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2 Journal of Mathematical Psychology 53 (2009) Contents lists available at ScienceDirect Journal of Mathematical Psychology journal homepage: Reconsidering the two-second decay hypothesis in verbal working memory Shane T. Mueller a,, Adam Krawitz b a Klein Associates Division, ARA Inc., Fairborn, OH 45434, USA b Department of Psychological and Brain Sciences, Indiana University, Bloomington, IN 47405, USA a r t i c l e i n f o a b s t r a c t Article history: Received 12 November 2007 Received in revised form 22 October 2008 Available online 9 January 2009 Keywords: Working memory Short-term memory Decay Immediate serial recall A common belief in the study of short-term memory is that the verbal trace decays around two seconds after it is encoded. This belief is typically assumed to follow from the finding that in immediate serial recall, the time required to rapidly articulate a span-length list is around two seconds. Empirically, this belief is in opposition to a broad set of findings across a number of domains that establish mean decay times to be longer than two seconds. Theoretically, the available computational and mathematical models of immediate serial recall do not address this issue directly, because they typically rely on other mechanisms in addition to decay to account for forgetting. As such, they may show that decay times can be longer than two seconds, but they fail to show that they cannot be as short as two seconds. We address the issue directly and set a lower bound on mean trace decay times, even under the limiting assumption that all forgetting is due to trace decay. We do this by presenting a simple item-based model of trace decay that allows us to estimate values of mean trace duration. For a set of words whose span-length lists can be rapidly articulated in about two seconds, the model offers a conservative estimate for their mean decay times of around four seconds. Both the experimental and theoretical evidence show that items in verbal working memory decay considerably slower than the two-second decay hypothesis claims Elsevier Inc. All rights reserved. In their influential paper on verbal short-term memory, Baddeley, Thomson, and Buchanan (1975) reported the articulatory duration effect, in which lists comprised of shorter words are recalled more accurately than equal-length lists of longer words. They noted that the number of words that could be remembered (out of five) was equal to the number that could be rapidly articulated in about 1.5 s (across experiments and participants, their estimates ranged between 0.93 and 1.95 s). This result was later bolstered by the results of Standing, Bond, Smith, and Iseley (1980), who showed that across a wide variety of stimuli and experimental conditions, the time it took to subvocalize memory-span length lists was between about 1.7 and 2.1 s. These results suggest that memory traces for individual words in short-term memory decay with time, and provide evidence for the articulatory loop model of verbal working memory. While these findings were not initially used to infer a direct estimate for the duration of the verbal memory trace, such a link was soon made. Indeed, Baddeley and Lewis (1984) were probably the first to conjecture the two-second decay hypothesis: that the duration of the verbal short-term memory trace was Sponsored in part by United States Office of Naval Research grant N J-1173 to the University of Michigan. Corresponding author. address: smueller@ara.com (S.T. Mueller). roughly equal to the duration of a span-length list, a matter of about two seconds. A mathematical model of trace decay was soon proposed by Schweickert and Boruff (1986) which demonstrated impressive fits to immediate serial recall data from a variety of stimuli. The duration of the verbal trace was consistently found to be around two seconds, providing validation for the two-second decay hypothesis. Since then, the two-second decay hypothesis has entered the experimental psychology literature, appearing in introductory textbooks and research articles alike. 1 However, we will show that this hypothesis (i.e., that the memory traces for individual words in short-term memory decay in an average of two seconds) is wrong. 2 Specifically, it has been confused with another proposition, that intact lists survive for an average of two seconds. We will present a simple model that illustrates this point, and demonstrate its ability to fit empirical data, producing a new lower-bound estimate for 1 For example, a current best-selling cognitive psychology textbook, Sternberg (2006), states that The working-memory model is probably the most widely used and accepted today (p. 191), and then goes on to state that without articulatory rehearsal, acoustic information decays after about 2 seconds (p. 193). 2 A more generous interpretation of the two-second decay hypothesis would be that the memory traces responsible for maintaining the order of words last for about two seconds, as most errors in the immediate serial recall task stem from recalling words from the presented list in the incorrect order. Consequently, the memory traces for the identities of words could be more durable /$ see front matter 2008 Elsevier Inc. All rights reserved. doi: /j.jmp

3 S.T. Mueller, A. Krawitz / Journal of Mathematical Psychology 53 (2009) the duration of the memory trace of individual words in verbal working memory that is greater than two seconds. 