Hostname: page-component-745bb68f8f-cphqk Total loading time: 0 Render date: 2025-01-08T05:39:54.127Z Has data issue: false hasContentIssue false
  • English
  • Français

The Lognormal Race: A Cognitive-Process Model of Choice and Latency with Desirable Psychometric Properties

Published online by Cambridge University Press:  01 January 2025

Jeffrey N. Rouder ,
Jordan M. Province ,
Richard D. Morey ,
Pablo Gomez  and
Andrew Heathcote
Jeffrey N. Rouder*
Affiliation:
University of Missouri
Jordan M. Province
Affiliation:
University of Missouri
Richard D. Morey
Affiliation:
University of Groningen
Pablo Gomez
Affiliation:
Depaul University
Andrew Heathcote
Affiliation:
University of Newcastle
*
Requests for reprints should be sent to Jeffrey N. Rouder, University of Missouri, Columbia, MO, USA. E-mail: rouderj@missouri.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a cognitive process model of response choice and response time performance data that has excellent psychometric properties and may be used in a wide variety of contexts. In the model there is an accumulator associated with each response option. These accumulators have bounds, and the first accumulator to reach its bound determines the response time and response choice. The times at which accumulator reaches its bound is assumed to be lognormally distributed, hence the model is race or minima process among lognormal variables. A key property of the model is that it is relatively straightforward to place a wide variety of models on the logarithm of these finishing times including linear models, structural equation models, autoregressive models, growth-curve models, etc. Consequently, the model has excellent statistical and psychometric properties and can be used in a wide range of contexts, from laboratory experiments to high-stakes testing, to assess performance. We provide a Bayesian hierarchical analysis of the model, and illustrate its flexibility with an application in testing and one in lexical decision making, a reading skill.

