Immediate Recall of Serial Numbers With or Without Multiple Item Repetitions

Abstract

Immediate recall of lists of items in random serial order has been examined in thousands of studies throughout the history of experimental psychology. In most studies, though, there have been no repetitions of items within a list, or occasionally a single repetition. These stimuli differ from the common uses of item series, which often include multiple repetitions (e.g., identification numbers; orders of multiple individuals at a restaurant table). To begin to understand such cases we presented lists that, in some trial blocks, were constructed with no restrictions on repetitions. Specifically, we examined immediate serial recall of visually-presented, nine-digit lists, either spatially separated into three separate groups of three digits (Experiment 1) or undivided (Experiment 2). Many of the lists included single or multiple repetitions of digits, with repeated digits either adjacent or non-adjacent in an unpredictable manner. We assessed theoretical expectations derived from prior research. Effects of repetition were often helpful but, when repetitions favoured a grouping that conflicted with the presented grouping into threes in Experiment 1, repetition was disadvantageous. We suggest a theoretical analysis in which participants can use presented grouping cues or, when those cues are absent, create their own groupings to exploit repetitions among the stimuli.

Since the onset of experimental psychology, immediate serial recall of lists has been avidly studied. For example, Nipher (1878) examined serial position effects and Ebbinghaus (1885/1913) discussed the “first fleeting grasp” (p. 33) of sufficiently short lists of nonword syllables. This kind of recall was designed to examine the structure of immediate memory while minimizing contributions of real-world knowledge. By now, quite a bit has been learned from this type of study. (For the entry “immediate serial recall” in April, 2021, Google Scholar showed 4,950 results. For a recent review of this kind of recall, see Oberauer et al., 2018.) However, the findings have been almost entirely based on lists in which there are no repetitions of items. In striking contrast, repetitions abound in lists found in modern life, which include many within-list repetitions (e.g., telephone numbers, social security numbers, other personal identification strings, serial numbers of electronic equipment, passcodes, and so on). Repetition is also a compelling part of the natural environment more generally; consider for example repetitions in the color categories of houses along a street or in the kinds of trees distributed along a stretch of woods.

Here we introduce the complexity of repetitions into series of digits to be recalled. Given the long-understood propensity of grouping in serial recall (e.g., the grouping of U.S. social security numbers into a group of three, a group of two, and then a group of four digits), in Experiment 1 we presented lists of nine digits grouped in a way thought to be helpful to recall, as three sets of three (Hitch et al., 1996; Ryan, 1969).

In Experiment 2, we dispensed with the grouping. For visual stimuli such as ours, participants could be able to create their own grouping. For example, Cowan et al. (2002) found scalloped serial position effects for lists of nine printed digits, whether the presented lists were temporally divided into subsets of three digits or not.

In the present work, when we included repetitions among the stimuli, we wondered if participants might benefit from an absence of grouping cues, which might allow them to group stimuli in an opportunistic manner that conformed to any perceived patterns in the stimuli. For example, if the list comprised the nine-digit series 499927334, a good grouping might be 4-999-2733-4 to allow the triplet of 9’s to form a group and to allow the 4 at the beginning and at the end to be noted. In such a case, presenting the stimuli in groups of three might be counterproductive, so the grouping cues were omitted in Experiment 2.

The most likely reason for the absence of repetitions in lists within memory experiments is that there are many configurations of repetitions to be evaluated. They have been omitted to permit an understanding other factors in recall with less uncontrolled complexity. Repetitions are used in identification strings, probably because they allow many more unique strings. For example, in 7-digit numbers that can include 0 as a digit, without repetitions there would be 10*9*8*7*6*5*4=604,800 phone numbers possible; with repetition, 10⁷=10,000,000 phone numbers possible.

Theoretical Expectations for Lists with Repetitions

Repetitions are likely to make up perceivable patterns in the stimuli but there are many kinds of patterns that could be perceived, such as runs of consecutive items, (e.g., 234 or 432) or familiar series (e.g., a pair of digits matching one’s current age). The general expectation is that patterns are perceived as unified objects or chunks, which tend to lower the number of items that need to be retained independently (e.g., Burtis, 1982; Cowan, 2019; Chen & Cowan, 2009; Cowan et al., 2012; Norris et al., 2020; Miller, 1956) and compress the representation of the list into a shorter abstract description to be remembered (Chekaf et al., 2016; Mathy & Feldman, 2012). Repetitions could contribute to a usable pattern, as does the digit 8 in the series 848-858-828, or in the series 488857963, in which the triplet reduces the number of independent digits to be remembered. The issue of repetitions is a vast enough topic that we consider other bases of patterns (e.g., consecutive runs and known series) to be beyond the scope of our work.

Although perceived patterns based on repetitions would be expected to be generally helpful to recall, the overall effect of repetitions also could be negative in certain circumstances, for several reasons. The serial position of each item must be remembered, and serial positions could be more difficult to remember for a repeated item. As an extreme example, it seems difficult to remember the order of digits in the string 122121221 because of the difficulty of discerning a simple, orderly pattern for the two digits. As one potential factor, it has long been clear that phonological similarity is detrimental to the retention of ordered verbal lists (e.g., Conrad, 1964) and the identity of two items is an extreme kind of phonological similarity between them. On the other hand, a typical problem with phonological similarity is the tendency to switch the locations of similar items, and there is no such problem with identical items. Redundancies in the phonological record nevertheless could cause a problem by eliminating serial order cues. In the example, the digit 2 sometimes is followed by a 1 and other times is followed by another 2, introducing uncertainty as to which digit comes next at each such position. That kind of arrangement probably limits the temporal distinctiveness of list items (e.g., Unsworth et al., 2008). As a less extreme example, suppose one is trying to recall the sequence 284726. The first instance of the digit 2 may be associated with the subsequent digit, 8. However, the second instance of the digit 2 may be associated with the digit following it, 6. When an item has more than one associate and the two associates are in conflict, the result is a fan effect in which there is degradation in the use of the associations (e.g., Radvansky, 1999).

Providing one basis for theoretical expectations for unconstrained repetition, there has been considerable work on recall with a single repetition, a well-controlled situation (for reviews see Henson, 1998; Kahana & Jacobs, 2000). The general finding of that literature is that repetitions in adjacent serial positions are helpful for recall, whereas repetitions in non-adjacent serial positions are harmful. Note, though, that participants in these experiments would be aware that they should look for at most a single repetition per list, and we do not know whether the results would generalize to lists in which more repetitions might be present.

One reason to expect that repetition could have a harmful effect is that many theories of working memory include a response suppression mechanism to ensure that an item that has been spoken is not again spoken (e.g., Brown et al., 2000; Burgess & Hitch, 1999; Farrell & Lewandowsky, 2002; Henson, 1998). This mechanism might lead to the prediction that the second instance of a digit could be suppressed and omitted. However, it is not clear if that suppression occurs even when frequent repetitions occur, making suppression inappropriate; and whether any such suppression would build up further over more than two identical items. It is possible that any such suppression could be outweighed by positive aspects of perceiving a pattern of repetition.

In sum, we expect a tension between helpful and harmful effects of repetition. The helpful effect of repetitions that lead to an easy-to-perceive pattern, as when several successive items within a group are identical (Burtis, 1982), is likely to be in tension with the just-described, deleterious effects of repetition. Repetitions that are further away from one another within a short list tend to cause more interference with recall (Henson, 1989), probably because, when repetitions are further away, a coherent pattern is not observed as often, so the helpful effects do not occur as often. These findings are based on a single list repetition but could be expected to yield comparable results for unrestricted repetitions.

