Positional and temporal clustering in serial order memory

Transcription

1 Positional and temporal clustering in serial order memory Alec Solway Princeton University Bennet B. Murdock University of Toronto Michael J. Kahana University of Pennsylvania The well-known finding that errors in serial recall tend to be clustered around the position of the target item has bolstered positional-coding theories of serial order memory. Here we show that this well-known effect is confounded with the tendency for the item recalled prior to an error to come from the correct list position. When the standard analysis of positional clustering is conditioned on the positional-accuracy of the previously recalled item, one can see very strong effects of temporal contiguity, with errors being distributed near the position of the item following the prior recall. We show that a simple associative chaining model with asymmetric neighboring, remote associations, and a primacy gradient can account for both the standard positional clustering effect and the contiguity effect demonstrated here. Using the same parameters, the model produces reasonable serial position curves and captures the change in these curves across study-test trials as well as the gains and losses of item and order information across trials. Keywords: Serial recall, serial order, association, clustering. Associative chaining and positional coding consistute the two classic models of serial order memory. Although associative chaining was the implicit theory in Ebbinghaus (1885) seminal studies of serial learning, early scholars also recognized that people can remember the positions of list items, and that this positional information was often used to aid the learning process (Ladd & Woodworth, 1911). During the 195 s and 196 s a great deal of research was focused on identifying the functional stimulus in serial learning: the prior item, as predicted by chaining theory, or the item s position, as predicted by positional theory. This work was generally inconclusive, with various experiments lending partial support to one or the other account (see Young, 1968, for a review). In recent decades, as the emphasis shifted from the study of multitrial serial learning to the study of immediate serial recall, theorists have largely rejected the possibility that chaining plays an important role in serial-order memory (e.g., Botvinick & Plaut, 26; Henson, Norris, Page, & Baddeley, 1996; Farrell & Lewandowsky, 22). Instead, most modern accounts of serial order memory emphasize the importance of positional information as the major retrieval cue. One of the major sources of evidence for the positional coding view is the phenomenon of positional clustering (also The authors acknowledge support from National Institutes of Health grant MH55687 and from the Dana Foundation. Correspondence concerning this article may be addressed to either Michael J. Kahana (kahana@psych.upenn.edu) or Alec Solway (asolway@princeton.edu) known as positional gradients, and the locality constraint). Whereas most items that are recalled appear in their correct order, list items that are recalled out of order tend to occur in neighboring positions more often than in distant positions (e.g. Estes, 1972; Lee & Estes, 1977; Nairne, 1992). This phenomenon would be expected if the retrieval cue is a memorial representation of each item s list position. When the positional cue fails to correctly retrieve the appropriate item it will tend to activate items from neighboring list positions. In addition to positional clustering, other critical evidence for position-based models comes from findings of protrusions (erroneous recall of an item from the same position in a prior list), and confusions among phonologically simlar items occuring in distinct list positions (e.g., Baddeley, 1968). Although some experiments have demonstrated a role for associative processes in serial learning (e.g., Serra & Nairne, 2; Kahana, Mollison, & Addis, 21) the striking effects of positional clustering seen in all serial recall tasks suggests that any contributions of associative chaining to serial order memory are secondary to the more prominent role of positional information. In the present paper we reevaluate the phenomenon of positional clustering by looking at the distribution of errors conditional on the accuracy of the preceding responses. Specifically, we compare cases where the preceding response was a list item recalled in the correct vs. the incorrect list position. We also analyze the effect of temporal clustering in serial recall. Whereas positional clustering is observed when recalled items cluster around their correct output positions, temporal clustering is observed when recalled items cluster

