Quantitative prediction of spacing effects in learning

JOURNAL OF VERBALLEARNING AND VERBALBEHAVIOR16, 693-698 (1977)

Quantitative Prediction of Spacing Effects in Learning ADAM V. REED

The Rockefeller University Two parametric theories, the Markov model of J. L. Young (1971, Journal of Mathematical Psychology 8, 58-81) and A. V. Reed's (1976a, Memory and Cognition 4, 453-458) version of the strength-resistance theory (Wickelgren, W. A., 1972, Journal of Mathematical Psychology 9, 418-455; Wickelgren, W. A., 1974a, Contemporary developments in mathematiealpsychology, Freeman, New York; Wickelgren, W. A., 1974b, Memory and Cognition 2, 775-780), were tested against the spacing effects data of A. M. Glenberg (1976, Journal of Verbal Learning and Verbal Behavior 15, 1-16). Under the assumption that the position of items in the strength distribution about the mean is due to random variability in perceptual processes at the time of retrieval, the strength-resistance theory fitted the data quantitatively without significant deviations, the Z2 value falling between the median and the 90th percentile of the applicable Z2 distribution. The strength-resistance theory also accounts for qualitative observations left unexplained by other theoretical schemes.

Although quantitative theories of memory date back at least to Ebbinghaus (1885), the size of experiments necessary to check such theories discouraged later workers, so that qualitative approaches have been, and still are, the rule in investigations of memory. The introduction of laboratory computers has changed this situation; the present note will use data from a recent qualitative study (Glenberg, 1976) to evaluate two contemporary quantitative theories.

STRENGTH-RESISTANCE THEORY

The key concept of the strength-resistance theory of memory dynamics (Wickelgren, 1972, 1974a) is trace strength; retrieval occurs only if this variable exceeds a threshold. This threshold depends on the type of retrieval process, so that, for example, recall requires a higher minimum-strength /~---a higher

threshold--than ordinary recognition. Items presented under a particular set of conditions, t, for example, t seconds before retrieval, will have a Gaussian distribution of strength, with mean S ( t ) and a standard deviation tr whose dependence on mean strength may be approximated by a = e cs(t) (1) The basic axioms of the strength-resistance theory concern the relationship between S(t) and a resistance variable, r(t), whose value depends on the history of the trace. The strength of an item is assumed to be dissipated according to the equation d S(T)/dt = -k[S(t)/r(t)], (2) that is, in a manner analogous to the discharge of a capacitor through a resistor (hence the name). This resistance is variable, and a function of time: Specifically, r(t) = t I-v, (3) with the exponent of the power function between 0 and 1. If an item is presented twice, and recognized on the second presentation, the strength of the item is brought up to the original or a slightly higher value; but the resistance continues to grow without interruption as a function of time since the item was first stored.

This work was supported by Alcohol, Drug Abuse, and Mental Health Administration National Research Service Award 5F32MH05028-02 from the National Institute of Mental Health. The author is grateful to Arthur Glenberg for his helpful comments on an earlier version of this manuscript. Send reprint requests to Adam V. Reed, The Rockefeller University, 1230 York Avenue, New York, New York 10021. Copyright~) 1977by AcademicPress,Inc. 693 Allrightsofreproductionin any formreserved. Printedin GreatBritain

ISSN0022-5371

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ADAM V. REED

The original version of the strength-resistance theory assumed the simultaneous presence of several memory traces for each item, with growth of resistance and decay of strength starting in all cases at the moment of presentation (Wickelgren, 1972, 1974a). Subsequent data called for a reassessment of the multiple-trace hypothesis (Wickelgren, 1974b; Reed, 1976a), and the present paper, following Reed (1976a), will assume that there is only one passive memory trace, following the assumptions of the strength-resistance theory, per item. This version of the strength-resistance theory further assumes that the growth of resistance, as well as the decay of strength, begins at the moment a trace is transferred into passive memory from an active, limitedcapacity (fixed-span) attentional buffer. Items still in the buffer are distributed with mean strength S O. It may be readily shown that, if the time is measured from the start of resistance growth, the strength of an item stored at time t o with a strength S 0, and retrieved at time tr, will be S(t r) = S O e -°(try-to').

(4)

If the growth of resistance starts at t -- r, then (t - r) would be substituted for t in the above. (p, 2,, c, and k in Eqs. (1)-(4) are free parameters, presumably determined by experimental conditions. If N ( x ) is the cumulative integral of the normal distribution, the probability of retrieval is Pr = 1 -- N{[S(tr)

~ fl]/tr}.

