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

Logan, Gordon D.

doi:10.1037/0033-295x.102.4.751

Cited by 38 publications

(34 citation statements)

References 29 publications

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“…The Wald distribution arises from a diffusion process with one absorbing boundary and has been used to explain simple (detection) behavior (Emerson, 1970;Wald, 1947), as well as to describe the firing pattern of simple neurons (Rudd, 1996). Finally, the Weibull model describes RT (under certain conditions; see Colonius, 1995) when a large number of parallel processes compete to produce a response (Logan, 1988(Logan, , 1992(Logan, , 1995.…”

Section: Methodsmentioning

confidence: 99%

How to fit a response time distribution

2000

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Among the most valuable tools in behavioral science is statistically fitting mathematical models of cognition to data-response time distributions, in particular. However, techniques for fitting distributions vary widely, and little is known about the efficacy of different techniques. In this article, we assess several fitting techniques by simulating six widely cited models of response time and using the fitting procedures to recover model parameters. The techniques include the maximization of likelihood and least squares fits of the theoretical distributions to different empirical estimates of the simulated distributions. Arunning example is used to illustrate the different estimation and fittingprocedures. The simulation studies reveal that empirical density estimates are biased even for very large sample sizes. Some fitting techniques yield more accurate and less variable parameter estimates than do others. Methods that involve least squares fits to density estimates generally yield very poor parameter estimates.The importance ofconsidering the entire response time (RT) distribution in testing formal models ofcognition is now widely appreciated. Fitting a model to mean RT alone can mask important details of the data that examination of the entire distribution would reveal, such as the behavior of fast and slow responses across the conditions ofan experiment (e.g., Heathcote, Popiel, & Mewhort, 1991), the extent of facilitation between perceptual channels (Miller, 1982), and the effects ofpractice on RT quantiles (Logan, 1992). Techniques for testing hypotheses based on the RT distribution have been developed (Townsend, 1990). In addition, the RT distribution provides an important meeting ground between theory and data; the ability ofa model to predict the observed shape ofthe RT distribution is seen as a critical test ofthat model (Luce, 1986).Many models state explicitly the characteristics ofRT by specifying it as a random variable. All ofthe information about a random variable is contained in its probability density function (density, for short) or cumulative distribution function (CDF).l The density represents the likelihood that an RT is observed within some arbitrarily small window of time, whereas the CDF represents the probability that an RT is less than or equal to some specific time. Most models ofRT predict CDFs that are ogival: monotonic, nondecreasing S-shaped functions that begin at zero and asymptote at one. The RT densities predicted by most models are, in contrast, bell shaped: nonPortions of this article were presented at

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Section: Methodsmentioning

confidence: 99%

How to fit a response time distribution

2000

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“…The theory most closely tied to a power practice function is Logan's (1988Logan's ( , 1992) theory based on the minimum time for race among instance retrievals. As Logan (1995) has acknowledged, "A major goal in developing the theory was to account for the power function speedup" (p. 751). Logan's theory uses a weak learning mechanism, in comparison with other theories of skill acquisition.…”

Section: Theoretical Implicationsmentioning

confidence: 99%

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

Heathcote

Brown

Mewhort

2000

Psychonomic Bulletin & Review

624

565

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The power function is treated as the law relating response time to practice trials. However, the evidence for a power law is flawed, because it is based on averaged data. We report a survey that assessed the fonn ofthe practice function for individual learners and learning conditions in paradigms that have shaped theories of skill acquisition. We fit power and exponential functions to 40 sets of data representing 7,910 learning series from 475 subjects in 24 experiments. The exponential function fit better than the power function in all the unaveraged data sets. Averaging produced a bias in favor of the power function. A new practice function based on the exponential, the APEX function, fit better than a power function with an extra, preexperimental practice parameter. Clearly, the best candidate for the law of practice is the exponential or APEX function, not the generally accepted power function. The theoretical implications are discussed.Curve fitting without benefit of a model is notoriously a black art.

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“…Clearly, there may be alternative ways of considering a variable's influence on RT distributions that are preferable (see Van Zandt, 2000, for several examples). For example, RT distributions have also been modeled with the Weibull (Logan, 1995) and ex-Wald (Schwarz, 2001) distributions. In this study, we have demonstrated how ex-Gaussian analyses allow mean differences to be partitioned into two components: one that reflects distributional shifting and one that reflects distributional skewing.…”

Section: Utility Of Distributional Analyses and Cross-task Convergencementioning

confidence: 99%

Additive and interactive effects on response time distributions in visual word recognition.

Yap

Balota

2007

Journal of Experimental Psychology: Learning, Memory, and Cogni

214

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Across 3 different word recognition tasks, distributional analyses were used to examine the joint effects of stimulus quality and word frequency on underlying response time distributions. Consistent with the extant literature, stimulus quality and word frequency produced additive effects in lexical decision, not only in the means but also in the shape of the response time distributions, supporting an early normalization process that is separate from processes influenced by word frequency. In contrast, speeded pronunciation and semantic classification produced interactive influences of word frequency and stimulus quality, which is a fundamental prediction from interactive activation models of lexical processing. These findings suggest that stimulus normalization is specific to lexical decision and is driven by the task's emphasis on familiarity-based information.Keywords: distributional analysis, task-specific effects, stimulus quality, word frequency, visual word recognition Beginning with Donders (1868/1969), a central goal of understanding human cognition has been to isolate constituent subprocesses through the use of mental chronometry. As there were problems with the insertion procedure Donders advocated, Sternberg (1969a) developed additive-factors logic in which one can provide leverage on the manner in which stages of information processing are organized. Specifically, one can use response time (RT) data from factorial experiments to make inferences about the modules associated with a mental process. For example, Sternberg argued that in an experiment in which two variables are manipulated, additive effects of both variables (i.e., main effects for both variables and no interaction) suggest that the variables influence separately modifiable processing stages. In contrast, interactive effects are more consistent with the variables influencing at least one stage in common.In a classic application of additive-factors logic, stimulus quality (intact vs. degraded) and set size (number of items in memory) were manipulated in a memory search task, and these two factors were found to be additive (Sternberg, 1967(Sternberg, , 1969b. These additive effects were interpreted as being consistent with a stage model of memory search, in which stimulus quality influences an early encoding stage, and set size influences a subsequent serial comparison stage (see Figure 1). In contrast, factors that interact are assumed to influence a common processing locus. For example, Becker (1979) investigated the effects of word frequency (high vs. low frequency) and semantic context (related vs. unrelated context) on word recognition and reported that word frequency interacts with semantic context. This suggests that word frequency and semantic context influence a common stage.Following this early classic work, demonstrations of clear additivity have been observed across diverse studies (see Sternberg, 1998, for an extensive review; see also Roberts, 1987;Sanders, 1990), supporting the claim that additive effects reflect discrete sta...

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

Cited by 38 publications

References 29 publications

How to fit a response time distribution

How to fit a response time distribution

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

Additive and interactive effects on response time distributions in visual word recognition.

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