Modelling Working Memory Capacity: Is the Magical Number Four, Seven, or Does it Depend on What You Are Counting?

The CSVI and the Bose-Einstein distribution

The original version of the CSVI (Pascual-Leone, 1970) was designed for children of different ages and did not use computer technology; the stimuli were presented on cards and participants responded with different motor actions. More recently, however, computerized versions of the CSVI have been developed (Hitzig, 2008; see also Morra, 2015) and used with adults. In this task, a participant views stimuli with a variable number of features and must respond to each of the relevant features that are present in a stimulus.

Consistent with the neo-Piagetian assumption that schemes are the units of cognition, the CSVI involves a training phase in which participants learn a set of nine stimulus-response pairs, the stimulus being a single visual feature (e.g. red, square, etc.), and the response an action defined as correct for that feature (i.e., in the computerized version, presses of different keys on a special response box). In this training phase each figure has only one relevant feature, and training continues until a participant shows errorless performance over a long sequence of trials. When a participant has learned thoroughly those nine “schemes” or s-r pairs, testing starts with compound stimuli (i.e., figures with 2 to 8 relevant features); the participant is not informed of how many relevant features are present in any compound stimulus and must respond to all of the features he/she detected. Clearly, the number of correct responses to a compound stimulus is a random variable that can take all integer values up to the number of features present in that figure.

In order to relate CSVI performance to attentional capacity, Pascual-Leone made a number of assumptions. The first is that a person’s attentional energy has a limited capacity k (k being an integer), which can simultaneously activate no more than k schemes. The second assumption is that, if plenty of time is available for looking at the stimulus and responding, participants will attend repeatedly to the stimulus (having k units of capacity available on each attending act). More specifically, it was assumed that after each attending act a participant evaluates whether he/she observed the stimulus well enough, and after k attending acts the participant has a feeling of attentional saturation and stops exploring the figure. Thus, with long stimulus presentation and unlimited response time, a participant will attend to each compound stimulus k times, each time with a capacity of activating up to k schemes, for a total of k² units of capacity.

The third assumption made by Pascual-Leone is that the probability distribution of the number x of correct responses to items with n relevant features is a Bose-Einstein distribution. This is widely used in physics to model the distribution of particles (bosons) across different states; it represents the probability that a variable number x of states are occupied, under the assumptions that a set of r undistinguishable particles is randomly distributed over a set of n distinguishable states, and that more particles can occupy the same state. Phenomena studied in different fields of science have been modelled with this distribution (Feller, 1968; see also Ijiri & Simon, 1975; Tuckwell & Koziol, 1987). Its probability mass function is:

and, as a more intuitive metaphor, one can think of r undistinguishable balls thrown to a set of n distinguishable boxes, with the random variable x representing the number of boxes that turn out to be occupied by at least one ball. Referring to the CSVI, n is the number of (distinct) features in a compound stimulus, r is the number of (undistinguishable) units of attentional capacity allocated to the stimulus, which takes the value of k² for the reasons given above, and x is the random variable that expresses the number of features detected and responded to. A detailed task analysis of the CSVI task was presented by Pascual-Leone (1970). Here, suffice it to say that it seems reasonable to assume that the units of attention allocated by a participant to a certain stimulus are not individually distinguishable, because the participant’s response is informative on whether a certain feature was detected or not, but it gives no clues to whether only one or more units of attention were allocated to a certain feature, and we cannot label each unit of attention and list the particular unit(s) that were allocated to each detected feature. Thus, we can regard the Bose-Einstein distribution as a conceptually plausible model for the CSVI task.

One could ask why assuming that participants perform the task this way, instead of following a straightforward strategy of scanning mentally a list of all possible features and responding to those that are present in a stimulus. Our answer is that to discover and use this strategy one needs in the first place a very large working memory capacity, able to hold the representations of all possible features. Very few adults seem to use this efficient strategy. This could cause some error variance in estimating with the Bose-Einstein distribution their capacity; however, setting a ceiling of k = 9 (i.e., the number of potentially present features) in individual participants’ parameter estimation can keep this source of error to a minimum.

Extensive research (Globerson, 1983; Johnson, Fabian & Pascual-Leone, 1989; Johnson, Im-Bolter & Pascual-Leone, 2003; Miller, Pascual-Leone, Campbell & Juckes, 1989; Morra, 2015; Pascual-Leone, 1970, 1978; Pascual-Leone & Johnson, 2005; Pascual-Leone & Sparkman, 1980; de Ribaupierre et al., 1990) provided evidence supporting the Bose-Einstein model for the CSVI, usually with long stimulus presentation (5 sec), assuming k² capacity units available for each stimulus, and found values of k increasing with age during childhood and adolescence as specified in the TCO.

A possible problem, however, is that the stimulus presentation time could affect capacity estimation in unknown ways, for instance because long presentation might elicit using more refined encoding or maintenance strategies (Cowan, 2001b). Consequently, it seems necessary to clarify whether and in which ways presentation time affects CSVI performance, and whether capacity estimates from the CSVI are robust to manipulation of presentation time.

Following Pascual-Leone’s line of reasoning, we assume that presenting the compound stimulus for a short time, such that saccades or repeated looks are not possible, should reduce the number of correct responses, because of reduced opportunity to attend to the stimulus and thus detect its relevant features. More specifically, we assume that with brief presentation the participant cannot attend to the stimulus k times, but only fewer times (according to the presentation conditions). Consequently, we predict that – although the total number of correct responses will be larger with long than brief presentation – the Bose-Einstein distribution will account for performance in both conditions, only changing the number r of “balls” from k² with long presentation to 2k, k, and 3k, respectively with the brief presentation conditions of three different experiments. If the CSVI provides a valid estimate of attentional capacity, then the estimated value of k should be similar with both brief and long presentation. These predictions are tested in Experiments 1a–1c reported below.

