& Cognition 1993, 21 (2), 168-175 A multinomial processing tree model for degradation and redintegration in immediate

RICHARD SCHWEICKERT Purdue University. West Lafayette. Indiana

When items are presented for immediate recall, a verbal trace is formed and degrades quickly, becoming useless after about 2 sec. The span for items such as digits equals the number of items that can be pronounced in the available time. The length of the items affects span by affecting pronunciation rate. Other properties, such as phonological similarity and lexicality, can affect span without affecting pronunciation rate. These properties change the trace's useful lifetime by affecting redintegration. An analogy is drawn between trace reconstruction and repair of er­ rors in speech. When a trace is degraded, one process attempts to form a string, and another process attempts to form a . The two processes are autonomous and can be selec­ tively influenced by lexicality and phonological similarity. The resulting processing tree models make simple predictions that depend on whether or not the influenced processes are sequential. The results are illustrated with data from experiments by Besner and Davelaar (1982).

It has been known since at least the time of Oliver Wen­ correct if it is completed before the trace degrades to an dell Holmes (1871, p. 101) that a person cannot repeat unusable state. The trace can be refreshed by rehearsing verbatim a list of unrelated items immediately after pre­ the material, provided the time required to pronounce the sentation, unless the list is short. To this day, we do not items internally is less than the trace degradation time. know what determines how long the list can be. The num­ Memory span is often defined as the number of items ber of items in the list is not limited per se, because mem­ in a list that, half of the time, can be recalled in order ory span differs for different materials such as shape immediately after presentation. Then, the time to pro­ names and color names (Brener, 1940). Furthermore, the nounce a list of span length equals the time at which the span for a single material (e.g., digits) can differ when trace has become unusable. Ordinarily, this time is about the same individual uses different languages (e.g., Ellis 2 sec. & HenneUey, 1980). These facts challenge Miller's (1956) A particular version of this hypothesis is given in the idea that the span is limited to a certain number ofchunks model of Schweickert and Boruff (1986). It accounts for or meaningful units (according to this hypothesis, why differences in memory span for different materials in should the number of chunks in a list of digits change when terms of a race between recall and degradation of the the language is changed?). trace. According to the model, a list will be recalled cor­ A leading hypothesis is that short-term memory is not rectly if the time T, required to emit the list is less than limited to a number of items or chunks but is limited by the duration Tv of the verbal trace. Then, the probability the time for which a trace endures (Baddeley, Thomson, of correct recall is P = P[T, :5 Tvl. & Buchanan, 1975; Brown, 1958; Mackworth, 1963; The model leads to an equation originally proposed by Muter, 1980; Peterson & Peterson, 1959; Schweickert Baddeley et al. (1975) relating memory span, s, and pro­ & Boruff, 1986). According to this hypothesis, when nunciation rate, r, items are presented, a trace is formed and begins to de­ (1) grade, perhaps by decaying over time or by becoming s = rt + c, noisy because of interference. The useful lifetime of the in which t is the time required to pronounce a list of span trace will depend on what it is used for. Recall will be length. The intercept c is sometimes interpreted as the number of items recalled with the assistance of long-term knowledge (e.g., Hulme, Maughan, & Brown, 1991), al­ I thank William H. Batchelder, Gordon Brown, Richard Chechile, though Schweickert, Guentert, and Hersberger (1990) ar­ Nelson Cowan, Xiangen Hu, Randi Martin, and Mary C. Potter for gue that nonzero intercept values may be caused by an their helpful comments. Portions of this work were supported by NSF artifact. Grant DBS-9123865. Data in Tables I and 2 are copyrighted 1982 by Because one way to define the pronunciation rate r is the Canadian Psychological Association and are reprinted with permis­ with this equation, the fact that it holds is not surprising, sion. Correspondence should be addressed to the author at the Depart­ ment of Psychological Sciences, Purdue University, West Lafayette, nor is the fact that word length changes the pronuncia­ IN 47907. tion rate. What is surprising is that under a wide variety

Copyright 1993 Psychonomic Society, Inc. 168 DEGRADATION AND REDINTEGRATION 169

of circumstances, the time t is about the same, approxi­ The results of the experiments just described were in­ mately 2 sec, regardless of word length (Baddeley et al., terpreted by the investigators to mean that the trace du­ 1975; Hulme, Thomson, Muir, & Lawrence, 1984; Mack­ ration can be extended by the reconstruction of partially worth, 1963; Schweickert & Boruff, 1986). The time t degraded traces. Reconstruction is easier if the items are can be interpreted as the mean trace duration. The invari­ and if they are phonologically dissimilar (Sperling ance of this time over materials initially suggested that & Speelman, 1970). the trace duration is relatively constant and that the limit Cowan (in press) gives a detailed treatment of the role on memory span could be explained in terms of a limit of trace reconstruction. He proposed that trace reactiva­ on the time