The power law repealed: The case for an exponential law of practice

Heathcote, Andrew; Brown, Scott; Mewhort, D. J. K.

doi:10.3758/bf03212979

Cited by 624 publications

(618 citation statements)

References 53 publications

Supporting

Mentioning

585

Contrasting

Unclassified

Order By: Relevance

“…A particularly notable alternative is the three parameter exponential function (hereafter referred to as the exponential function) E( y j ) = θ (1) exp(−θ (2) (x j − 1)) + θ (3) . Heathcote et al (2000) compared the power and exponential functions across a wide range of datasets. Importantly, they noted that most previous evaluations of the power function had been performed on group-level data rather than individual-level data.…”

Section: Overall Task Learningmentioning

confidence: 99%

See 1 more Smart Citation

Hierarchical Bayesian models of subtask learning.

Anglim¹,

Wynton²

2015

Journal of Experimental Psychology: Learning, Memory, and Cogni

View full text Add to dashboard Cite

Abstract. The current study used Bayesian hierarchical methods to challenge and extend previous work on subtask learning consistency. A general model of individual-level subtask learning was proposed focusing on power and exponential functions with constraints to test for inconsistency. To study subtask learning, we developed a novel computer-based booking task, which logged participant actions, enabling measurement of strategy use and subtask performance. Model comparison was performed using DIC, posterior predictive checks, plots of model fits, and model recovery simulations. Results showed that while learning tended to be monotonically decreasing and decelerating, and approaching an asymptote for all subtasks, there was substantial inconsistency in learning curves both at the group-and individual-levels. This inconsistency was most apparent when constraining both the rate and the ratio of learning to asymptote to be equal across subtasks, thereby giving learning curves only one parameter for scaling. The inclusion of six strategy covariates provided improved prediction of subtask performance capturing different subtask learning processes and subtask trade-offs. In addition, strategy use partially explained the inconsistency in subtask learning. Overall, the model provided a more nuanced representation of how complex tasks can be decomposed in terms of simpler learning mechanisms.

show abstract

Section: Overall Task Learningmentioning

confidence: 99%

“…First, their study analyzed group-level data, when individual-level data is more psychologically meaningful (for discussion, see Heathcote et al, 2000). In particular, group-level data is well known to smooth over a wide range of variation at the individual-level.…”

Section: Subtask Learningmentioning

confidence: 99%

Hierarchical Bayesian models of subtask learning.

Anglim¹,

Wynton²

2015

Journal of Experimental Psychology: Learning, Memory, and Cogni

View full text Add to dashboard Cite

show abstract

“…As this last possibility would imply difficulties in averaging over Modeling Lexical Decision 24 participants in the analysis of these models (see for instance, Heathcote et al, 2002), analysis at the level of individual participants will be necessary to test properly or otherwise explore models with nonlinear parameters (a rank model has no nonlinear parameters, although it is linear in neither word frequency nor contextual diversity).…”

Section: Modeling Lexical Decision 23mentioning

confidence: 99%

Modeling lexical decision: The form of frequency and diversity effects.

Adelman

Brown

2008

Psychological Review

View full text Add to dashboard Cite

What is the root cause of word frequency effects on lexical decision times? argued that such effects are linear in rank frequency, consistent with a serial search model of lexical access. This paper (i) describes a method of testing models of such effects that takes into account the possibility of parametric overfitting; (ii) illustrates the effect of corpus choice on estimates of rank frequency; (iii) gives derivations of nine functional forms as predictions of models of lexical decision; (iv) details the assessment of these models and the rank model against existing data regarding the functional form of frequency effects; and (v) reports further assessments using contextual diversity, a factor confounded with word frequency. The relationship between the occurrence distribution of words and lexical decision latencies to those words does not appear compatible with the rank hypothesis, undermining the case for serial search models of lexical access. Three transformations of contextual diversity based on extensions of instance models do however remain as plausible explanations of the effect. Modeling Lexical Decision 3Modeling Lexical Decision: The Form of Frequency and Diversity EffectsThe effect of word frequency on lexical processes is both ubiquitous and large. It is evident in a wide range of both tasks (e.g., lexical decision, word naming, picture naming and memorial tasks) and measures (especially response times and eye fixation times, but also errors in the aforementioned tasks). It therefore seems reasonable to require models of lexical tasks to account not only for the existence of frequency effects, but also for their form. However, few existing models provide such an account. took up the challenge in the context of lexical decision times. Specifically, they showed that their bin model, a frequency-ordered serial-search model of lexical access, predicts that lexical decision times will be determined by the rank ordering of word frequency. In comparisons between rank and logarithmic transformations, they argued that just this pattern could be seen in the data, and (inter alia) interpreted this result as evidence for serial search models. leading to the prediction that expected search time will be linear in rank word frequency.We agree with Murray and Forster that future progress will require the derivation and test of specific predictions of fully explicated models such as the one described in their paper.In particular, the evaluation of whether the ranked frequency of a word, rather than some other (e.g., logarithmic, power, or exponential) transformation of frequency, predicts lexical decision time would appear to have wide implications for quite general classes of lexical access model. Modeling Lexical Decision 4In this paper we consider several different classes of model architecture and derive their specific predictions regarding the functional form of the word frequency effect (e.g., logarithmic, power and exponential from Bayesian, instance and contextual cue models respectively). Thi...