The VAT and its different methods for capacity estimation

On a typical VAT trial the participant is briefly presented with an array of colored squares, followed by a blank screen and then a test array, which either is identical to the first (no-change trials), or has one square in a different color (change trials). One of the squares in the test array is probed (in change trials the probed square is the one that changed color) and the participant must decide whether that square’s color changed or not. The set size (i.e., the number of squares in an array) varies across trials from few squares to a number that is far beyond capacity. Cowan (2001b) proposed a simple formula to estimate capacity from this task; assuming that a participant has the capacity to keep activated the representations of k squares during the retention interval, and the set size N is larger than k, then the participant will respond correctly if the probed square is one of the k squares maintained in working memory, and guess the correct response with probability .5 in case the probed square is one of the other N – k. It follows that, for N > k, the formula

is a valid estimate of participants’ capacity (Cowan, 2001b; see also Rouder et al., 2011). In this formula, N is the set size, p(H) is the probability of a hit, i.e., detecting a change when it takes place, and p(CR) is the probability of a correct rejection, i.e., recognizing that the probed square was actually unchanged. This estimate of k must be calculated separately for each set size, and has the property of being fairly constant across set sizes, provided that N > k.

Studies with young adult participants often found average estimates of k between 3 and 4 (see Cowan, 2001a, 2001b; Rouder et al, 2011) or even lower (Xu et al., 2018). Higher k values, however, were also reported sometimes; namely, both Awh et al. (2007, exp. 1) and Saults and Cowan (2007, single-task condition of exp. 1) found mean k values of 4.9, Morey and Cowan (2005, single-task condition of exp. 1) found a mean k of 5.14, and Cowan et al. (2005) found mean k values of 5.67 in exp. 1 and 5.77 in exp. 2.

Cowan’s (2001b) formula proved to be a simple and useful way to estimate the number of items held in working memory in the VAT, but some problematic assumptions of this formula have also been noted (Morey, 2011; Morey & Morey, 2011; Rouder et al, 2011). First, in case hits are fewer than false alarms, the formula yields negative values of k; although this case is unlikely in most circumstances, in certain experimental designs (e.g., dual tasks) it can turn into a real problem. Second, when the set size N < k, error-free performance is assumed but, actually, occasional errors do occur; moreover, valid estimates of k can only be obtained when N is greater than the true value of k, but sometimes it may be difficult to decide which threshold of N should be used. Third, there could be individual differences in guessing whether a change occurred or not. Fourth, the assumption that participants always perform at their full capacity, with no lapses of attention, seems unrealistic (see also Rouder et al., 2008).

Because of these potential problems, Morey (2011; Morey & Morey, 2011) proposed an alternative method to estimate capacity from the VAT. This is based on three parameters: the size k of attentional capacity, the probability z that a participant attends to a stimulus, and the probability g that, in the lack of information about the probed item, a participant guesses that a change occurred. It is assumed that, in case the participant attends to an array of N items, k of these items will be held in working memory (or all of them will be held in working memory in case N ≤ k), and if the probed item is one of these k, a correct response will be produced; but in case the stimulus was unattended, or in case the stimulus was attended but the probed item is one of the N – k that exceed capacity, then the participant will guess. The three parameters k, z, and g can be estimated for the sample means, for individual participants, and for specific experimental conditions if relevant to the research design. Differently from Cowan’s (2001b) formula, which must be computed separately for each set size, with this method the parameter estimates are computed across set sizes. For details of the underlying mathematical model, see Morey (2011). The estimation process can be carried out with an open-source software, called Working Memory Modelling using Bayesian Analysis Techniques (WoMMBAT), described by Morey and Morey (2011).

We agree that WoMMBAT has several advantages with respect to more traditional methods, and therefore we use this method in Experiments 2 and 3 reported below. However, due to the more widespread use of Cowan’s (2001b) formula in the literature, we also report estimates obtained with that formula for the sake of comparison.

A basic assumption of all methods for estimating working memory capacity from the VAT is that grouping, chunking, recoding, rote or elaborative rehearsal, or other space-saving strategies are not used by the participants. Early research (Luck & Vogel, 1997) ruled out the hypothesis of verbal coding, and also found that varying presentation time of the memory array from 100 to 500 msec had no effect on performance, thus excluding that extending presentation up to half a second enables participants to use more efficient strategies. However, in view of Cowan’s (2001b) argument on a possible role of long presentation time in artifactual overestimation of capacity, we still need investigating whether a longer presentation time (such as the 5 sec presentation typically used for the CSVI) yields an inflated estimate of k in the VAT.

The current study

This study is aimed at comparing and possibly integrating Cowan’s and Pascual-Leone’s views on limited capacity, by focusing on two tasks (the CSVI and the VAT) often used by either research group and considering some methodological and theoretical issues that emerged as potentially relevant in the controversies on capacity limits. In particular, we consider presentation time and attention to operative schemes as potential explanations of the discrepancy between Cowan’s and Pascual-Leone’s proposed estimates.

Our first goal is to determine whether the CSVI and the VAT tap, at least to some extent, the same capacity limitation. The correlation between capacity measures obtained from these two tasks has never been examined, because each task was used so far in separate lines of research. Our experiments 2 and 3 provide the relevant results.

Another goal of this study is examining the effect of presentation time on both tasks and its possible role in causing different capacity estimates from them. In agreement with Cowan (2001b), we expect that a 5-sec presentation time in the VAT enables participants to recode or chunk the stimulus items and use strategies, which would yield (artifactually) higher capacity estimates. We think that this prediction does not necessarily hold also for the CSVI, however. For the reasons explained above, we hypothesize that a drastic reduction of presentation time in the CSVI may not alter the capacity estimates obtained from it, provided that one sets to theoretically justified multiples of k (instead of k²) the r parameter in the Bose-Einstein distribution. This specific hypothesis is tested in Experiments 1a–1c.

The third goal of this study is examining the difference between the mean capacity estimates obtained from the CSVI and the VAT. The debate, so far, concerned results obtained in separate studies with different samples of participants. Although it is difficult to test directly the hypothesis that the VAT yields a lower capacity estimate because part of the capacity has to be allocated to operative schemes (i.e., procedural knowledge), determining the difference between the k estimates obtained from the two tasks can provide indirect evidence that could be used to shape more precisely an explicit model of the operative schemes that may be needed in the VAT. In this regard, our reasoning can only be indirect, because it seems difficult to make a direct comparison by devising a measure of the CSVI based on Cowan’s theory, or a VAT measure based on Pascual-Leone’s theory. To the best of our knowledge, in the literature there is no available model or task analysis of the CSVI based on Cowan’s theory, and regarding the VAT we have only Pascual-Leone’s (2001) generic remark that also operative schemes could be involved. However, we think that “translating” from one theory to the other is possible, because chunks of declarative knowledge in classical cognitive theories (including Cowan’s) correspond to Piagetian (and neo-Piagetian) figurative schemes, and Piagetian operative schemes correspond to units of procedural knowledge, although the latter are not regarded as part of the working memory content in Cowan’s model. Thus, we take an exploratory approach, comparing the established capacity estimates obtained from the “prototypical task” of each theoretical model, which have never administered so far to the same sample of participants. The results can provide information about the actual quantitative difference between these measures, and with this, at least a clue to which processes may account for that difference. Experiments 1a–1c test the impact of presentation time manipulation on capacity estimation in the CSVI. Experiment 2 turns to a direct comparison between CSVI and VAT, and in addition, tests the impact of presentation time manipulation on capacity estimation in the VAT. Experiment 3 replicates experiment 2 with some modifications for additional control.