2 2 SOLWAY, MURDOCK & KAHANA Table 1 Summary of the studies that were analyzed Presentation Response Lists Length Trials Kahana& Caplan (22, Exp. 2) Spoken Spoken criterion Kahana et al. (21) Spoken Spoken criterion Spoken Spoken criterion around the position of the just-recalled item. Here one finds that order errors are frequently items from list positions near that of the just recalled item, regardless of that item s serial position. This type of temporal clustering was characterized by Kahana (1996) in free recall, and has subsequently been shown to occur in serial recall by a number of investigators (Bhatarah, Ward, & Tan, 26, 28; Golomb, Peelle, Addis, Kahana, & Wingfield, 28; Klein, Addis, & Kahana, 25). Methods To reevaluate clustering effects in serial recall, we focused on two large studies (Kahana & Caplan, 22, Exp. 2; Kahana et al., 21). Both studies used modern experimental methods, including: common words rather than nonsense syllables, the study-test method rather than anticipation learning, and computer controlled randomization and timing of stimulus presentations. Furthermore, rather than writing their responses into a set of pre-allocated positions, participants in both studies vocally recalled the lists. Table 1 contains a summary of each data set. For details, see Appendix B. In analyzing the patterns of positional and temporal clustering, we focused on the middle list items (4 16 for 19-word lists, and 4 1 for 13-word lists). Edge items were excluded because they cannot appear at all distances and lags. Results Estes (1972) reported that recalls made in the wrong serial position still cluster closely around their correct position. This result is replicated in the top row of Figure 1, which shows that the probability of recalling an item decreases as a function of distance from the correct position. The probability of seeing an item at each distance was conditioned on the number of times it was possible to recall an item at that distance. To make this clustering measure concrete, consider the example list ABSENCE HOLLOW PUPIL RIBBON DARLING RAILWAY WIN- DOW. If a participant recalls ABSENCE HOLLOW PUPIL, then PUPIL is at distance, because it appears in the third position both in the list and in the recall sequence. On the other hand, if a participant recalls ABSENCE HOLLOW RIBBON, then RIBBON is at distance 1 because it appears one position early in the recall sequence compared to the list. Likewise, HOLLOW in the recall sequence ABSENCE PUPIL HOLLOW appears one position late, and is at distance+1. By contrast, the top row of Figure 2 shows the degree of clustering around preceding responses (i.e. lag-crp, Bhatarah et al., 26, 28; Golomb et al., 28; Klein et al., 25). Here, a distance of+1 means the item had the same predecessor both in the recall sequence and in the list. A distance of+2 means there was one item in the list separating the item and its predecessor, and so on. Negative distances correspond to filling in items that were skipped over earlier in the recall sequence. Finally, a distance of means that an item was repeated twice in a row. This did not occur in the data we analyzed. The probability of seeing an item at each lag was further conditioned on the lag s availability. The top row of Figure 2 illustrates the strong contiguity effect demonstrated in other serial recall experiments (Bhatarah et al., 26, 28; Golomb et al., 28; Klein et al., 25). Our joint findings of strong positional and temporal clustering effects may seem surprising given that the two phenomena have been associated with very different theoretical models (i.e., associative chaining and positional coding). However, it is important to recognize a critical confound in this comparison. Specifically, in most serial recall studies the majority of responses appear in their correct output positions. Consider, for instance, the correct recall sequence ABSENCE HOLLOW PUPIL from our example list. If the next response is RIBBON, then its distance from the correct output position is, because RIBBON appears in the fourth list position. Its distance from the preceding response, PUPIL, is+1. Correct responses appearing at the beginning of each recall sequence account for the majority of responses made, and they all fall into the bin of the positional clustering measure and the+1 bin of the temporal clustering measure. To our knowledge, this important confound has not been considered in previous analyses of positional clustering. Fortunately, one can eliminate the confound described above by restricting the clustering analyses to items following the first order error on each trial. For example, in the recall sequence ABSENCE HOLLOW RAILWAY WINDOW, WINDOW is included because RAILWAY was recalled early. Plots of positional and temporal clustering for such items are displayed in the bottom rows of Figures 1 and 2 respectively. Figure 1 shows that the probability of recalling an item no longer decreases as a function of distance from the item s correct position. On the other hand, Figure 2 shows that, although weaker than before, the contiguity effect is preserved. After making their first order error, participants tend to pick up with the item that followed the erroneous recall in the list. Strength-based associative chaining model We next asked whether an associative chaining model that lacks any direct representation of positional information can account for the positional and temporal clustering effects described above. In previous work, associative chaining models (e.g., Lewandowsky & Murdock, 1989) have been successfully applied to several key features of serial learning and se-

