A framework for ML estimation of parameters of (mixtures of) common reaction time distributions given optional truncation or censoring

  • Conor V. Dolan
  • Han L. J. van der Maas
  • Peter C. M. Molenaar


We present a framework for distributional reaction time (RT) analysis, based on maximum likelihood (ML) estimation. Given certain information relating to chosen distribution functions, one can estimate the parameters of these distributions and of finite mixtures of these distributions. In addition, left and/or right censoring or truncation may be imposed. Censoring and truncation are useful methods by which to accommodate outlying observations, which are a pervasive problem in RT research. We consider five RT distributions: the Weibull, the ex-Gaussian, the gamma, the log-normal, and the Wald. We employ quasi-Newton optimization to obtain ML estimates. Multicase distributional analyses can be carried out, which enable one to conduct detailed (across or within subjects) comparisons of RT data by means of loglikelihood difference tests. Parameters may be freely estimated, estimated subject to boundary constraints, constrained to be equal (within or over cases), or fixed. To demonstrate the feasibility of ML estimation and to illustrate some of the possibilities offered by the present approach, we present three small simulation studies. In addition, we present three illustrative analyses of real data.


  1. Akaike, H. (1974). A new look at statistical model identification.IEEE Transactions on Automatic Control,AU-19, 719–722.Google Scholar
  2. Azzelini, A. (1996).Statistical inference based on the likelihood. London: Chapman and Hall.Google Scholar
  3. Browne, M. W., &Cudeck, R. (1993). Alternative ways of assessing model fit. In K. A. Bollen & J. Scott Long (Eds.),Testing structural equation models. Newbury Park, CA: Sage.Google Scholar
  4. Burbeck, S. L., &Luce, R. D. (1982). Evidence from auditory simple reaction times for both change and level detectors.Perception & Psychophysics,32, 117–133.CrossRefGoogle Scholar
  5. Colonius, H. (1995). The instance theory of automaticity: Why the Weibull?Psychological Bulletin,102, 744–750.Google Scholar
  6. Cousineau, D., &Larochelle, S. (1997). PASTIS: A program for curve and distribution analysis.Behavior Research Methods, Instruments, & Computers,29, 542–548.CrossRefGoogle Scholar
  7. Dennis, J. E., &Schnabel, R. B. (1983).Numerical methods for unconstra in ed optimization and non-standard constraints. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
  8. Dolan, C. V. (2000).DISFIT: FORTRAN 77 program for distributional RT analysis: Program documentation. Unpublished manuscript, University of Amsterdam, Department of Psychology.Google Scholar
  9. Dolan, C. V., &Molenaar, P. C. M. (1991). A comparison of 4 methods of calculating standard errors in the analysis of covariance structure using normal theory ML estimation.British Journal of Mathematical & Statistical Psychology,47, 359–368.Google Scholar
  10. Evans, M., Hastings, N., &Peacock, B. (2000).Statistical distributions (3rd ed.). New York: Wiley.Google Scholar
  11. Everitt, B. S., &Hand, D. J. (1981).Finite mixture distributions. London: Chapman and Hall.Google Scholar
  12. Gill, P. E., Murray, W., Saunders, M. A., &Wright, M. H. (1986).User’s guide for NPSOL (version 5.0-2). (Tech. rep. SOL 86-2). Stanford, CA: Stanford University, Department of Operations Research.Google Scholar
  13. Gill, P. E., Murray, W., &Wright, M. H. (1981).Practical optimization. London: Academic Press.Google Scholar
  14. Greene, W. H. (1993).Econometric analysis (2nd ed.). New York: Macmillan.Google Scholar
  15. Greenwood, P. E., &Nikulin, M. N. (1996).A guide to chi-square testing. New York: Wiley.Google Scholar
  16. Heathcote, A. (1996). RTSYS: A DOS application for the analysis of reaction time data.Behavior Research Methods, Instruments, & Computers,28, 427–445.CrossRefGoogle Scholar
  17. Heathcote, A., Popiel, S. J., &Mewhort, D. J. K. (1991). Analysis of response time distributions: An example using the stroop task.Psychological Bulletin,109, 340–347.CrossRefGoogle Scholar
