

$$Var[U(n)]=nVar[U(1)]$$ (8)

(see Palmer, et al., 1993 for the derivation).

Given the use of standardized distributions such as f(x), this change in the variance can be described as a scaling of the x values by the standard deviation of the distribution. Specifically, let f(x) describe the representation when all samples being directed at a single feature, when the samples are equally distributed over n features the representation becomes $f(x/\sqrt{n})$. Thus the key prediction of Equation (7) becomes the following for n relevant features:

$$P(\text{correct})=0.5\left[1-{\int }_{-\infty }^{\infty }f\left(\frac{x}{\sqrt{n}}\right)F\left(\frac{x-w\delta }{\sqrt{n}}\right)dx\right]+0.5\left[{\int }_{-\infty }^{\infty }f\left(\frac{x}{\sqrt{n}}\right)F\left(\frac{x+w\delta }{\sqrt{n}}\right)dx\right].$$ (9)

To provide a numerical example, assume Gaussian distributions and wδ = 1, then the calculated probability correct for set size 1 is 0.760 and for set size 4 is 0.638. The complete set of predictions is shown in Figure 8. It plots the proportion correct for set size 4 against the proportion correct for set size 1. The positive diagonal denotes equal performance in the two conditions as predicted by an unlimited-capacity model. The dashed curve denotes the predictions for the fixed-capacity model. This curve is generated by varying wδ from 0 to 5 and calculating the probability correct from Equation (9) for the two set sizes. Also shown are the results of the two experiments.

Figure 8.

This parametric plot shows the full range of predictions for the unlimited- and fixed-capacity models. It plots the proportion correct for set size 4 against the proportion correct for set size 1. The positive diagonal shows equal performance as predicted by the unlimited-capacity model. The dashed curve shows the set-size effect predicted by the fixed-capacity model. Different points along the curve correspond to tasks of different difficulty. Also shown are the results of the two experiments.