Abstract
We describe and test quantile maximum probability estimator (QMPE), an open-source ANSI Fortran 90 program for response time distribution estimation.1 QMPE enables users to estimate parameters for the ex-Gaussian and Gumbel (1958) distributions, along with three “shifted” distributions (i.e., distributions with a parameter-dependent lower bound): the Lognormal, Wald, and Weibull distributions. Estimation can be performed using either the standard continuous maximum likelihood (CML) method or quantile maximum probability (QMP; Heathcote & Brown, in press). We review the properties of each distribution and the theoretical evidence showing that CML estimates fail for some cases with shifted distributions, whereas QMP estimates do not. In cases in which CML does not fail, a Monte Carlo investigation showed that QMP estimates were usually as good, and in some cases better, than CML estimates. However, the Monte Carlo study also uncovered problems that can occur with both CML and QMP estimates, particularly when samples are small and skew is low, highlighting the difficulties of estimating distributions with parameter-dependent lower bounds.
Article PDF
Similar content being viewed by others
References
Brown, S., &Heathcote, A. (2003). QMLE: Fast, robust, and efficient estimation of distribution functions based on quantiles.Behavior Research Methods, Instruments, & Computers,35,485–492.
Breukelen, G. J. P. (1995). Parallel information processing models compatible with lognormally distributed response times.Journal of Mathematical Psychology,39,396–399.
Cheng, R. C. H., &Amin, N. A. K. (1981). Maximum likelihood estimation of parameters in the inverse Gaussian distribution with unknown origin.Technometrics,23, 257–263.
Cheng, R. C. H., &Amin, N. A. K. (1983). Estimating parameters in continuous univariate distributions with a shifted origin.Journal of the Royal Statistical Society: Series B,45, 394–403.
Cheng, R. C. H., &Iles, T. C. (1987). Corrected maximum likelihood in nonregular problems.Journal of the Royal Statistical Society: Series B,49, 95–101.
Cheng, R. C. H., &Iles, T. C. (1990). Embedded models in three-parameter distributions and their estimation.Journal of the Royal Statistical Society: Series B,52, 135–149.
Cheng, R. C. H., &Traylor, L. (1995). Non-regular maximum likelihood problems.Journal of the Royal Statistical Society: Series B,57, 3–44.
Cousineau, D., Brown, S., & Heathcote, A. (2004).Fitting distributions using maximum likelihood: Methods and packages. Manuscript submitted for publication.
Cousineau, D., Goodman, V., &Shiffrin, R. M. (2002). Extending statistics of extremes to distributions varying on position and scale, and implication for race models.Journal of Mathematical Psychology,46,431–454.
Dolan, C. V., van der Maas, H. L. J., &Molenaar, P. C. M. (2002). A framework for ML estimation of parameters (mixtures of) common reaction time distributions given optional truncation or censoring.Behavior Research Methods, Instruments, & Computers,34,304–323.
Ekström, M. (2001). Consistency of generalized maximum spacing estimates.Scandinavian Journal of Statistics,28,343–354.
Giesbrecht, F., &Kempthorne, O. (1976). Maximum likelihood estimation in the three-parameter lognormal distribution.Journal of the Royal Statistical Society: Series B,38, 257–264.
Gumbel, E. J. (1958).The statistics of extremes. New York: Columbia University Press.
Heathcote, A. (1996). RTSYS: A DOS application for the analysis of reaction time data.Behavior Research Methods, Instruments, & Computers,28,427–445.
Heathcote, A. (in press). Fitting the Wald and ex-Wald distributions to response time data.Behavior Research Methods, Instruments, & Computers.
Heathcote, A., &Brown, S. (in press). Reply to Speckman and Rouder: A theoretical basis for QML.Psychonomic Bulletin & Review.
Heathcote, A., Brown, S., &Mewhort, D. J. K. (2002). Quantile maximum likelihood estimation of response time distributions.Psychonomic Bulletin & Review,9,394–401.
Hyndman, R. J., &Fan, Y. (1996). Sample quantiles in statistical packages.American Statistician,50, 361–365.
McClelland, J. L. (1979). On the time relations of mental processes: An examination of systems of processes in cascade.Psychological Review,86,287–330.
McGill, W. J. (1963). Stochastic latency mechanisms. In R. D. Luce, R. R. Bush, & E. Galanter (Eds.),Handbook of mathematical psychology (pp. 193–199). Oxford: Wiley.
Ranneby, B. (1984). The maximum spacing method: An estimation method related to the maximum likelihood method.Scandinavian Journal of Statistics,11, 93–112.
Ratcliff, R. (1978). A theory of memory retrieval.Psychological Review,85,59–108.
Ratcliff, R., &Murdock, B. B. (1976). Retrieval processes in recognition memory.Psychological Review,83,190–214.
Rouder, J. N., Lu, J., Speckman, P., Sun, D., & Jiang, Y. (in press). A hierarchical model for estimating response time distributions.Psychonomic Bulletin & Review.
Speckman, P. L., & Rouder, J. N. (in press). A comment on Heathcote, Brown, and Mewhort’s QMLE estimation method for response time distributions.Psychonomic Bulletin & Review.
Titterington, D. M. (1985). Comment on “Estimating parameters in continuous univariate distributions.”Journal of the Royal Statistical Society: Series B,47, 115–116.
Ulrich, R., &Miller, J. (1994). Effects of outlier exclusion on reaction time analysis.Journal of Experimental Psychology: General,123,34–80.
Wald, A. (1947).Sequential analysis. New York: Wiley.
Weibull, W. (1951). A statistical distribution function of wide applicability.Journal of Applied Mechanics,18,292–297.
West, B. J., &Shlesinger, M. (1990). The noise in natural phenomena.American Scientist,78, 40–45.
Woodworth, R. S., &Schlosberg, H. (1954).Experimental psychology. New York: Holt.
Van Zandt, T. (2000). How to fit a response time distribution.Psychonomic Bulletin & Review,7,424–465.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by Australian Research Council grants to S. Andrews and A.H. and to A.H., B. Hayes, and D. J. K. Mewhort.
Rights and permissions
About this article
Cite this article
Heathcote, A., Brown, S. & Cousineau, D. QMPE: Estimating Lognormal, Wald, and Weibull RT distributions with a parameter-dependent lower bound. Behavior Research Methods, Instruments, & Computers 36, 277–290 (2004). https://doi.org/10.3758/BF03195574
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.3758/BF03195574