Research Strategy

Despite such a long history of serial recall experiments, relatively little has been done to examine effects of regularities within lists because so many different helpful and harmful regularities are possible. Even within the domain of list repetitions, there are too many possibilities to do a complete investigation. With that point in mind, as we explain next, we simplified the stimuli from what was possible to ensure a manageable set of issues that could be examined; issues regarding twice-presented, thrice-presented, and to a lesser extent more-often-repeated items.

List Pattern Constraints

We ran some trial blocks that contained no repetitions of items to approximate, in those blocks, the mindset of ordinary serial recall tasks. In the remaining trial blocks, we allowed repetitions in a semi-controlled, yet still quite variable manner to create some complexity to be examined. In those trial blocks, participants received repetitions in every trial but there were two subsets of trials designed to examine different kinds of patterns. Some trials were designed to ensure that we could examine effects of digits presented twice at any two serial positions. These trials included at least one matching pair, at any two serial positions, with the other positions randomly determined except with no digit presented three or more times. In other trials, a different constraint was used. At least one digit was presented at least three times (at any three serial positions) to ensure that enough of those trials were present. In these trials, three serial positions were assigned the same digit and then other serial positions were assigned any digit, randomly chosen with replacement. As a result, repetitions were quite variable, but we had enough trials to assess effects of items appearing two, three, and more than three times within haphazard lists somewhat resembling those fully randomized with replacement or resembling those often seen in identification numbers. Given practical constraints in time per participant, we did not include any lists with no repetitions intermixed in the trial blocks with repetitions. The distribution of resulting list profiles is summarized in the leftmost two columns of Table 1.

Table 1.

Profile of the Repetitions in a List and Frequency of the Profile, Items Correct, and Proportion of Trials Correct for Each Profile in Both Experiments

Profile	Trial Frequency		Items Correct				Proportion of Trials Correct
	Exp.1	Exp.2	Exp.1		Exp.2		Exp.1		Exp.2
	Total	Total	Mean	SD	Mean	SD	Mean	SD	Mean	SD
900000000	2145	2600	6.91	2.07	6.77	2.20	0.34	0.47	0.34	0.47
520000000	952	1161	6.78	1.95	6.72	2.21	0.27	0.44	0.31	0.46
411000000	826	984	6.79	2.06	6.80	2.07	0.30	0.46	0.30	0.46
330000000	796	976	6.74	2.04	6.81	2.12	0.26	0.44	0.32	0.47
221000000	468	640	6.87	1.98	6.95	2.06	0.30	0.46	0.34	0.47
710000000	275	329	6.83	1.96	6.98	2.13	0.27	0.45	0.37	0.48
601000000	223	273	6.97	1.96	6.92	2.18	0.31	0.46	0.34	0.48
310100000	173	195	7.31	1.70	7.47	1.95	0.35	0.48	0.46	0.50
302000000	139	169	7.01	1.87	6.85	2.15	0.32	0.47	0.31	0.46
140000000	122	134	6.89	1.96	6.90	2.04	0.30	0.46	0.29	0.46
500100000	95	94	7.18	1.99	6.79	2.35	0.36	0.48	0.35	0.48
120100000	54	42	6.89	2.03	7.00	2.04	0.37	0.49	0.38	0.49
112000000	52	78	6.98	1.42	7.37	1.73	0.21	0.41	0.41	0.50
201100000	37	26	6.41	2.41	7.31	2.13	0.24	0.43	0.42	0.50
031000000	32	46	6.94	1.78	6.93	2.33	0.28	0.46	0.35	0.48
400010000	22	17	8.00	1.31	7.65	1.58	0.50	0.51	0.47	0.51
210010000	10	19	7.30	1.06	6.21	2.20	0.10	0.32	0.26	0.45
011100000	4	4	7.75	1.50	5.25	2.36	0.50	0.58	0.00	0.00
003000000	2	3	8.50	0.71	6.67	1.15	0.50	0.71	0.00	0.00
101010000	2	4	9.00	0.00	8.50	1.00	1.00	0.00	0.75	0.50
110001000	2	1	9.00	0.00	9.00	n.a.	1.00	0.00	1.00	n.a.
300001000	2	1	9.00	0.00	9.00	n.a.	1.00	0.00	1.00	n.a.
020010000	1	2	6.00	n.a.	6.00	2.83	0.00	n.a.	0.00	0.00
100200000	1	2	6.00	n.a.	7.50	0.71	0.00	n.a.	0.00	0.00

Note. The first digit in the profile indicates the number of presented-once digits in the list; the second digit indicates the number of presented-twice digits in the list; the third digit, the number of presented-thrice digits, and so on. For example, 601000000 indicates that six digits were presented once each and a single digit was presented three times; 112000000 indicates a digit presented once, a digit presented two times, and two digits presented three times for the total of 1+2+3+3 =9 digits in the list, with only four unique digits. The list profile 900000 is exclusively from the trial block in which no repetitions were allowed. All profiles with non-zero digits only within the leftmost two places of the profile come from one trial restriction, and all that include nonzero digits in the third place or beyond come from the other trial restriction. Trial frequency is summed across 33 participants in Experiment 1 and 40 participants in Experiment 2.

SD=standard deviation based on individual-trial scores pooled across participants.

Despite these constraints, it must be realized that the situation is too daunting for a complete analysis. For example, the three presentations of a digit could occur in 504 different combinations of serial positions within nine-item lists, and there is no way for us to examine all of them separately. By examining a variety of repetition types with and without grouping of the lists, we can provide the first evidence of how multiple repetitions can be handled based on the use of stimulus-based, spatial grouping cues (Experiment 1) and without them (Experiment 2). The two experiments produce results similar enough to warrant some cross-experiment analyses, but with some important differences between experiments as well, based on the grouping cues.

Repetition Effects and List Grouping

We presented lists of nine digits for immediate serial recall, a length on the high end of what people can manage (Miller, 1956). Three items per group, like the stimuli presented in Experiment 1, is known to be a group size particularly helpful for memory (Hitch et al., 1996; Ryan, 1969; Wickelgren, 1964). Although participants sometimes impose their own grouping or change the grouping from what was presented (e.g., Cowan et al., 2002; McNicol & Heathcote, 1986), by presenting very participant-friendly grouping in Experiment 1, we hoped to minimize departures from the use of the presented grouping, in the process minimizing the variability among responses to each stimulus in that experiment. However, it might make other groupings difficult and, in Experiment 2, we considered whether participants might be able to make better use of other repetitions among the stimuli, presumably because they could adjust the size of mental groups from trial to trial based on the stimulus repetitions present in each list.

Specific Research Questions

Given that it is not possible to examine every aspect of lists generated with multiple repetitions, we formulated several questions that can be examined both with and without grouping of the lists, at the intersection of what is possible to examine and what is theoretically interesting. We can examine some a general characteristic of lists (e.g., the number of unique digits) and some specific qualities within list (e.g., effects of a particular digit replicated at two different serial positions). We can also examine another extensive quality thought to be especially influential, yet reasonably common within our stimuli: series of identical digits in a row, which we term a train of items. The effect of a train may depend on the starting serial position. Last, a series may be meaningless on its own (e.g., 394) but that pattern might repeat elsewhere in the list. Based on these factors, we formed seven general hypotheses, often with some with variants, as follows.

Hypothesis 1: Effect of the number of unique digits in the list.

When the number of unique digits to be remembered increases within the range of our stimuli, there should be an inverted-U-shaped effect on accuracy overall. This expectation is based on the point that recall requires both correct items and correct placement of those items. A list with many unique items may be difficult to remember because of the need to retain the items. However, a list with very few unique items may be difficult to remember because of the complication of ordering these items (e.g., in the list 4344254345). Results should apply to both lists recalled completely correctly and mean number of digits correct per list.

Hypothesis 2: Items repeated at two serial positions.