3 Conditional response probability.6.6 CLUSTERING IN SERIAL RECALL 3 A B C D E F Distance from correct position Figure 1. Probability of recalling an item as a function of distance from its correct position. Negative values correspond to recalling an item early, while positive values correspond to recalling an item late. Error bars indicate 95% confidence intervals computed using the method of Loftus & Masson (1994). Panels in the top row were computed based on all recalls, while those in the bottom row were computed based only on recalls following the first order error. Panels are based on the first trial of the following studies: A and D. Kahana & Caplan (22, Exp. 2). B and E. Kahana et al. (21, 13-word lists). C and F. Kahana et al. (21, 19-word lists) rial recall data, but not to the positional clustering effects that have been a critical source of evidence for positional coding theories. To address this question, we developed a reduced form model incorporating the two classic assumptions of associative chaining theory: 1) that the strength of association between items is an (exponentially) decreasing function of their distance, and 2) that forward associations (i.e., those between earlier and later items in the chain) are encoded more strongly than backward associations (Ebbinghaus, 1885/1913; Kahana & Caplan, 22; Raskin & Cook, 1937). Consistent with other models, we assume a primacy gradient in item encoding strength to simulate participants tendency to allocate greater attention to early list items (Brown, Preece, & Hulme, 2; Henson, 1998; Jensen, 1962; Lewandowsky & Murdock, 1989; Page & Norris, 1998). At test, the model simulates recall probabilistically according to a Luce choice rule (Luce, 1959), and can choose to stop at any position. As in other strengthbased recall models, we use matrices to store the strengths of inter-item associations (e.g., Kimball, Smith, & Kahana, 27; Raaijmakers & Shiffrin, 198; Sirotin, Kimball, & Kahana, 25); we do not explicitly model item representations themselves. In addition to asking whether the associative chaining model can account for the conditional positional and temporal clustering effects described above, we sought to examine whether the model could simultaneously account for two other critical aspects of serial recall and learning data: specifically, the multi-trial serial position curves (Ward, 1937) and the gains and losses of item and order information across trials Addis and Kahana (24). In most modeling studies, different experiments from the literature are used to illustrate different empirical phenomena. As such, separate model parameters are estimated for each experiment and one cannot be sure whether it is the model s mechanisms or its free parameters that are doing the work of fitting the empirical regularities. In order to assess whether the model can account for each of the findings using a single set of parameter values, we fit data from a single experiment (Kahana & Caplan, 22, Exp. 2). Pariticipants in this experiment learned the lists to a criterion of one perfect recall (see AppendixB), and as such the number of trials to criterion varied from list to list and from participant to participant. To analyze these lists together, we focused on the first three trials of each list. Lists that were learned in fewer than three trials were excluded. Summary of Parameters As an illustrative example of the model s dynamics, consider the list in Figure 3. During study, each item is bound to all of the preceding items. The strength of associations between neighboring items has a Gaussian distribution with

4 4 SOLWAY, MURDOCK & KAHANA Conditional response probability.6.6 A B C D E F Distance from preceding item Figure 2. Probability of recalling an item as a function of distance from preceding recall. Negative values correspond to recalling an earlier item from the list, while positive values correspond to recalling a later item from the list. Error bars indicate 95% confidence intervals computed using the method of Loftus & Masson (1994). Panels in the top row were computed based on all recalls, while those in the bottom row were computed based only on recalls following the first order error. Panels are based on the first trial of the studies listed in Figure 1. $START$ ABSENCE HOLLOW PUPIL Figure 3. An illustrative example of the chaining model. an exponentially decaying mean (the primacy gradient; c s controls the gain, d s the decay rate, and min as the lower bound) and with varianceσ as. The primacy gradient allows the model to mimic the way in which participants learn lists: items from the beginning of the list are remembered on the first trial, and items from later in the list are progressively added on subsequent trials (Slamecka, 1964). The strength of associations between non-adjacent items decays exponentially with distance. The rate of the decay is Gaussian with parametersµ bs andσ bs (note the increasingly lighter shade of gray between non-adjacent items in Figure 3). Remote associations allow the model to skip ahead and later return to omitted items. Without remote associations, the only type of errors the model would be able to produce are errors of omission. Strengths of backward associations are weighed by the parameter w b. In addition to being bound to one another, list items are bound to an additional non-list item whose serial position precedes the first. During test, this start marker serves as the initial retrieval cue. Retrieval is probabilistic and follows a Luce choice rule with softmax parameterγ. Whenγ=1, the probability of retrieving an