Skip to main content

How to fit a response time distribution

Psychonomic Bulletin & Review volume 7, pages 424–465 (2000)Cite this article

Abstract

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. A running example is used to illustrate the different estimation and fitting procedures. 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.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price includes VAT (USA)
Tax calculation will be finalised during checkout.

References

  • Ashby, F. G., Tein, J.-Y., &Balakrishnan, J. D. (1993). Response time distributions in memory scanning.Journal of Mathematical Psychology,37, 526–555.

    Google Scholar 

  • Balota, D. A., &Spieler, D. H. (1999). Word frequency, repetition, and lexicality effects in word recognition tasks: Beyond measures of central tendency.Journal of Experimental Psychology: General,128, 32–55.

    Google Scholar 

  • Bickel, P. J., &Doksum, K. A. (1977).Mathematical statistics. San Francisco: Holden-Day.

    Google Scholar 

  • Blanco, M. J., &Alvarez, A. A. (1994). Psychometric intelligence and visual focused attention: Relationships in nonsearch tasks.Intelligence,18, 77–106.

    Google Scholar 

  • Blough, D. S. (1988). Quantitative relations between visual search speed and target-distractor similarity.Perception & Psychophysics,43, 57–71.

    Google Scholar 

  • Bloxom, B. (1984). Estimating response time hazard functions: An exposition and extension.Journal of Mathematical Psychology,28, 401–420.

    Google Scholar 

  • Bloxom, B. (1985). A constrained spline estimator of a hazard function.Psychometrika,50, 301–321.

    Google Scholar 

  • Burbeck, S. L., &Luce, R. D. (1982). Evidence from auditory simple reaction times for both change and level detectors.Perception & Psychophysics,32, 117–133.

    Google Scholar 

  • Chechile, R. A. (1998). Reexamining the goodness-of-fit problem for interval-scale scores.Behavior Research Methods, Instruments, & Computers,30, 227–231.

    Google Scholar 

  • Colonius, H. (1995). The instance theory of automaticity: Why the Weibull?Psychological Review,102, 744–750.

    Google Scholar 

  • Cousineau, D., &Larochelle, S. (1997). PASTIS: A program for curve and distribution analyses.Behavior Research Methods, Instruments, & Computers,29, 542–548.

    Google Scholar 

  • Dawson, M. R. W. (1988). Fitting the ex-Gaussian equation to reaction time distributions.Behavior Research Methods, Instruments, & Computers,20, 54–57.

    Google Scholar 

  • Devroye, L. (1987).A course in density estimation. Boston: Birkhäuser.

    Google Scholar 

  • Emerson, P. L. (1970). Simple reaction time with Markovian evolution of Gaussian discriminal processes.Psychometrika,35, 99–109.

    Google Scholar 

  • Feller, W. (1968).An introduction to probability theory and its applications (Vol. 1). New York: Wiley.

    Google Scholar 

  • Gallant, A. R. (1987).Nonlinear statistical models. New York: Wiley.

    Google Scholar 

  • Golden, R. M. (1999). Statistical tests for comparing possibly misspecified and non-nested models.Journal of Mathematical Psychology,44, 153–170.

    Google Scholar 

  • Green, D. M., &Luce, R. D. (1971). Detection of auditory signals presented at random times: III.Perception & Psychophysics,9, 257–268.

    Google Scholar 

  • Green, D. M., &Smith, A. F. (1982). Detection of auditory signals occurring at random times: Intensity and duration.Perception & Psychophysics,31, 117–127.

    Google Scholar 

  • Heathcote, A. (1996). RTSYS: A DOS application for the analysis of reaction time data.Behavior Research Methods, Instruments, & Computers,28, 427–445.

    Google Scholar 

  • Heathcote, A., Popiel, S. J., &Mewhort, D. J. (1991). Analysis of response time distributions: An example using the Stroop task.Psychological Bulletin,109, 340–347.

    Google Scholar 

  • Hockley, W. E. (1984). Analysis of response time distributions in the study of cognitive processes.Journal of Experimental Psychology: Learning, Memory, & Cognition,10, 598–615.

    Google Scholar 

  • Hockley, W. E., &Murdock, B. (1987). A decision model for accuracy and response latency in recognition memory.Psychological Review,94, 341–358.

    Google Scholar 

  • Hohle, R. H. (1965). Inferred components of reaction times as a function of foreperiod duration.Journal of Experimental Psychology,69, 382–386.