According to evidence on the Ranschburg effect (Henson, 1989), item repetitions tend to be helpful when they are adjacent and can become harmful when they are separated. The helpful effect can come from grouping them together, and the harmful effect can come from the recall of an item leading to its post-recall suppression (e.g., Henson, 1998), interfering with recalling it again later in the same list. We expect these factors to differ somewhat for the two experiments. For Experiment 1 (grouped presentation), Hypothesis 2a is that repetition should show a benefit if the digits fall within a group or at the same within-group location of two different groups (e.g., Positions 1 and 4 out of 9). The basis of this prediction is the assumption that participants form a hierarchical, grouped associative structure for the list (e.g., Farrell et al., 2011; Lee & Estes, 1981). Identical items that should be associated with one another should make the associative process easier. In Experiment 2 (ungrouped presentation), however, Hypothesis 2b is that repetitions of nearby items should be helpful and form a usable unit even when they do not fall within the same group of three or at the same within-group location of different groups of three. For example, there might be an advantage of a repetition in Positions 3-4 (e.g., 389925486), inasmuch as participants could form groups of different sizes to capitalize on repetitions that occur in the list (e.g., in the example, perhaps 3899-25-486). On the other hand, the ability to form ad hoc groups may be limited if the process of forming the groups comes with a cost to capacity (Norris et al., 2020).

Hypothesis 3: Maximum length of a train.

One form of grouping that is relatively easy to examine is what we call a train, a situation in which a digit is presented consecutively two or more times (e.g., the digit 6 in the list 936666528). Longer trains should be more helpful because they reduce the number of remaining digits to be recalled. We were able to examine the effect of the maximum length of a train.

Hypothesis 4: Accuracy for lists with three-item trains at different positions.

Three-item trains of successive, identical stimuli are the longest ones that we could examine reliably at all serial positions because the longer trains were too infrequent in the stimuli. Three-item trains are expected to be a powerful source of list grouping. In Experiment 1 (grouped presentation), Hypothesis 4a states that these trains should be most helpful when they fall within a group, but less so when they fall between groups. The assumed associative structure that could be formed based on the provided grouping into triads is violated by identical items crossing group boundaries. Hypothesis 4b states that, for Experiment 2 (with ungrouped presentation), trains of three identical digits should be of use regardless of their placement, inasmuch as the participant is free to establish a particular grouping at will to capitalize on the location of cues in the stimuli.

Hypothesis 5: Serial position effects for three-item trains at different list positions.

It is possible to gain more insight into the grouping cues that are presumably the most prominent among cues we can examine, the trains of three identical items, by tracing performance across the entire list as a serial position function. Any benefit of a train of three identical digits should extend beyond the location of that repetition to other serial positions in the list because the repetition reduces the total memory load (cf. Norris et al., 2020).

Hypothesis 6: Chaining and fan effects.

There are two main theories about how serial order information is represented and used in immediate memory tasks. First, serial order could be represented in terms of associations between items and their serial positions, or in a hierarchical version of that, also between groups of items and serially ordered group nodes (Lee & Estes, 1981). Second, in the chaining hypothesis, the information is represented at least partly in terms of associations between each item and the next one (for a review of these competing models see Kalm & Norris, 2014). The chaining hypothesis makes a special prediction for repeated items when combined with the fan effect (Radvansky, 1999). If a digit is repeated in two locations, then there should be a degradation in the performance on the items following the instances of the repeated digit because the chain or association would be diluted by the fan effect. For example, if the digit 8 were repeated in two locations in a list (e.g., 382954861) then there would be conflicting chained associations from the repeated digit (e.g., 82 and then 86). By checking for this possible degradation of performance, we indirectly evaluate the chaining hypothesis. If one alternatively proposes that chaining does not occur, or that fan effects do not apply in this circumstance, then these effects would not be expected.

Hypothesis 7: Multi-Item Sequence Repetition.

In a more complex type of repetition, a random series repeats, as for example in the sequence 654398439, in which the series 439 occurs twice. In Hypothesis 7a, it could be expected that the repetition might be noticed and might strengthen the recall of that series. Presentations of the series that are farther apart may benefit from the spacing effect, in which learning is superior when repetitions of the learned material are further apart (Cepeda et al., 2006; Dempster, 1988; Ebbinghaus, 1885/1913), though that effect has been examined primarily in longer-term recall. However, to the contrary, in Hypothesis 7b, a disadvantage of series repetition could occur because of suppression of the second repetition (e.g., Henson, 1998) or because the first and second presentation of the series are surrounded by different digits and are associated with different serial positions, diluting the use of these associations through a fan effect (Radvansky, 1999).

For comparison between experiments, most of the data figures show results for both experiments. For ease of exposition, we therefore present the methods of both experiments followed by the results, organized by hypotheses.

Method

Experiment 1

Participants

The participants were 33 college students and other young adults (9 male, 24 female) who received course credit or payment for their participation and were native speakers of English with normal or corrected-to-normal vision.

Apparatus, Stimuli, and Procedure

Apparatus, stimuli, and timing.

The experiment was conducted in sound-attenuated booths. Participants studied 9-digit numbers on a computer screen (white numbers on a gray background) and then typed in their responses after a delay. The numbers were presented in a single row with spaces between groups, as 3 groups of 3 digits. Each digit fit in a rectangle approximately 0.8 cm tall and 0.5 cm wide, with 0.1 cm between digits and 0.5 cm between groups of three digits within the list, viewed from about 50 cm away for a list width of about 7 degrees in visual angle. There was 1 s before each list, the list was presented with all 9 digits concurrently for 3.86 s, and there were 2 s between the list and the response cue.

Responses and practice.

Participants typed responses with the keypad and were allowed to backspace but were to attempt to reproduce the presented order before indicating with a final key press that the response was complete. These capabilities were included because we were interested in participants’ judgment without that judgment being obscured by any key-in errors they might make, just as false starts can be corrected in most oral recall methods. Repetitions in the response were allowed even when there were none in the stimulus list.

Participants typed in their responses on the 10-key pad on a standard keyboard, except that the digits were renumbered so that they were like a phone keypad (i.e., 1 is in the top left). Participants were able to choose to use the number pad as stickered or, if they were more comfortable with the standard 10-key entry, they could use the standard keypad numbering, which was available on a second keyboard. After making an initial keypad selection, participants were given 3 practice trials, after which they could make another selection. If the participant made the same selection twice in a row, that was their final selection that they used for the entire experiment. After making their final keyboard selection, they did 3 more practice trials.

Test trial order.

Every list contained digits selected randomly with restrictions. Specifically, there were three types of lists that differed in terms of the pattern of digit repetitions: (1) no digit repetitions, (2) at least one digit repeating exactly twice (not necessarily adjacent to one another) and no other digit repeating more than twice, and (3) at least one digit repeating three or more times (again, not necessarily adjacent to one another). Not including practice trials, there were three blocks of 65 test trials, or 195 test trials in all. One trial block included only trials of the first type (no repetitions) and the other two blocks included an equal number of the other two types (which always included repetitions) intermixed in approximately equal numbers. The order of the blocks was counterbalanced. Thus, a third of the students received the non-repetition block of trials first, a third received it second, and a third received it last among the three trial blocks.

List distribution.

Table 1 shows the different profiles of lists presented in the experiments, with the frequency of occurrence of each profile and overall performance statistics by profile. These profiles are defined here in terms of the number of digits appearing only once in the list, the number of digits appearing twice, and so on, regardless of the serial positions of each repetition of a digit. the frequencies of occurrence are a natural consequence of the restrictions on randomization that we used; thus, the frequencies of occurrence in the two blocks of trials with repetitions were correlated at r=.998.