available for rehearsal or recall. This hypoth­ tion takes place in the pauses between items during spoken esis is contradicted by recent experiments, described in recall and that decay of the traces of items takes place the next section, in which the time t was malleable. Recon­ during actual speaking. The model in the following sec­ sideration of Equation 1 shows that a variable affecting tion can be thought of as sketching in a few details about memory span can do so by affecting the pronunciation what goes on during the reconstruction postulated in Co­ rate r, the trace duration t, the intercept c, or some com­ wan's model. Since the following model is not necessar­ bination of these. If variables affecting memory span al­ ily tied to the model of Cowan, I use the terms degrada­ ways affected all of the parameters, the equation would tion and redintegration in lieu of his terms decay and provide no resolving power. But the studies mentioned reactivation. above have shown that word length affects the pronunci­ ation rate r, without affecting the trace duration t. This A Model for Trace Degradation and Redintegration leads to the question of whether there are variables capa­ The challenge in modeling short-term memory trace ble of affecting memory span in another way, by chang­ redintegration is that so little is known about the nature ing t or c without changing r. of degraded traces that it is premature to specify details about how missing or noisy features are reconstituted. Phonological Similarity Some knowledge about degraded traces is available; for Hulme and Tordoff(1989) and, independently, Schweick­ example, Cowan and Morse (1986) provide a detailed ert et al. (1990) carried out experiments to investigate the analysis of vowel changes in short-term memory. But until locus of the effect of phonological similarity in serial or­ more is known, modeling with a moderate level of detail dered recall. The experiments differed in several respects, seems most likely to succeed, and multinomial processing but each supported the conclusion that phonological sim­ trees are useful structures for this purpose (Batchelder & ilarity can affect memory span without changing speak­ Riefer, 1986, 1990; Chechile, 1987; Chechile & Meyer, ing rate. That is, phonologically similar items have a 1976; Hu & Batchelder, 1992; Riefer & Batchelder, smaller memory span because they have a shorter useful 1988). In a multinomial processing tree (e.g., Figure 1), trace duration. an attempt to recall an item is carried out with a sequence of cognitive processes represented by branches on a path Lexicality from the root of the tree to one of the terminal nodes. Memory span is typically greater for words than for The outcome of the recall attempt depends on which ter­ nonwords. In the experiment by Schweickert and Boruff minal node is reached. (1986), the nonwords were pronounced more slowly than The model in Figure 1 represents the act of immediate the words, suggesting that the effect of lexicality could recall of a single item in a randomly ordered list. A com­ simply be a by-product of an effect on pronunciation rate. plete account of immediate recall would model aspects However, Hulme et al. (1991) found that when words and such as serial position effects and memory for the order nonwords were selected so that speech rates were equated, of the items (see, e.g., Lee & Estes, 1977; Nairne, 1991; memory span for the words was greater than memory span Shiffrin & Cook, 1978); a complete account would also for the nonwords. They concluded that, for their data, lexicality affected the intercept c in Equation 1, but did c not affect r. Support from an independent experiment was reported by Schweickert, Ansel, and McDaniel (1991). The words and nonwords used as stimuli were closely matched in several ways. Each list of items was presented visually and rehearsed aloud once; during this rehearsal, words and nonwords were spoken at the same rate. Nonethe­ less, the memory span was longer for words than for non­ words, supporting the hypothesis that the memory trace had a longer useful lifetime for words than for nonwords. The conditions used in the experiment did not provide an E indication of whether lexicality affects either the time t F'JgUre 1. Recall is correct (C) ifthe trace is intact or recomtructed; or the intercept c in Equation 1. otherwise, there is an error (E). 170 SCHWEICKERT

model recall of sentences. These are important but are Although the effects of factors selectively influencing beyond the scope of this paper. processes in a processing tree may not be simply addi­ The model in Figure 1 illustrates a commonly held idea, tive, the effects are systematic, and they are informative articulated by Estes (1991), that subjects carry out im­ about the structure of the processing tree. The tree in Fig­ mediate recall by first attempting direct readout and then ure 1 predicts that a factor influencing probability [ and guessing if that fails. With probability l, the trace for the another factor influencing probability R would have inter­ item is intact, and the item is recalled unhesitatingly and active effects on the probability of a correct response. If [ correctly. With probability 1 - l, the trace is so degraded is increased by M and R is increased by M, the interaction as to be ambiguous. In that case, readout cannot be com­ is predicted to equal their negative product, - txl»: M (see pleted, and other slower and more error-prone processes the Appendix). Because the combined effect of increas­ attempt to reconstruct the item on the basis of the remain­ ing both probabilities is less than the sum of the individ­ ing