Abstract
We present a statistical model for inference with response time (RT) distributions. The model has the following features. First, it provides a means of estimating the shape, scale, and location (shift) of RT distributions. Second, it is hierarchical and models between-subjects and within-subjects variability simultaneously. Third, inference with the model is Bayesian and provides a principled and efficient means of pooling information across disparate data from different individuals. Because the model efficiently pools information across individuals, it is particularly well suited for those common cases in which the researcher collects a limited number of observations from several participants. Monte Carlo simulations reveal that the hierarchical Bayesian model provides more accurate estimates than several popular competitors do. We illustrate the model by providing an analysis of the symbolic distance effect in which participants can more quickly ascertain the relationship between nonadjacent digits than that between adjacent digits.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Andrews, S., &Heathcote, A. (2001). Distinguishing common and task-specific processes in word identification: A matter of some moment.Journal of Experimental Psychology: Learning, Memory, & Cognition,27, 514–544.
Ashby, F. G., Maddox, W. T., &Lee, W. W. (1994). On the dangers of averaging across subjects when using multidimensional scaling or the similarity-choice model.Psychological Science,5, 144–151.
Ashby, F. G., Tien, J.-Y., &Balakrishnan, J. D. (1993). Response time distributions in memory scanning.Journal of Mathematical Psychology,37, 526–555.
Ashby, F. G., &Townsend, J. T. (1980). Decomposing the reaction time distribution: Pure insertion and selective influences revisited.Journal of Mathematical Psychology,21, 93–123.
Atkinson, R. C., Holmgren, J. E., &Juola, J. F. (1969). Processing time as influenced by the number of elements in a visual display.Perception & Psychophysics,6, 321–326.
Balota, D. A., &Chumbley, J. I. (1984). Are lexical decisions a good measure of lexical access? The role of word frequency in the neglected decision stage.Journal of Experimental Psychology: Human Perception & Performance,10, 340–357.
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.
Batchelder, W. H., &Riefer, D. M. (1999). Theoretical and empirical review of multinomial process tree modeling.Psychonomic Bulletin & Review,6, 57–86.
Brown, S., &Heathcote, A. (2003). Averaging learning curves across and within participants.Behavior Research Methods, Instruments, & Computers,35, 11–21.
Busemeyer, J. R., &Townsend, J. T. (1993). Decision field theory: A dynamic-cognitive approach to decision making in an uncertain environment.Psychological Review,100, 432–459.
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.
Clark, H. H. (1973). The language-as-fixed-effect fallacy: A critique of language statistics in psychological research.Journal of Verbal Learning & Verbal Behavior,12, 335–359.
Colonius, H. (1995). The instance theory of automaticity: Why the Weibull?Psychological Review,102, 744–750.
Cousineau, D., Goodman, V. W., &Shiffrin, R. M. (2002). Extending statistics of extremes to distributions varying in position and scale and the implications for race models.Journal of Mathematical Psychology,46, 431–454.
Curran, T. C., &Hintzman, D. G. (1995). Violations of the independence assumption in process dissociation.Journal of Experimental Psychology: Learning, Memory, & Cognition,21, 531–547.
Dey, D., Ghosh, S., &Mallick, B. (2000).Generalized linear models: A Bayesian perspective. New York: Dekker.
Dolan, C. V., van der Maas, H. L. J., &Molenaar, P. C. M. (2002). A framework for ML estimation of parameters of (mixtures of) common reaction time distributions given optional truncation or censoring.Behavior Research Methods, Instruments, & Computers,34, 304–323.
Dzhafarov, E. N. (1992). The structure of simple reaction time to stepfunction signals.Journal of Mathematical Psychology,36, 235–268.
Edwards, W. (1965). Optimal strategies for seeking information: Models for statistics, choice reaction times, and human information processing.Journal of Mathematical Psychology,2, 312–329.
Edwards, W., Lindman, H., &Savage, L. J. (1963). Bayesian statistical inference for psychological research.Psychological Review,70, 193–242.
Estes, W. K. (1956). The problem of inference from curves based on grouped data.Psychological Bulletin,53, 134–140.
Fox, J.-P., &Glas, C. A. W. (2001). Bayesian estimation of a multilevel IRT model using Gibbs sampling.Psychometrika,66, 271–288.
Gelfand, A., &Smith, A. F. M. (1990). Sampling based approaches to calculating marginal densities.Journal of the American Statistical Association,85, 398–409.
Gelman, A., Carlin, J. B., Stern, H. S., &Rubin, D. B. (1995).Bayesian data analysis. London: Chapman & Hall.