Analyses.

In our analyses, we carried out conventional t tests and analyses of variance to provide descriptive measures of F and partial eta squared (η_p²), the proportion of variance relevant to an effect that was accounted for by the effect as opposed to error variance. For inferential purposes, however, we rely on Bayes Factor, BF, defined here as the likelihood of a statistical model that has the relevant effect included divided by the likelihood of a control model that omits the effect (JASP Team, 2019). This control model in the case of a t test was the point null. In the case of a multifactor ANOVA, JASP weighed all relevant comparisons. For example, if an analysis includes A and B effects then, to examine the B effect, the analysis would compare the model including A and B to the one including only A, and also would compare a model with only B to a model with neither A nor B. These Bayes factors would be weighted. The program limits models with an interaction to those in which all main effects and any lower-level interactions using the same factors are also included. Values of BF above 3.0 are considered moderate evidence of the hypothesis in question, reflecting a likelihood of the model with the effect in question three times as high as the comparable model that omits the effect. Values of BF below 0.33 (i.e., below the reciprocal of 3.0) are considered moderate evidence in favor of the null hypothesis (Wagenmakers et al., 2018).

Experiment 2

Participants

Allowing further experimentation during COVID-19, we collected data online, increasing the sample size to 40 participants (29 female, 10 male, and 1 unreported) in case variability was greater than in the laboratory. Participants were paid through an online service (Prolific) that administered the test using an experimenter-provided program. Inclusion criteria for the study were (1) being between 18 and 30 years old, (2) being a native speaker of English, (3) being British, American, or Canadian, (4) having normal or corrected-to-normal vision, (5) having no reported cognitive impairment or dementia, (6) having no reported language-related disorders, and (7) having an approval rating of at least 90% on prior submissions at Prolific. Two studies indicate that online data collection yields results similar to the laboratory (Germine et al., 2012; Peer et al., 2017).

Apparatus, Stimuli, and Procedure

The procedural details were the same as in Experiment 1 except that (1) the study was run on participants’ personal devices at a location of their choice, and (2) there were no spatial gaps between triads; each list now comprised an uninterrupted string of nine digits.

Results

Materials and data from this project are available at https://osf.io/mycwz/. Table 1 shows that performance overall was very similar for the two experiments. The mean items correct as a function of profile were correlated between experiments, r(22)=.58, BF=9.75, and the mean lists correct as a function of profile were correlated even more strongly, r(22)=.79, BF=2,147. Overall, there is no obvious advantage or disadvantage for recall of digits presented a particular number of times (Table 2), but there are patterns to be observed when looking more closely at the distribution of repeated items, which we examine according to the hypotheses presented above.

Table 2.

Performance on Items Presented Different Numbers of Times Within Each Profile, in Two Experiments

Profile	Frequency	Once-Presented items				Twice-Presented Items				Thrice-Presented Items				Items Presented 4 Times
		Exp.1		Exp.2		Exp.1		Exp.2		Exp.1		Exp.2		Exp.1		Exp.2
		Mean	SD	Mean	SD	Mean	SD	Mean	SD	Mean	SD	Mean	SD	Mean	SD	Mean	SD
900000000	2145	0.77	0.23	0.75	0.24
520000000	952	0.75	0.24	0.74	0.28	0.75	0.26	0.76	0.28
411000000	826	0.75	0.28	0.75	0.28	0.75	0.34	0.76	0.33	0.76	0.29	0.76	0.28
330000000	796	0.72	0.30	0.74	0.30	0.76	0.24	0.76	0.25
221000000	468	0.76	0.34	0.75	0.34	0.77	0.25	0.77	0.27	0.76	0.28	0.78	0.28
710000000	275	0.75	0.23	0.77	0.25	0.78	0.33	0.79	0.32
601000000	223	0.76	0.23	0.76	0.26					0.80	0.28	0.79	0.29
310100000	173	0.81	0.25	0.83	0.25	0.81	0.29	0.83	0.30					0.82	0.23	0.83	0.24
302000000	139	0.77	0.28	0.74	0.30					0.78	0.23	0.77	0.25
140000000	122	0.72	0.45	0.73	0.44	0.77	0.22	0.77	0.23
500100000	95	0.79	0.24	0.76	0.28									0.80	0.27	0.75	0.29
120100000	54	0.83	0.38	0.79	0.42	0.78	0.25	0.76	0.25					0.73	0.27	0.79	0.25
112000000	52	0.79	0.41	0.82	0.39	0.70	0.28	0.83	0.26	0.80	0.17	0.81	0.20
201100000	37	0.68	0.32	0.79	0.32					0.72	0.32	0.81	0.29	0.72	0.32	0.83	0.26
031000000	32					0.75	0.24	0.76	0.27	0.81	0.28	0.80	0.32
400010000	22	0.90	0.17	0.79	0.25
210010000	10	0.95	0.16	0.76	0.39	0.75	0.26	0.79	0.25
011100000	4					0.88	0.25	0.63	0.48	0.83	0.19	0.50	0.43	0.88	0.14	0.63	0.14
003000000	2									0.94	0.08	0.74	0.13
101010000	2	1.00	0.00	1.00	0.00					1.00	0.00	0.92	0.17
110001000	2	1.00	0.00	1.00		1.00	0.00	1.00
300001000	2	1.00	0.00	1.00
020010000	1					0.75		0.50	0.35
100200000	1	0.00		1.00	0.00									0.75		0.81	0.09

Note. The first digit in the profile number indicates the number of unrepeated digits in the list; the second digit indicates the number of presented-twice digits in the list; the third digit, the number of presented-thrice digits, and so on. For example, 601000000 indicates six digits that were presented once each and a single digit that was presented three times; 112000000 indicates a digit presented once, a digit presented two times, and two digits presented three times for the total of 1+2+3+3 =9 digits in the list but with only five unique digits. Trial frequency is summed across 33 participants in Experiment 1 and 40 participants in Experiment 2. SD=standard deviation based on individual-trial scores pooled across participants.

Hypothesis 1: Effect of the Number of Unique Digits in the List

According to the hypothesis, it should be harmful to have either too few or too many unique digits in the list. Lists with only 3 unique digits happened to be presented to only 10 participants in Experiment 1 and 14 participants in Experiment 2, too few to analyze, and there were no cases with only 2 or 1 unique digit (Table 1). We were, however, able to investigate this question for 4 to 9 unique digits per list, as shown in Figure 1. It shows the results in both experiments for two dependent variables: the number of lists correctly recalled (left-hand panel) and the number of items per list recalled in the correct serial position (right-hand panel).

Figure 1.

Performance for different numbers of unique digits per list (x axis) in both experiments (graph parameter). Left-hand panel, proportion of lists recalled correctly; right-hand panel, number of items per list recalled in the correct serial position. The lists were spatially divided into groups of three in Experiment 1 but not in Experiment 2. Lists with 9 unique digits come entirely from the trial block with no repetitions. Error bars represent plus and minus one standard error of the mean.

In Experiment 1, in an ANOVA with the number of unique digits (4-9) as a repeated measure and lists correctly recalled as the dependent variable, there was no effect, n=32, F(5, 155)= 1.40, η_p²=0.04, BF=0.13. That was the case as well for a comparable ANOVA with the number of items correct per list as the dependent variable, F(5, 155)=0.76, η_p²=0.02, BF=0.04. In an ANOVA in Experiment 2 on lists correctly recalled, there was a potential effect, but in an indeterminate range, n=40, F(5, 195)= 2.87, η_p²=0.07, BF=1.46. However, in a fourth ANOVA, there were clear findings of an effect of the number of unique digits on the number of items correct, F(5, 195)=4.04 , η_p²=0.09, BF=13.05. Bayesian pairwise tests indicated that the differences occurred between 4 unique items and 6 items (BF=8.65), 7 items (BF=11.36), or 9 items (BF=19.99), and between 5 and 7 unique items (BF=3.33). Fewer unique items led to better performance. The function was not U-shaped as had been expected.

