The ex-Wald distribution as a descriptive model of response times



We propose a new quantitative model of response times (RTs) that combines some advantages of substantive, process-oriented models and descriptive, statistically oriented accounts. The ex-Wald model assumes that RT may be represented as a convolution of an exponential and a Wald-distributed random variable. The model accounts well for the skew, shape, and hazard function of typical RT distributions. The model is based on two broad information-processing concepts: (1) a data-driven processing rate describing the speed of information accumulation, and (2) strategic response criterion setting. These concepts allow for principled expectations about how experimental factors such as stimulus saliency or response probability might influence RT on a distributional level. We present a factorial experiment involving mental digit comparisons to illustrate the application of the model, and to explain how substantive hypotheses about selective factor effects can be tested via likelihood ratio tests.


  1. Ashby, F. G. (1982). Testing the assumptions of exponential, additive reaction time models.Memory & Cognition,10, 125–134.Google Scholar
  2. Ashby, F. G., &Townsend, J. T. (1980). Decomposing the reaction time distribution: Pure insertion and selective influence revisited.Journal of Mathematical Psychology,21, 93–123.CrossRefGoogle Scholar
  3. 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 Psycholog y: General,128, 32–55.CrossRefGoogle Scholar
  4. Blanco, M. J., &Alvarez, A. A. (1994). Psychometric intelligence and visual focussed attention: Relationships in nonsearch tasks.Intelligence,18, 77–106.CrossRefGoogle Scholar
  5. 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
  6. Chhikara, R. S., &Folks, J. L. (1989).The inverse Gaussian distribution. New York: Marcel Dekker.Google Scholar
  7. Cousineau, D., &Larochelle, S. (1997). PASTIS: A program for curve and distribution analyses.Behavior Research Methods, Instruments, & Computers,29, 542–548.Google Scholar
  8. Cox, D. R., &Miller, H. D. (1965).The theory of stochastic processes. London: Chapman & Hall.Google Scholar
  9. Dawson, M. R. W. (1988). Fitting the ex-Gaussian equation to reaction time distributions.Behavior Research Methods, Instruments, & Computers,20, 54–57.Google Scholar
  10. Dehaene, S. (1997).The number sense. Oxford: Oxford University Press.Google Scholar
  11. Derenzo, S. E. (1977). Approximations for hand calculators using small integer coefficients.Mathematics of Computation,31, 214–225.CrossRefGoogle Scholar
  12. Emerson, P. L. (1970). Simple reaction time with Markovian evolution of Gaussian discriminal processes.Psychometrika,35, 99–109.CrossRefGoogle Scholar
  13. Feller, W. (1966).An introduction to probability theory and its applications (Vol. II). New York: Wiley.Google Scholar
  14. Heath, R. A. (1992). A general nonstationary diffusion model for two-choice decision making.Mathematical Social Sciences,23, 283–309.CrossRefGoogle Scholar
  15. Heathcote, A. (1996). RTSYS: A DOS application for the analysis of reaction time data.Behavior Research Methods, Instruments, & Computers,28, 427–445.Google Scholar
  16. Heathcote, A., Popiel, S., &Mewhort, D. J. K. (1991). Analysis of response time distributions: An example using the Stroop task.Psychological Bulletin,109, 340–347.CrossRefGoogle Scholar
  17. Hockley, W. E., &Murdock, B. B., Jr. (1987). A decision model for accuracy and response latency in recognition memory.Psychological Review,94, 341–358.CrossRefGoogle Scholar
  18. Hoel, P. G. (1971).Introduction to mathematical statistics (4th ed.). New York: Wiley.Google Scholar
  19. Hohle, R. H. (1965). Inferred components of reaction times as functions of foreperiod duration.Journal of Experimental Psychology,69, 382–386.CrossRefPubMedGoogle Scholar
  20. Johnson, N. L., Kotz, S., &Balakrishnan, N. (1994). Inverse Gaussian (Wald) distributions. In N. L. Johnson, S. Kotz, & N. Balakrishnan (Eds.),Continuous univariate distributions (2nd ed., Vol. 1, pp. 259–297). New York: Wiley.Google Scholar
  21. Juhel, J. (1993). Should we take the shape of reaction time distributions into account when studying the relationship between RT and psycho-metric intelligence?Personality & Individual Differences,15, 357–360.CrossRefGoogle Scholar
