The Power Integration Diffusion Model for Production Breaks

Transcription

1 Journal of Experimental Psychology: Applied Copyright 2002 by the American Psychological Association, Inc. 2002, Vol. 8, No. 2, X/02/$5.00 DOI: // X The Power Integration Diffusion Model for Production Breaks Sverker Sikström Stockholm University Mohamad Y. Jaber Ryerson University The learning curve during repetitive production and the associated forgetting during production breaks are fundamental issues in the understanding of behavior. A model is suggested that combines 3 basic findings, namely, that single memory traces decay according to a power function of the retention interval, that aggregated memory traces can be combined by integration, and that the time to produce a unit can be described by a diffusion process on the memory trace. This power integration diffusion model is validated with empirical data, and the result fits better than 14 other published forgetting models. The learning forgetting process has captured the attention of researchers in two scientific disciplines: industrial engineering and experimental psychology. A fundamental aspect of the learning forgetting process is how breaks in learning influence performance. In this article, we suggest a theory of this process and evaluate it in an applied setting where the main focus is how breaks in production influence performance. Forgetting curves have been extensively studied in the field of cognitive psychology. One of the first attempts to describe forgetting was suggested as early as 1897 in Jost s law (Jost, 1897, cited in Hovland, 1951). Jost stated that if two associations were of equal strength at a given point in time, then the older one would lose strength more slowly with further passage of time. Later research has suggested quantitative descriptions of the strength of a memory as a function of the retention interval. Crovitz and Schiffman (1974) suggested that forgetting could be described as a power function of the retention interval. This assumption means that a fraction of the strength of a memory trace is lost every time the retention interval is doubled. Several other forgetting functions have also been suggested. For example, Rubin and Wenzel (1996) gathered a database consisting of 210 published sets of forgetting curves and fitted it with 105 different forgetting functions. They found that the power function fits better than other suggested models in literature, in particular the exponential function. However, it was difficult on empirical grounds to differentiate between the power function and a few other functions, because the experimental data were not sufficiently precise. Power function forgetting curves have been found for a variety of different cognitive tasks. A convenient method of collecting Sverker Sikström, Department of Psychology, Stockholm University, Stockholm, Sweden; Mohamad Y. Jaber, Department of Mechanical, Aerospace and Industrial Engineering, Ryerson University, Toronto, Ontario, Canada. The research of Sverker Sikström was supported by a grant from HSFR (Humanistiska Samhällsvetenskapliga Forskningsrådet). The research of Mohamad Y. Jaber was supported in part by a grant from NSERC (Natural Sciences and Engineering Research Council of Canada). Correspondence concerning this article should be addressed to Sverker Sikström, Department of Psychology, S Stockholm University, Stockholm, Sweden. sverker@psych.utoronto.ca data on forgetting is autobiographic memory: collecting memories by instructing participants to freely generate events from their life experience and to specify at what times in their lives these events occurred. Frequencies of occurrence for these autobiographic events conformed well to a power function across huge time spans, with retention intervals ranging from 1 hr to 18 years (Rubin, 1982). Data on forgetting functions have also been collected in more controlled experimental conditions, for example, under conditions in which participants study a list of words and at different retention intervals are asked to recognize or recall words from the list (Rubin, Hinton, & Wenzel, 1999; Wixted & Ebbesen, 1991). The time taken to perform a task also decreases as a power function practice, often referred to as Fitt s law (for a review, see Newell & Rosenbloom, 1981). The debate over whether forgetting is really a power law as opposed to an exponential (exp) one has become a captivating topic to researchers in recent years. Wixted and Ebbesen (1991) conducted three experiments, two involving human participants and one involving pigeons, to describe the course of forgetting functions over time, with those involving humans being relevant to this article. They investigated six potential forgetting functions: linear, exponential, hyperbolic, logarithmic, power, and exppower. Two experiments were conducted. The first involved encoding and retrieval of a list of words, and the second experiment involved face recognition followed by retention intervals. Results from the first experiment of Wixted and Ebbesen (1991) showed that the exponential function, which describes so many natural processes, from population growth to radioactive decay, did not improve much