To understand more about why fewer unique items were better in Experiment 2, we examined the characteristics of the responses as a function of the number of unique items in the stimulus (Figure 2). These results are quite similar across the two experiments. The number of unique items in the response is comparable to the number of unique items in the stimulus, overshooting the stimulus number a bit when there are few unique items and undershooting it a bit when there are more unique items. This pattern is accompanied by a shift from intrusion errors when there are few unique digits to omission errors when there are many. However, the pattern, together with Figure 1, suggests that most errors involve reordering of digits, and not omitted digits or intruding digits that do not belong in the list.

Figure 2.

Number of unique digits bearing several relations to the response, as a function of the number of unique digits in the list (x axis). Left-hand panel, Experiment 1; right-hand panel, Experiment 2. The y axis indicates the mean number of unique digits per list for each response outcome (graph parameter). Standard error of the mean for each condition was at most 0.13 in Experiment 1 and at most 0.12 in Experiment 2, small enough that they have been omitted from the figure.

In sum, the evidence available suggests an advantage for lists with only 4 unique items compared to lists with more unique items, and not a potential disadvantage as suggested. Four unique items may be a “sweet spot” for making nine-item lists most memorable, although grouping divisions in Experiment 1 may prevent this advantage from materializing fully in terms of avoiding order errors.

Some errors might occur because of over-regularization of a lists to conform to a perceived structure. The next section provides evidence consistent with that notion.

Hypothesis 2: Items Repeated at Two Serial Positions

We examined an issue that is like the Ranschburg effect (Henson, 1989) except within the context of lists with multiple repetitions. In a series of analyses, each list was classified as to whether the first and second items were identical, whether the first and third items were identical, and so on, all the way to the eighth and ninth items. For each case, we examined the number of list items correct when the two items were identical versus when they were not and calculated a difference score for each pair, for each individual. Hypothesis 2a was an expectation that results would conform to a triad grouping in Experiment 1, and Hypothesis 2b was the expectation that in Experiment 2, participant could make better use of identical pairs even when they violated the triad grouping structure.

Difference scores from pairs at the same positions that were or were not identical is shown in the three panels of Figure 3. The top panel shows the benefit of identical pairs of items from the same list triad, which are supposed to be helpful in the formation of associations between items in a triad, possibly more so when the stimuli are grouped (Experiment 1). The panel shows that identical items of this sort were consistently helpful, about equally overall for the two experiments.

Figure 3.

Benefit of a match between items in two different serial positions (x axis) for the number of items correct in the list, in both experiments (bar color). Pair numbering: Pair Number 15 indicates that identical items occurred in Positions 1 and 5, and so on. Top panel, digits from the same triad; middle panel, digits from the same equivalent within different triads; bottom panel, digit pairs with neither of these potential advantages. Error bars indicate plus and minus one standard error of the mean. *= BF>3.0 experiment difference. += BF>2.0 (Pair 37).

The middle panel of Figure 3 shows the benefit of identical pairs of items from the same locations within different triads (e.g., Positions 1 & 4, the first digits in their respective groups). Here, these matches were of more use in Experiment 1, with significant differences between experiments favoring that experiment in t tests. Apparently, the stimulus grouping provided in Experiment 1 facilitated the process of participants noticing and using to their advantage the correspondence between items filling the same slots within groups, as expected.

The bottom panel of Figure 3 shows all remaining matches, neither in the same triad nor in the same position within their respective triads. In this situation, the larger benefit of the matches occurred for Experiment 2 in 10 of 12 significant cases, as expected. The effects of these matches were predominantly negative (harmful) in Experiment 1 and helpful in Experiment 2. As expected, the absence of grouping cues in Experiment 2 seems to have allowed participants flexibility in finding a mental structure that best suited the string of digits presented on a trial. The greatest benefit for Experiment 2 occurred for pairs of digits that were consecutive but straddling a group boundary, i.e., matches between Positions 3 and 4 and between Positions 6 and 7, whereas those identities did not help in Experiment 1. Thus, despite the overall similarity of performance in the two experiments, there are clear differences in line with expectations.

Further insight into why these differences between experiments emerged can be gleaned from Figure 4, which shows the effects of trains of two identical items on recall of the two items themselves. In Experiment 1 (top panel), many times the recall of the second item in the pair was harmed by the pairing. This was not the case in Experiment 2 (bottom panel). This is not a difference one would expect if the harm to the second item in a pair came from suppression of the output, as that suppression should tend to be lifted when moving from one group of three items to another. Instead, it may be that, in Experiment 1, the pairing interferes with the hierarchical associative structure in which trains of two items do not help unless they are items in the same group, or at the same within-group position of successive groups. Thus, in Experiment 1, a Position 1-4, 1-7, and 4-7 match, in which both items begin their group, seemed helpful or, at least, not harmful. None of the below-zero, harmful effects of pairing occurred for pairs that occupied the same group or the same within-group positions. In Experiment 2, in which grouping was not specified by stimulus cues, there were few harmful effects at any serial positions.

Figure 4.

Proportion benefit of a match (match minus no match) across participants, specific to the first and second serial positions potentially involved in the match (graph parameter). Pair numbering: Pair Number 12 indicates that identical items occurred in Positions 1 and 2, and so on. Top panel, Experiment 1; bottom panel, Experiment 2. Each two-digit number on the X axis represents the two serial positions at which the identical digits occur.

Hypothesis 3: Length of a Train

A clear, simple prediction was that longer trains of identical items presented at consecutive serial positions would be helpful for recall. The result in this regard was clear and simple, as shown in Figure 5. There was little difference for the two experiments, and we used both in a common analysis. The finding was reinforced by four different analyses with the maximum train length as a repeated measure. For lists correctly recalled, we examined the results in two ways. We carried out an ANOVA across trains of 1, 2, 3, and 4-6 digits in the train. We averaged each individual across identical trains with 4 through 6 items because few participants had trains of all three lengths. Many participants did not have any of these train lengths, so we also carried out an analysis of train lengths 1-3, which most participants did have. The same two analyses were carried out on items correct. Given the similarity of results for the two experiments (Figure 5), we report analyses combined across experiments and did not find an effect of experiment.

Figure 5.

Performance as a function of the maximum train length (x axis) within the list in each experiment (graph parameter). The lists were spatially divided into groups of three in Experiment 1 but not in Experiment 2. Maximum Train Length 1 comes entirely from the trial block with no repetitions. Left-hand panel, proportion of lists correct; right-hand panel, number of items correct per list. Error bars indicate plus and minus one standard error of the mean.

All of the analyses converged on comparable effects. For lists correct, using all levels of train length, the data included 17 participants from Experiment 1 and 15 from Experiment 2. The effect of maximum train length was F(3, 90)=25.82 , η_p²=0.46. BF>100,000. Bayesian paired tests showed that there was no difference between maximum train lengths of 1 (i.e., no trains) and 2, BF=0.29; whereas the other pairs were all different from one another with BF greater than 28. Using only maximum train lengths of 1 through 3 all participants were included, and an effect was again obtained, F(2, 142)= 20.89, η_p²=0.23, BF>100,000. Again, post-hoc analyses showed no difference between maximum train lengths of 1 and 2, BF=0.13, and differences between all other lengths, BF>900 in each case. For the number of items correct per list, using all train lengths, the effect was again positive, F(3, 90)=25.87, η_p²=0.46, BF>100,000, and post-hoc tests showed an indeterminate finding for 1 versus 2, BF=1.11, whereas all other pairs of values were different, BF>31 in each case. The finding was strongest for number of items correct using only maximum train lengths of 1 through 3, for which all participants were included, F(2, 142)= 29.91, η_p²=0.30, BF>100,000. In this case, there was a difference between maximum train lengths of 1 and 2, BF=3.53, as well as between the other pairs of lengths, BF>6,000 in each case. Clearly, as expected, the maximum length of a train is an important factor and, overall, it plays a similarly important role with or without grouping of the lists.