1. Experimental evidence about the duration of the verbal trace Historically, a wide range of results have been taken to indicate the duration of the auditory, verbal, or articulatory trace. The assumption that information decays from memory at all is controversial, and experiments designed to determine whether information does in fact decay have produced differing results (cf. Reitman (1971, 1974) and Shiffrin (1973)). Critics of trace decay theory have often supposed that other factors can account for some or all of the effects that appear to support time-based decay (e.g., Lewandowsky and Oberauer (2008), Nairne (2002), Neath and Nairne (1995)) and that the concept of decay is unnecessary, while others argue that the debate has not yet been settled (Cowan & AuBuchon, 2008; Portrat, Barrouillet, & Camos, 2008). But if we take seriously the notion that information decays from short-term memory with time, then even in the face of these other limitations, past techniques may provide reasonable estimates of its average decay time. Next, we will summarize the findings from a number of domains that have attempted to determine the duration of auditory, verbal, or articulatory memory Echoic and short auditory memory In a review paper on auditory memory, Cowan (1984) concluded that there exists a short-duration store for auditory information. In many of the reviewed experiments, participants were required to make fine discriminations between simple auditory stimuli. Results typically showed that masking and interference effects decayed with time and disappeared after about 300 ms, suggesting that the information being used only lasted an average of 150 ms. Research we will review later shows that not all auditory information disappears this rapidly, but clearly some aspects of the auditory trace begin to decay almost immediately after perception. Yet because this information only lasts for about as long as a single word takes to be spoken, it cannot support memory for ordered lists. Consequently, other methods may provide more relevant estimates. Crowder s (1976) textbook on human memory devoted a chapter to attempts to measure the decay properties of echoic memory. Several experiments he reviewed used recognition methods to determine the duration of the auditory memory trace. For example, Guttman and Julesz (1963) tried to determine whether participants could detect repeated patterns of noise. They found that one second was the slowest rate for detectable repetition, suggesting that the auditory store could retain about one second of auditory signal. Similarly, Treisman (1964) asked participants to detect when a message played in one ear was identical to a delayed message in the other ear. Detection failed at lags of about 1.5 s, suggesting that auditory information disappeared in less than two seconds. These techniques produce estimates with similar magnitude to the two-second decay hypothesis. However, a number of issues limit the applicability of these results to the duration of the verbal trace in immediate serial recall. Both techniques tested recognition, which likely requires less information than recall. Guttman and Julesz (1963) did not use verbal information. And Treisman s (1964) technique required monitoring two simultaneous streams, which probably led to poorer encoding and strong interference effects. Thus, these procedures may not provide good estimates of the duration of the verbal memory trace The partial report procedure and the stimulus suffix effect A technique used by Darwin, Turvey, and Crowder (1972) and Treisman and Rostron (1972) to infer the duration of short-term memory was a modified version of Sperling s (1960) partial report procedure. Auditory information was presented simultaneously in multiple locations followed by a visual post-stimulus cue indicating which channel to recall. The delay between stimuli and cue was varied. A recall advantage relative to full report was typically found with a short delay, but as the delay increased, the advantage diminished. When the stimuli were tones and participants were given recognition tests (e.g., Treisman and Rostron (1972)), the partial report advantage lasted for delays up to 1.6 s. When stimuli were letters and recall was required (e.g., Darwin et al. (1972)), roughly half of the partial report advantage was gone with a delay of two seconds, and a small advantage was still present after four seconds. Because the stimuli were presented in a one-second window, this indicates that the verbal trace lasted on average three seconds and was mostly gone after five seconds. Yet, this estimate may not be accurate, because the procedure requires divided attention to multiple interfering auditory streams. Another empirical phenomenon used to infer the duration of the verbal trace is the stimulus suffix effect. In immediate serial recall, an irrelevant suffix word at the end of a list reduces the recency effect. Crowder (1969) and Morton, Crowder, and Prussin (1971) found that the impact of the suffix disappears when it occurs with a delay of more than two seconds, which seems to suggest that the information decays within about two seconds. However, the information responsible for the suffix effect cannot be the same type of information that is responsible for immediate serial recall accuracy. According to Crowder (1976), the information responsible for the suffix effect is subject to substantial interference, and so only helps the last item on the list. Thus, whatever information is being used to maintain the rest of the list must have different properties. In addition, recent research by Nicholls and Jones (2002) has challenged the assumption that the suffix effect demonstrates information decay at all. Rather, the suffix appears to only have an effect when the stream of events is such that it is perceived as part of the list, regardless of its delay following the last item. Consequently, results from stimulus suffix experiments may not provide informative estimates of decay times The Brown Peterson task Over the years, a number of studies have used variations of the Brown Peterson task (Brown, 1958; Peterson & Peterson, 1959) to determine the decay properties of verbal information in shortterm memory. For example, in Peterson and Peterson s (1959) original study, participants were presented with letter trigrams which had to be recalled after delays between 3 and 18 s. Results showed that memory accuracy was still roughly 80% after three seconds, 50% after six seconds, and had reached an asymptote of less than 10% within 18 s. Later researchers adapted the technique to fit parametric decay functions. For example, Chechile (1987) estimated Weibull decay distributions from data in a version of the Brown Peterson task. The modes of the fitted decay distributions ranged from 5 s to 100 s, with the vast majority under 30 s. However, the presence of interfering activity in the task, and the reliance on interference to account for storage dynamics in the model, makes these estimates difficult to interpret. Nevertheless, results from the Brown Peterson task typically estimate the decay times of verbal information to be considerably longer than two seconds.

