Cognitive models of the decision process provide greater insight into response time and accuracy than do standard ANOVA techniques. However, such models can be mathematically and computationally difficult to apply. We provide instructions and computer code for three methods for estimating the parameters of the linear ballistic accumulator (LBA), a new and computationally tractable model of decisions between two or more choices. These methods-a Microsoft Excel worksheet, scripts for the statistical program R, and code for implementation of the LBA into the Bayesian sampling software WinBUGS-vary in their flexibility and user accessibility. We also provide scripts in R that produce a graphical summary of the data and model predictions. In a simulation study, we explored the effect of sample size on parameter recovery for each method. The materials discussed in this article may be downloaded as a supplement from http://brm.psychonomic-journals.org/content/supplemental.

Identification accuracy for sets of perceptually discriminable stimuli ordered on a single dimension (e.g., line length) is remarkably low, indicating a fundamental limit on information processing capacity. This surprising limit has naturally led to a focus on measuring and modeling choice probability in absolute identification research. We show that choice response time (RT) results can enrich our understanding of absolute identification by investigating dissociation between RT and accuracy as a function of stimulus spacing. The dissociation is predicted by the SAMBA model of absolute identification (Brown, Marley, Dockin,& Heathcote, 2008), but cannot easily be accommodated by other theories. We show that SAMBA provides an accurate, parameter free, account of the dissociation that emerges from the architecture of the model and the physical attributes of the stimuli, rather than through numerical adjustment. This violation of the pervasive monotonic relationship between RT and accuracy has implications for model development, which are discussed.

Most models of choice response time base decisions on evidence accumulated over time. A fundamental distinction among these models concerns whether each piece of evidence is equally weighted (lossless accumulation) or unequally weighted (leaky accumulation). The authors tested a hypothesis derived from A. Heathcote and S. Brown's (2002) self-exciting expert competitor (SEEXC) model of skill acquisition: that evidence accumulation becomes less leaky with practice. The hypothesis was supported by observation that the effects of prime stimuli increased with practice. The authors used metacontrast masked primes, which could not be consciously discriminated by most participants, to avoid methodological problems associated with conscious strategy changes. The form of the law of practice in the data is also shown to be consistent with the SEEXC model.

The most powerful tests of response time (RT) models often involve the whole shape of the RT distribution, thus avoiding mimicking that can occur at the level of RT means and variances. Nonparametric distribution estimation is, in principle, the most appropriate approach, but such estimators are sometimes difficult to obtain. On the other hand, distribution fitting, given an algebraic function, is both easy and compact. We review the general approach to performing distribution fitting with maximum likelihood (ML) and a method based on quantiles (quantile maximum probability, QMP). We show that QMP has both small bias and good efficiency when used with common distribution functions (the ex-Gaussian, Gumbel, lognormal, Wald, and Weibull distributions). In addition, we review some software packages performing ML (PASTIS, QMPE, DISFIT, and MATHEMATICA) and compare their results. In general, the differences between packages have little influence on the optimal solution found, but the form of the distribution function has: Both the lognormal and the Wald distributions have non-linear dependencies between the parameter estimates that tend to increase the overall bias in parameter recovery and to decrease efficiency. We conclude by laying out a few pointers on how to relate descriptive models of RT to cognitive models of RT. A program that generated the random deviates used in our studies may be downloaded from www.psychonomic.org/archive/.

We describe and test quantile maximum probability estimator (QMPE), an open-source ANSI Fortran 90 program for response time distribution estimation. QMPE enables users to estimate parameters for the ex-Gaussian and Gumbel (1958) distributions, along with three "shifted" distributions (i.e., distributions with a parameter-dependent lower bound): the Lognormal, Wald, and Weibul distributions. Estimation can be performed using either the standard continuous maximum likelihood (CML) method or quantile maximum probability (QMP; Heathcote & Brown, in press). We review the properties of each distribution and the theoretical evidence showing that CML estimates fail for some cases with shifted distributions, whereas QMP estimates do not. In cases in which CML does not fail, a Monte Carlo investigation showed that QMP estimates were usually as good, and in some cases better, than CML estimates. However, the Monte Carlo study also uncovered problems that can occur with both CML and QMP estimates, particularly when samples are small and skew is low, highlighting the difficulties of estimating distributions with parameter-dependent lower bounds.