(5)

SPACING (LAG) EFFECTS

Glenberg's (1976) Experiment 1, as well as the other experiments referred to in this section, deal with the following situation: Let a stimulus, such as an item pair in a paired-associates experiment, be presented twice, with an interpresentation lag of nl items, equivalent to spacing the two presentations t 1 seconds apart. After a retention interval of an additional n 2 items or t 2 seconds following the second presentation, the retention of the stimulus is tested, for example, by requiring

recall of the second item in the PA pair in response to the first. The data consist of accuracy measures, generally percentage correct in traditional PA experiments, as a function o f n I (or tl) and n2 (or t2). Studies using the Brown-Peterson distractor method (Peterson, 1963) and free recall (D'Agostino and DeRemer, 1973; Gartman and Johnson, 1972; Madigan, 1969; Melton, 1970), as well as paired associates with sufficiently long retention intervals after the second presentation (Glenberg, 1976)have found that the likelihood of correct recall on the final test increases monotonically with increasing t 1 within the range of tl values used in those studies. However, two conspicuous nonmonotonic effects of t I emerge with short values of t 2. First, while, for long t 2, recall with nonzero values of na is generally superior to performance for n I = 0, this relationship is conspicuously reversed for short t 2 (Peterson, Hillner, & Saltzman, 1962; Glenberg, 1976). Thus, for n 2 = 2 events (i.e., t 2 -- 9 see), Glenberg (1976) found that as n I changed from 0 to 1, Pr decreased by 5 percentage points. Second, for short values of t2, Pr increased with subsequent increases in n 1 only until it reached an optimum, and then declined [Peterson, Wampler, Kirkpatrick, & Saltzman, 1963; Atkinson and Shiffrin, 1968 (Brelsford and Atkinson experiment); Young, 1971; Glenberg, 1976]. Moreover, the optimal nl appears to increase with n 2, thus opening the possibility that the effect of n I is always nonmonotonic, but, in the case of long n2, the optimal n~ lies beyond the range of n I values explored in reported experiments. For a more detailed discussion of these phenomena, see Glenberg (1976). FITTING OF PARAMETRIC HYPOTHESES The standard error of probabilistic measurements is ap = [p(1 - p ) / m ] ~ , where m is the number of observations and p is the observed proportion. Thus a standard error of 2% requires, in the neighborhood of p = .5, a minimum of 625 observations per point. The data from Experiment 1 of Glenberg (1976),

SPACING EFFECTS

with 540 observations per point, are only marginally adequate for testing quantitative theories; but they do come close enough to be worth the attempt. [Experiments 2 and 3 of the same study use even fewer observations per point, and cannot be evaluated quantitatively. Experiment 3 found effects analogous to those of Experiment 1, using recognition in place of recall; this result is consistent with the strength theory, which postulates that the only difference between the requirements of recall and those of recognition is a lower fl for the latter. Since the strength-resistance theory postulates that the parameters governing decay and resistance depend on experimental conditions, the results of Experiment 2 (which found that short-term monotonicity could be reestablished by a similarity manipulation) are also explicable in strength-resistance terms.] A variety of qualitative theories and their relationship to the data were discussed in Glenberg (1976), and that discussion will not be repeated here. However, at least two quantitative theories, neither of which had been explicitly considered by Glenberg (1976), appeared potentially capable of explaining the results of that study: a five-state Markov model developed by Young (1971) specifically for the task of accounting for nonmonotonic spacing effects found in that study, and the strength-resistance theory (Wickelgren, 1972, 1974a, 1974b; Reed, 1976a) discussed earlier. The strengthresistance theory had been developed in the context of detection-theoretic studies of recognition and had not been quantitatively tested in any previous study against data from cued recall of paired associates. Six specific hypotheses were fitted to the data of Experiment 1 of Glenberg (1976). Hypothesis 1 incorporated the five-state Markov model of Young (1971). The other five hypotheses were derived from the strengthresistance theory, and differed in the specific assumptions applying the concepts of the strength theory to the PA cued.recall situation. Hypotheses 2 and 3 treated the concept of strength as applicable to each item in a pair