4 16 S.T. Mueller, A. Krawitz / Journal of Mathematical Psychology 53 (2009) Long auditory memory In addition to the short auditory store discussed earlier, Cowan (1984) identified a second type of auditory information, which he called the long auditory store, which lasts at least several seconds and perhaps as long as twenty seconds (cf. Watkins and Todres (1980)). The supporting evidence typically come from what Crowder (1976) termed sampling experiments, in which participants engage in a primary task while ignoring secondary information, with intermittent probes that test for the ability to detect recently ignored information. Eriksen and Johnson (1964) showed that even with delays of 10.5 s, unattended tones were detected about 25% of the time. Similar decay characteristics have also been found measuring brain activity: Sams, Hari, Rif, and Knuutila (1993) showed that measured neuromagnetic responses to tones lasted at least 10 s. While these studies may place an upper bound on the duration of the auditory trace, since they deal with the memory for tones, they do not provide a good estimate for the decay of verbal information. More relevant results were produced by both Glucksberg and Cowen (1970) and Norman (1969), who conducted experiments similar to Eriksen and Johnson (1964), but presented numbers instead of tones. To prevent rehearsal, they used a dichotic listening task in which one stream was shadowed. Results showed that the ignored information was accessible for two to five seconds after presentation. However, this shorter estimate than Eriksen and Johnson (1964) might have stemmed from interference produced by the dichotic shadowing task. To address this possibility, Wickelgren (1970) performed a similar study in which he simply instructed participants not to rehearse, rather than using dichotic shadowing. His results showed that after 10 s, the hit rate was still 30%, with a false alarm rate of 15%. These studies, which focused on recognizing the identity of individual stimuli, suggest that some auditory information may remain for on the order of 10 s. However, immediate serial recall requires maintenance of order information as well. Byrnes and Wingfield (1979) examined this issue specifically. Their results showed that after two seconds, at least 75% of the information remained intact, and substantial order information lasted for at least four seconds. Together, these results suggest that a reasonable lower bound for the mean decay duration must be greater than two seconds Recall times from immediate serial recall A final type of evidence about the duration of the verbal memory trace comes from experiments reporting the recall times from immediate serial recall. This putatively provides an estimate of the duration of the verbal memory trace because it must last for at least the duration of lists that can be recalled correctly. Relevant data were provided by Corballis (1969) who showed that participants normally take between three and six seconds to recall lists of eight items. Similarly, Dosher and Ma (1998) and Dosher (1999) have noted that output delays during recall often extend to four to six seconds for a span-length list. Such data might appear at odds with the earlier observations that span-length lists can be rapidly articulated in about two seconds. But rapid articulation differs from recall (a fact discussed by Mueller, Seymour, Kieras, and Meyer (2003)). Typically, recall can incorporate pauses, guessing, and fairly complex reasoning, and so can take longer than the measured duration of a rapidly articulated, correctly-recalled list. But to the extent that recall takes longer, this means that the verbal trace must be more durable. A possible way for the two-second decay hypothesis to explain these results would be if participants were engaging in iterative rehearsal during recall, and were performing this rehearsal at a much faster rate than recall itself. However, previous research has shown that subvocal speech does not happen considerably faster than overt speech (Landauer, 1962). Furthermore, recall is only modestly impacted by articulatory suppression (cf. Baddeley et al. (1975)), when it should be made nearly impossible if participants depended on iterative rehearsal during recall Summary of empirical estimates of trace decay duration The varied tasks used to assess the decay properties of auditory or verbal information have led to a wide range of conclusions. Depending upon the type of information being assessed and the method used to assess it, estimates for information loss range anywhere from 150 ms up to 20 s after presentation. In general, the studies discussed either found decay times on the order of hundreds of milliseconds, corresponding to Cowan s (1984) short auditory store (sometimes referred to as echoic memory), or they found decay times on the order of 5 20 s, corresponding to Cowan s (1984) long auditory store. Few of the empirical paradigms provide support for the two-second decay hypothesis, and these few were typically indirect measures based on the influence of interfering activity, and were for isolated information rather than ordered sequences. Furthermore, the experiments differ in whether they measure the time until just enough information is forgotten to be measured (as in immediate serial recall), or they measure the point where almost no information is retained (as in the partial report procedures). Most of the techniques that attempted to measure the decay properties of verbal information found that information lasted longer than two seconds on the order of 5 10 s. In contrast, the strongest support for the two-second decay hypothesis does not come directly from empirical results, but from conclusions based on Schweickert and Boruff s (1986) mathematical model of immediate serial recall. Because such models may be the only way to derive reasonable lower bounds on the decay time of verbal working memory, we examine them next. 2. Models of immediate serial recall Models of short-term memory have adopted the notion of decay since early proposals by Broadbent (1957) and Conrad and Hille (1958). In the past few decades, a number of complex models of short-term memory and immediate serial recall have been published which attempt to account for a wide variety of effects. However, few of these models have relied solely on timebased decay as a memory limitation. Table 1 reviews a set of prominent mathematical and computational models of immediate serial recall, and summarizes them according to four common limiting factors incorporated in the models. The limiting factors described in Table 1 require a bit of explanation. Theories of memory limitations have adopted numerous distinct classification systems. For example, Crowder (1976) described memory theories in terms of decay, displacement, and interference. In this table, we use four primary factors: decay, capacity, encoding interference and output interference. Memory limitations stemming from the passage of time are called decay. Limitations occurring because the storage mechanisms can hold or discriminate a limited amount of information are called capacity limitations. 3 Limitations stemming from specific activity 3 Capacity is similar to Crowder s (1976) notion of displacement, which tended to describe slot-based models having a fixed capacity. The term displacement has similar connotations to interference, and so we use the term capacity as a more general descriptor that might be computed using Shannon s (1948) entropy.