Quantile maximum likelihood (QML) is an estimation technique, proposed by Heathcote, Brown, and Mewhort (2002), that provides robust and efficient estimates of distribution parameters, typically for response time data, in sample sizes as small as 40 observations. In view of the computational difficulty inherent in implementing QML, we provide open-source Fortran 90 code that calculates QML estimates for parameters of the ex-Gaussian distribution, as well as standard maximum likelihood estimates. We show that parameter estimates from QML are asymptotically unbiased and normally distributed. Our software provides asymptotically correct standard error and parameter intercorrelation estimates, as well as producing the outputs required for constructing quantile-quantile plots. The code is parallelizable and can easily be modified to estimate parameters from other distributions. Compiled binaries, as well as the source code, example analysis files, and a detailed manual, are available for free on the Internet.

Myung, Kim, and Pitt (2000) demonstrated that simple power functions almost always provide a better fit to purely random data than do simple exponential functions. This result has important implications, because it suggests that high noise levels, which are common in psychological experiments, may cause a bias favoring power functions. We replicate their result and extend it by showing strong bias for more realistic sample sizes. We also show that biases occur for data that contain both random and systematic components, as may be expected in real data. We then demonstrate that these biases disappear for two- or three-parameter functions that include linear parameters (in at least one parameterization). Our results suggest that one should exercise caution when proposing simple power and exponential functions as models of learning. More generally, our results suggest that linear parameters should be estimated rather than fixed when one is comparing the fit of nonlinear models to noisy data.

We examine recent concerns that averaged learning curves can present a distorted picture of individual learning. Analyses of practice curve data from a range of paradigms demonstrate that such concerns are well founded for fits of power and exponential functions when the arithmetic average is computed over participants. We also demonstrate that geometric averaging over participants does not, in general, avoid distortion. By contrast, we show that block averages of individual curves and similar smoothing techniques cause little or no distortion of functional form, while still providing the noise reduction benefits that motivate the use of averages. Our analyses are concerned mainly with the effects of averaging on the fit of exponential and power functions, but we also define general conditions that must be met by any set of functions to avoid distortion from averaging.

We introduce and evaluate via a Monte Carlo study a robust new estimation technique that fits distribution functions to grouped response time (RT) data, where the grouping is determined by sample quantiles. The new estimator, quantile maximum likelihood (QML), is more efficient and less biased than the best alternative estimation technique when fitting the commonly used ex-Gaussian distribution. Limitations of the Monte Carlo results are discussed and guidance provided for the practical application of the new technique. Because QML estimation can be computationally costly, we make fast open source code for fitting available that can be easily modified to use QML in the estimation of any distribution function.

We present a new decision-making model that can account for trial-by-trial variability induced by a process ("pre-process") that occurs before an explicit sensory signal specifying a later motor response. A process after explicit sensory signals, referred to herein as the "post-process", has been investigated by a variety of so-called rise-to-threshold models including the LATER model. The LATER model formulates post-process variability but treats the pre-process as fixed within a block of an experiment. We propose an extension of the LATER model, which we call the extended LATER (ELATER) model, to account for trial-by-trial variability of both pre- and post-processes together. We present the mathematical formulation of the ELATER model and analyze its characteristics, including numerical examples and an example of saccade latency data in reward-manipulated conditions with caudate activity. The ELATER model is useful for investigating decision making by taking account of trial-by-trial variability of both pre- and post-processes.

Preference orderings among a set of options may depend on the elicitation method (e.g., choice or pricing); these preference reversals challenge traditional decision theories. Previous attempts to explain these reversals have relied on allowing utility of the options to change across elicitation methods by changing the decision weights, the attribute values, or the combination of this information--still, no theory has successfully accounted for all the phenomena. In this article, the authors present a new computational model that accounts for the empirical trends without changing decision weights, values, or combination rules. Rather, the current model specifies a dynamic evaluation and response process that correctly predicts preference orderings across 6 elicitation methods, retains stable evaluations across methods, and makes novel predictions regarding response distributions and response times.((c) 2005 APA, all rights reserved).