on the linear function. Similarly, the hyperbolic function, which describes the mathematical form of some learning curves, as in Mazur and Hastie (1978), did not fit well. The exp-power function was reasonably accurate, although it was outperformed by the power function. Wixted and Ebbesen s (1991) second experiment produced similar results. Anderson and Tweney (1997) argued that the power law of forgetting may be an artifact of arithmetically averaging individual participant forgetting functions that are truly exponential in form and that geometric averaging would avoid this potential problem (see also Heathcote & Brown, 2000a, 2000b, for similar arguments regarding reaction times). Wixted and Ebbesen (1991, 1997), using data from their second experiment, answered Anderson and Tweney s argument 118

2 POWER INTEGRATION DIFFUSION MODEL 119 by showing that their conclusions about the form of the forgetting function remain unchanged whether arithmetic or geometric averaging is used. Furthermore, Wixted and Ebbesen (1997) conducted an analysis of individual participants forgetting functions showing that they too are better described by a power function than by an exponential function. Lots of effort has been put into cognitive experimental studies of forgetting curves and ways of summarizing these data on mathematical functions. However, few models of memory have been suggested for forgetting; for exceptions, see the search of associative memory (SAM; Mensink & Raaijmakers, 1988) and the extended version of the theory of distributed associative memory (TODAM2; Murdock, 1993). These models account for forgetting by assuming that the context changes between retrieval and encoding. Recently, Sikström (1999, in press-a) suggested a connectionist model for forgetting of correct recall and recognition performance. This model assumes that information is stored in weights attached to connections between nodes. Sikström suggested that the negative slope of the forgetting curve occurs because the weights are bounded; that is, the weights are cut off so that they cannot exceed an upper boundary or drop below a lower boundary. After encoding, the expected value of a single weight decays at an exponential rate. Power function forgetting curves are found by introducing variability in the learning rates (hence, in this model, power function is a true phenomenon that occurs because of fundamental properties in neural systems and not an artifact due to how the data are summarized). Sikström (in press-a) showed that to achieve forgetting functions that precisely follow a power function the learning rate must be generated by a power function. However, different from other models, this model predicts a power function with a lag-one displacement; that is, the forgetting is a power function of the retention interval plus one lag. This prediction has an advantage in that forgetting is defined at immediate testing (which is not the case for models that use a nondisplaced power function). Furthermore, the forgetting function proposed by Sikström (in press-a) fits better than any other suggested twoparameter function when tested on what were claimed to be the most precise data on recognition and recall accuracy as a function of retention interval: the data from Rubin et al. s (1999) study. However, Sikström s (in press-a) model did not account for production time and breaks, which are the focal points of this article. An issue related to forgetting functions is how repeated tasks influence memory. This issue relates to a fundamental principle in models of memory, in particular to distributed or to connectionist models namely, that two or more memory traces can be aggregated or superimposed. For example, connectionist models typically calculate weight changes for each item to be memorized and simply calculate new weights by adding these weight changes to the old weights. Then an aggregated memory trace can be formed by integrating single memory traces over time. The improvement in performance on repeated tasks is often measured by reaction time. Perhaps the most developed model for reaction times is the diffusion model (Ratcliff, 1978; Ratcliff, Zandt, & McKoon, 1999). This model starts with the premise that a weak signal may be detected in the presence of noise by accumulation of evidences. In this context, the signal may be seen as the strength of a memory trace. The accumulator is first initialized to zero. Noise plus signal is added to the accumulator at each time step. A decision is made at the first time the accumulated evidence reaches an upper or lower boundary; the boundaries typically are placed symmetrical to the starting point of the diffusion process. The accumulation starts some time after the onset of the stimuli. This starting time is a free parameter that reflects the time for the stimuli to reach the decision stage as well as the time for the output from the decision stage. In recognition tests, a typical value for this parameter is 400 ms. The diffusion model has been successfully applied to a large number of findings regarding reaction time, such as the reaction time distribution and the finding that errors generally are made more slowly than correct responses (as well as exceptions from this finding). The diffusion model has also been used in a distributed-memory model; Sikström (2002) suggested that firing of neural cells could be seen as diffusion processes. If a lot of cells are active, then the diffusion process tends to go to the upper boundary; otherwise, it goes to the lower boundary. Furthermore, several phenomena, such as fast errors for easy tasks when time is prioritized over accuracy, can be modeled by letting the boundary depend on the neural noise. Wright (1936) is believed to be the first to investigate learning in an industrial setting. However, to industrial engineers, the forgetting phenomenon is fairly new. It was not until the 1960s that the first attempt was made to investigate the opposite phenomenon (i.e., forgetting). Researchers in both disciplines unanimously agree that the learning process is negatively affected when participants are interrupted for a significant period of time, during which some of the knowledge acquired in earlier learning sessions is lost. Cross-training of the workforce, strikes, and production line breakdowns are common causes of production breaks in manufacturing environments. In industrial engineering literature, Hancock (1967) is believed to be the first to investigate the effect of production breaks on workers performance. His study indicated that very short breaks, such as coffee breaks, have no effect on learning, whereas overnight and longer breaks definitely retard the learning process. Cochran (1973) and Anderlohr (1969) agreed that the length of the production break has a direct effect on the degree to which humans forget. They also considered other factors, such as the availability of the same personnel, tooling, and methods. Hoffman (1968), and later Adler and Nanda (1974), presented refined mathematical techniques for incorporating the effects of production breaks into production planning and control models. Carlson and Rowe (1976) hypothesized that the intercept of the forgetting curve representing the time to produce a unit varies after each interruption. Globerson and Levin (1987) developed a conceptual model of organizational forgetting as well as the development of a mathematical model that incorporated forgetting into learning. Unlike Carlson and Rowe, Elmaghraby (1990) assumed that the intercept of the forgetting is of a fixed value for all production cycles. Jaber and Bonney (1996) developed the learn forget curve model (LFCM). The LFCM uses a forgetting slope that is dependent on the minimum production break over which total forgetting is assumed to occur, the learning slope, and the amount of equivalent units of experience remembered at the point of interruption. This allowed the forgetting slope to vary from cycle to cycle. A second group of researchers conducted laboratory experiments resulting in a better understanding of the learning forgetting relationship. Globerson, Levin, and Shtub (1989) indicated that the degree of forgetting is a function of the level of experience gained

3 120 SIKSTRÖM AND JABER prior to interruption and of the length of interruption. Bailey (1989) concluded that the forgetting rate is related to learning rate, relearning rate is not correlated with the learning rate, and learning rate is highly correlated with the time taken to complete the first unit. Shtub, Levin, and Globerson (1993) used the forgetting model and the data of Globerson et al. (1989) to partially validate Bailey s hypothesis. In an experiment similar to that of Globerson et al. (1989), Sparks and Yearout (1990) analyzed the learning process after the break and the amount of forgetting during the break and concluded that traditional curves developed for mechanical tasks do not reflect the forgetting that occurs in modern industrial tasks. Hewitt, Sprague, Yearout, Lisnerski, and Sparks (1992), who conducted laboratory experiment of participants performing one of two simulated industrial tasks of varying cognitive contents, concluded that one type of learning forgetting model could not be applied to all tasks. Unlike the above-mentioned researchers, Dar-El, Ayas, and Gilad (1995a) suggested that learning an industrial task consists of two parallel phases: cognitive and motor. They collected data from an experimental apparatus that involved assembling an electric matrix board and electronic components that involved one complex task and another simple one. Dar-El et al. (1995a) concluded that learning is reduced as experience is gained. In a subsequent article, Dar-El, Ayas, and Gilad (1995b) defined forgetting as a consequence of a specific subtask reappearing in the next cycle after a whole cycle time of other activities is completed. Dar-El et al. (1995b) empirically determined that forgetting is a function of the learning constant and interruption length. Arzi and Shtub (1997) compared learning and forgetting of cognitive and motor tasks and concluded that forgetting intensity is affected only by the rate of learning before the break. Globerson, Nahumi, and Ellis (1998) used flight simulators to study cognitive and simple motor tasks in a condition where one long break ( days) separated two learning sessions. The results demonstrated