Hypothesis 4: Accuracy for Lists with Three-item Trains at Different Positions

The number of participants who happened to receive three-item trains at each possible list position (Positions 1-3, 2-4, etc., through 7-9) were, for Experiment 1: 25, 26, 22, 27, 25, 28, and 31, respectively; for Experiment 2: 25, 28, 29, 28, 28, 28, and 25, respectively. As expected, and as shown in Figure 6, trains of three were a great benefit at most serial positions. Separately for each starting serial position of the train, we carried out an ANOVA of items correct throughout the list, with the experiment as a between-participant factor and with the presence or absence of a three-item train and serial position (1-9) as within-participant factors. The evidence for the helpfulness of the three-item trains was strong, with main effects of the presence versus absence of the train clear except in Positions 3-5 [Positions 1-3, F(1,48)=22.63, η_p²=0.32, BF>100,00; Positions 2-4, F(1,52)=3.90, η_p²=0.07, BF=9.79; Positions 3-5, F(1,49)=2.81, η_p²=0.05, BF=2.57; Positions 4-6, F(1,53)=9.57, η_p²=0.15, BF=64,759; Positions 5-7, F(1,51)=17.97, η_p²=0.26, BF>1000,000; Positions 6-8, F(1,54)=5.36, η_p²=0.09, BF=53.46; Positions 7-9, F(1,54)=12.76, η_p²=0.19, BF>100,000.]

Figure 6.

Mean proportion of lists recalled correctly as a function of the location of a train of three identical digits in each experiment. Left-hand panel, proportion of lists correct; right-hand panel, number of items correct per list. Lists with repetitions at other locations in the list were included but not lists with more than one train of three or more. Error bars indicate plus and minus one standard error of the mean.

A more specific aspect of the hypothesis under investigation is that within-triad trains would be more helpful in Experiment 1 (Hypothesis 4a), whereas trains that extend between triads will be more helpful in Experiment 2 (Hypothesis 4b). These expectations were confirmed by the interaction of experiment with the presence or absence of the train, favoring Experiment 2 for trains at Serial Positions 2-4, F(1,52)=4.47, η_p²=0.08, BF>16.92, but favoring Experiment 1 for Serial Positions 7-9, F(1,54)=2.49, η_p²=0.04, BF>7.83 (for means see Figure 6). These results were as expected but the interaction did not materialize for analyses with trains starting at other serial positions (Positions 1-3, BF=0.26; Positions 3-5, BF=0.15; Positions 4-6, BF=0.20; Positions 5-7, BF=0.51; and Positions 6-8, BF=0.26).

Hypothesis 5: Serial Position Effects for 3-Item Trains at Different List Positions

The fifth hypothesis was that any benefit of a 3-item train should extend beyond the serial positions involved in the train, improving performance at other list positions as well. This hypothesis was examined using the analyses mentioned above. The first seven panels of Figure 7 show that, for various train positions, the presence of a train was superior to its absence (for each possible train position, respectively, BF>100,000; 9.79; 2.57; 64,759; >100,000; 53.46; and >100,000). For the first three such positions (top row of the figure), this appeared to be the case, as the presence or absence of a train did not interact with the serial position of the list (for a train in Positions 1-3, F(8,384)=3.01, η_p²=0.06, BF=0.23; in positions 2-4, F(8,416)=0.63, η_p²=0.01, BF=0.00; and in Positions 3-5, F(8,392)=1.03, η_p²=0.02, BF=0.01). For the last four train positions, however, the benefit was much larger for the latter part of the list (a train in Positions 4-6, F(8,424)=4.27, η_p²=0.08, BF=5.51; in Positions 5-7, F(8,408)=2.84, η_p²=0.05, BF=47.02; in Positions 6-8, F(8,432)=6.78, η_p²=0.11, BF=896.36; and in Positions 7-9, F(8,432)=15.64, η_p²=0.23, BF>100,000.

Figure 7.

Serial position curves for trials with a train of three identical digits at each of seven possible starting positions, trials without a train beginning at that position, and trials with no repetitions at all for comparison (graph parameter). See the graph parameter labels for the starting positions. The first seven panels show the data collapsed across experiments. The last two panels in the bottom row show the train (solid-line) and no-train (dashed-line) trials for the two starting serial positions in which the effect of the train differed reliably between the two experiments. Error bars indicate plus and minus one standard error of the mean.

For most of these train positions, there was no statistical difference between experiments, the only exception being the effects for a train of identical items in Positions 2-4 and 7-9, as noted above. As shown in the next-to-last panel of Figure 7, there was no benefit of a particular train that crossed triads (specifically, Positions 2-4) in Experiment 1, but there was a benefit for it in Experiment 2, most markedly in recall of the early serial positions. It presumably occurred due to the greater flexibility of mental grouping of the list in that experiment. In the last panel of the figure, we see that a train in Positions 7-9 improved memory for the end of the list in both experiments, but at an apparent cost in memory for early list items in Experiment 2. This cost could occur because of the need to determine the best grouping in Experiment 2. None of the three-way, train presence x serial position x experiment interactions were reliable.

Overall, the detailed pattern shown in Figure 7 provides mixed support for Hypothesis 5, in that the benefit of trains of three identical items was extended across the list. The data also provide further detail to the findings examined for Hypothesis 4, suggesting that the benefit specific to one experiment or another also went beyond the specific serial positions in which a train occurred.

Hypothesis 6: Chaining and Fan Effects

In chaining, associations should be formed between Positions N-1 and N in the list, which should contribute to the recall of Item N. In fan effects, if the item in Position N-1 is not unique, it should provide a poorer cue to the recall of the item in Position N. We examined chaining and fan effects by finding the location of the first error in the list for each Serial Position N from 2 to 9 and recording the number of repetitions of the digit in Position N-1 throughout the list. For example, if the list 493844083 is recalled as 493722083, the first error was made at Position 4 (i.e., N=4) and the preceding Position 3 contains a digit that was presented twice in the list (i.e., the result in this trial is 2). This kind of result was compared to the number of repetitions of the digit in Position N-1 within trials in which there was no error at least through Position N. (We excluded trials in which Positions N-1 and N contained the same digit, signifying repetition effects that we examine below.) The use of this measure eliminates “knock-on” effects in which an error at a certain position leads to further, less interpretable errors later in the list.

Figure 8 shows the result separately for the two experiments. Chaining and fan effects were clearly present, as shown by the higher level of repetition of the N-1 item in trials in which the first error was made at a particular serial position, compared to trials with no error up through that position. In our statistical analysis of the effect, it was not possible to include all serial positions in the same analysis because the relevant data for each position are not independent within the control condition; trials with no error at least through Position 3 include trials with no error at least through Position 2, and so on. The Bayes Factors in Table 3 confirm that there generally were chaining and fan effects, indicated by the larger number of Position N-1 item repetitions in trials with the first error at Position N, compared to no error. There are some interesting exceptions in the case of items at the end of a presented triad in Experiment 1; the status as the last item in a group may override the use of chaining.

Figure 8.