The authors present a dynamical multilevel model that captures changes over time in the bidirectional, potentially asymmetric influence of 2 cyclical processes. S. M. Boker and J. Graham's (1998) differential structural equation modeling approach was expanded to the case of a nonlinear coupled oscillator that is common in bimanual coordination studies in which participants swing hand-held pendulums but is also applicable to social systems in general. The authors' nonlinear coupled oscillator model decomposed the fluctuations into a competitive component, unique to each individual variable, and a cooperative component that captured bidirectional influence. The authors' model also generated an index of the symmetry/asymmetry of bidirectional influence. Together, the models are useful quantitative tools for the study of interacting, changing processes.

Recent theories of dynamic attention have renewed the interest in temporal context as a determinant of attention. The mechanism of dynamic attention remains unclear, and both stochastic time perception processes and deterministic oscillators are possible. The results of Experiment 1 demonstrate that attention can be guided by isochronous series of warning stimuli and that elapsed time cannot fully account for this effect. Experiment 2 indicates that temporal structure can be used over a limited range of time. The results of Experiment 3 indicate that temporal pattern, rather than variability, is a determinant of temporally focused attention. The results of Experiment 4 demonstrate that a coupled oscillator is a better predictor of reaction time than a stochastic timing mechanism is, under certain assumptions.

A key parameter in the understanding of renal hemodynamics is the gain of the feedback function in the tubuloglomerular feedback mechanism. A dynamic model of autoregulation of renal blood flow and glomerular filtration rate has been extended to include a stochastic differential equations model of one of the main parameters that determines feedback gain. The model reproduces fluctuations and irregularities in the tubular pressure oscillations that the former deterministic models failed to describe. This approach assumes that the gain exhibits spontaneous erratic variations that can be explained by a variety of influences, which change over time (blood pressure, hormone levels, etc.). To estimate the key parameters of the model we have developed a new estimation method based on the oscillatory behavior of the data. The dynamics is characterized by the spectral density, which has been estimated for the observed time series, and numerically approximated for the model. The parameters have then been estimated by the least squares distance between data and model spectral densities. To evaluate the estimation procedure measurements of the proximal tubular pressure from 35 nephrons in 16 rat kidneys have been analyzed, and the parameters characterizing the gain and the delay have been estimated. There was good agreement between the estimated values, and the values obtained for the same parameters in independent, previously published experiments.

D. von Winterfeldt, N.-K. Chung, R. D. Luce, and Y. Cho (1997) provided several tests for consequence monotonicity of choice or judgment, using certainty equivalents of gambles. The authors reaxiomatized consequence monotonicity in a probabilistic framework and reanalyzed von Winterfeldt et al.'s main experiment via a bootstrap method. Their application offers new insights into consequence monotonicity as well as into von Winterfeldt et al.'s 3 experimental paradigms: judged certainty equivalents (JCE), QUICKINDIFF, and parameter estimation by sequential testing (PEST). For QUICKINDIFF, the authors found no indication of violations of "random consequence monotonicity." This sharply contrasts the findings of von Winterfeldt et al., who concluded that axiom violations were the most pronounced under that procedure. The authors found potential evidence for violations in JCE and certainty equivalents derived from PEST.

An algorithm is described to efficiently compute the cumulative distribution and probability density functions of the diffusion process (Ratcliff, 1978) with trial-to-trial variability in mean drift rate, starting point, and residual reaction time. Some, but not all, of the integrals appearing in the model's equations have closed-form solutions, and thus we can avoid computationally expensive numerical approximations. Depending on the number of quadrature nodes used for the remaining numerical integrations, the final algorithm is at least 10 times faster than a classical algorithm using only numerical integration, and the accuracy is slightly higher. Next, we discuss some special cases with an alternative distribution for the residual reaction time or with fewer than three parameters exhibiting trial-to-trial variability.

M. Usher and J. L. McClelland (2004) recently proposed a new connectionist type of model to explain context effects on preferential choice including the similarity, attraction, and compromise effects. They compared their model with an earlier connectionist type model for these same effects proposed by R. Roe, J. R. Busemeyer, and J. T. Townsend (2001) and raised several new issues. The authors address these issues and point out the main theoretical differences between the 2 explanations for context effects.