a strong effect of break length on the three measures of forgetting and strongly supported the use of the power model to construct forgetting curves. Finally, participants tended to forget the cognitive tasks more than the motor tasks. We recommend Dar-El s (2000) book for additional reading on learning and forgetting processes. The aforementioned experimental studies assumed a single break against which Nembhard (2000) and Nembhard and Uzumeri (2000) cautioned. They have suggested that in industrial settings workers experience multiple breaks in the learning process, in which the length of the production break varies from one instance to another. Nembhard and Osothsilp (2001) empirically compared 14 existing forgetting functions listed in the 14 first rows of Table 1: 7 forgetting models (Globerson Levin Shtub; GLS1-7) in Globerson et al. (1989), the exponential (EX) and s-shaped (SS) models Table 1 Summary of Models for Production Breaks and Their Fit to Nembhard and Osothsilp s (2001) Empirical Data Model name Equation MAD STD STDER GLS1 T f C 0 C 1 T nf GLS2 T f C 0 C 1 T nf C 2 B GLS3 T f C 1 T nf C 2 B C1 GLS4 T f C 0 T nf GLS5 C1 T f C 0 T nf B C GLS6 C1 T f T nf B C GLS7 T f C 0 C Tnf B 1 C GNE T f T nf C 1 B C EX T f T(1) [T(1) T nf ]e ab SS T f T(1) [T(1) T nf ](ab 1)e ab x (t i t 0 ) i 1 RC T f L(xR x ), R x x(t x t 0 ) VRIF Tˆ x Tˆ ix f VRVF Tˆ x Tˆ ix f LFCM Tˆ x Tˆ ix f PID N T S 0 [ e 1 t e,1 a t e,2 a ] 1 t Note. See the text for details about and references for the various models. The denotations in the equations are as follows (see appropriate references in the text for details): T nf is the performance time of the first unit after a break in the case when no forgetting has occurred; T f is the performance time of the first unit after a break when forgetting has occurred; B is the break length; T (1) is the performance time of the first unit when learning starts; C 0,C 1,C 2, f, and a are forgetting parameters; L () is the learning function (e.g., Wright s model); Tˆ x is the time for the xth unit of lost experience on the forgetting curve (where Tˆ i is the first unit); and x is the number of the unit. See the text for details of the power integration diffusion (PID) model. In the learn forget curve model (LFCM) f and Tˆ i varies, and in the Variable Regression Variable Forgetting (VRVF) model Tˆ i varies. MAD mean absolute percentage deviation; STD the standard deviation of the percentage deviation; and STDER the standard error of the percentage deviation; GLS Globerson Levin-Shtub; GNE Globerson Nahumi Ellis; EX exponential; SS s-shaped; RC recency; VRIF Variable Regression Invariant model.

4 POWER INTEGRATION DIFFUSION MODEL 121 in Globerson and Levin (1987), the Globerson Nahumi Ellis (GNE) model (Globerson et al., 1998), the Recency (RC) model in Nembhard and Uzumeri (2000), the Variable Regression Variable Forgetting (VRVF) model in Carlson and Rowe (1976), the Variable Regression Invariant (VRIF) model in Elmaghraby (1990), and the LFCM in Jaber and Bonney (1996). Globerson and Levin (1987) presented and differentiated between two conceptual learning forgetting models, the EX model and the SS model, that describe individual and organizational memory. They hypothesized that the forgetting process is a function of several factors, including turnover, communication, and documentation. Globerson et al. (1989) simulated a data entry office environment in a microcomputer laboratory in two sessions, with breaks between sessions that varied from a single day to 82 days. The data obtained were fitted to seven forgetting models (which were provided with little theoretical justification), of which the power model was found to be best. The RC model developed by Nembhard and Uzumeri (2000) was an extension of, and improvement on, that of Mazur and Hastie (1978). Nembhard and Uzumeri (2000) introduced a modification to the three-parameter hyperbolic learning function by including a measure termed recency of experimental learning (R). They stated that for each unit of cumulative production x, one can determine the corresponding recency measure, Rx, by computing the ratio of the average elapsed time. The remaining three forgetting models the VRVF, VRIF, and the LFCM were compared by Jaber and Bonney (1997). The VRIF and the VRVF use a fixed and externally specified forgetting rate; however, the former uses a variable intercept for the forgetting curve, whereas the latter uses a fixed intercept for the forgetting curve. In contrast, the LFCM uses both variable forgetting rate and a variable intercept for the forgetting curve. The three models hypothesize two important relationships between learning and forgetting: (a) When total forgetting occurs, the time to process reverts to the time required to process the first unit with no prior experience; and (b) the performance time on the learning curve equals that on the forgetting curve at the point of interruption. Nembhard and Osothsilp (2001) evaluated these 14 models by fitting them to empirical data. The data were collected during a 6-month period of final inspections of car radios. There were six models of radios under inspection; each required approximately 130 similar inspection