Keywords

cognitive psychometricsresponse-times modelsrace models
Type
Original Paper
Information
Psychometrika , Volume 80 , Issue 2 , June 2015 , pp. 491 - 513
Copyright
Copyright © 2014 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anderson, J.R., & Lebiere, C. (1998). The atomic component of thought. Mahwah: Lawrence Erlbaum Associates.Google Scholar
Audley, R.J., & Pike, A.R. (1965). Some alternative stochastic models of choice. British Journal of Mathematical and Statistical Psychology, 18, 207–225.CrossRefGoogle Scholar
Bertelson, P. (1961). Sequential redundancy and speed in a serial two choice responding task. Quarterly Journal of Experimental Psychology, 13, 290–292.CrossRefGoogle Scholar
Brown, S.D., & Heathcote, A. (2008). The simplest complete model of choice reaction time: linear ballistic accumulation. Cognitive Psychology, 57, 153–178.CrossRefGoogle ScholarPubMed
Craigmile, P.F., Peruggia, M., & Van Zandt, T. (2010). Hierarchical Bayes models for response time. Psychometrika, 75, 613–632.CrossRefGoogle Scholar
Davis, C. (2010). The spatial coding model of visual word identification. Psychological Review, 117(3), 713.CrossRefGoogle ScholarPubMed
Diederich, A., & Busemeyer, J.R. (2003). Simple matrix methods for analyzing diffusion models of choice probability, choice response time and simple response time. Journal of Mathematical Psychology, 47, 304–322.CrossRefGoogle Scholar
Dufau, S., Grainger, J., & Ziegler, J.C. (2012). How to say “no” to a nonword: a leaky competing accumulator model of lexical decision. Journal of Experimental Psychology: Learning, Memory, and Cognition, 38, 1117–1128.Google Scholar
Dzhafarov, E.N. (1992). The structure of simple reaction time to step-function signals. Journal of Mathematical Psychology, 36, 235–268.CrossRefGoogle Scholar
Embretson, S.E. (1991). A multidimensional latent trait model for measuring learning and change. Psychometrika, 56, 495–516.CrossRefGoogle Scholar
Falmagne, J.C., Cohen, S.P., & Dwivedi, A. (1975). Two choice reactions as an ordered memory scanning process. In Rabbit, P.M.A., & Dornic, S. (Eds.), Attention and performance V (pp. 296–344). New York: Academic Press.Google Scholar
Gelfand, A., & Smith, A.F.M. (1990). Sampling based approaches to calculating marginal densities. Journal of the American Statistical Association, 85, 398–409.CrossRefGoogle Scholar
Gelman, A., Carlin, J.B., Stern, H.S., & Rubin, D.B. (2004). Bayesian data analysis (2nd ed.). London: Chapman and Hall.Google Scholar
Gold, J.I., & Shadlen, M.N. (2007). The neural basis of decision making. Annual Review of Neuroscience, 30, 535–574.CrossRefGoogle ScholarPubMed
Gomez, P., Ratcliff, R., & Perea, M. (2008). The overlap model: a model of letter position coding. Psychological Review, 115(3), 577.CrossRefGoogle Scholar
Greenwald, A.G., Draine, S.C., & Abrams, R.L. (1996). Three cognitive markers of unconscious semantic activation. Science, 273(5282), 1699–1702.CrossRefGoogle ScholarPubMed
Grice, G.R. (1968). Stimulus intensity and response evocation. Psychological Review, 75, 359–373.CrossRefGoogle ScholarPubMed
Hanes, D.P., & Schall, J.D. (1996). Neural control of voluntary movement initiation. Science, 274(5286), 427–430.CrossRefGoogle ScholarPubMed
Heathcote, A., & Love, J. (2012). Linear deterministic accumulator models of simple choice. Frontiers in Cognitive Science, 3, 292. http://www.frontiersin.org/Cognitive_Science/10.3389/fpsyg.2012.0029.Google ScholarPubMed
Huk, A., & Shadlen, M.N. (2005). Neural activity in macaque parietal cortex reflects temporal integration of visual motion signals during perceptual decision making. Journal of Neuroscience, 25, 10420–10436.CrossRefGoogle ScholarPubMed
Jackman, S. (2009). Bayesian analysis for the social sciences. Chichester: Wiley.CrossRefGoogle Scholar
Kass, R.E. (1993). Bayes factors in practice. The Statistician, 42, 551–560.CrossRefGoogle Scholar
Kass, R.E., & Raftery, A.E. (1995). Bayes factors. Journal of the American Statistical Association, 90, 773–795.CrossRefGoogle Scholar
Kucera, H., & Francis, W.N. (1967). Computational analysis of present-day American English. Providence: Brown University Press.Google Scholar
Lee, S.-Y. (2007). Structural equation modelling: a Bayesian approach. New York: Wiley.CrossRefGoogle Scholar
Link, S.W. (1992). Wave theory of difference and similarity. Hillsdale: Erlbaum.Google Scholar
Luce, R.D. (1986). Response times. New York: Oxford University Press.Google Scholar
McClelland, J.L. (1993). Toward a theory of information processing in graded, random, and interactive networks. In Meyer, D.E., & Kornblum, S. (Eds.), Attention & performance XIV: synergies in experimental psychology, artificial intelligence and cognitive neuroscience. Cambridge: MIT Press.Google Scholar
McClelland, J.L., & Rumelhart, D.E. (1981). An interactive activation model of context effects in letter perception: I. An account of basic findings. Psychological Review, 88, 375–407.CrossRefGoogle Scholar
McGill, W. (1963). Stochastic latency mechanism. In Luce, R.D., & Galanter, E. (Eds.), Handbook of mathematical psychology (pp. 309–360). New York: Wiley.Google Scholar
Morey, R.D., Rouder, J.N., Pratte, M.S., & Speckman, P.L. (2011). Using MCMC chain outputs to efficiently estimate Bayes factors. Journal of Mathematical Psychology, 55, 368–378. http://dx.doi.org/10.1016/j.jmp.2011.06.004.CrossRefGoogle Scholar
Peruggia, M., Van Zandt, T., & Chen, M. (2002). Was it a car or a cat I saw? An analysis of response times for word recognition. Case Studies in Bayesian Statistics, 6, 319–334.CrossRefGoogle Scholar
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research.Google Scholar
Ratcliff, R. (1978). A theory of memory retrieval. Psychological Review, 85, 59–108.CrossRefGoogle Scholar