show abstract

“…For more than a century, psychologists have sought a simple and universal empirical law of practice, focusing mostly on the function describing speedup (for reviews, see Newell & Rosenbloom, 1981;Heathcote, Brown, & Mewhort, 2000). A primary motivation behind this work has been to guide theory development, and a number of theories of skill learning have been influenced heavily by the prevailing view of what function best describes speedup (Anderson, 1982(Anderson, , 1993Anderson & Schooler, 1991;Cohen, Dunbar, & McClelland, 1990;Logan, 1988Logan, , 1995Newell & Rosenbloom, 1981;Palmeri, 1997;Rickard, 1997).…”

Section: Implications For the Empirical Law Of Learningmentioning

confidence: 99%

Forgetting and learning potentiation: Dual consequences of between-session delays in cognitive skill learning.

Rickard

2007

Journal of Experimental Psychology: Learning, Memory, and Cogni

View full text Add to dashboard Cite

In the cognitive skill literature, between-session delays have been treated either as having a negligible effect on performance or as causing forgetting. In contrast, in the procedural skill literature, overnight between-session delays can result in performance gains. In 5 multi-session data sets, the author demonstrates that neither of these 2 models holds for the case of cognitive skill learning. Instead, the delay between sessions appeared to yield both forgetting and enhanced potential for new learning. Two candidate classes of explanation are considered, and implications for the empirical law of learning are discussed.Keywords: skill, learning law, forgetting, fatigue, consolidation Most naturalistic skill acquisition requires multiple practice sessions over many days. A complete theory of human skill must therefore account for the effects of the delays between sessions on learning and performance. Yet, in the cognitive skill acquisition literature, these effects are typically ignored, with the implicit assumption that they have negligible impact on performance. One important exception is work by Anderson, Fincham, and Douglass (1999). In a multisession problem-solving task, they showed that response times (RTs) exhibit the classic decelerating speedup function within each practice session but exhibit slowing at the start of each new session, yielding a scalloped speedup pattern over sessions. The simplest case model of this effect would assume that the (nonlinear) learning rate is constant over repetitions and that the forgetting rate is constant over time. Anderson et al. (1999) provide one possible implementation of that model in their latency equation (p. 1122). It follows from that equation that the mean RTs that are predicted by extrapolating from the fit to one session must fall below the population mean RTs of all subsequent sessions, as represented in Figure 1. Stated differently, the RTs after a delay must be above the RTs that would have been expected had all of the practice occurred within a single session.There are two opposing traditions about the effects of delays between sessions in the procedural skill literature, in which commonly studied tasks include repetition of a sequence of finger movements (e.g., Walker, Brakefield, Morgan, Hobson, & Stickgold, 2002), movement of a stylus or cursor between specified points (e.g., Brashers-Krug, Shadmehr, & Bizzi, 1996), and rotary pursuit (e.g. Bourne & Archer, 1956). On one hand, in the older literature exploring the phenomena of reminiscence and warm-up decrements, the general finding is similar to that reached by Anderson et al. (1999); provided that practice within the first session is distributed (e.g., a 40-s break between each 10-s trial), an overnight delay between sessions can result in worsened performance (e.g., Adams, 1952;Digman, 1959). On the other hand, in the more recent literature on procedural consolidation (for reviews, see Stickgold, 2005;Walker & Stickgold, 2006), the delay between practice sessions has been shown to result in p...

show abstract

The power law repealed: The case for an exponential law of practice

Cited by 624 publications

References 53 publications

Hierarchical Bayesian models of subtask learning.

Hierarchical Bayesian models of subtask learning.

Modeling lexical decision: The form of frequency and diversity effects.

Forgetting and learning potentiation: Dual consequences of between-session delays in cognitive skill learning.

Contact Info

Product

Resources

About