steps. Inspection time data were collected for car radios, and the data were based on 71 permanent workers. The manual inspection time of each radio was recorded in an automated fashion. The inspection procedure is procedural, dominated by mental or cognitive steps (e.g., visual inspection or decisions of whether the radio is defective), and contains some simple manual steps (e.g., turning the radio on or placing a tested unit on a pallet). Nembhard and Osothsilp claimed that forgetting in this setting is most likely to occur because of failure to recall certain steps, or the sequence of steps, and needing to refer to documentation. Less than 0.1% of the radios failed the inspection. Nembhard and Osothsilp found no statistical difference regarding the inspection time per unit, the learning rate, and the forgetting rate of the six radio models. Hence, the results were averaged over the six radio models. For a more precise description of the data collection, see Nembhard and Osothsilp s article. The dots in Figure 1 show the averaged production times from these data (where each data point is the average production time Figure 1. Production times as a function of cumulative units of production. The empirical data, shown as small diamonds, is an adapted version from Nembhard and Osothsilp s Figure 1 (2001). Each data point is the average production time over 20 units as a function of the cumulative units of production. The predicted data, shown as the solid line, is from the power integration diffusion model. There are seven breaks with lengths of 3.7 days, 6.5 days, 11.6 days, 3.8 days, 4.7 days, 3.7 days, and 11.2 days listed in chronological order. The breaks are shown as peaks in the solid line. Notice that the break length of 6.5 days and the break length of 11.6 days around 2800 cumulative unit is difficult to see in the figure because they occur close to each other. over 20 units) as a function of the cumulative units of production, where break lengths longer than 50 hr are included in the analysis. There were seven break lengths that exceeded this criterion. (See the figure caption for details. Nembhard and Osothsilp [2001] also analyzed break lengths longer than 20 hr and 40 hr, with similar results.) Nembhard and Osothsilp (2001) concluded that the RC model performed best in terms of efficiency, stability, and balance. The 14 first rows of Table 1 list the name of the 14 models tested by Nembhard and Osothsilp, the mathematical equations describing these models, the mean absolute percentage deviation (MAD) score, the standard deviation of the percentage deviation, and the standard error of the percentage deviation. The purpose of this article is to combine the models for forgetting curves (Sikström, in press-a) and reaction time (Ratcliff, 1978; Ratcliff et al., 1999; Sikström, 2002) to predict data on production breaks. The suggested model, referred to as the power integration diffusion (PID) model, is based on three fundamental findings: that (a) single memory traces decay as a power function of the retention interval, (b) the memory traces can be integrated over time, and (c) the reaction time can be calculated by a diffusion process. The rest of this article is organized as follows: First we discuss mathematical modeling of the PID model. Then we provide an illustrative numerical example. We then test the fit of the PID model to the data of Nembhard and Osothsilp (2001), and summarize the limitations of the RC model (Nembhard & Uzumeri, 2000). Next, we discuss the generality of the PID model and describe alternative methods for formulating the PID model; last, we summarize the article and discuss our conclusions. The PID Model An important concept in the PID model is the strength of the memory trace. This is a concept that is proportional to a measure-

5 122 SIKSTRÖM AND JABER ment of performance accuracy that is free from ceiling effects. The exact way of measuring the memory strength may vary for different tasks. For example, in item recognition the memory strength could be measured by d (whereas percentage correct is less appropriate, because it depends on the participant s criteria and may also be influenced by ceiling effects near 100% correct). However, in this article the memory strength is not directly measured, because we are instead interested in reaction times or production times. Nevertheless, the memory strength becomes an important theoretical concept for justifying the assumptions of the PID model. The basic idea of PID is that every time a task is performed a single memory trace is formed. The strength of this single memory trace decays as a power function over time. When the same task is repeated, an aggregated memory trace can be found by integrating the strength of the single memory trace over the time interval that the task is repeated. The integral of a power function is a power function; therefore, the aggregated memory strength of an uninterrupted set of repetitions can be described as the difference between the power function of the retention interval at the start of the repetitions and a power function of the retention interval at the end of repetitions. The time it takes to perform a task is determined by a diffusion