695

separately, although transfer of resistance still required recognition of both members of a pair on its second presentation; cued recall required recognition of the first item and recall of the second. Hypotheses 4, 5, and 6, on the other hand, treated the pair as a single unit characterized by a specific value of strength. Hypotheses 2 and 4 assumed that the variability of strength around the mean value was due to variability in retention conditions and did not depend on perceptual processes during recognition. Hence, these hypotheses assumed that items not recognized on their second presentation would form brand new traces without transfer of resistance. Hypotheses 3 and 5 assumed that this variability was due to perceptual variations of the time of retrieval, so that if a pair failed to be recognized it would also fail to be stored in a new trace. Finally, hypothesis 6 assumed that the location of each paired associate in its strength distribution relative to the distribution's mean was fixed, due to the inherent relatedness of the two member terms. Hypothesis 1 uses seven parameters, defined as in Young (1971, pp. 72-77). The five hypotheses based on the strength-resistance theory may be formulated with either seven or six parameters. The seven parameters are ~, ~0, and c, defined in Eqs. (1) and (4); S 1 and S 2, the initial strengths after the first and second presentations, respectively; and fll and f12, the respective strength thresholds for recognition recall. The six-parameter fits assume that S 2 = Sl; that is, that second presentation restores item strength exactly to its original value. PROCEDURE

The modular program package MODPAC, described in Reed (1967b), was used for all parameter fittings. Linear error measure, computed by subroutine ERLIN of that package, was used throughout. A separate dependent-variable computing routine (function DEVAR) was used for each of the six hypotheses. Each model was initially fitted two times, starting with different parameter values

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on each session. If the results of the two fitting sessions did not agree, further fitting runs were performed, starting with parameter values extrapolated from the best fits to date, until absolute convergence was achieved. X2 calculations used a standard error estimate of .0215, corresponding to 540 observations per point in the neighborhood ofp = .5. The Reed (1976a) version of the strengthresistance theory assumes that the decay of strength and the growth of resistance start when the capacity of the attentional buffer (often called "span") is exceeded, and the trace is transferred to passive memory; in the Glenberg procedure, this would occur on the presentation of the third consecutive paired associate, that is, at r = 6 see. This assumption was incorporated in DEVAR programs for hypotheses 2 through 6. It was tested by additional fitting runs in which r was fitted as a free parameter. The fitted values of r were all in the range of 5.99990 to 6.00009, indicating that the above assumption was justified. The closeness of the fit is surprising; it may be that Glenberg's procedure determines transfer time more consistently than the procedures of experiments examined in Reed (1976a). RESULTS AND DISCUSSION

The fitted parameters and significance measures for all six hypotheses are shown in Table 1. In the table, P3 is the probability that an item was recognized on the second presentation; P4 and P5 are final recognition and recall probabilities, respectively, for pairs recognized on the second presentation; P~ and P2 are recognition and recall probabilities on the basis of the last presentation only. The best-fitting curves, those of hypothesis 3, are compared with the data of Glenberg (1976) in Fig. 1. The most significant departures from data occurred with hypotheses 1 (model of Young, 1971) and 6 (strength-resistance model with position of particular pairs relative to the distribution mean inherently fixed). The departures from the model were marginally significant (.10 > p > .05) for hypotheses 2

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and 4, which assumed that the position of pairs in their distributions had been determined by variability in storage. Hypotheses 3 and 5, which assumed that strength distributions about the mean were due to perceptual variability in retrieval, produced curves that did not depart significantly from the data, with ~f2 values falling between the median and the 90th percentile of the Z2 distribution for d f = 17. The fits for hypotheses assuming that each pair was stored as a unit did not differ from those for hypotheses assuming separate storage of the two words in each pair in any way that would allow the rejection of either alternative. The optimal parameters for all strengthresistance hypotheses conformed to the expected constraints. Thus, for all fits, the recall threshold t2 was considerably higher than the recognition threshold fir Similarly, in all cases, the second presentation starting strength S 2 was slightly higher than the first presentation starting strength $1; and the two starting strengths were close enough to equality to permit the fitting of each hypothesis, under a fixed starting strength assumption, with only a slight increase in relative fitting error. The values of all fitted parameters are within the range expected from previous studies (Wickelgren, 1972, 1974a, 1974b). One reason for the relatively poor performance of the Young (1971) model is its failure to predict the decrement in performance from