Method

Materials and procedure: Experiment 1a

The CSVI requires responding to multiple features of a visual stimulus by pressing different keys on a special response box. The stimuli were presented on a 15-inch CRT monitor with a refresh rate of 75 ms; the participant was comfortably sitting at a viewing distance of approximately 70 cm. The relevant features were square shape, red color, large size, dashed contour, presence of a frame around the figure, presence of an X in the centre, presence of an O in the centre, presence of a bar under the figure, and purple background. The response box had 12 keys, clearly distinguishable by shape and color. Nine keys were associated each to one of the relevant features, two keys were dummy fillers, and a larger red key was an “enter” key to be pressed by the participant to signal that s/he had finished responding to a trial.

The training stimuli were 72 figures, eight of which were used to train participants on each of the nine relevant features; in each set of eight figures, the intended feature was present in four of them and absent in the other four. For each feature, one at a time, the experimenter told the participant which key was associated to it and required the participant to respond to the eight figures in the set.

The practice stimuli were 51 figures, 50 of which were divided in five blocks of 10; in each block, nine figures presented a single relevant feature (requiring the participant to press a single key followed by “enter”) and 1 presented no relevant feature (requiring to press only the “enter” key). The computer provided accuracy feedback after response to each practice stimulus. In block #1 the stimulus remained on the screen until response. Block #1 was presented again in case the participant did not produce at least 80% correct responses. In the following blocks, in case the participant had not responded within 5 s, the screen turned white but the participant was allowed to respond to the stimulus also after it disappeared. Each of blocks #2, #3, and #4 was presented again in case the participant did not produce all 10 correct responses, until perfect performance on each of these blocks. Block #5 was presented only to the participants who made one or more errors on any of the previous three blocks and was presented again in case the participant did not produce all 10 correct responses. This procedure ensured that all participants had learned thoroughly which key was associated to each feature. Finally, there was one more practice stimulus with 3 relevant features; the participant was required to press the keys associated to all relevant features of this stimulus, followed by the “enter” key. No feedback was given on this final practice stimulus.

There were 56 test stimuli, i.e., eight trials for each level from 2 to 8, where a “level” is defined as the number of relevant features present in a stimulus. Figure 1 presents an example of a level-7 stimulus. Presentation time in the long condition was five seconds as in Pascual-Leone (1970), and in the short condition 120 msec, which prevents saccades during presentation (Spotorno, Tatler & Faure, 2013). We assume that 120 msec presentation affords only two attending acts (one while the stimulus is present and one while the iconic memory is still fully vivid).

Figure 1

The test stimuli were arranged in a pseudo-random fixed sequence (the same for all participants), divided into four blocks, each of which included two trials of each level. Half participants viewed each stimulus in the first and third block of for 5 s, and those in the second and fourth block for 120 ms; the presentation time of the blocks was reversed for the other half participants.¹ A short pause was allowed after each block.

The experiment was programmed in e-prime (Schneider, Eschman & Zuccolotto, 2002). The session lasted approximately 50 minutes.

Materials and procedure: Experiments 1b and 1c

In experiment 1b everything was identical to experiment 1a, except that in the short condition each stimulus was presented for 80 msec and followed by a mask for 500 msec. We assume that this presentation affords only one attending act. To prevent habituation to the mask, we created six different complex masks and presented one at random after each stimulus in the short condition; Figure 2 presents one of the masks.

Figure 2

In experiment 1c everything was identical to experiment 1b, except that in the short condition each stimulus was presented three times in a row,² always for 80 msec and followed each time by a different mask for 500 msec. We assume that this presentation affords three attending acts; the participant’s responses were pooled over the three presentations.

Results and discussion

Combined experiments 1a–1c

In a 3 × 2 mixed ANOVA of the k scores, with experiments as the between factor and presentation time (short vs long) as within factor, neither effect was significant, F(2, 57) = 0.50, p > 0.61, η_p² = 0.02 for experiment, F(1, 57) = 2.84, p = 0.097, η_p² = 0.047 for presentation time, and F(2, 57) = 0.02, p > .98, η_p² = .001 for the interaction (see Figure 3). A Bayesian ANOVA with the same design, carried out with the JASP default priors, confirmed that the null model was the best fitting one, BF_M = 3.14. The overall means were 6.55 (s.d. = 1.58) and 6.90 (s.d. = 2.18) with long and short presentation, respectively. A Bayesian paired-t test, with one-tailed H₁ long > short and the prior for H₁ set as a Cauchy distribution with scale = 0.316, supported the null hypothesis, BF₀₁ = 8.38. It can also be noted that all confidence intervals overlapped with one another. These results can be regarded as further confirmation of the robustness of the CSVI k scores, invariant through experiments and presentation times (provided that adequate assumptions are made regarding the attending acts occurring in each experimental condition).

Figure 3

It can be concluded that the mean capacity estimates of about six or seven units are not artifacts due to long presentation or to any peculiarity of the mathematical model, such as the assumption that in the long condition the participant attends k times to a figure.

Table 1 reports the mean number of correct responses per trial on each level of the CSVI in the long and short conditions of each experiment. It also reports the corresponding means expected from the Bose-Einstein model, under the simplifying assumption that all participants had a capacity of k = 7. Obviously, the means of correct responses increased with increasing levels because of the larger number of features one could respond to, and they were higher in the long than in the short condition. The interesting observation is that the ratio of correct responses in the short to the long condition tended to decrease with increasing levels. This decrease was quite dramatic in experiment 1b (where only one attending act was posited in the short condition), moderate in experiment 1a (where two attending acts were posited), and minimal in experiment 1c (where three attending acts were posited). This pattern of decreasing short-to-long observed ratios mirrored well the expected pattern predicted by the Bose-Einstein model. The correlation between the vectors of observed and predicted ratios was r(5) = 0.89, p < .01 in experiment 1a, r(5) = 0.98, p < .001 in experiment 1b, and r(5) = 0.84, p < .01 in experiment 1c; Bayesian analyses with the JASP defaults yielded BF₁₀ = 16.52, BF₁₀ = 433.56, and BF₁₀ = 5.64, respectively in each experiment. Considering all three experiments jointly, the correlation was r(19) = 0.94, p < .001, BF₁₀ = 3.82 × 10⁷. These results, consistent with the theoretical predictions, lend further support to the Bose-Einstein model.