Chaining producing fan effects. In each experiment (left and right panels), for each serial position N from 2 to 9 (x axis), the y axis shows the mean number of presentations throughout the list of the N-1 item. The result is shown separately when N was the position of the first error in the list (solid lines) versus when the list recall was correct up through Position N (dashed lines). Error bars are standard errors, too small to see for some positions in the dashed lines. These results exclude trials in which Positions N-1 and N were the same digit. The larger numbers of repetitions of the item in Position N-1 for errors indicate that the the item in Position N-1 must affect recall of Position N (chaining) and that less distinct Position N-1 to N chaining cues hurt performance (fan effects).

Table 3.

Bayes Factors from Analyses of Fan Effects

Serial Position N	First Error Versus None Effect	Experiment Effect	Interaction
2	3633.56	0.25	0.28
3	10.18	1.37	4.30
4	1.517e +6	0.21	0.24
5	31757.87	0.07	0.24
6	6.92	7.01	14.91
7	2.766e +8	0.31	5.19
8	41724.77	9.94	1.59
9	0.53	0.53	0.23

Note. The dependent measure was the number of repetitions throughout the list of the item at Serial Position N-1. At the displayed Serial Position N, either the first error in the recall of the list occurred or (in the control condition) the list was recalled correctly at least through that position. The results exclude trials in which the digits in Positions N-1 and N were identical. Each serial position was analyzed separately in a Bayesian ANOVA with Experiment between participants and the condition (first error or none) within participants. Bolded effects are those considered to provide evidence substantially in favor of a non-null finding.

Hypothesis 7: Multi-Item Sequence Repetition

Not only digits can repeat, but also digit series. For example, the series 583965822 has the series 58 in two locations. We had enough data to examined series of two or three items repeating at two points in a list. Caution must be taken to ensure that the effect of repetition is not confounded with the serial positions in which the repeated items occur. For 2-item series repeated, we looked separately at each possible starting place for the first occurrence and second occurrence of the pattern. In each case, it was compared to the condition in which there were no repetitions in the list. Trials in which there was more than one repetition in the list were excluded. The possible starting positions for each two-digit series, examined in separate analyses, were 1&3, 1&4, 1&5, 1&6, 1&7, 1&8, 2&4, 2&5, 2&6, 2&7, 2&8, 3&5, 3&6, 3&7, 3&8, 4&6, 4&7, 4&8, 5&7, 5&8, and 6&8. Each comparison was done in an ANOVA with repetition condition (no repetition versus a specific series repetition) within participants and with the experiment between participants.

Among these analyses, the only one that yielded an effect was for starting serial positions 3&6, the condition in which a series (e.g., 58) occupied Positions 3 and 4, and also Positions 6 and 7. Notice that this condition is the only one that has the same series bridging the first two triads of the list and bridging the second two triads. There were 23 participants from Experiment 1 and 25 participants from Experiment 2 who had this condition. This analysis yielded no clear main effect of the repetition condition, F(1,46)= 2.50, η_p²=0.05, BF=0.51, or of the experiment, F(1,46)= 3.50, η_p²=0.07, BF=1.28, but did show an interaction between the condition and experiment, F(1,46)= 5.26, η_p²=0.10, BF=10.72. In Experiment 1, the proportion of list items correct in the condition with no repetitions (M=0.77, SEM=0.02) was exceeded by the condition with the repeated pattern (M=0.86, SEM=0.03). In Experiment 2, however, the no-repetitions (M=0.74, SEM=0.03) and repeated pattern (M=0.72, SEM=0.05) conditions did not differ. Figure 9 shows that for this comparison, in Experiment 1 the effect seems distributed across the list. Apparently, when the items were presented grouped into triads in Experiment 1, a consistent bridge between groups was of special use. No effects were obtained for repeating series of 3 items.

Figure 9.

Proportion of items correct in each serial position for participants in Experiment 1 (left) and Experiment 2 (right) in the no-repetition condition and in the condition in which a two-digit series at Serial Positions 3 and 4 was repeated at Serial Positions 6 and 7, spanning between triads of items in the list (graph parameter), as in the series 295 845 867. Error bars indicate plus and minus one standard error of the mean.

Our finding does not lend support either to the specific possibility we raised that repetition of a series would be of general use because of learning of the pattern, especially with more space between patterns (Hypothesis 7a), or to the contrary specific possibility we raised that this repetition would be harmful because of the occurrence of the pattern in two different contexts that interfere with one another, perhaps due to fan effects (Hypothesis 7b). Instead, only a more limited use of repetition was found, allowing participants in Experiment 1 to bridge between triad groups when the presentation included spatial gaps between triads.

Discussion

The present paper contrasts with the long, rich research literature on serial recall of lists without repetition (see Oberauer et al., 2018 for a review), or with a few carefully controlled repetitive features (e.g., Burtis, 1982; Henson, 1989; Mathy & Feldman, 2012). Here, we have introduced quasi-random repetitions of items to examine conditions more like the practical uses of strings of characters for the identification of people or products, and to begin to assess the kinds of effects of various patterns introduced through repetitions. This topic also allows a comparison of various hypothetical mechanisms that may play a role in serial recall.

Outcome of Hypothesis Testing

The analyses were organized around seven hypotheses. Hypothesis 1 had to do with the number of unique digits in the list. It was expected that the number could be too small or too large, resulting in too many errors from ordering difficulties, on one hand, or from an absence of structure that repetition can bring, on the other hand. Instead, there was some evidence that lists with 4 unique digits produced better recall than lists with less repetition and therefore more unique digits. This finding (Figures 1–2) is the initial evidence that repetition is sometimes of use. We had too few trials to test lists with only 2 or 3 unique digits, so it is possible that some of the predicted additional difficulty from serial order confusion would emerge in those cases. That is an important goal for future research.

Hypothesis 2 had to do with items repeated at two serial positions. The top panel of Figure 3 shows that when the two digits fall within the same triad of items, it is generally helpful for recall, whether the items are presented in a spatially grouped (as in Experiment 1) or ungrouped (as in Experiment 2) manner. The middle panel shows that participants also make associations between identical items presented in the same intra-triad location, such as the fourth and seventh serial positions, both starting a triad. This kind of more complex correspondence between items was more helpful to participants in Experiment 1 than Experiment 2, showing that there is some extra benefit of the grouping in perceiving the intended structure when the digits correspond to that structure.

There is, however, a cost to that perception of structure. The bottom panel of Figure 3 shows that, for identical pairs of digits that did not occur within a triad or in comparable positions within different triads, there was a discrepancy between experiments; in many of these instances, repetition was harmful in Experiment 1, with its grouped presentation, but helpful in Experiment 2, in which the list was not grouped. This pattern was as predicted. Figure 4 shows that, more specifically, harm in the first experiment accrued to the second item in a repeating pair, when the identical items in the pair did not match the presented grouping. This harm can be attributed to the use of the presented grouping but is contrary to certain other principles. For example, the principle of suppression of a second presentation of an item would be expected to occur in both experiments. Emphasis on this principle might be less in experiments with repetitions inasmuch as the expectation of repetitions could induce participants to reduce the amount of suppression, voluntarily or otherwise, when it is counterproductive.

Hypothesis 3 was about the length of a train of the same item multiple times in immediate succession, clearly a potent cue to grouping as expected. Longer trains were more beneficial, with quite similar outcomes of this measure in the two experiments (Figure 5). Apparently, different benefits and drawbacks of presentation methods in the two experiments may balance out overall, so that there is not a clear benefit or drawback of presenting lists in a spatially grouped manner in this regard.

Hypothesis 4 has to do with the accuracy of recall for lists that include, specifically, trains of three items in a row. This reflects presumably the most consequential form of repetition that we are able to study with a sufficient number of trials. Figure 6 shows that these trains were generally helpful to recall, especially when they fell within a triad but to some extent even when they crossed triad boundaries, in both experiments. With trains of three, participants may have little uncertainty regarding where the train takes place, allowing a benefit rather than a hinderance even if the train crosses a group boundary.