    PubMed  Google Scholar 

  • Juhel, J. (1993). Should we take the shape of reaction time distributions into account when studying the relationship between RT and psychometric intelligence?Personality & Individual Differences,15, 357–360.

    Google Scholar 

  • Knuth, D. E. (1981).Seminumerical algorithms. Reading, MA: Addison-Wesley.

    Google Scholar 

  • Leth-Steenson, C., King Elbaz, Z., &Douglas, V. I. (2000). Mean response times, variability, and skew in the responding of ADHD children: A response time distributional approach.Acta Psychologica,104, 167–190.

    Google Scholar 

  • Logan, G. D. (1988). Toward an instance theory of automatization.Psychological Review,95, 492–527.

    Google Scholar 

  • Logan, G. D. (1992). Shapes of reaction-time distributions and shapes of learning curves: A test of the instance theory of automaticity.Journal of Experimental Psychology: Learning, Memory, & Cognition,18, 883–914.

    Google Scholar 

  • Logan, G. D. (1995). The Weibull distribution, the power law, and the instance theory of automaticity.Psychological Review,102, 751–756.

    Google Scholar 

  • Luce, R. D. (1986).Response times: Their role in inferring elementary mental organization. New York: Oxford University Press.

    Google Scholar 

  • Madden, D., Gottlob, L., Denny, L., Turkington, T., Provenzale, J., Hawk, T., &Coleman, R. (1999). Aging and recognition memory: Changes in regional cerebral blood flow associated with components of reaction time distributions.Journal of Cognitive Neuroscience,11, 511–520.

    PubMed  Google Scholar 

  • McClelland, J. (1979). On the time relations of mental processes: An examination of systems of processes in cascade.Psychological Review,86, 287–330.

    Google Scholar 

  • McElree, B. (1998). Attended and non-attended states in working memory: Accessing categorized structures.Journal of Memory & Language,37, 225–252.

    Google Scholar 

  • McElree, B., &Dosher, B. A. (1993). Serial retrieval processes in the recovery of order information.Journal of Experimental Psychology: General,122, 291–315.

    Google Scholar 

  • McGill, W. J. (1963). Stochastic latency mechanisms. In R. D. Luce & R. R. Bush (Eds.),Handbook of mathematical psychology (Vol. 1, pp. 309–360). New York: Wiley.

    Google Scholar 

  • Miller, J. (1982). Divided attention: Evidence for coactivation with redundant signals.Cognitive Psychology,14, 247–279.

    PubMed  Google Scholar 

  • Myung, I. J. (1999). The importance of complexity in model selection.Journal of Mathematical Psychology,44, 190–204.

    Google Scholar 

  • Nelder, J. A., &Mead, R. (1965). A simplex method for function minimization.Computer Journal,7, 308–313.

    Google Scholar 

  • Nosofsky, R. M., &Palmeri, T. J. (1997). Comparing exemplarretrieval and decision-bound models of speeded perceptual classification.Perception & Psychophysics,59, 1027–1048.

    Google Scholar 

  • Parzen, E. (1962). On estimation of a probability density function and mode.Annals of Mathematical Statistics,33, 1065–1076.

    Google Scholar 

  • Pike, R. (1973). Response latency models for signal detection.Psychological Review,80, 53–68.

    PubMed  Google Scholar 

  • Plourde, C., &Besner, D. (1997). On the locus of the word frequency effect in visual word recognition.Canadian Journal of Experimental Psychology,51, 181–194.

    Google Scholar 

  • Possamai, C. (1991). A responding hand effect in a simple-RT precueing experiment: Evidence for a late locus of facilitation.Acta Psychologica,77, 47–63.

    PubMed  Google Scholar 

  • Press, W. H., Teukolsky, S. A., Vetterling, W. T., &Flannery, B. P. (1992).Numerical recipes in FORTRAN: The art of scientific computing (2nd ed.). New York: Cambridge University Press.

    Google Scholar 

  • Ratcliff, R. (1978). A theory of memory retrieval.Psychological Review,85, 59–108.

    Google Scholar 

  • Ratcliff, R. (1979). Group reaction time distributions and an analysis of distribution statistics.Psychological Bulletin,86, 446–461.

    PubMed  Google Scholar 

  • Ratcliff, R. (1988). Continuous versus discrete information processing: Modeling the accumulation of partial information.Psychological Review,95, 238–255.