item is proportional to the strength of the forward associations between the retrieval cue and the item. Asγapproaches, only the the item with the strongest association can be retrieved. If a list item is successfully retrieved, it then serves as the retrieval cue for the subsequent position. Recall may also terminate at any list position if a competing non-list item is selected (the strength of this item is controlled by the parameter stop and is independent of the retrieval cue). Table 2 provides a summary of the model s parameters along with their corresponding best fit values. A formal description of the model can be found in Appendix A. Modeling positional and temporal clustering Figures 4 and 5 show that the chaining model captures the major qualitative features of the data on both temporal and positional clustering, computed in the standard way (top rows), and conditional on the prior recall being in the wrong position (bottom rows). The parameter values used in the simulation (Table 2) have a simple and direct mapping to these results. The mean rate with which the strength of remote associations is reduced

5 CLUSTERING IN SERIAL RECALL 5 Conditional response probability.6.6 A B C Data Model D E Distance from correct position F Figure 4. Probabilities of recalling an item at various distances from its correct position. Negative values correspond to recalling an item early, while positive values correspond to recalling an item late. The solid curve was computed from Kahana & Caplan (22, Exp. 2) and the dotted curve was computed from the simulation. Model parameters for the simulation are given in Table 2. Error bars indicate 95% confidence intervals computed using the method of Loftus & Masson (1994). The three columns correspond to the first three trials of the empirical and simulated data. All plots are aggregated across mid-list items (4-16 of a 19 word list). A-C. Based on all items. D-F. Based only on items following the first order error in the recall sequence. Table 2 Model parameters and their corresponding values Parameter Value c s.945 d s.318 min as. σ as.645 µ bs 7. σ bs 9.93 w b.134 stop.1 γ as a function of distance (µ bs ) is high throughout the list. On average, the strength between an item and another item more than two positions away is very low (<.1). Combined with strong forward asymmetry (w b ), these values explain the large contiguity effect typical of serial recall. The model slightly overpredicts the number of items recalled early (Figure 4) for this same reason. Modeling serial position curves over trials Serial position curves for the data are shown in Figure 6A. Values for these curves were computed using relative order scoring (i.e. a recalled item is considered correct if the previous recall was the item s immediate predecessor in the list). This scoring method is well suited to the experimental paradigm that we are modeling. Rather than being forced to make a response at each serial position (e.g. saying pass for serial positions that cannot be recalled), participants were free to recall only the words that came to mind. Using relative order scoring allows items from later serial positions to be correctly recalled even if items from earlier serial positions were omitted (a common occurrence for long lists using spoken recall). As shown in Figure 6B, the chaining model captures the extended primacy effect and the change in both the level and the shape of the serial position curve across trials. The primacy effect is a result of: 1) the inherent interdependencies that exist between items, and 2) the gradient in encoding strengths. On the first trial, mid-list items exhibit low levels of recall because the stop item acts as a formidable contender. This effect is progressively reduced in later trials as inter-item associations are reinforced. Learning is a result of the variability in encoding strength (e.g.σ as ) coupled with a

6 6 SOLWAY, MURDOCK & KAHANA Conditional response probability.6.6 A B C Data Model D E Distance from preceding item F Figure 5. Probabilities of recalling an item at various distances from the preceding recall. Negative values correspond to recalling an earlier item from the list, while positive values correspond to recalling a later item from the list. The solid curve was computed from Kahana & Caplan (22, Experiment 2) and the dotted curve was computed from the simulation. Model parameters for the simulation are given in Table 2. Error bars indicate 95% confidence intervals computed using the method of Loftus & Masson (1994). The three columns correspond to the first three trials of the empirical and simulated data. All plots are aggregated across mid-list items (4-16 of a 19 word list). A-C. Based on all items. D-F. Based only on items following the first order error in the recall sequence. Recall probability 1.6 A Serial position Trial 1 Trial 2 Trial 3 B Figure 6. Simulating the change in the serial position curve across study-test trials. A. Serial position curves for Trials 1-3 of a 19 word list (Kahana & Caplan, 22, Experiment 2). Trials 1-3 are indicated by solid, dashed and dotted lines respectively. Error bars indicate 95% confidence intervals computed using the method of Loftus & Masson (1994). B. Simulated values obtained using the chaining model. Model parameters are given in Table 2.