Table 1

Observed and expected means of correct responses per trial, by level for short and long presentation.

	LEVEL	OBS. LONG	OBS. SHORT	OBS.SH/L RATIO	EXP. LONG	EXP. SHORT	EXP. SH/L RATIO
(a) Experiment 1a
	2	1.99	1.89	0.95	1.96	1.87	0.95
	3	2.86	2.78	0.97	2.88	2.69	0.91
	4	3.88	3.46	0.89	3.77	3.29	0.87
	5	4.53	3.84	0.85	4.62	3.89	0.84
	6	5.39	4.44	0.82	5.44	4.42	0.81
	7	6.08	5.09	0.84	6.24	4.90	0.79
	8	6.73	5.30	0.79	6.47	4.18	0.65
(b) Experiment 1b
	2	2.00	1.64	0.82	1.96	1.75	0.89
	3	2.93	2.31	0.79	2.88	2.33	0.81
	4	3.79	2.71	0.72	3.77	2.80	0.74
	5	4.59	3.09	0.67	4.62	3.18	0.69
	6	5.31	3.48	0.65	5.44	3.50	0.64
	7	6.11	3.91	0.64	6.24	3.77	0.60
	8	6.63	4.14	0.62	6.47	3.69	0.57
(c) Experiment 1c
	2	1.96	1.89	0.96	1.96	1.91	0.97
	3	2.88	2.66	0.93	2.88	2.74	0.95
	4	3.71	3.44	0.93	3.77	3.50	0.93
	5	4.55	3.96	0.87	4.62	4.20	0.91
	6	5.20	4.71	0.91	5.44	4.85	0.89
	7	5.99	5.25	0.88	6.24	5.44	0.87
	8	6.54	5.88	0.90	6.47	6.00	0.93

[i] Note: Obs. Long = Observed mean in the long condition. Obs. Short = Observed mean in the short condition. Obs. Sh/L Ratio = Ratio between observed means in the short and long conditions. Exp. Long = Expected mean from the B-E model (with k = 7) in the long condition. Exp. Short = Expected mean from the B-E model (with k = 7) in the short condition. Exp. Sh/L Ratio = Ratio between expected means in the short and long conditions.

Finally, we examined whether the Bose-Einstein distribution accounted well for the actual distribution of participants’ responses. To this effect, we considered the 240 trials performed at each level in the long condition (i.e., 4 trials per level for 60 participants) and we compared the distribution of correct responses on each level with the distribution expected from the Bose-Einstein model, and with a potentially alternative binomial model. The results provided additional support for the Bose-Einstein model. They are fully reported in the supplementary materials (see Supplementary file 1: Distributions of CSVI correct responses per level, Figure S1-1 and Tables S1-1 and S1-2). Furthermore, we examined in detail the responses to each particular feature and their temporal order (see Supplementary file 2: Responses to single CSVI features).

Method

Materials and procedure

The CSVI materials were the same as in Experiments 1a–1c. Also procedure was the same, except that all test stimuli were presented for 5 s. We considered that using all 56 items in a single condition can yield more reliable estimates than using only 28, and we used long presentation because in the literature it is better established for the CSVI.

The VAT requires participants to decide, by pressing one of two keys, whether two consecutive arrangements of colored squares are identical, or the color of one of the squares was changed. The stimuli were presented on a 15-inch CRT monitor with a refresh rate of 75 ms; the participant was comfortably sitting at a viewing distance of approximately 70 cm. There were 320 test trials (80 of each set size = 6, 8, 10, and 12), in half of which the memory array was presented for 120 ms, and for 5 s in the other half; in addition, there were 16 practice items. The subsets of stimuli presented for 120 ms or 5 s were balanced over participants. Each stimulus had a grey background and included a given number (i.e., its set size) of colored squares. The squares were randomly placed within an area of 9.8° × 7.3° of visual angle at the centre of the screen; the size of each square was .75° × .75° and its color was randomly drawn with replacement from a list of 7 (white, black, red, blue, green, yellow, violet) with the only constraint that a color could not appear more than three times in a stimulus. The position of each square was randomized with the only constraint of a distance of at least 2° between any two squares. Each memory array was followed by a probe, which (with a probability of .5) was either identical to it or had a single square changed in color; one of the squares in the probe was surrounded by a circle with a diameter of 1.5°. If a square in the probe had a different color from the memory array, then the circle surrounded the changed square; otherwise, it surrounded a randomly selected square. The participant was instructed to press the keys m or z, respectively to indicate change or no change. Only accuracy was required, not speed.

A trial was started by the participant pressing a key. A fixation stimulus (a black cross in the centre of the screen) was presented for 1000 ms, followed by the memory array for 120 ms or 5 s according to the condition; then the screen remained empty for 1000 ms. At that point the probe was presented and remained visible until the participant responded. A feedback on accuracy was presented for 500 ms, then the subject was required to press a key to start the next item (see Figure 4 for an example).

Figure 4

The 320 test trials were divided into four blocks of 80 (20 per set size, 10 of which had a color changed in the probe). Items with different set size were randomly mixed in each block. Half participants viewed for 120 ms the memory arrays of the first and third block, and for 5 s those of the second and fourth block; the presentation time was reversed for the other half participants.

A VAT session started with two practice blocks of 8 trials each, one with 5 s and the other with 120 ms presentation. Then the four blocks of test trials followed. Pauses between blocks were allowed and the participants were also encouraged to pause within a block in case they felt tired. On average the VAT session lasted about 75 minutes, including pauses.

Both tasks were programmed in e-prime. The CSVI was administered in the first session and the VAT in the second session, with an interval of one or a few days.