Ratcliff, R., & McKoon, G.M. (2008). The diffusion decision model: theory and data for two-choice decision tasks. Neural Computation, 20, 873–922.CrossRefGoogle ScholarPubMed
Ratcliff, R., & Rouder, J.N. (1998). Modeling response times for decisions between two choices. Psychological Science, 9, 347–356.CrossRefGoogle Scholar
Ratcliff, R., Thapar, A., & McKoon, G. (2003). A diffusion model analysis of the effects of aging on brightness discrimination. Perception and Psychophysics, 65, 523–535.CrossRefGoogle ScholarPubMed
Ratcliff, R., Gomez, P., & McKoon, G.M. (2004). A diffusion model account of the lexical decision task. Psychological Review, 111, 159–182.CrossRefGoogle ScholarPubMed
Reddi, B., & Carpenter, R. (2003). Accuracy, information and response time in a saccadic decision task. Journal of Neurophysiology, 90, 3538–3546.CrossRefGoogle Scholar
Riefer, D.M., Knapp, B.R., Batchelder, W.H., Bamber, D., & Manifold, V. (2002). Cognitive psychometrics: assessing storage and retrieval deficits in special populations with multinomial processing tree models. Psychological Assessment, 14, 184–201.CrossRefGoogle ScholarPubMed
Roberts, G.O., & Sahu, S.K. (1997). Updating schemes, correlation structure, blocking and parameterization for the Gibbs sampler. Journal of the Royal Statistical Society, Series B, Methodological, 59, 291–317.CrossRefGoogle Scholar
Roberts, S., & Pashler, H. (2000). How persuasive is a good fit? A comment on theory testing. Psychological Review, 107, 358–367.CrossRefGoogle Scholar
Roitman, J.D., & Shadlen, M.N. (2002). Response of neurons in the lateral intraparietal area during a combined visual discrimination reaction time task. Journal of Neuroscience, 22, 9475–9489.CrossRefGoogle ScholarPubMed
Rouder, J.N. (2005). Are unshifted distributional models appropriate for response time?. Psychometrika, 70, 377–381.CrossRefGoogle Scholar
Rouder, J.N., Sun, D., Speckman, P.L., Lu, J., & Zhou, D. (2003). A hierarchical Bayesian statistical framework for response time distributions. Psychometrika, 68, 587–604.CrossRefGoogle Scholar
Rumelhart, D.E., & McClelland, J.L. (1982). An interactive activation model of context effects in letter perception: II. The contextual enhancement effect and some tests and extensions of the model. Psychological Review, 89, 60–94.CrossRefGoogle Scholar
Schall, J.D. (2003). Neural correlates of decision processes: neural and mental chronometry. Current Opinion in Neurobiology, 13, 182–186.CrossRefGoogle ScholarPubMed
Skrondal, A., & Rabe-Hesketh, S. (2004). Generalized latent variable modeling: multilevel, longitudinal, and structural equation models. Boca Raton: CRC Press.CrossRefGoogle Scholar
Smith, P.L. (2000). Stochastic dynamic models of response time and accuracy: a foundational primer. Journal of Mathematical Psychology, 44, 408–463.CrossRefGoogle ScholarPubMed
Spiegelhalter, D.J., Best, N.G., Carlin, B.P., & van der Linde, A. (2002). Bayesian measures of model complexity and fit (with discussion). Journal of the Royal Statistical Society, Series B, Statistical Methodology, 64, 583–639.CrossRefGoogle Scholar
Thissen, D. (1983). Timed testing: an approach using item response theory. In Weiss, D.J. (Ed.), New horizons in testing: latent trait test theory and computerized adaptive testing (pp. 179–203). New York: Academic Press.Google Scholar
Tuerlickx, F., & De Boek, P. (2005). Two interpretations of the discrimination parameter. Psychometrika, 70, 629–650.CrossRefGoogle Scholar
Ulrich, R., & Miller, J.O. (1993). Information processing models generating lognormally distributed reaction times. Journal of Mathematical Psychology, 37, 513–525.CrossRefGoogle Scholar
Usher, M., & McClelland, J.L. (2001). On the time course of perceptual choice: the leaky competing accumulator model. Psychological Review, 108, 550–592.CrossRefGoogle ScholarPubMed
van Breukelen, G.J.P. (2005). Psychometric modeling of response time and accuracy with mixed and conditional regression. Psychometrika.CrossRefGoogle Scholar
Vandekerckhove, J., Tuerlinckx, F., & Lee, M.D. (2011). Hierarchical diffusion models for two-choice response time. Psychological Methods, 16, 44–62.CrossRefGoogle Scholar
Vandekerckhove, J., Verheyen, S., & Tuerlinckx, F. (2010). A cross random effects diffusion model for speeded semantic categorization decisions. Acta Psychologica, 133, 269–282.CrossRefGoogle ScholarPubMed
van der Linden, W.J. (2007). A hierarchical framework for modeling speed and accuracy on test items. Psychometrika, 72, 287–308.CrossRefGoogle Scholar
van der Linden, W.J. (2009). Conceptual issues in response-time modeling. Journal of Educational Measurement, 14, 247–272.CrossRefGoogle Scholar
van der Linden, W.J., Scrams, D.J., & Schnipke, D.L. (1999). Using response-time constraints to control for differential speededness in computerized adaptive testing. Applied Psychological Measurement, 23, 195–210.CrossRefGoogle Scholar
van der Maas, H.L.J., Molenaar, D., Maris, G., Kievit, R.A., & Borsboom, D. (2001). Cognitive psychology meets psychometric theory: on the relation between process models for decision making and latent variable models for individual differences. Psychological Review, 118, 339–356.CrossRefGoogle Scholar
Wagenmakers, E.-J., van der Maas, H.L.J., & Grasman, R.P.P.P. (2007). An EZ-diffusion model for response time and accuracy. Psychonomic Bulletin & Review, 14, 3–22.CrossRefGoogle ScholarPubMed
Whitney, C., Bertrand, D., & Grainger, J. (2011). On coding the position of letters in words . Experimental Psychology (formerly Zeitschrift für Experimentelle Psychologie), 1–6.Google Scholar
Wild, P., & Gilks, W.R. (1993). Adaptive rejection sampling from log-concave density functions. Applied Statistics, 42, 701–708.CrossRefGoogle Scholar