process in which the strength of the aggregated memory constitutes the signal. To simplify the calculation, the noise in the diffusion process is disregarded, and the time to perform a task is the inverse of the aggregated memory strength plus a constant reflecting the start time of the diffusion process. Next, we mathematically describe the PID model. The following notations are used: The strength of a single memory trace at time t r following encoding during a short time interval (dt 3 0) starting at time t and ending at time t dt is S (t, t r ). The strength of an aggregated memory trace at time t r following encoding during a time interval from t e,1 to t e,2 is S(t e,1, t e,2, t r ), where t e,1 t e,2 t r. The following parameters are used: (a) the forgetting parameter 0 a 1, the asymptotic time to produce a unit t 0 0 (or the start time of the diffusion process), and (b) a scaling parameter S 0 0. (See the Appendix for the expression and derivation of the time to produce the first unit.) The strength of a single memory trace at time t r that is encoded at time t follows a power function of the retention interval since training: S (t, t r ) S 0 (t r t) a dt. The strength of the aggregated memory trace at time t r following encoding during a time interval from t e,1 to t e,2 is the integral over this time period, and it is determined as: S(t e,1, t e,2, t r ) te,1 t e,2 S (t, t r ) S 0 1 a t r t e,1 ) 1 a (t r t e,2 ) 1 a ]. (1) The strength of the aggregated memory trace at time t r following encoding during N time intervals is simply the sum over these intervals, and it is determined from Equation 1 as: S(t e,1, t e,2, t r ) S N 0 1 a [(t r t e,1 ) 1 a (t r t e,2 ) 1 a ]. (2) e 1 The time to produce a unit is calculated with a diffusion model. The strength of the aggregated memory trace is conceived of as the signal. To simplify the calculation, the noise in the diffusion process is ignored or set to zero. The time to produce a unit is then simply the inverse of the aggregated memory strength, and it is given from Equation 2. The start time of the diffusion process constitutes a constant (t 0 ) that is added to the total time to produce a unit: T(t e,1, t e,2, t r ) S(t e,1, t e,2, t r ) 1 t 0 1 a S 0 Equation 3 could be written as: N [ (t r t e,1 ) 1 a (t r t e,2 ) 1 a ] 1 t 0. (3) e 1 N T(t e,1, t e,2, t r ) S 0 [ t e,1 a t e,2 a ] 1 t 0, (4) e 1 where t e,1 is the retention interval from the start of encoding of interval e to the time of retrieval (t r t e,1 ), t e,2 is the retention interval from the stop of encoding of interval e to the time of retrieval (t r t e,2 ), 0 a 1 is the rescaling of a, and S 0 is a rescaling of S 0. The rescaling of the parameters is introduced for convenience to simplify the final expression. In the Appendix it is shown by integrating the memory strength that the accumulated time to produce x units without production breaks is [xs 0 (1 a )] 1/(1 a ) given that t 0 0. Thus, the time to produce unit x without production breaks is T x xs 0 (1 a )] 1/(1 a ) [(x 1) S 0 (1 a )] 1/(1 a ). In particular, the time to produce the first unit is T 1 S 0 (1 a )] 1/(1 a ) given that t 0 0. Numerical Example The predictions of PID are easy to calculate. The aggregated memory strength is the sum of the time elapsed since the start of each production raised to a constant, minus the sum of the time elapsed since the end of each production raised to a constant. The prediction is proportional to the inverse of the aggregated memory strength plus a constant. For example, assume that t 0 1, S 0 2, and a 0.5 (corresponding to S and a 0.5). The chosen value of a is typically found when fitting to empirical data, compared with the fits below where a Similar rates of forgetting have also been found in other studies (Alchain, 1963; Hirsch, 1952, 1956; Wright, 1936). Let workers start a production at t 1,1 10 and stop this production at t 1,2 20. Then there is a production break. The workers resume their work at t 2,1 40 and stop their work at t 2,2 60. How long does it take to produce a unit according to PID at a time of 50? Notice first that the time of encoding must always be less or equal to the time of retrieval. Therefore, t 2,2 is set to 50 (even if the production eventually stops at 60). The predicted production time per unit is computed from Equation 4 as:

6 POWER INTEGRATION DIFFUSION MODEL 123 T 2/{[(50 10) 0.5 (50 20) 0.5 ] [(50 40) 0.5 (50 50) 0.5 ]} 1 2/{[ ] [ ]} 1 2/{[ ] [3.1 0]} Fits As reviewed earlier, Nembhard and Osothsilp (2001) evaluated 14 different published models of forgetting breaks (7 GLS models; the GNE, EX SS, RC, VRIF, VRVF models, and the LFCM) on empirical data of inspections of car radios. The PID model was fitted to the same empirical data as Nembhard and Osothsilp. The fit was made by minimizing the MAD score by adjusting t , a 0.51, and S (corresponding to a 0.49 and S ). The seven break times (t e,1, t e,2 ) and the retrieval times (t r ) were estimated from the production time per unit and cumulative units of production in the experimental data (see Figure 1 in Nembhard & Osothsilp). The results of the PID model are shown in the last row of Table 1 and as the solid line in Figure 1. The degrees of