Results and discussion

Each CSVI item was scored as in Experiment 1. The participants’ total number of correct responses ranged from 209 to 272 (mean = 249.9, s.d. = 14.7) out of a maximum possible = 280, replicating well (given the doubled number of items) the results of Experiments 1a–1c with long presentation. The mean number of correct responses per trial, on levels 2–8, was 1.97, 2.91, 3.77, 4.50, 5.36, 6.00, 6.63, respectively (also similar to experiments 1a–1c). Using the Bose-Einstein model, the mean estimated value of k was 6.48 (s.d. = 1.54), close to the value of 6.55 found across Experiments 1a–1c.

We also examined whether the Bose-Einstein distribution accounted well for the distributions of correct responses, considering the 400 trials performed at each level. Also in this experiment, the results favored the Bose-Einstein model over the binomial model, as reported in the supplementary materials (Supplementary file 1: Distributions of CSVI correct responses per level, Figure S1-2 and Tables S1-3 and S1-4).

The VAT data were analyzed in different ways. First, for comparison with previous studies that used Cowan’s (2001b) formula, we applied this formula to the data of each participant in each presentation condition for each set size, and the k values thus obtained were submitted to a 2 (presentation) x 4 (set size) repeated-measures ANOVA. There were significant effects of presentation time, F(1, 49) = 296.74, p < 0.001, η_p² = .86, MSE = 2.10, set size, F(3, 147) = 17.12, p < 0.001, η_p² = 0.26, MSE = 1.75, and the interaction, F(3, 147) = 21.97, p < 0.001, η_p² = 0.31, MSE = 1.33 (see Figure 5). A Bayesian ANOVA with the same design, carried out with the JASP default priors, confirmed that the best fitting model included both factors as well as their interaction, BF_M = 2.42 × 10¹¹.

Figure 5

Separate analyses for short and long presentation showed that with short presentation the mean k was fairly constant across set sizes, F(3, 147) = 2.53, p > 0.06, η_p² = 0.05, MSE = 1.67. A Bayesian ANOVA with the same design confirmed that the null model was the best fitting one, BF_M = 1.86. The overall mean of k, pooled across set sizes, was 4.22 (s.d. = 1.34); this value is not significantly different from 4, t(49) = 1.15, n.s. (p > 0.25), with BF₀₁ = 3.51 in a Bayesian analysis. The finding of a capacity of about 4 units, often reported from VAT studies with short presentation time and using Cowan’s formula, was replicated.

With long presentation, instead, the mean estimate of k varied with set size, F(3, 147) = 39.20, p < 0.001, η_p² = 0.44, MSE = 1.40; a Bayesian ANOVA with the same design confirmed the existence of the set size effect, BF_M = 4.91 × 10¹⁵. Contrasts between adjacent set sizes showed an increase from set size 6 to 8, F(1, 49) = 87.00, p < .001, η_p² = 0.64, MSE =0.83 (BF₁₀ = 4.44 × 10⁹), and from set size 8 to 10, F(1, 49) = 17.07, p < 0.001, η_p² = .26, MSE = 2.17 (BF₁₀ = 168.7); only at that point k leveled, the contrast between set sizes 10 and 12 yielding F(1, 49) = 1.00, p > 0.32, η_p² = .02, MSE = 3.75 (BF₀₁ = 4.05). The mean estimate of k, averaging set sizes 10 and 12, was 7.51 (s.d. = 1.98).

These results suggest that in the VAT, contrary to the CSVI, the estimate of k is strongly affected by presentation time. This does not imply that the participants’ capacity actually varies over presentation conditions; rather, chunking and other capacity-saving strategies become possible with long presentation. During informal interviews at the end of the session, several participants reported having used strategies in the long presentation condition, such as naming the colors that were repeated in a memory array, recoding groups of squares as national flags (e.g., blue-white-red = France) or shirts of football teams, and other, more idiosyncratic strategies; they also reported that they were unable to use such strategies with short presentation. No such strategies were reported in the long presentation condition of the CSVI; a small minority of participants reported going through the list of features while selecting their responses to the CSVI, but this strategy could be used with any presentation time, because it takes place during response selection. So far, it seems that Cowan’s (2001b) suggestion that presentation time can affect capacity estimates is true of the VAT.

However, as noted in the introduction, R. Morey (2011) proposed a Bayesian method for capacity estimation that can be regarded as a methodological improvement over the classical formula. We analyzed the data separately for the short and long presentation conditions, using the WoMMBAT software (Morey & Morey, 2011), which yields parameter estimates using a Markov chain Monte Carlo method. We ran 10,000 iterations (with burn-in set at 1,000) with lower ɛ = .035 and upper ɛ = .05 for the short presentation data, and 10,000 iterations (with burn-in set at 1,000) with lower ɛ = .025 and upper ɛ = .04 for the long presentation data, thus obtaining good quality Markov chains, as indicated by the acceptance rates (.72 and .70 for short and long presentation, respectively) and by the approximately flat lines obtained for all parameters in the graphic interface, which suggest good convergence of the estimates; Geweke’s test did not reject the null hypothesis of convergence for any of the three estimated parameters in either data set.

Table 2 presents the mean and the s.d. of the individual participants’ parameter estimates obtained with this analysis. For the goals of this study it is particularly important to compare the estimates of k obtained in the CSVI and in the two conditions of the VAT. As reported above, participants obtained larger k estimates in the long than in the short VAT.

Participants obtained larger k estimates in the long VAT than in the CSVI, t(49) = 8.66, p < 0.001, s.e. = 0.23, d = 1.47. A Bayesian two-tailed paired-t test, with the H₁ prior set as a Cauchy distribution with scale = 0.316, confirmed that these means were different, BF₁₀ = 2.84 × 10⁸ (in this analysis we used the same scale for H₁ as in Experiments 1a–1c; however, here we used a two-tailed test because both tasks were administered with the same long presentation time and there were no previous results or clear theoretical predictions for an expected mean score in the long VAT). Also this result is consistent with the view that a long presentation of the VAT stimuli yields an overestimation of participants’ capacity, because participants can use chunking and recoding strategies in this condition.

The mean estimate of k in the short VAT was somewhat higher than the value of 4, often obtained (also in this study) with Cowan’s (2001b) formula. Nevertheless, the mean estimate of k in the short VAT was lower than in the CSVI, t(49) = 4.66, p < 0.001, s.e. = 0.19, d = 0.67. A Bayesian paired-t test, with one-tailed H₁ CSVI > short VAT and the prior for H₁ set as a Cauchy distribution with scale = 0.316, supported H₁, BF₁₀ = 1420.6 (also in this analysis we used the same scale for H₁ as in Experiments 1a–1c; here we used a one-tailed test because extensive previous results indicated higher mean capacity estimates in the CSVI than in the short VAT).