7 CLUSTERING IN SERIAL RECALL 7 Figure 7. State diagram representing the possible gains and losses of item and order information between consecutive trials. The three states correspond to not recalling an item at all, recalling an item out of order, and recalling an item in its correct order. Items transition between states in response to changes in item and order information. It is also possible that no change takes place, in which case an item would stay in the same state (not shown). closed-loop learning rule (see Appendix A). Modeling Gains and Losses of Item and Order (GLIO) If a list item is recalled at test, we say that corresponding item information is present, regardless of the recall s output position. If the recall follows the prior list item, then order information is present as well. On each trial, an item may have no corresponding information (if it was omitted), only item information (if it was recalled out of order), or both item and order information. Figure 7 illustrates the possible changes in information between two consecutive trials. Each circle represents an item s state at trial k-1, and following one of the arrows leads to the state at trial k. All items start in state None. It is also possible that there is no change in information, in which case the item stays in the same state. Such self-referencing transitions are not shown in Figure 7, but are valid. For example, if an item is recalled on the first trial in an incorrect output position (using relative order scoring), it follows transition +I and ends up in state Item. Likewise, if an item is recalled in it s correct position, it follows transition +IO and ends up in state Item&Order. If item information, order information, or both are lost in a later trial, the item transitions back to one of its previous states. In all, there are six possible transitions of interest: +IO Gaining item and order information +I Gaining item information, incorrect order information the parametersµ bs andσ bs, which control the strength of re- associations, and the parameter γ, which controls the +O Gaining order information, maintaining item informationmote -IO Losing item and order information frequency with which weakly associated items are selected -I Losing item information, did not have order information for recall. The low value of w b in our simulation, combined -O Losing order information, maintaining item information Figure 8A shows the probability of gaining item and order information together (+IO) at each serial position of the first three trials. The data show a large primacy effect and small Table 3 Probabilities of gaining and losing item and order information Trial Beh. Data Simulation +IO I O IO I O recency effect on Trial 1 (this is equivalent to the Trial 1 serial position curve in Figure 6A). In later trials, a primacy effect is no longer apparent 1 and instead there is an increase for groups of items from progressively later in the list. The model is able to capture the pattern of combined item and order gains of beginning and mid-list items (cf. Figure 8B). As before, the primacy items are learned first because of the interdependent nature of inter-item associations and the gradient in encoding strength. Further strength increments on later trials allow the mid-list items to be learned. The probability of gaining item and order information separately falls below fifteen percent at almost all serial positions. We summarize these results in Table 3, aggregating across all positions. Here we see a trade-off occur over trials, with some items gained out of order in early trials and supplemented with order information in later trials. As shown in Table 3, the model is able to capture the low level of separate item and order gains compared to the higher level of combined gains. The proportion of combined gains relative to separate gains is controlled by the parameter w b, which determines the likelihood of backward transitions, 1 We did not condition on the availability of particular transitions at each serial position and trial, since it would make it appear as if more items are gained in later trials than in earlier ones. For a full discussion, see Addis and Kahana (24).

8 8 SOLWAY, MURDOCK & KAHANA A B Recall probability 1 1 Serial position Trial Figure 8. Simulating combined gains of item and order information across serial positions and trials. A. The probabilities of gaining item and order information together at each serial position on Trials 1-3 of a 19 word list (Kahana & Caplan, 22, Experiment 2). B. Simulated values obtained using the chaining model. Model parameters are given in Table 2. with the high values ofµ bs andσ bs, and the high value ofγ together make combined gains more frequent. The probability of losing information of any type is below five percent at all serial positions, and so we again aggregate the data across all positions and summarize the results in Table 3. Once an item is placed in its correct position, that position is seldom lost. In the model, this corresponds to an item accumulating enough associative strength from its neighbor. The same parameter values that make combined item and order gains most frequent also ensure that there is almost no chance of retrieving a remote item once the associations between nearest neighbors are sufficiently strong. Table 3 shows that item information by itself is also seldom lost and is instead supplemented with order information on a later trial. In the model, the probability of recalling all items is increased on every trial as inter-item associations are reinforced. General Discussion In recalling sequences of items, people frequently remember items out of their correct order. When making these