The Weibull distribution, the power law, and the instance theory of automaticity.

Logan, Gordon D. · 1995 · Crossref

DOI: 10.1037/0033-295x.102.4.751

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Summary

This paper addresses the mathematical foundations of the instance theory of automaticity, specifically responding to critiques by Colonius (1995) regarding the justification for using the Weibull distribution to model reaction times. The instance theory posits that automaticity arises from a race between memory traces retrieved from past instances, rather than resource-based mechanisms. A central prediction of this theory is the power function speedup, where performance improves as a power function of practice. Logan clarifies the theoretical arguments for assuming that retrieval times follow a Weibull distribution, correcting errors in his previous asymptotic reasoning and evaluating Colonius’s alternative nonasymptotic arguments. The analysis focuses on the statistical properties of extreme values, as the theory assumes performance is governed by the fastest retrieved trace (the minimum of a sample). Logan acknowledges that his previous claim—that distributions of minima converge asymptotically to the Weibull distribution—was mathematically flawed because asymptotic distributions require linear transformations of the original variable, which are not justified by the theory’s assumptions. Instead, he validates the nonasymptotic argument: the Weibull distribution is "min-stable," meaning the distribution of minima sampled from a Weibull parent remains Weibull, merely changing in scale according to a power function. This property provides a principled basis for the power function speedup prediction. Logan also discusses Colonius’s suggestion that observing a power function speedup in means implies a Weibull distribution, citing Huang’s (1989) theorem, but notes practical limitations in testing this via comparisons of only two practice levels. Logan argues that the best empirical test involves fitting constrained Weibull distributions to data across a wide range of practice, ensuring the shape parameter remains constant while the scale reduces according to the power function. He highlights that the power function speedup in the entire distribution, not just the mean, is a unique prediction of the instance theory that distinguishes it from mixture models. However, he expresses caution regarding the strict assumption that all traces have identical, independent retrieval time distributions (the iid assumption). He notes that real-world factors, such as primacy/recency effects, context-dependent encoding, and variability in perceptual or motor processes ("intercept" processes), may violate this assumption. Simulations suggest the power function predictions are robust to some violations of the iid assumption, but future mathematical development should aim to separate central retrieval processes from peripheral encoding and response processes. The significance of this work lies in refining the mathematical rigor of the instance theory while maintaining its viability as an account of skill acquisition. By correcting the asymptotic argument and clarifying the role of the Weibull distribution’s min-stable property, Logan strengthens the theoretical justification for the power law. The paper also outlines future directions for the field, emphasizing the need to address the "intercept problem" and relax the iid assumption to make the theory more realistic. Ultimately, the instance theory remains a compelling memory-based explanation for automaticity, offering distinct predictions about the distribution of reaction times that differentiate it from alternative models of skill acquisition.

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• automaticity skill acquisition