freedom were 275. The MAD score of PID was 0.103, which is smaller than the MAD score for the 14 other models. The MAD score for the RC model is The seven GLS models performed next best, with MAD scores ranging from to The EX model had a MAD score of 0.208, and the remaining models had MAD scores above The PID model also shows a better performance than the RC model in terms of the other measurements: The standard deviation of the percentage of deviation and the standard error of the percentage of deviation are lower for the PID model compared with the RC model (see Table 1 for details), and the ratio of underpredictions to overpredictions are 55% 45% for the PID model and 42% 58% for the RC model. However, it should be noted that these fits are limited to data aggregated over participants; how the models would fit to data from individual participants is not known, because these data are not available. Limitations of the RC Model As shown above, PID shows a better fit than the RC model to data from Nembhard and Osothsilp (2001). In this section we scrutinize the RC model and show that some of the predictions of the model are inconsistent with fundamental findings in the memory literature. The RC model determines the effect that the recency of experience has on each operator by weighting the cumulative work x by a factor R x a, where R x is a measure of how recently an individual s practice was obtained. The equations for the RC model are x (t i t 0 ) T f T 0 (xr a x ) b i 1, where R x 2, x(t x t 0 ) where x is the accumulated number of produced units, t x is the time when unit x is produced, t 0 is the time when the first unit is produced, t i is the time when unit i is produced, T 0 is the performance of the first unit, and a and b are forgetting and learning parameters. Consider Production Sequence A and Production Sequence B, each consisting of the production of four units. The production sequences are identical in all respects except the following. Production Unit 2 is one time step younger in Sequence A compared with Sequence B, whereas Production Unit 3 is one time step older in Sequence A compared with Sequence B. In both sequences, Unit 1 and Unit 4 are produced at the same time. The 2RC model is based on the average time of production, which is identical for Sequences A and B. Thus, the RC model predicts the same production time for Unit 4. This prediction contradicts Jost s law, or the empirical finding that a newer memory decays faster than an old memory (Jost, 1897, cited in Hovland, 1951, p. 649). That is, Production Sequence B should have better performance than Production Sequence A, because the change in memory strength is greater for the younger Unit 3 compared with the older Unit 2. Therefore, the RC model is inconsistent with Jost s law of forgetting, whereas PID is consistent with this law. Another limitation of the RC model is that it is very sensitive to the time of the first unit. This assumption causes the predictions of the RC model to degenerate when the time of the first unit is much earlier than the time of the other units; that is the R x factor converges to a constant of 2 as t 0 approaches negative infinity for all finite values of t i 0, i 0, yielding T f T 0 (x2 a ) b. The Generality of PID The general formulation of PID given earlier describes how interruptions affect learning, forgetting, the strength of the memory traces, and the production times. Uninterrupted learning and forgetting following a brief learning period are special cases of this. In this section we describe these special cases. PID suggests that uninterrupted learning measured by production time, uninterrupted forgetting following a brief learning period measured by production time, uninterrupted encoding measured by aggregated memory strength, and uninterrupted forgetting following a brief encoding measured by aggregated memory strength can be described by power functions. Furthermore, the learning and forgetting parameters in these functions follow lawful relationships, listed below. Production time is learned according to a power function. Wright s law suggests that the time to produce a unit can be described as a power function of the number of units produced. In the Appendix it is shown that Wright s law is a special case of PID. When there is no production break, PID simply predicts a power function of the time of production consist with Wright s law T [(1 a)/s 0 ]t e,1 a 1 t 0 ]. Production time is forgotten according to a power function. Wright s law does not specify the forgetting function. In this case, PID is more general then Wright s law. The simplest production time forgetting occurs after a brief period of learning (T ; that is, the learning time must be much shorter than the retention interval. Compare this with the definition of S.) Thus, the production time increases as a power function of the retention interval (T S 1 0 t e,1 a t 0 ). The production time forgetting parameter a equals the production time learning parameter a 1plus1(a a 1 1). The single memory strength is forgotten according to a power function. PID not only predicts the time to produce a unit, but it also predicts the memory strength, or accuracy, in performance.