Actually, the difference between the mean k estimates obtained from the CSVI (6.48) and the short VAT (5.59) was close to 1 unit; testing the hypothesis that the difference between these means is equal to 1 yielded t(49) = –0.58, n.s. (p > .56), s.e. = 0.19, d = 0.08; a Bayesian analysis carried out with the JASP default priors yielded BF₀₁ = 5.55. This would be consistent with the view that performance on the CSVI and the short VAT is limited by capacity of the same resource and in the short VAT one unit of capacity is allocated to an operative scheme.

However, before drawing strong conclusions on this point we also need to examine correlational evidence. Table 3 presents the correlations between parameter estimates obtained from the VAT and the CSVI. The correlations between different estimates of k are particularly relevant to the goal of this study. The k parameters obtained from the CSVI and the short VAT were positively correlated, r(48) = 0.48, p < .001, 95% CI [0.24, 0.67]; a Bayesian analysis with the JASP default priors yielded BF₁₀ = 167.1. This, too, is consistent with the hypothesis that the same limited-capacity resource is involved in both tasks.

The k estimates obtained from the CSVI and the long VAT were also positively correlated, although to a lesser extent, r(48) = 0.29, p < 0.04, 95% CI [0.01, 0.52]; in a Bayesian analysis, BF₁₀ = 2.46. The correlation between CSVI and short VAT was still significant when long VAT was partialled out, r(47) = 0.41, p < 0.01, 95% CI [.14, .62], and in a Bayesian analysis, BF₁₀ = 15.49, whereas the correlation between CSVI and long VAT was no longer significant when short CSVI was partialled out, r(47) = 0.02, p > 0.91, 95% CI [–0.27, 0.30), and in a Bayesian analysis, BF₀₁ = 7.66. This suggests that the source of variance shared by CSVI and short VAT is shared only to a lesser extent by CSVI and long VAT. This point is notable because in this experiment both CSVI and long VAT involved 5-s stimulus presentation, but it was the short VAT (with only 120-ms presentation) that correlated higher with the CSVI, strengthening the conclusion that they both tap the same limited capacity.

It can also be noted in Table 3 that the estimates of k and z were positively correlated. This replicates and extends Morey’s (2011) finding of a substantial positive correlation between these two aspects of the attentional system. The estimates of g in both conditions correlated with each other but were uncorrelated to the other parameters.

However, a note of caution is necessary with these results. The Markov chain Monte Carlo method implemented with WoMMBAT assumes, for accurate estimation of z, to have at least one set size below capacity and one above capacity (Morey and Morey, 2011). Our data in the long presentation condition met this requirement, because set size ranged from 6 to 12 and the mean estimate of k was 8.47; in particular, with set size = 6, performance was near ceiling with an overall proportion of 0.94 accurate responses. However, our data in the short presentation condition did not meet this requirement because the mean estimate of k, i.e. 5.59, falls out of the 6–12 range; in particular, with set size = 6, the overall proportion correct was .84, and only 18 participants out of 50 produced at least 90% correct responses. Hence, performance on the smallest set size could not be regarded as near ceiling, which implies possible distortion or bias in the z estimates.

The requirement of at least one set size below capacity is only needed for estimating z, not k; however, biased estimates of z would in turn affect the estimates of k (i.e., underestimating z would cause an overestimate of k, and vice versa). A way to control for possible bias due to assumption violation is not modelling z but setting it at a fixed value, reasonably high to be consistent with previous research (C. C. Morey, personal communication).

Consequently, we ran WoMMBAT with the short VAT data, setting z at a fixed value of 0.95, and obtained a mean estimate of k = 4.51 (s.d. = 1.14). This mean k was lower than that obtained from the CSVI, t(49) = 9.82, p < 0.001, s.e. = 0.20, d = 1.42. A Bayesian paired-t test, with one-tailed H₁ CSVI > short VAT and the prior for H₁ set as a Cauchy distribution with scale = 0.316, supported H₁, BF₁₀ = 2.37 × 10¹⁰. Individual participants’ k estimates obtained in this way correlated positively with those obtained from the CSVI, r(48) = 0.47, p < 0.001, 95% CI [0.22, 0.66]; a Bayesian analysis with the JASP default priors yielded BF₁₀ = 62.60, and this correlation remained significant when k estimates from the long VAT were partialled out, r(47) = 0.39, p < 0.01, 95% CI [0.12, 0.60], BF₁₀ = 3.50.⁴ The difference between the mean estimate of k obtained from this analysis and that obtained from the CSVI was greater than 1 unit, t(49) = 4.83, p < 0.001, s.e. = 0.20, d = 0.68, and in a Bayesian analysis with the JASP default priors, BF₁₀ = 1409.0; actually, it was close to 2 units, t(49) = –0.16, n.s. (p > 0.87), s.e. = 0.20, d = 0.02, and in a Bayesian analysis, BF₀₁ = 6.42.

To summarize: when analyzing the short VAT data with WoMMBAT leaving z as a free parameter, we found mean estimates of k = 5.59 and z = 0.813; this value of k was lower than that obtained in the CSVI by approximately 1 unit, and the individual estimates of k from the CSVI and the short VAT were positively correlated. In a control analysis, with z fixed at .95 to control for its possible underestimation, we found a mean estimate of k = 4.51, and we replicated the findings that the k estimates from the short VAT correlated positively with those from the CSVI, and the mean k in the short VAT was lower than in the CSVI. These main findings, therefore, are robust against possible assumption violation in estimating z; the qualitative pattern of results was the same in both analyses. The only substantive difference between the results of the two analyses is that, leaving z as a free parameter, the short VAT mean k was approximately 1 unit less than the CSVI mean k, whereas this difference turned out to be approximately 2 units when z was fixed at .95. Experiment 3 was carried out to clarify whether a difference of 1 or 2 units is the most correct quantitative estimate.