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ADAM V. REED

n I = 0 to nl = 1 with low values of r/2. N o published theory other than the strength-resistance theory predicts this phenomenon; Glenberg (1976) wrote that "this interaction has yet to be satisfactorily explained." The strength-resistance theory explains this interaction quite readily. According to the Reed (1976a) version of the theory, a first presentation will cause the storage of the item in a limited-capacity attentional buffer, from which it is transferred to passive memory only later. A n item which is presented only once, as the n~ = 0 items were in the Glenberg (1976) experiment, will not be transferred into passive memory until displaced by subsequent items. On the other hand, an item which is presented a second time while still in the attentional buffer, as was the case for n~ = 1 items, will be transferred to passive memory only if its strength exceeds the recognition threshold--but, if it does, it will be transferred immediately, and will begin to build up resistance earlier than comparable n, = 0 items. Thus, with short retention times, occasional failures of recognition on the second presentation will make n 1 = 1 items less likely to be correctly retrieved than nl = 0 items. But, with longer retention intervals, the additional resistance will compensate for this by slower decay later on. Thus, under the assumption that strength variability about the mean of the strength distribution is due to perceptual variability at the time of retrieval, the Reed (1976a) version of the strength-resistance theory is adequate to account, quantitatively as well as qualitatively, for the spacing effect data obtained by Glenberg (1976). The finding that this theory is capable of accounting for data obtained in an experiment not originally designed for parametric verification of a quantitative model, and of doing it better than the qualitative theories the experiment had been designed to test, augers well for its applicability in still other contexts. REFERENCES ATKINSON,R. C., & SHIFFmN,R. M. Human memory: A proposed system and its control processes. In K. E. Spence & J. T. Spence (Eds.), The Psychology of

learning and motivation: Research and theory. New York: Academic Press, 1968. Vol. II. D'AGosT~o, P. R., & DEREMER, P. Repetition effects as a function of rehearsal and encoding variability. Journal of Verbal Learning and VerbalBehavior, 1973, 12, 103-113. EBBINGHAUS,H. Ober das Geddchtnis. Leipzig: 1885. English translation, H. A. Ruger & C. E. Bussenius, Memory: A contribution to experimental psychology, New York: 1913. GARTMAN,L. M., & JOHNSON,N. F. Massed versus distributed repetitions of homographs: A test of the differential-encoding hypothesis. Journal of Verbal Learning and Verbal Behavior, 1972, 11, 801808. GLENBERG, A. M. Monotonic and nonmonotonic lag effects in paired-associate and recognition memory paradigms. Journal of Verbal Learning and VerbalBehavior, 1976, 15, 1-16. MADMAN, S. A. Intraserial repetition and coding processes in free recall. Journal of Verbal Learning and Verbal Behavior, 1969, 8, 828-835. MELTON,A. W. The situation with respect to the spacing of repetitions and memory. Journal of Verbal Learning and Verbal Behavior, 1970, 9, 596-606. PETERSON,L. R. Imme&ate memory: Data and theory. In C. N. Cofer & B. S. Musgrave (Eds.), Verbal behavior and learning: Problems and processes. New York: McGraw-Hill, 1963. PETERSON, L. R., HILLNER, K., & SALTZMAN, T. Supplementary report: Time between pairings and short-term retention. Journal of Experimental Psychology, 1962, 64, 550-551. PETERSON, L. R., WAMPLER,R., KIRKPATRICK,M., & SALTZMAN,D. Effect of spacing presemations on retention of a paired associate over short intervals. Journal of Experimental Psychology, 1963, 66, 206-209. REED, A. V. Recognition memory: One-component strength functions in the "short-term to long-term transition region." Memory and Cognition, 1976, 4, 453-458. (a) REED, A. V. MODPAC: A modular package of programs for fitting model parameters to data and plotting fitted curves. Behavior Research Methods and Instrumentation, 1976, 8, 375-377. (b) WICKELGREN,W. A. Trace resistance and the decay of long-term memory. Journal of Mathematical Psychology. 1972, 9, 418-455. WICKEL~m~N,W. A. Strength-resistance theory of the dynamics of memory storage. In D. H. Krantz, R. C. Atkinson, R. D. Lute, & P. Suppes (Eds.), Contemporary developments in mathematical psychology. New York: Freeman, 1974 (a) WICKELGREN, W. A. Single-trace fragility theory of memory dynamics. Memory and Cognition, 1974, 2, 775-780. (b) Yotmo, J. L. Reinforcement-test intervals in pairedassociate learning. Journal of Mathematical Psychology, 1971, 8, 58-81.