5 S.T. Mueller, A. Krawitz / Journal of Mathematical Psychology 53 (2009) Table 1 Summary of the factors that limit retention in a number of recent models of short-term memory. Model Decay Capacity Encoding interference Output interference Lee and Estes (1977, 1981): Perturbation model + Shiffrin and Cook (1978) + + Schweickert and Boruff (1986) + Lewandowsky and Murdock (1989): TODAM + + o Burgess and Hitch (1992, 1999): Network model Neath and Nairne (1995): Feature model + + Brown and Hulme (1995) o o o Anderson and Matessa (1997): ACT-R Dosher and Ma (1998) and Dosher (1999) + + Page and Norris (1998): The Primacy model Henson (1998): Start-End model Byrne (1998): SPAN Kieras et al. (1999): EPIC + Brown et al. (2000): OSCAR Mueller (2002): Modified EPIC o o o o Farrell and Lewandowsky (2002): SOB Brown et al. (2007): SIMPLE + + indicates the model does not incorporate this limitation, + indicates the limitation is present, and o indicates the limitation is present but treated as optional. are viewed as interference, which typically takes on two forms in these models. First, some models assume that the act of encoding a new word causes interference, either retroactively damaging previously-encoded information or proactively making it more difficult for new information to be encoded. Second, several models have mechanisms that can be described as output interference: information loss stemming from activity related to recalling or producing responses. Several of the models listed in Table 1 account for word length effects using means other than decay, (e.g., Brown and Hulme (1995) and Neath and Nairne (1995)), or do not account for the word length effect at all (e.g., OSCAR and SIMPLE). Several others incorporate decay in conjunction with other factors (e.g., the Primacy Model, ACT-R, SPAN, etc.) in ways that cannot be cleanly separated. These models cannot provide a lower bound estimate of the mean decay duration, because much of their forgetting stems from other processes. So, in order to estimate this lower bound, we need to look at models that incorporate decay alone or in separable ways. Only three models in Table 1 rely on time-based decay alone to produce word-length effects: Lee and Estes s (1977) Perturbation model, Schweickert and Boruff s (1986) model and the EPIC models of Kieras, Meyer, Mueller, and Seymour (1999) and Mueller (2002). Of these, the Perturbation Model only verbally describes a decay process, and so it cannot provide a formal estimate of decay for immediate serial recall. Furthermore, the two other models are seemingly at odds with one another: Schweickert and Boruff (1986) estimated mean decay time to be less than two seconds, whereas Kieras et al. (1999) estimated it to be between five and seven seconds (in greater concordance with the majority of empirical findings reviewed earlier). What might be the source of this divergence? At one point, Kieras et al. (1999) suggested that the two-second decay hypothesis failed to consider the complex executive control needed for implementing the articulatory loop. 4 While this may be true, in the next section we will show that the different estimates primarily occur because Schweickert and Boruff s model is a listbased model, in contrast to every other model listed in Table 1, which are item-based models. 4 Page and Henson (2001) made a different suggestion while noting that in the Primacy Model, the primacy gradient decayed to half its strength in approximately two seconds. They stated that this value was not directly related to the value obtained by multiplying the mean number of items recalled from a list by a list item s articulatory duration. Yet this does not address the mechanics of Schweickert and Boruff s model, which describes the relationship between memory span (the length of a list that can be recalled correctly 50% of the time) and the time required to rapidly articulate a list of memory-span length Schweickert and Boruff s list-based model Initial empirical observations of the relationship between speech rate and memory accuracy (Baddeley et al., 1975; Standing et al., 1980) noted that there appeared to be a linear relationship between the length of a list that can be recalled correctly and speech rate, embodied by the equation s = k r + c, where memory span (s) is a function of speech rate (r), with k as the slope of the function, and (c) as the intercept. The slope k has been interpreted as the duration of the verbal memory trace, and c as the contribution of other memory systems to performance (e.g., Hulme, Maughan, and Brown (1991)). When fit to empirical data, the slope k is normally around 2.0, and the intercept c is often close to 0, suggesting that the verbal trace decays within about two seconds, and that decay is the primary manner in which information is lost from short-term memory. Based on this observed relationship, Schweickert and Boruff (1986) proposed a simple model of trace decay that accounted for 95% of the variance in the psychophysical functions relating list length (in number of words) to the probability of correct list recall in the immediate serial recall task, using only spoken duration of the list (and not number of words) as a predictor. This model made a number of simplifying assumptions: a list of items takes T r seconds to rehearse or recall; the verbal trace of a list lasts for T v seconds; and if T r is less than T v, the list should be recalled correctly. This should hold even if the two times T r and T v are random variables, based on which Schweickert and Boruff derived psychophysical functions predicting the probability of correct recall for lists of differing lengths. These functions fit data well with estimates of decay times that were typically slightly less than two seconds. Schweickert and Boruff s (1986) model is both elegant in its simplicity and parsimonious in its ability to account for relationships between speaking times and list recall accuracies. Additionally, it has been used as a tool to investigate contributions to memory span by other factors, such as word frequency (e.g., Hulme et al. (1997)). However, it is also often misunderstood, because it does not describe the decay distribution of individual words in verbal short-term memory, but rather the functional decay distribution of an entire list of words taken as a single entity. In this sense, the model is at odds with all of the other models described in Table 1, which consider the forgetting properties of individual words. In Schweickert and Boruff s (1986) model, accuracy is a function of list articulation time and trace duration alone. Consequently, the list-length effect arises solely because lists with more words take longer to recall, and not simply because they contain more