information. Reconstruction explains the finding of ual effects, the interaction is said to be underadditive. Standing, Bond, Smith, and Isely (1980) that the smaller The predicted interaction might be small. For exam­ the alphabet from which the items are selected, the larger ple, if [ were increased by .10 and R were increased by the memory span. .20, the interaction of - .10x .20 = -.02 would be hard Suppose the reconstruction processes, whatever they to detect. Fortunately, the interactions are not necessar­ may be, are correct with probability R. Then the proba­ ily small. For example, if [ were increased by .4 and R bility of correctly recalling an item is were increased by .5, the predicted interaction of - .20 would easily be detected. Interactions are important for PC = [ + (1 - [)R. (2) testing multinomial processing tree models, and predic­ An equation of the same form is the basis of Waugh and tions based on selective influence are derived in the Ap­ Norman's (1965) model. The parameter [has essentially pendix for experiments in which one or both of the fac­ the same meaning here as in their model. However, in tors have more than one level. their model, R would represent the probability the item is recalled from long-term storage. Here, R represents the Applying the Model probability that a degraded trace for the item is correctly Word length and lexicality are factors likely to selec­ reconstructed; this is a long-term memory contribution, tively influence processes in the processing tree in Fig­ but only in the sense that it is done through knowledge ure 1. When the moment arrives for recalling a particular of the language. A processing tree model leading to Equa­ item, the trace for that item is more likely to be degraded tion 2 was proposed by Link (1982) for a more general to the point of ambiguity if the other items in the list are purpose-namely, correcting response times and response long. This does not mean that the quality of the trace at choices for guessing. a particular time, say 1 sec after an item is presented, de­ One approach to modeling with multinomial trees is to pends on whether the items are short or long. Rather, the estimate the parameters and to test hypotheses about how time at which a particular item is about to be recalled de­ they differ from group to group in an experimental de­ pends on whether the other items are short or long. With sign. Another approach is to use factorial experiments to this reasoning, word length affects the probability (1 - l) analyze the effects of manipulating experimental factors of trace degradation. The experiments reported by Hulme that selectively influence the branch probabilities (Batch­ et al. (1991) and by Schweickert et al. (1991) were inter­ elder & Riefer, 1986). This approach is analogous to the preted to mean that lexicality affects the probability of analysis of response times with Sternberg's (1969) addi­ trace reconstruction, R. According to Equation 2, as noted tive factor method. above, these factors should have an underadditive inter­ Sternberg (1969) proposed that processing is carried out action on the probability of correct recall. in a series of stages, and if each of two factors selectively From the data available at this time, I am aware of an influences a different stage, then the combined effect on experiment with relevant data, although the results are, response time of influencing the two stages would be the unfortunately, not conclusive. Resner and Davelaar (1982) sum of the individual effects. Techniques based on selec­ manipulated the two factors of word length and lexicality tive influence have been very useful in analyzing response in an experiment on immediate ordered recall. The de­ times, leading some to propose that they would also be sign of the experiment would be ideal for testing the model useful for analyzing the percentages of correct responses in Equation 2 except that, for the experimenter's purposes, (Batchelder & Riefer, 1986; Ollman, 1982; Schweickert, no more than two levels of any factor were needed, and 1985). since the interaction was not of particular interest, the sam­ Although additive effects are expected in reaction time ple size does not provide a powerful test of it. Nonethe­ experiments when factors selectively influence different less, the experiment illustrates the technique and provides serial stages, Batchelder and Riefer (1986) warned that useful, if provisional, conclusions. investigators should not expect additive effects for fac­ Items were either one or three syllables in length. Lex­ tors selectively influencing processes in a processing tree. ical items were pseudohomophones such as "churtch" They provide an example in which two such factors pro­ and "familly." Nonlexical items were pronounceable con­ duce a crossover interaction, rather than additivity, for trols such as "chitch" and "ramitty." Lists were pre­ percent correct recall. sented visually in blocks of items with the same number DEGRADATION AND REDINTEGRATION 171