There were experimental differences of the predicted types, but only selectively. With a three-item train at Positions 2-4, crossing triad boundaries, there was a benefit only in Experiment 2 whereas, with a three-item train at Positions 7-9, there was a benefit that was larger for Experiment 1, presumably because participants could benefit from the match between the triad grouping and the location of the train. Follow-up research is needed to determine why these effects were not found at all possible serial positions.

Hypothesis 5 also has to do with the effects of trains of identical items presented across consecutive serial positions. The hypothesis was that trains would help not only the serial positions involved in the train, but also the rest of the list by reducing the memory load (cf. Cowan et al., 2012; Norris et al., 2020). Figure 7 shows that this appears to be the case for three-item trains in the earlier serial positions (starting in one of the first three positions), whereas trains starting later in the list were of greater use in retrieval of list-final items. This difference might occur because difficulty in the earlier list positions causes interference that affects retention or appropriate recall of the later list items, and multi-item trains starting early in the list tend to ease the recall of that part of the list, resulting in less interference with recall of the remainder of the list.

Despite this general description, the last two panels of Figure 7 indicate that differences between experiments emerged that were localized primarily in the early portion of the list. Regarding the last panel of the figure, when the train is in Positions 7-9, the need to determine the grouping to use in Positions 1-6 in Experiment 2 may reduce scores in those positions compared to Experiment 1, when the triad grouping was presented. The topic of the possible use of capacity for determination of grouping merits further investigation but has come up before (Norris et al., 2020).

In sum, Hypothesis 5 was moderately successful in predicting widespread benefits of a train of three for recall throughout the list. The benefits were more ubiquitous across the list when the train occurred early in the list.

Hypothesis 6 examined whether associative chains between items would result in a fan effect, such that the recall of the items following a repeated item would suffer because the associative chain is weakened by a fan effect. Despite previous findings in proportion correct measures indicating no chaining effects (e.g., see Kalm & Norris, 2014), chaining and fan effects were obtained, speaking to the usefulness of identifying the first error in the list as a measure. This finding, along with evidence that serial position cues are heavily used (e.g., Lee & Estes, 1981), suggests that participants are able to combine multiple cues to the recall of items in order.

Last, Hypothesis 7 concerns repeated sequences such as 58 or 632 presented in two locations. Only one effect of repeated sequences was observed, in Experiment 1 when a repeated sequence of two digits helped bridge the spatial divide between groups. For example, in the list 325 895 864, the 5-8 transition would apparently facilitate performance greatly across the list (Figure 9). This effect was unanticipated, but it fits with the general notion that finding the structure in the lists is the primary cognitive work that participants must do (Miller, 1956) and that repetitions can ease that process in the right circumstances. We move from here to practical and then theoretical implications of the findings.

Practical Implications

Overall, the evidence is reassuring for the typical, practical usage of lists with haphazard repetition of digits, as in identification numbers and telephone numbers. There is just not much difference between the overall levels of fully correct list recall when digits do or do not repeat, with an overall advantage for repetitions. In situations in which particularly memorable digit lists are needed, it can help to place repetitions strategically in the same grouping or in the same location across groups.

The data present slightly more of a dilemma when it comes to the common practice of presenting numbers with spaces or dashes separating groups of two to four digits or characters. When repetitions are present, there appears to be a slight overall advantage of not presenting digit lists in discrete spatial groups, contrary to the common practice. The probable reason is that the absence of grouping allows participants to adopt an uneven grouping to try to make use of the repetitions (e.g., dividing the list, 824689805 into the groups 8246-89-805 to allow each group to begin with the digit 8). However, this finding with large digits presented on a computer screen in a memory trial might not be replicated with smaller print in real life. When the print is small, division of items into groups may be of use in perceiving the items, or it may modulate the way in which the digits are read aloud, which is likely to produce a larger effect favoring grouped presentation (Cowan et al., 2002). It may be possible to use digit combinations that somehow make use of the grouping structure (e.g., 404-232-939 or 888-74-222), at least when there are enough possible identification numbers to allow such restrictions in the assigned numbers.

The results were remarkably similar in Experiment 1, in which spaces occurred between triads in the list, and Experiment 2, in which no such spaces occurred. Nevertheless, the serial position functions in Figure 9 do show a subtle difference between experiments. In Experiment 1, the serial position function can be seen to form scallops separately for each triad, whereas that triad grouping structure cannot be seen in Experiment 2. This difference counteracts the finding for a study of temporal grouping (Cowan et al., 2002), in which participants tended to have scalloped serial position functions for visual stimuli regardless of whether the lists were grouped. Studies of temporal grouping in serial recall generally have produced some mixed results on the effects of this grouping (for a review see Grenfell-Essam et al., 2019). In our study, and in perhaps some others, regardless of the grouping cues used, the presence of grouping cue may have both advantages and disadvantages. One advantage is that the process of forming groups is less demanding of working memory capacity, and one disadvantage is that it prevents a more flexible grouping structure that might otherwise be established to make the best use of patterns that emerge in the stimuli, which would differ from trial to trial.

Theoretical Implications

The evidence is favorable to an approach in which an associative structure between list items is formed (e.g., Lee & Estes, 1981). In an ungrouped presentation, moreover, there may be the possibility of forming a different grouping depending on where repetitions occur. This possibility has not been explored previously, most likely because of a paucity of previous studies examining lists with multiple repetitions, for which the flexibility of grouping would be especially advantageous. The results are quite consistent with studies of list memory in which the list structure goes beyond discrete chunks, in that more complex structures can be used (e.g., lists in which each group starts with the same digit). Regularities or patterns can be perceived that allow a list to be retained with a shorter description length, reducing the memory load in a way termed compression, which includes, but is not limited to, separate, discrete chunks (Mathy & Feldman, 2012; Chekaf et al., 2016). One illustration of this principle is the result for repeated strings like 58…58, which facilitated the bridging between separated triads in Experiment 1 by making both transitions comparable 5-to-8 transitions (Figure 9 and accompanying text).

The absence of a more pervasive, detrimental effect of single repetitions in Experiment 2 (Figures 3 & 4) seems slightly at odds with the situation termed the Ranschburg effect (e.g., Henson, 1989). In that effect, a non-consecutive repetition of a single item is found to be somewhat detrimental to performance. Perhaps in that situation, given that most digits can be counted on not to repeat, it is difficult to keep in mind exactly which digit did repeat, as a statistical rarity compared to most digits. In the present procedure, in contrast, repetitions occurred widely and haphazardly, so that in the blocks of trials that included repetitions, no digit could be unexpected on the grounds that it had occurred in the list already. Therefore, the present study may represent the practical case better than the Ranschburg study and may be just as valid theoretically, though further research on the issue of the participant’s expectations of repetition seems vital.

Overall, in addition to supporting the notion of the use of hierarchical structure in lists (e.g., Farrell et al., 2011; Lee & Estes, 1981) and the use of pattern perception and compression processes (e.g., Mathy & Feldman, 2012), the research is in line with a notion of a core capacity limit of several items, with additional items held in the activated portion of long-term memory with rapid new learning (e.g., Cowan, 1988, 2001, 2019; Cowan et al., 2012). Repetitions often comprise a compelling factor that participants can use to perceive patterns that ease the memory load, allowing more of the list to fit within a limited capacity. In future work, more systematic study of some of the specific patterns of repetition can be examined more completely, along with factors such as the processing time available and the modality. It must be only wariness of excess complexity that has discouraged researchers from examining repetitions in most of the broad and voluminous literature on serial recall (Oberauer et al., 2018).