6 18 S.T. Mueller, A. Krawitz / Journal of Mathematical Psychology 53 (2009) words. Schweickert and Boruff s model is able to estimate a decay time for a list of words, but it does not estimate the decay time for individual words. Empirical evidence has indicated that these two quantities are not the same: indeed individual words are often recalled correctly, even when the entire list is not (e.g., Chen and Schweickert (2004) and Drewnowski and Murdock (1980)). Thus, two seconds may be a reasonable estimate of the average time until the first word is lost from memory. 5 Consequently, the list-based model s estimate of decay time is difficult to interpret. What does it mean for a list to decay after two seconds, when it takes two seconds to articulate, and is composed of multiple units which are known to often be reordered incorrectly during recall? The other models in Table 1 deal with these units directly, and so the EPIC models remain the only models in Table 1 that can give relevant lower-bound estimates for decay times of the individual words in immediate serial recall EPIC model of immediate serial recall The EPIC (Executive Process/Interactive Control) architecture (e.g., Meyer and Kieras (1997a,b)) has been used to implement a number of models of immediate serial recall. These models began by adopting a simple limitation (time-based decay) and hypothesized that many other aspects of serial recall data may stem from recall and guessing strategies, rather than low-level aspects of the memory code. The models of immediate serial recall have quite complex rule sets for managing rehearsal (Kieras et al., 1999); executive control (Meyer, Glass, Mueller, Seymour, & Kieras, 2001); retrieval instruction, guessing policy, and error recovery (Mueller, 2002); and list reorganization (Krawitz, 2007). In contrast to the approach of Schweickert and Boruff (1986), these models hypothesize that decay occurs at the item and serial order linkage level, rather than the list level. One of the insights of these EPIC models is that the procedural knowledge needed to implement and manage iterative rehearsal and a robust recall strategy are quite substantial and complex. This complexity presents the same problems for interpretation that many of the models in Table 1 exhibit: the models produce behavior that is an emergent property of a number of interacting factors. Consequently, in this paper we will define a simple mathematical formulation of the critical aspects of the EPIC models, assuming only trace decay as a limiting factor. This model will provide a tractable mathematical formulation in which a lower bound on mean trace decay time can be estimated, uncontaminated by other processes. 3. An item-based trace decay model The models listed in Table 1 are too complex (or perhaps too complicated) to offer a simple mathematical prediction of how time should affect memory span if item-based decay is the only limiting factor. In this section, we will present a simple mathematical model that will enable this relationship to be calculated. This new model makes several assumptions that simplify calculations and enable the decay distributions of individual items to be estimated without Monte Carlo simulation. In this model, the decaying information can represent specific order information, item information, or both; it does not matter which, because the model simply computes the probability of accurate recall of the entire list, which requires all information to remain intact until it is recalled. 6 We will first consider the 5 Mackworth (1963) measured the distribution of times to the first error in immediate serial recall, and it averaged 2.07 s. However, stimuli were presented simultaneously in a visual array, so this value may be coincidental. 6 Although the EPIC models described by Kieras et al. (1999) assumed that both item and order information are encapsulated and decay as a single unit, situation without articulatory rehearsal, and then we will consider the situation with rehearsal Item decay with rehearsal suppressed As an initial step towards modeling the decay of individual items, we consider a model that assumes rehearsal is eliminated through articulatory suppression. 7 Additionally, the model assumes that decay for each item is independent and identically distributed according to an arbitrary distribution with density f (t), where t indicates the time between complete item presentation and some point in the future. The distribution function of f (t) is denoted as F(t) = t f (t)dt, which indicates the probability of an 0 item having decayed after t seconds, and its survival function is denoted as S(t) = 1 F(t), which indicates the probability that an item remains intact after t seconds. In the model, I p indicates the time between complete presentation of consecutive items, I d indicates the delay between presentation and recall (often equal to I p ), I r indicates the time between onsets of consecutive recalled items (assuming that all items take the same amount of time to utter), and N indicates the length, in words, of the list. Based on these assumptions, the probability of recalling an item depends on the time elapsed since its presentation was completed. For an item in serial position n, this time is: I Tn = I p (N n) + I d + (n 1) I r (1) which can be refactored as: I Tn = (I d I r ) + N (I p ) + n (I r I p ), (2) where the first part is a constant for all positions of all lists using the same word set, the second part depends on the number of words in the list, and the third part depends on a word s serial position in the list. Although the relationship between memory accuracy and pronunciation time was initially shown as a proportion of items correctly recalled out of five (cf. Baddeley et al. (1975)), it is typically understood as a relation between the pronunciation time and memory span (the length at which probability of recalling the entire list correctly is.5). For an entire list to be recalled correctly, each of the individual items must be recalled correctly as well. Consequently, an N-item list should be recalled correctly with probability: N S(I Tn ). (3) n=1 This formula is unable to predict partial recall accuracy (i.e., the number of words in a list that are recalled in the correct position), and is thus not appropriate for predicting serial position functions. However, Schweickert, Chen, and Poirier (1999) found empirical support for the assumption that the product of conditional probabilities of consecutive items accounts well for the probability of correct list recall. To conform to this finding, S(I Tn ) might be considered the conditional probability of correct recall, so that item n will be recalled correctly with probability S(I Tn ) if the previous item was recalled correctly. later versions by Mueller (2002) and Krawitz (2007) used a modified version of the architecture in which item and order information were lost independently. Reasonable fits to immediate serial recall data could be achieved while only allowing order information to decay, suggesting that item information is indeed more durable. 7 The assumption that articulatory suppression simply acts to eliminate rehearsal is controversial. However, it is part of the set of basic assumptions used by Baddeley and others to infer the decay time of items from working memory.