Table 1 or communicate continually (see Martin, Weisberg, & Experiment 2: Percent Correct Recall Saffran, 1989, for a brief review). Autonomous models in Data of Resner and Davelaar (1982) ---~--- ._--- for speech production include those ofGarrett (1975) and Not Lexical Lexical Levelt (1983). Nonautonomous models include those of 3 syllable items .578 .824 Dell (1986) and Stemberger (1985), the TRACE model I syllable items .729 .901 (McClelland & Elman, 1986), and the Boltzmann machine (Hinton & Sejnowski, 1986). ofsyllables and the same lexical status. Immediately after The issue of autonomy has not been resolved empiri­ each list was presented, the subject attempted to recall cally in experiments on overt speech, so it is unlikely to the items in order. Table 1 gives the probability of re­ be resolved in immediate recall tasks, which inevitably calling an item in its correct serial position. involve some covert speech. But various secondary con­ Each factor had a significant main effect. The inter­ siderations favor the study of autonomous models. Au­ action was not significant, but, as predicted, the combined tonomous models are simpler, because there is less com­ effect of lexicality and word length was numerically less munication. Furthermore, it is unlikely that an than the sum of their individual effects. The simple main experimenter would be able to selectively influence a sin­ effect oflexicality is .824 - .578 = .246; the simple main gle level in a nonautonomous model. But, by definition, effect of word length was.729 .:.... .578 = .151. Their an autonomous process can be selectively influenced (see, combined effect was .901 - .578 = .323, which is .074 e.g., Cutler, Mehler, Norris, & Segui, 1987, p. 146). less than the sum of their individual effects. The value Therefore, tests of nonautonomous models are not as ofthis interaction contrast, - .074, is close in magnitude straightforward as are tests of autonomous models, thus to the value predicted by the product of the simple main the autonomous models can be more easily disposed of. effects of each factor, -.037 = -(.824 - .578) x (.729 Within the class ofautonomous models, there is fairly - .578). The experiment demonstrates that testing the good agreement that separate processes operate at, model is feasible, and the result is promising. roughly, the phonetic, lexical, semantic, and syntactic levels. The agreement is not so good about how the pro­ Processes for Redintegration cesses are arranged, but if we restrict our consideration It is worth specifying trace reconstruction in more de­ to only the phonetic and lexical levels, those most rele­ tail. Suggestions are readily available from models of a vant for ordered recall of lists of items, only a few possi­ closely related activity- namely, monitoring speech and bilities have been proposed. correcting it for errors. According to one prominent the­ The autonomous processes are serial in the models of ory (Levelt, 1983), when a speaker formulates an intended Garrett (1975) and Levelt (1983); serial processes have message, the message passes through , also been proposed for the task of phoneme monitoring where it is phenomenologically available to the speaker (see, e.g., Fodor, Bever, & Garrett, 1974). Parallel au­ as a string of in inner speech. The speaker mon­ tonomous processes were proposed for phoneme monitor­ itors the inner speech for errors in about the same way ing by Cutler and Norris (1979). In a model of speech as he or she would monitor someone else's overt speech. production by Martin et al. (1989, Figure 1), pairs of au­ When trouble is detected, a correction is attempted on the tonomous processes are in series, and the pairs themselves basis of all the information the speaker has centrally avail­ are executed in parallel. The authors call this arrange­ able. The ability to carry out a correction is limited be­ ment serial, but it has both serial and parallel components. cause of the limited capacity of working memory. Because the arrangements are stated so explicitly, the From this point ofview, many immediate recall errors autonomous models are testable by manipulating factors are analogous to, if not synonymous with, speech errors. that selectively influence the processes. Processes analo­ Mechanisms for repairing speech are available for recon­ gous to, if not the same as, those proposed for repair of structing short-term memory traces, and vice versa. The speech would be useful for the job of reconstructing a major difference is that in immediate recall, the intended degraded memory trace. As before, if the trace for an item message is a list of recently presented items that have al­ is not degraded, the item is recalled directly from the ready been formulated into language, whereas during trace. Otherwise, the item is reconstructed from a speech production, some parts of the intended message degraded trace through two processes. One process at­ have been formulated into language while other parts have tempts to transform the degraded representation into a not. Consequently, portions of the theory of speech repairs word or other higher level entity. Such a process is postu­ are concerned with such matters as whether or not a para­ lated to account for the result, discussed in the introduc­ phrase retains the speaker's intentions; these portions are tion, that memory span for words is greater than that for not too important for immediate recall of items in a ran­ nonwords and that the difference cannot be explained in domly prescribed order. terms ofa difference in the length ofthe items. The other Errors in speech can occur at many levels, including process attempts to transform the degraded representa­ phonetic, lexical, and semantic levels. Models of speech tion into a string of phonemes. At first it may seem super­ commonly propose processes responsible for each level. fluous for the system to have a phonemic process in ad­ The models can be divided into two broad classes, de­ dition to a lexical process. But, to the contrary, some of pending on whether the processes proceed autonomously the most important practical uses for short-term verbal 172 SCHWEICKERT