  22. Laming, D. R. J. (1968).Information theory of choice reaction time. New York: Wiley.Google Scholar
  23. Link, S. W. (1992).The wave theory of difference and similarity. Hillsdale, NJ: Erlbaum.Google Scholar
  24. Luce, R. D. (1986).Response times: Their role in inferring elementary mental organization. Oxford: Oxford University Press.Google Scholar
  25. Macmillan, N. A., &Creelman, C. D. (1991).Detection theory: A user’s guide. Cambridge: Cambridge University Press.Google Scholar
  26. Maddox, W. T., Ashby, F. G., &Gottlob, L. R. (1998). Response time distributions in multidimensional perceptual categorization.Perception & Psychophysics,60, 620–637.Google Scholar
  27. McGill, W. J., &Gibbon, J. (1965). The general gamma distribution and reaction times.Journal of Mathematical Psychology,2, 1–18.CrossRefGoogle Scholar
  28. Plourde, C. E., &Besner, D. (1997). On the locus of the word frequency effect in word recognition.Canadian Journal of Psychology,51, 181–194.Google Scholar
  29. Ratcliff, R. (1978). A theory of memory retrieval.Psychological Review,85, 59–108.CrossRefGoogle Scholar
  30. Ratcliff, R., &Murdock, B. B. (1976). Retrieval processes in recognition memory.Psychological Review,83, 190–214.CrossRefGoogle Scholar
  31. Ratcliff, R., &Rouder, J. N. (1998). Modeling response times for two-choice decisions.Psychological Science,9, 347–356.CrossRefGoogle Scholar
  32. Rohrer, D., &Wixted, J. T. (1994). An analysis of latency and inter-response time in free recall.Memory & Cognition,22, 511–524.Google Scholar
  33. Schwarz, W. (1990). Stochastic accumulation of information in discrete time: Comparing exact results and Wald approximations.Journal of Mathematical Psychology,34, 229–236.CrossRefGoogle Scholar
  34. Schwarz, W. (1992). The Wiener process between a reflecting and an absorbing barrier.Journal of Applied Probability,29, 597–604.CrossRefGoogle Scholar
  35. Schwarz, W. (1993). A diffusion model of early visual search: Theoretical analysis and experimental results.Psychological Research,55, 200–207.CrossRefPubMedGoogle Scholar
  36. Schwarz, W. (1994). Diffusion, superposition, and the redundant-targets effect.Journal of Mathematical Psychology,38, 504–520.CrossRefGoogle Scholar
  37. Schwarz, W., &Stein, F. (1998). On the temporal dynamics of digit comparison processes.Journal of Experimental Psychology: Learning, Memory, & Cognition,24, 1275–1293.CrossRefGoogle Scholar
  38. Seshadri, V. (1993).The inverse Gaussian distribution. Oxford: Oxford University Press, Clarendon Press.Google Scholar
  39. Smith, D. G., &Mewhort, D. J. K. (1998). The distribution of latencies constrains theories of decision times: A test of the random-walk using numerical comparisons.Australian Journal of Psychology,50, 149–154.CrossRefGoogle Scholar
  40. Smith, P. L. (1995). Psychophysically principled models of visual simple reaction time.Psychological Review,102, 567–593.CrossRefGoogle Scholar
  41. Sternberg, S. (1998). Discovering mental processing stages: The method of additive factors. In D. Scarborough & S. Sternberg (Eds.),An invitation to cognitive psychology: Vol. 4. Methods, models, and conceptual issues (pp. 703–863). Cambridge: Cambridge University Press.Google Scholar
  42. Stone, M. (1960). Models of choice reaction time.Psychometrika,25, 251–260.CrossRefGoogle Scholar
  43. 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.CrossRefGoogle Scholar
  44. Stuart, A., &Ord, J. K. (1987).Kendall’s advanced theory of statistics: Vol. 1. Distribution theory (5th ed.). London: Griffin.Google Scholar
  45. Ueno, T. (1992). Sustained and transient properties of chromatic and luminance systems.Vision Research,32, 1055–1065.CrossRefPubMedGoogle Scholar
  46. Wixted, H. T., &Rohrer, D. (1993). Proactive interference and the dynamics of free recall.Journal of Experimental Psychology: Learning, Memory, & Cognition,19, 1024–1039.CrossRefGoogle Scholar

Copyright information

© Psychonomic Society, Inc. 2001

Authors and Affiliations

  1. 1.NICIUniversity of NijmegenHE NijmegenThe Netherlands

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