order errors, people show a strong tendency to cluster these incorrectly recalled items near their proper list positions. This positional clustering effect has been well documented in many recall paradigms, and has been used to support the view that people store the position of each list item and then later use that positional information to guide recall. We have shown that an associative chaining model can exhibit significant positional clustering (see Figure 4A-C), providing a good fit to the levels observed in the data. Because the model lacks a positional coding mechanism, its ability to exhibit appropriate levels of positional clustering suggests that the measures of positional and temporal clustering may be confounded. Conditioning these analyses on an incorrect prior recall allowed us to separately examine the effects of positional and temporal clustering. We found that although attenuated, the asymmetrical temporal contiguity effect typically seen in serial recall persists following order errors, whereas the positional clustering effect collapses. After making an error, participants were most likely to recall the item on the list following the just recalled item and not the item that appeared in the next position. Because chaining theory assumes that each recall acts as the retrieval cue for the next position, our model is able to capture these conditional clustering effects (see Figures 4D-F and 5D-F). Although we have not attempted to simulate conditional clustering effects with a positional coding model, it is worth considering how such a model might account for these results. Consider for example a model in which items are bound to a slowly varying list context (e.g., Brown et al., 2; Burgess & Hitch, 1999, 26). At test, the list context is reset to its state at the start of the list as a cue for the first item. Context is then played back to the network, successively cueing the retreiveal of each subsequent item. Because the context signal is temporally autocorrelated, contexts associated with neighboring items will be similar, thereby producing positional gradients. However, unless recall of an item retrieves its own previously associated context, the model will incorrectly predict that the most likely response following an error is the item that belongs in the serial position of the current context cue rather than the serial position of the item that followed the recall error. As our conditional analysis of recall errors reveals, the responses tend to cluster around the position following the just recalled item. The present findings point to a non-positional contiguitybased associative mechanism as the primary factor underlying positional clustering effects in serial recall and in serial learning. We note, however, that if positional cues are re-

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10 1 SOLWAY, MURDOCK & KAHANA Study Phase Our chaining model uses two matrices to hold the strengths of associations between items, one for forward associations and one for backward associations. For simplicity we assume that lists words are not semantically related and that there are no associations across lists, and so we set the initial strengths to zero. Similar to the Lewandowsky and Murdock (1989) model, a start marker is used to simulate the way in which participants access the beginning of a list. The start marker acts as list item and is associated with each subsequent item. Following the study of list item i on trial T, we increment the strength of the forward association from the immediately preceding item, i 1, according to the storage equation: F(i 1, i) T = a s [1 F(i 1, i) T 1 ], (1) where F(i 1, i) T 1 is the strength of the association on the previous trial and a s is a random variable (Gaussian with parameters (µ as andσ as ) that controls the change in strength on the current trial. We do not allow the model to unlearn previous strength increments, and so we clamp negative values of a s to zero. We assume a primacy gradient in encoding strength and reduce the mean of a s exponentially across serial positions to an asymptotic minimum of min as : µ as = c s e d s(i 1) + min as, (2) where i indexes the serial position of the item.σ as in Equation 1 and c s, d s, and min as in Equation 2 are model parameters. Equation 1 implements a closed-loop learning rule (e.g. Lewandowsky & Murdock, 1989): the increment in strength is proportional to the amount of association already in memory. The strength of the backward association from item i to item i 1 is computed by scaling the forward increment by w b, a parameter that ranges from to 1: B(i, i 1) T = w b F(i 1, i) T (3) This allows the model to mimic the forward asymmetry that is typical of serial recall (Bhatarah et al., 26, 28; Golomb et al., 28; Klein et al., 25). In addition to forming associations between nearest neighbors, our model also forms remote associations by incrementing the strength between item i and each earlier list item, i x, according to the more general storage equation: F(i x, i) T = a s [1 F(i x, i) T 1 ]e b s x 1 Here, a s is the same as above, and b s is a random variable (Gaussian with parametersµ bs andσ bs ) that determines the strength of remote associations relative to nearest neighbors. The minimum value b s can take is clamped to one.