c cess produces the correct item, whereas another ex­ c perimental factor changes the probability S that the phonemic process produces the correct item. In Figure 3, at least one path through the tree contains branches labeled with L and other branches labeled with S. For factors changing these parameter values, an interaction is pre­ dicted (see the Appendix). In Figure 2, in which the pro­ E cesses are not sequential, there is no path through the tree containing branches labeled with both an L and an S. This c model predicts additive effects of the two factors on per­ cent correct recall (see the Appendix). An earlier example of a prediction of additivity is in the model of Martin et al. (1989) for the combined ef­ fects of semantic and phonological similarity. As noted E earlier, these authors call their arrangement of processes Figure 2. With probability I, the trace is intact and recall is cor­ "serial," resulting in their assertion that selective influ­ rect. H thetrace is too degraded for direct readout, a phoneme string ence of serial autonomous processes leads to additive ef­ is reconstructed with probability S, and a higher levelentity is recon­ fects. In the terminology used here, the influenced pro­ structed with probability L. cesses in their model (their Figure 1) would not be called serial (or sequential). The terminological difference is not c c important, but, unfortunately, Martin et al. go on to say that all autonomous models predict additivity, and they interpret all interactions as evidence for nonautonomous models. This conclusion is too strong, because Batchelder and Riefer (1986) have shown that a crossover interaction can result from factors selectively influencing two pro­ cesses in a processing tree. More details are in the E Appendix. Two factors suitable for investigatingthe models in Fig­ ures 2 and 3 are phonological similarity and lexicality, and these were manipulated in another experiment of Bes­ ner and Davelaar (1982). As before, only two levels of each factor were used, so only a limited test of the models Figure 3. A processing tree in which the phonological process, S, can be made. All stimuli were visually displayed non­ and the lexical process, L, are sequential. words. Pseudohomophones (e.g., "phood") were items that would be pronounced as words and are labeled lexi­ memory pertain to nonwords, such as learning a vocabu­ cal here. Control items (e.g., "thude") would not be pro­ lary in a foreignlanguage and children learning new words nounced as words. The control items would be lexical if (Baddeley, 1992). one or two letters were altered; consequently, the proba­ Ifthe material consists of words arranged in meaning­ bility of reconstructing an item through lexical knowledge ful phrases or sentences, then reconstruction may be aided may not be 0, even for the control items. Lists of items by processes concerned with syntax, semantics, and re­ with the same level of lexicality and phonological simi­ lations provided by context. These higher level processes larity were presented together in blocks. Examples of con­ may provide enough constraints on the possibilities to trol items in lists with high phonological similarity are change or eliminate lower level effects of phonological "thude" and "zewd"; examples of control items with similarity, mispronounced words as stimuli, and so on. low phonological similarity are "thawl" and "zede." Im­ These processes are beyond the scope of this paper, al­ mediately after each list was presented, each subject at­ though of course they are important for the immediate tempted to recall all the items in order. The probability recall of sentences. Readers interested in that topic are of correctly recalling each item in its correct serial posi­ referred to the work of Potter and Lombardi (1990). tion was calculated. An arrangement in which autonomous phonemic and It may seem at first that items with high phonological lexical processes are not sequential is represented as a tree similarity would have a good chance of being recon­ in Figure 2. In the figure, with probability A, the lexical structed from a partially degraded trace. Ifall the items process determines the response outcome, and with prob­ rhyme with food, and the initial phoneme for "thude" ability 1- A, the phonemic process determines the out­ is available, the entire item, "thude," can be recon­ come. A sequential arrangement of the two processes is structed. But the items must be recalled in order. If the represented in the tree in Figure 3. crucial initial phoneme is not available, there is a very What do these models predict? Suppose one experimen­ low chance of guessing the item from the remainder. A tal factor changes the probability L that the lexical pro- straightforward calculation shows that this low chance of DEGRADATION AND REDINTEGRATION 173

Table 2 BROWN, J. (1958). Some tests of the decay theory of immediate mem­ Experiment I: Percent Correct Recall ory. Quarterly Journal of Experimental Psychology, 10. 12-21. in Data of Resner & Davelaar (1982) CHECHlLE, R. (1987). Trace susceptibility theory. Journal ofExperimen­ Not Lexical Lexical tal Psychology: General, 110, 203-222. CHECHILE. R., & MEYER, D. L. (1976). A Baysean procedure for sep­ Phonologically similar .750 .602 arately estimating storage and retrieval components of forgetting. Jour­ Phonologically dissimilar .707 .896 nal of Mathematical Psychology. 