7 S.T. Mueller, A. Krawitz / Journal of Mathematical Psychology 53 (2009) Fig. 1. Timelines of events during immediate serial recall, for conditions where iterative rehearsal is present and where it is not present. Eq. (3) can be combined with Eq. (2) to produce a direct formula for the probability of correct list recall when participants are engaging in articulatory suppression: N P(I p, I r, I d, N) = S((I d I r ) + N (I p ) + n (I r I p )). (4) n=1 Although this equation is useful, many experiments do not require participants to engage in articulatory suppression, and so do not discourage iterative rehearsal. Eq. (4) would not be appropriate in these cases. Consequently, in the next section we will compute a similar formula for immediate serial recall accuracy when articulatory rehearsal is being used Articulatory rehearsal The processes involved in articulatory rehearsal are quite complex and may involve any number of strategies (e.g., Kieras et al. (1999) and Mueller (2002)) and probably cannot be modeled easily with a simple mathematical formula. However, some progress can be made if we assume that the adopted rehearsal strategy involves (1) iterative rehearsal occurring while the list is being presented, with (2) few errors made and with (3) the time between the last rehearsal of an item and recall of that item being approximately equal to the duration of the list. In the timeline shown in the bottom panel of Fig. 1, rehearsal of a complete list continues through the period I d. If this period is short or if recall can commence immediately after the last item, then, unlike what is shown in Fig. 1, the final iterative rehearsal might occur before the last item on the list is presented. The time between last rehearsal and recall for an item can be estimated to be roughly equal to I r (N 1) for each list item, because each item must exist for the duration of the N 1 other items on the list. Consequently, for this model, we assume that the inter-stimulus and post-stimulus intervals do not have much effect because rehearsal maintains the memory for the list. This assumption is most likely wrong, although Baddeley and Lewis (1984) showed that presentation rate has little effect on accuracy when articulatory rehearsal is allowed. Given these assumptions, the probability of recalling item n correctly is S(I r (N 1)), and the probability of recalling an entire list correctly is: N P(I r, N) = S(I r (N 1)) = S(I r (N 1)) N. (5) i=1 Note that if I r = I p and I d = 0, Eq. (4) is identical to Eq. (5). That is, if words are presented at the same rate they can be rehearsed at, and recall starts immediately, rehearsal has no impact because it is essentially impossible to perform Evaluating the item-based decay model Using the formulas developed so far it is simple to show that individual items in memory-span length lists (i.e., those at 50% accuracy) must be recalled with considerably greater accuracy than 50%. For example, under suppression, I Tn will have N different values on which S(t) is evaluated, and under rehearsal it will have N identical values. For both conditions, when a span-length list is being recalled, the product of S(t) evaluated at these N points must equal 0.5, and so a rough estimate of the probability of recalling any single item in a span-length list is.5 1/N. For lists whose memory span is 2, 3, 4, 5, 6, and 7, these values are.71,.79,.84,.87,.89, and.91, respectively. Thus, depending on the length of the span list, the survival function must be considerably greater than.5 when recall accuracy of the complete list is at.5. Interestingly, this demonstration is not sufficient to conclude that the mean of the decay distribution for individual items is greater than the list rehearsal time. As an example, for a tape loop decay distribution which retains all information for t seconds, after which it completely decays (i.e., a step function), the duration of a span-length list will be approximately t. Any list that can be recalled more quickly than t will always be recalled correctly, and any list that takes longer will never be recalled correctly. Interestingly, the tape loop was one metaphor that motivated Baddeley s phonological loop model, although it is probably misleading, because it is unrealistic to suppose that memory decay works as mechanistically as a tape loop suggests. For the model to make predictions about data, however, a decay distribution must be chosen Decay distributions A model that specifies a probability distribution function for decay is different from a model that specifies an underlying process by which activation decays. A decay distribution describes the probability that recall of an item will be correct as a function of time. This is distinct from a description of the underlying mental representation and the process by which it decays over time. In general, the function that describes the graded decay of an underlying representation need not be the same as the function that describes the resulting decay distribution for recall. A number of models assume that some activation or connection

