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.
References
Ashby, F. G., Tein, J.-Y., &Balakrishnan, J. D. (1993). Response time distributions in memory scanning.Journal of Mathematical Psychology,37, 526–555.
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.
Bickel, P. J., &Doksum, K. A. (1977).Mathematical statistics. San Francisco: Holden-Day.
Blanco, M. J., &Alvarez, A. A. (1994). Psychometric intelligence and visual focused attention: Relationships in nonsearch tasks.Intelligence,18, 77–106.
Blough, D. S. (1988). Quantitative relations between visual search speed and target-distractor similarity.Perception & Psychophysics,43, 57–71.
Bloxom, B. (1984). Estimating response time hazard functions: An exposition and extension.Journal of Mathematical Psychology,28, 401–420.
Bloxom, B. (1985). A constrained spline estimator of a hazard function.Psychometrika,50, 301–321.
Burbeck, S. L., &Luce, R. D. (1982). Evidence from auditory simple reaction times for both change and level detectors.Perception & Psychophysics,32, 117–133.
Chechile, R. A. (1998). Reexamining the goodness-of-fit problem for interval-scale scores.Behavior Research Methods, Instruments, & Computers,30, 227–231.
Colonius, H. (1995). The instance theory of automaticity: Why the Weibull?Psychological Review,102, 744–750.
Cousineau, D., &Larochelle, S. (1997). PASTIS: A program for curve and distribution analyses.Behavior Research Methods, Instruments, & Computers,29, 542–548.
Dawson, M. R. W. (1988). Fitting the ex-Gaussian equation to reaction time distributions.Behavior Research Methods, Instruments, & Computers,20, 54–57.
Devroye, L. (1987).A course in density estimation. Boston: Birkhäuser.
Emerson, P. L. (1970). Simple reaction time with Markovian evolution of Gaussian discriminal processes.Psychometrika,35, 99–109.
Feller, W. (1968).An introduction to probability theory and its applications (Vol. 1). New York: Wiley.
Gallant, A. R. (1987).Nonlinear statistical models. New York: Wiley.
Golden, R. M. (1999). Statistical tests for comparing possibly misspecified and non-nested models.Journal of Mathematical Psychology,44, 153–170.
Green, D. M., &Luce, R. D. (1971). Detection of auditory signals presented at random times: III.Perception & Psychophysics,9, 257–268.
Green, D. M., &Smith, A. F. (1982). Detection of auditory signals occurring at random times: Intensity and duration.Perception & Psychophysics,31, 117–127.
Heathcote, A. (1996). RTSYS: A DOS application for the analysis of reaction time data.Behavior Research Methods, Instruments, & Computers,28, 427–445.
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.
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.
Hockley, W. E., &Murdock, B. (1987). A decision model for accuracy and response latency in recognition memory.Psychological Review,94, 341–358.
Hohle, R. H. (1965). Inferred components of reaction times as a function of foreperiod duration.Journal of Experimental Psychology,69, 382–386.
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.
Knuth, D. E. (1981).Seminumerical algorithms. Reading, MA: Addison-Wesley.
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.
Logan, G. D. (1988). Toward an instance theory of automatization.Psychological Review,95, 492–527.
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.
Logan, G. D. (1995). The Weibull distribution, the power law, and the instance theory of automaticity.Psychological Review,102, 751–756.
Luce, R. D. (1986).Response times: Their role in inferring elementary mental organization. New York: Oxford University Press.
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.
McClelland, J. (1979). On the time relations of mental processes: An examination of systems of processes in cascade.Psychological Review,86, 287–330.
McElree, B. (1998). Attended and non-attended states in working memory: Accessing categorized structures.Journal of Memory & Language,37, 225–252.
McElree, B., &Dosher, B. A. (1993). Serial retrieval processes in the recovery of order information.Journal of Experimental Psychology: General,122, 291–315.
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.
Miller, J. (1982). Divided attention: Evidence for coactivation with redundant signals.Cognitive Psychology,14, 247–279.
Myung, I. J. (1999). The importance of complexity in model selection.Journal of Mathematical Psychology,44, 190–204.
Nelder, J. A., &Mead, R. (1965). A simplex method for function minimization.Computer Journal,7, 308–313.
Nosofsky, R. M., &Palmeri, T. J. (1997). Comparing exemplarretrieval and decision-bound models of speeded perceptual classification.Perception & Psychophysics,59, 1027–1048.
Parzen, E. (1962). On estimation of a probability density function and mode.Annals of Mathematical Statistics,33, 1065–1076.
Pike, R. (1973). Response latency models for signal detection.Psychological Review,80, 53–68.
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.
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.
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.
Ratcliff, R. (1978). A theory of memory retrieval.Psychological Review,85, 59–108.
Ratcliff, R. (1979). Group reaction time distributions and an analysis of distribution statistics.Psychological Bulletin,86, 446–461.
Ratcliff, R. (1988). Continuous versus discrete information processing: Modeling the accumulation of partial information.Psychological Review,95, 238–255.
Ratcliff, R. &Murdock, B. B., Jr. (1976). Retrieval processes in recognition memory.Psychological Review,83, 190–214.
Read, T. R. C., &Cressie, N. A. C. (1988).Goodness-of-fit statistics for discrete multivariate data. New York: Springer-Verlag.
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.
Rohrer, D., &Wixted, J. T. (1994). An analysis of latency and interresponse time in free recall.Memory & Cognition,22, 511–524.
Rudd, M. E. (1996). A neural timing model of visual threshold.Journal of Mathematical Psychology,40, 1–29.
Silverman, B. W. (1986).Density estimation for statistics and data analysis. London: Chapman & Hall.
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.
Smith, P. L., &Vickers, D. (1988). The accumulator model of twochoice discrimination.Journal of Mathematical Psychology,32, 135–168.
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.
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.
Tapia, R. A., &Thompson, J. R. (1978).Nonparametric probability density estimation. Baltimore: Johns Hopkins University Press.
Tolhurst, D. J. (1975). Reaction times in the detection of gratings by human observers: A probabilistic mechanism.Vision Research,15, 1143–1149.
Townsend, J. T. (1990). Truth and consequences of ordinal differences in statistical distributions: Toward a theory of hierarchical inference.Psychological Bulletin,108, 551–567.
Townsend, J. T., &Ashby, F. G. (1983).Stochastic modeling of elementary psychological processes. New York: Cambridge University Press.
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.
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.
Wald, A. (1947).Sequential analysis. New York: Wiley.
Wickens, T. D. (1982).Models for behavior: Stochastic processes in psychology. San Francisco: Freeman.
Wixted, J. T., &Rohrer, D. (1993). Proactive interference and the dynamics of free recall.Journal of Experimental Psychology: Learning, Memory, & Cognition,19, 1024–1039.
Author information
Authors and Affiliations
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
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3758/BF03214357