13, 269-295. COWAN, N. (in press). Verbal memory span and the timing of spoken recall. Journal of Memory & Language. guessing when the initial phoneme is lost far outweighs COWAN, N., & MORSE, P. A. (1986). The use of auditory and phonetic the improved chance of guessing when the later phonemes memory in vowel discrimination. Journal ofthe Acoustical Sociery are lost. Note also that if the initial phoneme of a control of America, 79, 500-507. item such as "zede" is degraded, very little lexical knowl­ CUTLER, A., MEHLER, J., NORRIS, D., & SEGUI, J. (1987). Phoneme edge would be of use in reconstruction, but detailed identification and the lexicon. , 19, 141-177. CUTLER, A., & NORRIS, D. (1979). Monitoring sentence comprehen­ knowledge of the features of phonemes in the context of sion. In W. E. Cooper & E. C. T. Walker (Eds.), Sentence process­ other phonemes would be useful. ing: Psycholinguistic studies presented to Merrill Garrett. Hillsdale, The data of Besner and Davelaar (1982) are listed in NJ: Erlbaum. Table 2. The main effect of each factor was significant, DELL, G. S. (1986). A spreading-activation theory of retrieval in sen­ but the interaction was not. This pattern is predicted by tence production. Psychological Review, 93, 283-321. ELLIS, N. C., & HENNELLEY, R. A. (1980). A bilingual word-length the model in Figure 2, in which the lexical and phono­ effect: Implications for intelligence testing and the relative ease of logical processes are not sequential. The alternative se­ mental calculation in Welsh and English. British Journal ofPsychol­ quential arrangement in Figure 3 predicts an interaction ogy, 71, 43-52. (see Appendix). Note again that an experiment specifi­ ESTES, W. K. (1991). On types of item coding and source of recall in short-term memory. In W. E. Hockley & S. Lewandowsky (Eds.), cally designed to test the tree models would have more Relating theory and data: Essays on human memory in honor ofBen­ factor levels and greater power. net B. Murdock (pp. 155-174). Hillsdale, NJ: Erlbaum. Besner and Davelaar (1982) also put forth an interest­ FODOR, J. A., BEVER, T. G., & GARIlETT, M. F. (1974). The psychol­ ing hypothesis. They proposed that the phonological code ogy of language. New York: McGraw-Hili. GARRETT, M. F. (1975). The analysis of sentence production. In G. H. used to maintain information in short-term memory is dif­ Bower (Ed.), The psychology of learning and motivation (Vol. 9, ferent from the code used for lexical access from printed pp. 133-177). New York: Academic Press. information. Their hypothesis was supported by a dissoci­ HINTON, G. E., & SEJNOWSKI, T. J. (1986). Learning and relearning ation of the effects of articulatory suppression. Those re­ in Boltzmann machines. In D. E. Rumelhart & J. L. McClelland sults will not be discussed here, because articulatory sup­ (Eds.), Parallel distributed processing: Explorations in the micro­ structure ofcognition (Vol. I, pp. 282-317). Cambridge, MA: MIT pression seems to produce wide-rangingeffects rather than Press. selective influence. HOLMES, O. W. (1871). Mechanism in thought and morals. Boston: In conclusion, experiments on memory span suggestthat Osgood. immediate recall can be increased by an item or two if Hu, X., & BATCHELDER, W. H. (1992). The statistical analysis ofgen­ eral processing tree models with the EM algorithm (Tech. Rep. No. degraded traces are easier to reconstruct. Reconstruction MBS 92-06). Irvine: University of California, Irvine Research Unit can be explained in terms of the autonomous processes in Mathematical Behavioral Sciences. postulated in some models for tasks such as the repair of HULME, C., MAUGHAN, S., & BROWN, G. D. A. (l99\). Memory for utterances and phoneme monitoring. These processes in­ familiar and unfamiliar words: Evidence for a long-term memory con­ clude one influenced by lexicality and another one influ­ tribution to short-term memory span. Journal ofMemory & Language, 30, 685-701. enced by phonological similarity. Multinomial process­ HULME, c., THOMSON, N., MUIR,c., & LAWRENCE, A. (1984). Speech ing trees provide a natural way to represent the outcomes rate and the development of short-term memory span. Journal ofex­ of the processing. The structure of the tree can be found perimental Child Psychology, 38, 241-253. if there are factors that selectively influence the proba­ HULME, C., & TORDOFF, V. (1989). Working memory development: The effects of speech rate, word length, and acoustic similarity on bilities associated with the branches in the tree. serial recall. Journal ofExperimental Child Psychology, 47, 72-87. LEE, C. L., & ESTES, W. K. (1977). Order and position in primary mem­ REFERENCES ory for letter strings. Journal ofVerbal Learning & Verbal Behavior, 16, 395-418. BADDELEY, A. D. (1992). Working memory. Science, 255, 556-559. LEVELT, W. J. M. (1983). Monitoring and self-repair in speech. Cog­ BADDELEY, A. D., THOMSON, N., & BUCHANAN, M. (l975). Word nition, 14,41-104. length and the structure of short-term memory. Journal of Verbal LINK, S. W. (1982). Correcting response measures for guessing and Learning & Verbal Behavior, 14, 575-589. partial information. Psychological Bulletin, 92, 469-486. BATCHELDER, W. H., & RiEFER, D. M. (1986). The statistical analysis MACKWORTH, J. F. (1963). The duration of the visual image. Cana­ of a model for storage and retrieval processes in human memory. Brit­ dian Journal of Psychology, 17,62-81. ish Journal ofMathematical & Statistical Psychology, 39, 129-149. MARTIN, N., WEISBERG, R. W., & SAFFRAN, E. M. (1989). Variables BATCHELDER, W. H., & RiEFER, D. M. (1990). Multinomial process­ influencing the occurrence of naming errors: Implications for models ing models of source monitoring. Psychological Review, 97, 548-564. of lexical retrieval. Journal ofMemory & Language, 28, 462-485. BESNER, D., & DAVELAAR, E. (1982). Basic processes in reading: Two MCCLELLAND, J. L., '" ELMAN, J. L. (1986). Interactive processes in phonological codes. Canadian Journal ofPsychology, 36, 701-711. speech perception: The TRACE model. In J. L. McClelland, D. E. BIlENER, R. (1940). An experimental investigation of memory span. Rumelhart, & The PDP Research Group (Eds.), Parallel distributed Journal of Experimental Psychology, 26, 467-482. processing: Exploration in the microstructure of cognition, Vol. 2: 174 SCHWEICKERT

