Refining the law of practice.

Evans, Nathan J.; Brown, Scott; Mewhort, D. J. K.; Heathcote, Andrew · 2018 · Psychological Review

DOI: 10.1037/rev0000105

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Summary

This theoretical note addresses the longstanding debate regarding the mathematical form of the "law of practice," which quantifies the relationship between mean response time (RT) and practice trials. While early research favored a power law, subsequent analyses using individual-level data suggested an exponential law was superior. However, both traditional models assume a monotonically decreasing rate of learning, failing to account for empirical observations of initial learning delays or S-shaped curves where performance improves slowly, then rapidly, before slowing again. The authors propose a refined "delayed" law that accommodates these initial delays and accounts for the entire distribution of RTs, not just the mean. The authors introduce two new functions: the delayed-power and delayed-exponential laws. These models add a single delay parameter ($\tau$) to the standard power and exponential equations. When $\tau = 0$, the functions reduce to the traditional forms; as $\tau$ increases, an initial period of slow improvement emerges, allowing the model to capture slower-faster-slower learning patterns. To evaluate these proposals, the authors employed hierarchical Bayesian modeling on data from a broad array of tasks. This approach pools data across participants to minimize averaging artifacts while accounting for individual differences. The analysis utilized inference procedures that adjust for differences in model flexibility, ensuring fair comparisons between the traditional and new laws. Additionally, the authors linked the practice laws to the Lognormal Race evidence accumulation model to explain RT variability and its decrease with practice. The results indicated that in a clear majority of paradigms, the delayed-exponential law provided the best fit to the data. This model successfully accounted for cases where participants exhibited an initial period of slow performance followed by rapid improvement, a pattern often attributed to transitions between algorithmic processing and direct retrieval. The traditional exponential function struggled to fit data with initial delays, often overestimating early response times and showing distortions in later regions. The delayed-exponential model, by contrast, accommodated standard exponential data as a special case while also capturing complex learning trajectories involving initial plateaus or strategy shifts. The significance of this work lies in providing a more flexible and theoretically robust measurement model for skill acquisition. By incorporating a delay parameter and modeling the full RT distribution, the proposed law resolves inconsistencies in previous findings and offers a unified framework for describing learning across diverse tasks. This refinement supports the view that learning is not always a simple monotonic process but can involve strategic transitions and initial delays. The findings suggest that the delayed-exponential law should replace the traditional power and exponential laws as the standard for quantifying practice effects, offering better descriptive accuracy and stronger links to process-based cognitive models.

Key finding

The delayed exponential law provided a superior fit to practice data across a majority of paradigms compared to standard power and exponential functions.

Methodology

modeling

Provenance

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Topics

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• automaticity skill acquisition