8 20 S.T. Mueller, A. Krawitz / Journal of Mathematical Psychology 53 (2009) weights decay with time (e.g. Anderson, Bothell, Lebiere, and Matessa (1998), Brown and Hulme (1995) and Wickelgren and Norman (1966)), but recall accuracy is often a non-linear function of that activation. Consequently, an exponential decay of activation will typically produce the S-shaped accuracy functions generally found by comparing list length to probability correct recall (cf. Wickelgren and Norman (1966)). The item-based model we present here adopts various probability distribution functions for decay, without making assumptions about the underlying process. Of those theories that assume a specific form of the decay distribution, some have suggested an exponential distribution (cf. Cowan (1984)), others have suggested a Weibull (Chechile, 1987), Schweickert and Boruff (1986) suggested a normal distribution, and Kieras et al. (1999) adopted a log-normal distribution. Recently, Chechile (2006) advocated hazard analysis to identify the properties of forgetting distributions, and showed how a two-process forgetting model is supported by this analysis. The exponential distribution is an attractive option, as it is prominent in physical decay processes and has an interpretation in terms of half-life that is sensible to many researchers. However, this distribution has the property that it is history-less: the time an item has existed in the past has no effect on its expected future lifetime. For such a distribution, the conditional decay distribution given that an item has survived to time t is the same as if it were newly-encoded at time t. This suggests that if words decayed from short-term memory according to an exponential distribution, rehearsal would not improve performance. Consequently, the exponential distribution is not a realistic candidate to consider for a decay time distribution if articulatory suppression reduces memory span by disrupting rehearsal. The choice of decay distribution will influence the estimates one obtains about the mean lifetime for items in short-term memory. For example, for a set of items whose memory span is seven, if items decay according to an exponential distribution, it must have a mean 5.7 times as long as the list articulation time (for a list that takes two seconds to articulate, the exponential decay distribution for which S(2.0) =.5 1/7 =.91 has a mean of 11.4 s). Another possible decay distribution is the uniform distribution with density f (t) = 1/T for t < T ; f (t) = 0 elsewhere, which has mean T/2. For this distribution, the mean decay time needs to be 5.3 times as large as the articulation time of the list (10.6 s). Many other distributions are possible, and distributions with longer tails and faster declines will typically estimate longer mean decay times for the same empirical data. To show how the lower estimate of decay time depends upon the choice of decay distribution, we will demonstrate the model s fits using several different decay distributions: the log-normal distribution, two exponential distributions, and a uniform distribution The log-normal decay distribution Schweickert and Boruff s (1986) use of a normal decay distribution is problematic because it suggests that there is some probability that a sampled decay time could be below zero, meaning there would be some lists that decayed before they were presented. Kieras et al. (1999) adopted a two-parameter version of the log-normal distribution (Hastings & Peacock, 1975), which we will consider here. The density function for the log-normal distribution is: f (t) = 1 tσ 2π exp { } (log(t/m)) 2 2σ 2 σ > 0; M, t 0; (6) and its survival function S(t) can be expressed as a function of the standard normal cumulative distribution function Φ, which can be approximated numerically by functions found in most scientific computing packages. S(t) = 1 Φ(log(t/M)/σ ), (7) This parameterization of the log-normal distribution is useful for representing decay times because it has no density below zero, and it has two parameters that are easily interpretable (M, the median in seconds, and σ the spread or shape of the distribution, which is independent of the timescale and so has no associated value label). Combinations of these parameters offer a considerable amount of flexibility in the shape and scale of the distribution: when σ is close to 0, the distribution approaches a step-like tape loop distribution, and when it grows large, the survival function decays slowly with a long tail. The mean of the log-normal distribution is M e σ 2 /2, which is typically slightly larger than the median. Other one-tailed distributions (such as the Weibull) could have been chosen as well, although it may be nearly impossible to distinguish between these distributions with empirical data Uniform decay distributions Another simple and flexible decay distribution is a 2-parameter uniform distribution. It assumes that items always remain for a minimum time period, and are always gone by a maximum time point, but their lifetimes are evenly distributed between those points. The density function for this two-parameter uniform distribution is: f (t) = 0; t < min (8) f (t) = 1/(max min); min t < max (9) f (t) = 0; t max. (10) Its mean and median are identical: (max + min)/ Exponential decay distributions Typically, exponential decay distributions have a single parameter, the decay rate. We will examine the model performance with exponential decay, but to give it similar flexibility to the lognormal, we will also examine a two-parameter version where the distribution is shifted to the right by a fixed value, which is also a free parameter. The density function for these distributions is: f (t) = 0; t < m (11) f (t) = λe λ(t m) ; t m. (12) Because the half-life of the one-parameter exponential decay distribution is 1/λ, the mean decay time for this distribution is m + 1/λ, and the median is m + ln(2)/λ. We fitted data with two versions of an exponential decay distribution, one in which both parameters (m and λ) were allowed to vary freely, and a second in which m was fixed to zero. 4. Modeling the relationship between list length and recall accuracy The utility of the list-based trace decay model of Schweickert and Boruff (1986) has been shown by its ability to fit to numerous data sets (e.g., six data sets in Schweickert and Boruff (1986); five data sets with two participants in Schweickert, Hayt, Hersberger, and Guentert (1996)). To demonstrate that the item-based decay model is able to make similar predictions about the psychophysical functions produced in immediate serial recall, the model was fit to these same 16 data sets. To estimate the average time required to rehearse each word, we estimated a single slope parameter of the relationship between list length and total recall time for