Psychological and biological models (pp. 58-121). Cambridge, MA: process, say process a, is executed; it is successful with proba­ MIT Press. bility p., and fails with probability I-p•. The branch cor­ MILLER, G. A. (1956). The magical number seven, plus or minus two: responding to the outcome of the process, success or failure, Some limits on our capacity for processing information. Psychologi­ is included in the path, and a new node is reached. The proba­ cal Review, 63, 81-97. bility that a path is followed is the product of the probabilities MUTER, P. (1980). Very rapid forgening. Memory & Cognition, 8, associated with the branches on the path. The probability of cor­ 174-179. NAIRNE, J. S. (1991). Positional uncertainty in long-term memory. Mem­ rect recall ofthe item is the sum of the probabilities of the paths ory & Cognition, 19, 332-340. ending at terminal nodes associated with correct recall. OLLMAN, R. (1982, August). Additive factors and the speed-accuracy A process may be executed as the result of more than one tradeoff. Paper presented at the meeting of the Society for Mathe­ chain of events. That is, for a given parameter p, there may matical Psychology, Princeton, NJ. be more than one node whose outgoing branches have the prob­ PETERSON, L. R., & PETERSON, M. J. (1959). Short-term retention of abilities p and I- p. Suppose this does not happen in such a way individual verbal items. Journal of Experimental Psychology, 58, that some path from the root of the tree to a terminal node en­ 193-198. counters the same parameter more than once; that is, the prod­ POTTER, M. C.; & LOMBARDI, L. (1990). Regeneration in the short­ uct of probabilities on the branches of a path through the tree term recall of sentences. Journal ofMemory & Language, 29, 633-654. RIEFER, D. M., & BATCHELDER, W. H. (1988). Multinomial modeling has no parameter with a power greater than I. and the measurement of cognitive processes. Psychological Review, The processing trees considered here have binomial nodes­ 95, 318-339. that is, a node a is the starting node for exactly two branches, SCHWEICKERT, R. (1985). Separable effects of factors on speed and ac­ and the parameter p. appears only on these two branches, once curacy: Memory scanning, lexical decision, and choice tasks. Psy­ as p, and once in the expression I-p•. An experimental factor chological Bulletin, 97, 530-546. is said to selectively influence process a if changing the level SCHWEICKERT, R., ANSEL, B., & McDANIEL, M. A. (1991, Novem­ of the factor changes the value p, and of I-p. but does not ber). Lexicality and generation effects on memory span. Paper pre­ change the probability associated with any other branch. sented at the 32nd Annual Meeting of the Psychonomic Society, San The pair of branches starting at a node contributes nothing Francisco. to the probability calculation if the two branches end in termi­ SCHWEICKERT, R., & BORUFF, B. (1986). Short-term memory capac­ ity: Magic number or magic spell? Journal ofExperimental Psychol­ nal nodes of the tree with the same outcome, o. Such pairs of ogy: Learning, Memory, & Cognition, 12,419-425. branches can be removed, so their starting node becomes a ter­ SCHWEICKERT, R., GUENTERT, L., & HERSBERGER, L. (1990). Phono­ minal node of the tree labeled with the outcome o. This proce­ logical similarity, pronunciation rate, and memory span. Psycholog­ dure can be repeated until in the tree remaining, each parame­ ical Science, 1, 74-77. ter occurs on at least one path associated with correct recall and SHIFFRlN, R. M., & COOK, J. R. (1978). Short-term forgetting of item on at least one path associated with incorrect recall. Changing and order information. Journal ofVerbal Learning & Verbal Behavior, the level of a factor selectively influencing the probability of 17, 189-218. a branch will then produce both a change in the probability of SPERUNG, G., & SPEELMAN, R. G. (1970). Acoustic similarity and audi­ tory short-term memory: Experiments and a model. In D. A. Norman correct recall and a change of equal magnitude but opposite sign (Ed.), Models ofhuman memory (pp. 152-202). New York: Academic in the probability of incorrect recall. Press. Suppose the levels of two factors are changed. In a factorial STANDING, L., BOND, B., SMITH, P., & ISELY, C. (1980). Is the im­ experiment, if the combined effect of changing the levels of both mediate memory span determined by subvocalization rate? British Jour­ factors is greater than the sum of the effects of changing them nal of Psychology, 71, 525-539. separately, the interaction is said to be overadditive. Additivity STEMBERGER, J. P. (1985). An interactive activation model oflanguage and underadditivity are defined analogously. production. In A. W. Ellis (Ed.), Progress in the psychology oflan­ Given two parameters p and q, it follows from the assump­ guage. (Vol. I). London: Erlbaum. tions that the probability of some paths through the tree may STERNBERG, S. (1969). Memory scanning: Mental processes revealed by reaction-time experiments. American Scientist, 57,421-457. involve only the parameter p, some may involve only q, some WAUGH, N. C., & NORMAN, D. A. (1965). Primary memory. Psycho­ may involve bothp and q, and some may involve neither. Then logical Review, 72, 89-104. the probability of a correct recall can be written as