Method

Results and discussion

Preliminary analyses were run to check that the changes introduced in this experiment actually had the intended consequences. The participants’ total number of correct responses in the CSVI ranged from 200 to 260 (mean = 235.0, s.d. = 17.0). The mean number of correct responses per trial, on levels 2–8, was 1.95, 2.77, 3.66, 4.37, 4.95, 5.64, 6.03, respectively (slightly lower than in the experiments with adults). Using the Bose-Einstein model, the mean estimated value of k was 5.29 (s.d. = 1.12), and the range of k was 4–7. Therefore, it seems that no participant in this experiment overperformed in the CSVI thanks to particularly sophisticated strategies. Moreover, considering the 192 trials performed at each level, also in this experiment the Bose-Einstein distribution accounted well for the distributions of correct responses, as reported in the supplementary materials (Supplementary file 1: Distributions of CSVI correct responses per level, Figure S1-3 and Tables S1-5 and S1-6).

The proportion of accurate responses on set size = 3 was .96 in the short and .99 in the long presentation conditions; in particular, in the short condition 23 participants out of 24 produced at least .90 correct responses, and the median was .975. Hence, performance on the smallest set size was near ceiling in both conditions (see also Figure 6), which ensures that all three WoMMBAT parameters can be modeled without biases or distortions.

Figure 6

The k values obtained from the VAT using Cowan’s (2001b) formula, presented in Figure 6, were submitted to a 2 (presentation) × 4 (set size) repeated-measures ANOVA. There were significant effects of presentation time, F(1, 23) = 84.84, p < 0.001, η_p² = 0.79, MSE = 1.53, set size, F(3, 69) = 71.19, p < 0.001, η_p² = 0.76, MSE = 1.18, and the interaction, F(3, 69) = 36.57, p < .001, η_p² = 0.61, MSE = 0.57. A Bayesian ANOVA with the same design, carried out with the JASP default priors, confirmed that the best fitting model included both factors as well as their interaction, BF_M = 6.19 × 10¹³.

Separate analyses were run for short and long presentation. With short presentation, the mean k estimate was not constant across set sizes, F(3, 69) = 12.17, p < 0.001, η_p² = 0.35, MSE = 0.99, and a Bayesian analysis yielded BF_M = 13945.7 for the presence of a set size effect; however, this was merely due to ceiling performance on set size = 3. When only the data from set sizes 6, 8, and 10 were analyzed, the result was nonsignificant, F(2, 46) = 0.65, p > 0.5, η_p² = 0.03, MSE = 1.06, with BF_M = 5.26 for the null model in a Bayesian analysis. The mean k obtained pooling these three set sizes was 4.17, which was not significantly different from 4, t(23) = 0.71, n.s. (p > 0.48), with BF₀₁ = 3.71 in a Bayesian analysis.

With long presentation, instead, the mean estimate of k varied with set size, F(3, 69) = 122.53, p < .001, η_p² = 0.84, MSE = 0.76, and a Bayesian analysis yielded BF_M = 1.29 × 10²⁶ for the presence of a set size effect; contrasts between adjacent set sizes showed increases from set size 3 to 6, F(1, 23) = 132.75, p < 0.001, η_p² = 0.88, MSE = 0.71 (BF₁₀ = 2.28 × 10⁹), from set size 6 to 8, F(1, 23) = 19.47, p < 0.001, η_p² = 0.46, MSE = 1.03 (BF₁₀ = 144.73), and from set size 8 to 10, F(1, 23) = 36.69, p < 0.001, η_p² = 0.62, MSE = 1.52 (BF₁₀ = 5613.8). Replicating a finding of experiment 2, these results show that in the VAT the estimate of k is strongly affected by presentation time. Again, this does not imply that the participants’ actual capacity varies over presentation conditions; rather, chunking and recoding are possible with long presentation. In the case of the VAT, this finding is consistent with Cowan’s (2001b) suggestion that presentation time can affect capacity estimates.

We analyzed the VAT data using the WoMMBAT software separately for the short and long presentation conditions. We ran 20,000 iterations (with burn-in set at 2,000) with lower ɛ = 0.035 and upper ɛ = 0.05 for the short presentation data, and 20,000 iterations (with burn-in set at 2,000) with lower ɛ = 0.03 and upper ɛ = 0.045 for the long presentation data, thus obtaining good quality Markov chains, as indicated by the acceptance rates (0.65 and 0.66 for short and long presentation, respectively) and by the approximately flat lines obtained for all parameters in the graphic interface, which suggest good convergence of the estimates. In particular, Geweke’s test did not reject the null hypothesis of convergence for any of the three estimated parameters in either data set.

Table 4 presents the mean and s.d. of the individual participants’ parameter estimates obtained with this analysis. The estimates of k obtained in the CSVI and in the two conditions of the VAT were compared. The participants achieved larger k estimates in the long VAT than in the CSVI, t(23) = 5.95, p < 0.001, s.e. = 0.29, d = 1.08. A Bayesian two-tailed paired-t test, with the H₁ prior set as a Cauchy distribution with scale = 0.316, confirmed that these means were different, BF₁₀ = 2653.8 (in this analysis we used the same scale for H₁ as in Experiments 1a–1c and a two-tailed test, as in the same analysis of Experiment 2). This is consistent with the view that a long presentation of the VAT stimuli yields an overestimation of participants’ capacity, because of chunking and recoding in this condition.

The mean k estimate in the short VAT was lower than in the CSVI, t(23) = 4.90, p < 0.001, s.e. = 0.18, d = 0.89. A Bayesian paired-t test, with one-tailed H₁ CSVI > short VAT and the prior for H₁ set as a Cauchy distribution with scale = 0.316, supported H₁, BF₁₀ = 575.0 (in this analysis we used the same scale for H₁ as in Experiments 1a–1c and a one-tailed test, as in the same analysis of Experiment 2). So far, the results replicate those obtained in Experiment 2.

Again, the difference between the mean k estimates obtained from the CSVI (5.29) and the short VAT (4.39) was close to 1 unit; this replicates well the result obtained in Experiment 2 using the WoMMBAT estimates in which all three parameters were modeled. Testing the hypothesis that the difference between these means is equal to 1 yielded t(23) = –0.52, n.s. (p > 0.60), s.e. = 0.18, d = 0.11; a Bayesian analysis carried out with the JASP default priors yielded BF₀₁ = 4.11. This would be consistent with the view that in the short VAT one unit of capacity is allocated to an operative scheme.