  18. Heck, A. (1997).Introduction to Maple (2nd ed.). New York: Springer-Verlag.Google Scholar
  19. 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.CrossRefGoogle Scholar
  20. Hohle, R. H. (1965). Inferred components of reaction times as functions of foreperiod duration.Journal of Experimental Psychology,69, 382–386.CrossRefPubMedGoogle Scholar
  21. Johnson, N. L., Kotz, S., &Balakrishnan, N. (1994).Continuous univariate distributions (Vol. 1, 2nd ed.). New York: Wiley.Google Scholar
  22. Jöreskog, K. G. (1993). Testing structural equation models. In K. A. Bollen & J. S. Long (Eds.),Testing structural equation models (pp. 294–316). Newbury Park, CA: Sage.Google Scholar
  23. Kendall, M. G., &Stuart, A. (1968).The advanced theory of statistics (Vol. 2, 3rd ed.). London: Griffin.Google Scholar
  24. 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.CrossRefGoogle Scholar
  25. Logan, G. D., &Cowan, W. B. (1984). On the ability to inhibit thought and action: A theory of an act of control.Psychological Review,91, 295–327.CrossRefGoogle Scholar
  26. Luce, R. D. (1986).Response times: Their role in inferring elementary mental organization. New York: Oxford University Press.Google Scholar
  27. Miller, I., &Miller, M. (1999).John E. Freund’s mathematical statistics (6th ed.). London: Prentice-Hall.Google Scholar
  28. Moore, D. S. (1986). Tests of chi-squared type. In R. B. D’Agostino & M. A. Stephens (Eds.),Goodness of fit techniques (pp. 63–95). New York: Dekker.Google Scholar
  29. Neale, M. C., &Miller, M. B. (1997). The use of likelihood-based confidence intervals in genetic models.Behavior Gen etics,27, 113–120.CrossRefGoogle Scholar
  30. Ratcliff, R. (1979). Group reaction time distributions and an analysis of distribution statistics.Psychological Bulletin,86, 190–214.CrossRefGoogle Scholar
  31. Ratcliff, R. (1993). Methods for dealing with reaction time outliers.Psychological Bulletin,114, 510–532.CrossRefPubMedGoogle Scholar
  32. Ratcliff, R., &Murdock, B. B., Jr. (1976). Retrieval processes in recognition memory.Psychological Review,83, 190–214.CrossRefGoogle Scholar
  33. Ratcliff, R., Van Zandt, T., &,McKoon, G. (1999). Connectionist and diffusion models of reaction times.Psychological Review,106, 261–300.CrossRefPubMedGoogle Scholar
  34. Schnipke, D. L., &Scrams, D. J. (1997). Modeling item response times with a two-state mixture model: A new method of measuring speededness.Journal of Educational Measurement,34, 213–232.CrossRefGoogle Scholar
  35. Schwarz, G. (1978). Estimating the dimension of a model.Annals of Statistics,6, 461–464.CrossRefGoogle Scholar
  36. Ulrich, R., &Miller, J. (1993). Information processing models generating lognormally distributed reaction times.Journal of Mathematical Psychology,37, 513–525.CrossRefGoogle Scholar
  37. Ulrich, R., &Miller, J. (1994). Effects of truncation of reaction time analysis.Journal of Experimental Psychology: General,123, 34–80.CrossRefGoogle Scholar
  38. Van Zandt, T. (2000). How to fit a response time distribution.Psychonomic Bulletin & Review,7, 424–465.CrossRefGoogle Scholar
  39. 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.CrossRefGoogle Scholar
  40. 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.CrossRefGoogle Scholar
  41. Venzon, D. J., &Moolgavkar, S. H. (1988). A method for computing profile-likelihood-based confidence intervals.Applied Statistics,37, 87–94.CrossRefGoogle Scholar
  42. Visser, I., Raijmakers, M. E. J., &Molenaar, P. C. M. (2000). Confidence intervals for hidden Markov model parameters.British Journal of Mathematical and Statistical Psychology,53, 317–327.CrossRefPubMedGoogle Scholar
  43. Yantis, S., Meyer, D. E., &Smith, J. E. K. (1991). Analyses of multinomial mixture distributions: New tests for stochastic models of cognition and action.Psychological Bulletin,110, 350–374.CrossRefPubMedGoogle Scholar

Copyright information

© Psychonomic Society, Inc. 2002

Authors and Affiliations

  • Conor V. Dolan
    • 1
  • Han L. J. van der Maas
    • 1
  • Peter C. M. Molenaar
    • 1
  1. 1.Developmental Psychology, Department of PsychologyUniversity of AmsterdamAmsterdamThe Netherlands

Personalised recommendations