APPENDIX PC = Ap + B(I - p) + Cq + D(I - q) + Epq + F(l - p)q Factors Selectively Influencing Processes in + Gp(l - q) + H(l - p)(1 - q) + I, (AI) Multinomial Processing Trees in which none of the expressions A, B, ..., I depend on p or q. Each node in a multinomial processing tree represents a pro­ The combined effects of selectively influencing the process­ cess used in attempts to recall an item. Each process has a finite ing at two different nodes a and b depends on how the nodes number of possible outcomes; for instance, one outcome of a are arranged in the processing tree. Consider, for example, the process might be that a remnant of a memory trace was suc­ effects of changing the parameters A and L in Figure 2. The cessfully identified as a word, whereas another outcome might probabilities of correct recall are given in the top part of Ta­ be the failure to do this. Each outcome is represented by a branch ble A I for three values of A and four values of L. The proba­ starting at the node representing the process. It is assumed here bility of correct recall increases monotonically with L at each that two branches start at each node; in principle, there could level of A, because a branch on a path leading to correct recall be more. Each terminal node of the tree represents the result is labeled with L. The effects of changing A are more compli­ of a recall attempt. For our purposes, each terminal node cor­ cated. For low values of L, the probability decreases with A, responds to either the correct or incorrect recall of an item. The whereas for high values of L, the probability increases. The attempt to recall a particular item on a particular trial is repre­ change in sign is produced because one path to correct recall sented by a path from the root of the tree to one of the terminal in Figure 2 goes through a branch labeled A, whereas another nodes. When a node is reached in the recall attempt, a certain path goes through a path labeled 1 - A. DEGRADAnON AND REDINTEGRAnON 175

Table Al ter p . note that paths contributing to the probability of correct Effects of Changing Parameters A and L in the Tree in Figure 3 recall may include p, or I - p ; or neither, and higher powers L of P do not occur. Consequently, probability of correct recall A .20 .40 .60 .80 as a function of p can be written, Probability of Correct Recall PC(p) = Jp + K(1 - p) + L, .25 .54 .58 .62 .66 in which J, K, and L are real numbers. When p is increased .50 .48 .56 .64 .72 by lip, the corresponding probability is .75 .42 .54 .66 .78 PC(p + lip) = J(p + lip) + K(1 - p - lip) + L, Interaction Contrasts .25 and the increase in probability of correct recall is .50 .04 .08 .12 .75 .08 .16 .24 PC(p + lip) - PC(p) = (J - K)lip. This can be calculated from

The interaction contrast for cell (i,)}, in which factor I is at PC(p+lip) - PC(p) = apC(p) lip. (A2) level i and factor 2 is at level), is PCij - PC} - PC" + PC". ap For example, the interaction contrast for the cell in row 3 and column 4 is .78 - .66 - .42 + .54 = .24. The interaction The interaction contrast for a factor increasing p by lip and contrasts are in the lower pan of Table AI. They are monoton­ another factor increasing q by liq can be derived by repeated ically increasing in both the rows and columns. application of Equation A2. In general, the effects of selectively influencingtwo processes, PC(p +lip,q+liq) - PC(p, q+liq) - PC(p+lip, q) +PC(p, q) a and b, in a processing tree meeting the assumptions leading to Equation A I can be summarized as follows. The levels of = apC(p,q+liq'l lip a~C;(p,q) lip the factors can be ordered so that for a fixed level of one of ap ap the factors, percent correct is monotonic with changes in the level of the other factor. As the levels of one factor (e.g., fac­ a [PC(p, q+liq) - PC(p, q)] lip tor I) increase, the relation between percent correct and the other ap factor may change, but not more than once, from monotoni­ cally increasing to monotonically decreasing. If the interaction --_ a apC(p,q) J.J.qJ.J.pAA contrasts are not all zero, they will all have the same sign and ap aq will either monotonically increase or decrease in the rows and 2pC(p,q) _AA a columns, with no change in the direction of monotonicity. The - J.J.pJ.J.q-. interactions can be classified in three ways, depending on how ap Bq the influenced processes are arranged in the tree: (I) The fac­ The interaction contrast will have the same sign as the second tors will have additive effects if and only if no path from the derivative in the equation above. This sign depends only on the root to a terminal node contains a branch with an expression arrangement of the nodes in the processing tree and not on the involving p and another branch with an expression involving values of the probabilities themselves. q; (2) if the factors have overadditive effects, then at least one To use the formula above to predict the interaction contrast path from the root to a terminal node includes branches labeled when I and R are changed in Equation 2, note that the partial with po and p., or branches labeled with I - p, and I - p.; derivative of PC with respect to R is I - I and that the deriva­ and (3) if the factors have underadditive effects, then at least tive of this with respect to I is - I. Thus, the interaction con­ one path from the root to a terminal node includes branches la­ trast is predicted to be - /ilfiR. beled with p; and I - Ph, or labeled with I - p, and Ph. The statements above can be established through straightfor­ ward calculations of the interactions. To calculate the effect on (Manuscript received May 9, 1992; the probability of correct recall of a change Ap in a parame- revision accepted for publication September 4, 1992.)