However, in Experiment 2 we also ran WoMMBAT with a fixed value for the parameter z; in this case, the mean k estimate obtained from the short VAT was about 2 units less than that obtained from the CSVI. In Experiment 2 that control analysis was deemed necessary, because the absence of a condition at ceiling could have introduced distortions in the parameter estimates, and therefore, although it was clear that the CSVI and short VAT estimates of k correlated with each other and the CSVI yielded a larger mean, Experiment 2 left undecided whether the mean difference was 1 or 2 units. In this experiment there was no risk of such confound; therefore, to clarify the size of the difference, we also tested the hypothesis that the difference between the mean k estimates obtained from the CSVI and the short VAT is equal to 2 units. This hypothesis, however, was rejected, t(23) = –5.95, p < 0.001, s.e. = 0.18, d = 1.21; a Bayesian analysis carried out with the JASP default priors yielded BF₁₀ = 4414.3. Therefore, we conclude that this experiment clarified the important point that the difference between the k estimates obtained from the CSVI and the short VAT is approximately 1 unit, not 2.

Table 5 presents the correlations between parameter estimates obtained from the VAT and the CSVI. The k parameters obtained from the CSVI and the short VAT were positively correlated, r(22) = 0.60, p = 0.002, 95% CI [0.26, 0.81]; a Bayesian analysis with the JASP default priors yielded BF₁₀ = 46.90. This replicates one of the main findings of Experiment 2; note that the CIs found in Experiments 2 and 3 for the correlation between these variables largely overlap. This positive correlation is consistent with the hypothesis that the same limited-capacity resource is involved in both tasks.

The k estimates obtained from the CSVI and the long VAT were also positively correlated, r(22) = 0.61, p = 0.002, 95% CI [0.27, 0.81], and in a Bayesian analysis, BF₁₀ = 52.90. The correlation between CSVI and short VAT was still significant with long VAT partialled out, r(21) = 0.42, p < 0.05, 95% CI [0.01, 0.71], and in a Bayesian analysis, BF₁₀ = 3.11. Also the correlation between CSVI and long VAT was significant with short VAT partialled out, r(21) = 0.43, p < 0.05, 95% CI [0.02, 0.72], and in a Bayesian analysis, BF₁₀ = 3.55. In this regard, this experiment did not fully replicate the findings of the previous one. In Experiment 2, the CSVI correlated higher with the short than the long VAT; in this one, instead, both correlations were equally high, perhaps because strategy use in the long VAT was less prominent in adolescents than in university students or graduates.

It can also be noted in Table 5 that, in this experiment, some but not all estimates of k and z were positively correlated. The z parameter in the short condition correlated significantly (p < 0.04) with the k parameter in the short condition, and the correlation between z in the short condition and the k estimate obtained from the CSVI fell just short of significance (p = 0.055). This adds further evidence to the finding of positive correlations between the aspects of the attentional system indexed by k and z. The z estimate in the long condition, instead, did not correlate significantly with any estimate of k, possibly because of a ceiling effect in the z estimates in the long condition (see Table 4). Finally, the estimates of g in both conditions correlated with each other but were uncorrelated with the other parameters.

Representations and resources

The debate on working memory capacity in change-detection tasks became soon intertwined with a debate on representation. Are whole objects or their distinct features the basic units? Luck and Vogel (1997) found that participants maintained in working memory an equal number of one-featured or four-featured objects, and therefore claimed that representations of whole objects are the basic units of visual working memory. However, subsequent research demonstrated that, in the case of complex objects in which multiple features are bound together, both objects and features have a role, the relevant representations being structured hierarchically (Brady et al., 2011; Cowan, Blume & Saults, 2013; Langerock et al., 2018). This seems consistent with both Cowan’s and Pascual-Leone’s theories, which assume that long-term memory representations (Cowan, 1988) or schemes (Pascual-Leone & Goodman, 1979) have a multi-level, hierarchical structure and any level of representation can be attended to. It is also consistent with neuropsychological evidence that both objects and their properties are represented in the brain during maintenance in working memory (e.g., Luria et al., 2010; Thyer et al., 2022; Wilson et al., 2012). However, the materials we used for the VAT had only one relevant feature (color); therefore, our capacity estimations do not need modelling both levels of a hierarchical representational structure of the objects in an array, and we can safely assume that each square counts as a single unit in the focus of attention. Similarly, the CSVI concerned the features of a single object; therefore, each relevant feature can be regarded as a distinct unit, with no need of modelling a hierarchical structure.

Accounting for the mean difference between CSVI and short VAT

Limitations and prospects

A possible limitation of our study is not having considered those versions of the VAT in which a continuous dimension (e.g., orientation, distance, or shade of color) of an object must be recalled (e.g., Bays, Catalao & Husain, 2009). Those experiments were often cited in support of theories (e.g., Ma et al., 2014) claiming that attention is distributed flexibly among all items, with no limit on their number. Hardman, Vergauwe & Ricker (2017), however, demonstrated that no more than one color is represented in working memory in a continuous fashion, the others being represented categorically. Future research could extend our approach to estimation of capacity for continuous and categorical representations.

An important limitation is having used only two tasks. We suggest extending this line of research in the future, also taking into consideration verbal tasks and the specific problems they pose for capacity estimation (e.g., Camos, Mora & Oberauer, 2011; Cowan et al., 2012).

Moreover, future research could explore further the methodological differences between the two tasks used in this study, for instance, by creating “hybrid” versions of the tasks. The VAT could be modified by presenting a single array and requiring the participants to detect all colors present in that array; one could test whether the Bose-Einstein distribution fits the data in that condition. In principle, one could also modify the CSVI by requiring participants to detect changes between two consecutive compound stimuli; however, this procedure might be less reliable, because capacity estimates in the CSVI are robust to violations of the assumption of feature equiprobability for the reasons given in the supplementary materials (Supplementary file 2: Responses to single CSVI features), but a change detection procedure might be too sensitive to salience differences among features.

Another limitation is having compared only two theories of capacity of the attentional resources used to activate a small number of chunks or schemes. Other working memory theories also posit limited attentional capacity, framed in terms of attentional control (Unsworth & Engle, 2007), time-based resource sharing (Barrouillet, Portrat & Camos, 2011), or interference removal (Oberauer et al., 2012). Future research could extend comparison to these theoretical frameworks, considering also tasks and experimental paradigms (e.g., complex span) used by these researchers.

Finally, it would be interesting to extend this research to developmental trends. Cowan (2016) and Pascual-Leone and Johnson (2005) agree that maturation has a primary role in the development of capacity, and Pascual-Leone (1970) posited a developmental pattern with an increase of one unit approximately every second year. It would be important to establish, if possible, a similar pattern in the context of Cowan’s theory, and compare the developmental patterns of the capacity measures used most often in either theoretical framework.