    PubMed  Google Scholar 

  • Ratcliff, R. &Murdock, B. B., Jr. (1976). Retrieval processes in recognition memory.Psychological Review,83, 190–214.

    Google Scholar 

  • Read, T. R. C., &Cressie, N. A. C. (1988).Goodness-of-fit statistics for discrete multivariate data. New York: Springer-Verlag.

    Google Scholar 

  • Reber, P. J., Alvarez, P., &Squire, L. R. (1997). Reaction time distributions across normal forgetting: Searching for markers of memory.Learning & Memory,4, 284–290.

    Google Scholar 

  • Rohrer, D., &Wixted, J. T. (1994). An analysis of latency and interresponse time in free recall.Memory & Cognition,22, 511–524.

    Google Scholar 

  • Rudd, M. E. (1996). A neural timing model of visual threshold.Journal of Mathematical Psychology,40, 1–29.

    Google Scholar 

  • Silverman, B. W. (1986).Density estimation for statistics and data analysis. London: Chapman & Hall.

    Google Scholar 

  • Smith, D., &Mewhort, D. (1998). The distribution of latencies constrains theories of decision time: A test of the random-walk model using numeric comparison.Australian Journal of Psychology,50, 149–156.

    Google Scholar 

  • Smith, P. L., &Vickers, D. (1988). The accumulator model of twochoice discrimination.Journal of Mathematical Psychology,32, 135–168.

    Google Scholar 

  • Spieler, D. H., Balota, D. A., &Faust, M. E. (1996). Stroop performance in healthy younger and older adults and in individuals with dementia of the Alzheimer’s type.Journal of Experimental Psychology: Human Perception & Performance,22, 461–479.

    Google Scholar 

  • Strayer, D. L., &Kramer, A. F. (1994). Strategies and automaticity: I. Basic findings and conceptual framework.Journal of Experimental Psychology: Learning, Memory, & Cognition,20, 318–341.

    Google Scholar 

  • Tapia, R. A., &Thompson, J. R. (1978).Nonparametric probability density estimation. Baltimore: Johns Hopkins University Press.

    Google Scholar 

  • Tolhurst, D. J. (1975). Reaction times in the detection of gratings by human observers: A probabilistic mechanism.Vision Research,15, 1143–1149.

    PubMed  Google Scholar 

  • Townsend, J. T. (1990). Truth and consequences of ordinal differences in statistical distributions: Toward a theory of hierarchical inference.Psychological Bulletin,108, 551–567.

    PubMed  Google Scholar 

  • Townsend, J. T., &Ashby, F. G. (1983).Stochastic modeling of elementary psychological processes. New York: Cambridge University Press.

    Google Scholar 

  • Van Zandt, T., Colonius, H., &Proctor, R. W. (2000). A comparison of two response time models applied to perceptual matching.Psychonomic Bulletin & Review,7, 208–256.

    Google Scholar 

  • Van Zandt, T., &Ratcliff, R. (1995). Statistical mimicking of reaction time data: Single-process models, parameter variability, and mixtures.Psychonomic Bulletin & Review,2, 20–54.

    Google Scholar 

  • Wald, A. (1947).Sequential analysis. New York: Wiley.

    Google Scholar 

  • Wickens, T. D. (1982).Models for behavior: Stochastic processes in psychology. San Francisco: Freeman.

    Google Scholar 

  • Wixted, J. T., &Rohrer, D. (1993). Proactive interference and the dynamics of free recall.Journal of Experimental Psychology: Learning, Memory, & Cognition,19, 1024–1039.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Johns Hopkins University, Baltimore, Maryland

    Trisha Van Zandt

Authors
  1. Trisha Van Zandt
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Portions of this article were presented at the 31st Annual Meeting of the Society for Mathematical Psychology, August 1998, Vanderbilt University, and at the 39th Annual Meeting of the Psychonomic Society, November 1998, Dallas. The project was funded by NSF Grant SBR-9702291. The author gratefully acknowledges the many contributions to this project by Steven Yantis. Thanks also are due Howard Egeth, Andrew Heathcote, and John Wixted for many helpful comments that greatly improved the paper.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Van Zandt, T. How to fit a response time distribution. Psychon Bull Rev 7, 424–465 (2000). https://doi.org/10.3758/BF03214357

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3758/BF03214357

Keywords

  • Cumulative Distribution Function
  • True Density
  • Race Model
  • Response Time Distribution
  • Histogram Estimate