7 124 SIKSTRÖM AND JABER However, this prediction is not empirically validated here. There are several ways of measuring accuracy in performance, and these measurements are dependent on the type of test used. For example, episodic memory accuracy can be measured by d in recognition memory. Empirical data on forgetting in these tests can be well described by a power function (e.g., Rubin & Wenzel, 1996). In PID, the simplest type of forgetting of a single memory trace occurs after a brief period of encoding. The strength of this trace is forgotten according to a power function (S S 0 t e,1 a). The single memory strength forgetting parameter a equals to the reaction time forgetting parameter a with the opposite sign. Aggregated memory strength is encoded according to a power function. The strength of a cumulative rehearsal increases the aggregated memory strength according to a power function [S [S 0 /(1 a)]t e,1 1 a]. The aggregated memory strength encoding parameter 1 a equals the reaction time learning parameter a 1 with the opposite sign. Generality of PID: Current Limitations and Possible Extensions There are a few different alternative formulations of PID. Some of the possible formulations are discussed below. Length of the retention interval. In PID, the aggregated memory strength is calculated by integrating a power function over the retention interval. However, how does one define the length of the retention interval? That is, does the retention interval start at beginning or the end of the presentation of an item? Sikström (in press-a) showed that a connectionist network with bounded weights and variability of learning rates would show a power function with a retention interval that starts at the beginning of the presentation. This prediction is consistent with the most precise empirical data on forgetting in recognition and recall (Rubin et al., 1999). These data show a systematic deviation from a power function when the retention interval is measured from the end of presentation. This deviation disappears when the retention interval starts at the beginning of the presentation. Thus, both the theoretical model and the empirical data indicate that the retention interval should start at the beginning of the presentation of a unit. This can be introduced in PID by setting the smallest retention interval to the time to produce a unit. It also has the convenient advantage that the power function is defined also for negative values of the exponent. However, this implementation was avoided in the development of PID because it complicates the predictions (the retention interval influences the time to produce a unit; therefore, the time to produce a unit must be solved by a recursive algorithm). Furthermore, the results from such an implementation differ nonsignificantly from the original version of PID (MAD rather than 0.103). Calculating the diffusion process. In PID, the diffusion process is simplified by assuming that the noise is zero. The advantage of this approximation is that the computation of the predictions is greatly simplified; also, there is no need to introduce a parameter for noise. However, the disadvantage is that some detailed predictions regarding the reaction time or the time to process a unit could be lost. For example, modeling the diffusion process yields information on the distribution of reaction times and on how accuracy can be traded with speed as the decision boundary is changed. However, during the development of PID the major goal was simplicity, and the exact diffusion process is therefore not modeled. Interested readers may refer to Ratcliff (1978), Ratcliff et al. (1999), or Sikström (2002) for detailed discussion on how to model a diffusion process. Proactive interference and novelty. A well-established principle is that an item that differs from the preceding items is better recalled than an item that is similar to the encoding context. Examples of this principle are the von Restorff effect (an item that differs from the other list items is better recalled), proactive interference (in which repeated items from a similar category decrease the ability to maintain items in short-term memory), and spaced versus massed repetition (items presented with intervening items are better recalled than items presented without intervening items; Underwood, 1969; Underwood & Freund, 1970). Generalizing from these findings, the strength of encoding should be stronger following a break than following several repetitions. A plausible mechanism that may account for this is that neural cells habituate over multiple presentations of similar stimuli, leading to a lower degree of synaptic plasticity. The current version of PID is limited in that it does not account for this type of phenomenon; however, see Sikström (2001, in press-b) for how the model may be extended with habituation to account for these effects. Retroactive interference. Several empirical phenomena indicate that forgetting occurs not only because of passage of time but also is influenced by the interference from encoding of other information during the retention interval (Jenkins & Dallenbach, 1924; Waugh & Norman, 1965). However, because of the difficulty of finding independent measurements of the amount of interference, the current version PID is limited in that it implements forgetting as a function of time rather than being dependent on interference. We believe that over extended time periods, such as the data being modeled in this article, time is a reasonable estimate of, or is strongly correlated with, the amount of interference. Nevertheless, a possible extension of PID may base forgetting on interference rather than time. One possible implementation of forgetting with interference is to use a connectionist network; in particular, see Sikström (1999, in press-a) for how power function forgetting curves can be implemented in connectionist networks. Summary and Conclusions In this article we have introduced PID to account for production time breaks. The model is based on three fundamental findings: (a) the experimental finding that memory traces can be described by power functions (Rubin et al., 1999; Wixted & Ebbesen, 1991), (b) the combined effect of several memory traces can be found by integrating several memory traces, and (c) the idea that decisions can be described by a diffusion process (Ratcliff, 1978; Ratcliff et al., 1999). PID is a useful tool for helping to understand how production breaks influence production time. The model has several advantages: A major advantage is that PID has a strong theoretical base. The assumptions of power function decay, integration, and a diffusion process are plausible and have empirical support. This also makes the model relatively simple, and the predictions from the model are straightforward. Finally, we have shown that PID fits better than the 14 other models of production breaks when tested on data from Nembhard and Osothsilp s (2001) study. We have introduced the PID model; however, the model may also have implications for future psychological theory. PID inte-

8 POWER INTEGRATION DIFFUSION MODEL 125 grates several different and difficult issues within one rather simple framework. Future work should carefully investigate several of these issues in greater detail for example, test how the rate of learning interacts with the rate of forgetting, investigate how reaction times (or production times) relate to accuracy (or memory strength), and elaborate on how production breaks can be planned so that productivity is optimized according to some specific criterion. Acquisition of skills over time, applied in sport, study, or work situations, requires repetitive training of tasks and involves breaks needed either for rest or for prioritizing other activities. From this perspective, PID may have an important application for human performance. The model specifies how potential breaks from training influences performance. 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