9 S.T. Mueller, A. Krawitz / Journal of Mathematical Psychology 53 (2009) Table 2 Summary of goodness-of-fit statistics for four different decay distributions, along with estimates for mean and median decay times in seconds. Distribution R 2 RMSE Max deviation Decay median Decay mean Log-normal Uniform parameter exponential parameter exponential Table 3 Summary of model parameters and mean or median decay times in seconds for different decay distributions. The one-parameter exponential decay distribution is not shown. Log-normal decay Uniform decay Exponential decay Median Spread Min Max Median Min λ Mean Schweickert & Boruff, 1986, digits Schweickert & Boruff, 1986, colors Schweickert & Boruff, 1986, consonants Schweickert & Boruff, 1986, nouns Schweickert & Boruff, 1986, shapes Schweickert & Boruff, 1986, CVCs Schweickert et al., 1996, part. 1, letters Schweickert et al., 1996, part. 1, words Schweickert et al., 1996, part. 1, prepositions Schweickert et al., 1996, part. 1, colors Schweickert et al., 1996, part. 1, shapes Schweickert et al., 1996, part. 2, letters Schweickert et al., 1996, part. 2, words Schweickert et al., 1996, part. 2, prepositions Schweickert et al., 1996, part. 2, colors Schweickert et al., 1996, part. 2, shapes each of the 16 data sets. Then, based on Eq. (4), we investigated the median decay estimates for four decay distributions (lognormal, uniform, and two exponential distributions). 8 We fit psychophysical functions relating list recall accuracy to list length for each experimental condition, using numerical techniques that minimized root mean squared error for each condition. Across data sets, the log-normal decay distribution fit best, with the uniform and two-parameter exponential fitting adequately but somewhat less accurately. The one-parameter exponential produced very poor fits, with maximum deviations from predicted that averaged 0.27 per experiment. It produced median decay time estimates that were especially long, so we will not present its results in greater detail. The mean decay times and goodness-of-fit statistics for these four decay distributions are shown in Table 2. The good fits are not entirely surprising, because the original psychophysical functions were very smooth and new parameters were fit for each condition. However, the choice of decay distribution does apparently matter, as the log-normal decay produced better fits than the others, and the 1-parameter exponential produced fits to data that were completely inadequate. As an illustration of the typical decay distributions obtained by each version of the model, Fig. 2 shows the mean survival function for each decay distribution obtained across all 16 experimental conditions under consideration (that is, the survival functions produced by the means of the parameters of the individual experiments). Table 3 shows the estimated parameters for each of the two-parameter decay distributions across the 16 data sets. Importantly, and as expected, the median decay parameters of this model were all longer than two seconds. Because the log-normal decay distribution had the best goodness of fit and the lowest estimated decay times, we will 8 The use of the suppression model is justified for these data because the experimental procedure used a fast self-paced presentation that precluded rehearsal, and in which recall began immediately following presentation. This procedure avoids the theoretical concerns about how rehearsal impacts the representations and durability of list items, and means that I r = I p and that I d = 0. In this case, Eqs. (4) and (5) are identical. Fig. 2. Typical survival functions produced for each decay distribution under consideration, formed by averaging the fitted parameters of the sixteen experimental conditions under consideration. use it for all subsequent analyses. Fig. 3 shows the the original data and model fits with the log-normal decay distribution. The corresponding survival functions of the estimated decay distributions are shown in Fig. 4. Overall, the log-normal decay distributions had median parameters M which ranged between 3.1 s and 4.5 s, with an average of 3.7 s and spread parameters σ which ranged between.27 and.52 with an average of.35. Across the 16 data sets, the fitted values of the two parameters had a correlation of.58, indicating that some trade-offs existed in the parameter values. Overall, the means of these distributions were between 3.4 and 4.8 s, with an average of 4.0 s, even though the span-length lists could be articulated in less than two seconds. In Schweickert and Boruff s list-based decay model, list decay times were 1.88 s on average, with a variance of s 2, which is less than half of the mean decay times produced in the item-based decay model. For comparison, the survival function of Schweickert and Boruff s (1986) distribution is shown as a dashed curve in Fig. 4. Note, however, that their distribution is intended to represent the decay of complete lists. Although the fitted lower-bound estimates of decay distributions provide good accounts of the data, there are a number of reasons why the decay times may be even longer. For example, because the model assumes mechanically perfect encoding and recall processes, it likely starts with a conservative estimate of