µ bs and σ bs are both model parameters. Note that when x=1, this equation reduces to Equation 1. During list study, a s is sampled once for each item and b s is sampled once for each list. The strength of remote associations is reduced exponentially (4) Table A1 Associative strengths following the first trial study phase of a sample simulation $ A B C D $ A B C D as a function of the lag, x, which ranges from the previously studied item (x=1) to the start marker (x=i). As with nearest neighbor associations, we assume that remote associations are formed in both the forward and backward directions, with the backward strength increment scaled by w b. Test Phase At test, the start marker serves as the first retrieval cue for each list. After the first position, each successfully retrieved item serves as the cue for the next position. Assuming the model recalled item i, the next item, j, is chosen by the Luce choice rule (Luce, 1959): P( j i)= S (i, j) γ k S (i, k) γ + stopγ, (5) where S = F+ B, k ranges over all unrecalled items, and stop sets the probability of retrieval failure. Both stop andγ are model parameters. A Numerical Example Consider the behavior of the model on the first trial of the list A B C D, using the following parameter values: w b =.15, min as =.1, d s =.55, c s =.6,µ bs = 1.2, stop=.2,γ=1. For simplicity, variability of encoding is omitted. All associations are set to zero at the beginning of the simulation. We begin the study phase by applying Equation 2 to determine the increment in associative strength between the start marker and the first item: µ as (1)=.6e.55() +.1=.7, Equation 4 is used to update the strength matrix. Since there is no encoding variability and no prior learning, the increment is exactly.7 in the forward direction and.15 in the backward direction (.7 w b ). The rest of the list is presented in a similar manner, and Table A1 shows the resulting associative strength matrices. Here we combine the forward (F, located in the upper triangle) and backward (B, located in the lower triangle) matrices for ease of presentation. At test, the probability of retrieving item A is computed using Equation 5: P(A $)= =.789

11 CLUSTERING IN SERIAL RECALL 11 The probability of retrieving each of the remaining items is computed in a similar fashion, and an item is selected by sampling from this distribution. If an item is successfully retrieved (i.e. the stop marker is not sampled), the next recall is selected by sampling from the distribution formed by conditioning the remaining items on the first recall. Recall proceeds in this manner until either the stop marker is selected or all of the items are recalled. In the case of choosing the stop marker, the study-test cycle is repeated (for a maximum of ten trials). If the list is correctly recalled, the strength matrix is reset to zero and simulation of the next list begins. Parameter Estimation We used a genetic algorithm (Mitchell, 1996) to find a set of parameters that minimized the mean root-mean squared deviation (RMSD) for the three aspects of the data described in the main text (the clustering measures, the serial position curves, and the gains and losses of item and order information). We first used a population of 2, random parameter sets for ten generations, and then reduced the size to 1, members for twenty more generations. The parameter values shown in Table 2 provided the best fit, and the simulations referenced in the main text were performed using these values. Appendix B Experiment Details Kahana and Caplan (22), Exp. 2. Sixty participants performed the experiment over the course of five sessions. The first session consisted of one 1-word, one 15-word, and two 19-word practice lists, and was not included in our analysis. The remaining four sessions each consisted of five 19- word lists. Words were drawn from the Toronto Word Pool (Friendly, Franklin, Hoffman, & Rubin, 1982) without replacement and presented aurally at a rate of 1.5 sec. After studying a list, participants were asked to vocally recall the list in the presented order. The cycle of study and test was repeated for each list until it was recalled perfectly. In order to avoid edge artifacts, we focused our analysis of the recall data on list positions 4 16 (see main text). Furthermore, in analyzing serial position curves and the gains and losses of item and order information across trials, we focused our attention on the first three trials of each list. Lists that were learned in fewer than three trials (about 17%) were excluded. Kahana et al. (21). Forty-two participants performed the experiment over the course of four sessions. Each list consisted of 7, 13, or 19 words drawn from the Toronto Word Pool (Friendly et al., 1982) without replacement and presented aurally at a rate of 1 sec. At test, participants were given 1 min. to vocally recall the list in the presented order, and the study and test cycle was was repeated for each list until it was recalled perfectly. The experiment consisted of two conditions. In the constant start condition, participants studied each list in the usual manner, starting at the same list position. In the spin-list condition, participants studied each list starting in a random position on each trial. After studying the last list item, participants would then see the first item, and the presentation would continue until reaching the position immediately prior to the start. The first session of the experiment consisted of six practice lists, one for each combination of list length and starting condition, and was not included in our analysis. Subsequent sessions consisted of three lists of each possible length under one of the two starting conditions (the order of conditions was counterbalanced across participants). Four subjects were excluded because they failed to learn any lists within a predetermined maximum number of trials. Our analysis includes only the data from the constant start condition. Furthermore, in order to avoid edge artifacts, we focused our analysis on positions 4 1 for lists of length 13 and positions 4 16 for lists of length 19. We excluded lists of length 7.