Green, D. M., Smith, A. F., &von Gierke, S. M. (1983). Choice reaction time with a random foreperiod.Perception & Psychophysics,34, 195–208.
Haider H., &Frensch, P. A. (2002). Why aggregated learning follows the power law of practice when individual learning does not: Comment on Rickard (1997, 1999), Delaney et al. (1998), and Palmeri (1999).Journal of Experimental Psychology: Learning, Memory, & Cognition,28, 392–406.
Hasher, L., &Zacks, R. T. (1979). Automatic and effortful processes in memory.Journal of Experimental Psychology: General,108, 356–388.
Heathcote, A. (1996). RTSYS: A DOS application for the analysis of reaction time data.Behavior Research Methods, Instruments, & Computers,28, 427–445.
Heathcote, A., Brown, S., &Mewhort, D. J. K. (2000). The power law repealed: The case for an exponential law of practice.Psychonomic Bulletin & Review,7, 185–207.
Heathcote, A., Brown, S., &Mewhort, D. J. K. (2002). Quantile maximum likelihood estimation of response time distributions.Psychonomic Bulletin & Review,9, 394–401.
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 reaction time distributions in the study of cognitive processes.Journal of Experimental Psychology: Learning, Memory, & Cognition,10, 598–615.
Hogg, R. V., &Craig, A. T. (1978).Introduction to mathematical statistics. New York: Macmillan.
Hohle, R. H. (1965). Inferred components of reaction time as a function of foreperiod duration.Journal of Experimental Psychology,69, 382–386.
Hsu, Y. F. (1999).Two studies on simple reaction times: I. On the psychophysics of the generalized Pieron’s law. II. On estimating minimum detection times using the time estimation paradigm. Unpublished doctoral dissertation, University of California, Irvine.
Jacoby, L. L. (1991). A process dissociation framework: Separating automatic from intentional uses of memory.Journal of Memory & Language,30, 513–541.
Jiang, Y. (2002).Statistical approaches to forming group RT distributions. Unpublished master’s thesis, University of Missouri, Columbia.
Jiang, Y., Rouder, J. N., &Speckman, P. L. (2004). A note on the sampling properties of the Vincentizing (quantile averaging) procedure.Journal of Mathematical Psychology,48, 186–195.
Johnson, N. L., Kotz, S., &Balakrishnan, N. (1994).Continuous univariate distributions (2nd ed., Vol. 2). New York: Wiley.
Kass, R., &Raftery, A. (1995). Bayes factors.Journal of the American Statistical Association,90, 773–795.
Kosslyn, S. M. (1975). Information representation in visual images.Cognitive Psychology,7, 341–370.
Kreft, I. G. G., &de Leeuw, J. (1998).Introducing multilevel modeling. London: Sage.
Lee, P. M. (1997).Bayesian statistics: An introduction. New York: Wiley.
Lehmann, E. L. (1991).Theory of point estimation. Pacific Grove, CA: Wadsworth & Brooks.
Link, S. W. (1975). The relative judgement theory of two choice response time.Journal of Mathematical Psychology,12, 114–135.
Link, S. W. (1990). Modeling imageless thought: The relative judgment theory of numerical comparisons.Journal of Mathematical Psychology,34, 2–41.
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.
Lu, J. (2004).Bayesian hierarchical models and applications in psychological research. Unpublished doctoral dissertation, University of Missouri, Columbia.
Lu, J., Sun, D., Speckman, P. L., & Rouder, J. N. (2005).Hierarchical models for practice-effects in response time. Manuscript submitted for publication.
Luce, R. D. (1959).Individual choice behavior. New York: Wiley.
Luce, R. D. (1986).Response times. New York: Oxford University Press.
Masson, M. E. J., &Loftus, G. R. (2003). Using confidence intervals for graphically based data interpretation.Canadian Journal of Experimental Psychology,57, 203–220.
McClelland, J. L., &Rumelhart, D. E. (1981). An interactive activation model of context effects in letter perception: I. An account of basic findings.Psychological Review,88, 375–407.
Meng, X., &Wong, W. (1996). Simulating ratios of normalizing constants via a simple identity: A theoretical exploration.Statistica Sinica,6, 831–860.
Moyer, R. S. (1973). Comparing objects in memory: Evidence suggesting an internal psychophysics.Perception & Psychophysics,13, 1080–1084.
Moyer, R. S., &Landauer, T. K. (1967). Time required for judgements of numerical inequality.Nature,215, 1519–1520.
Myung, I. J. (2003). Tutorial on maximum likelihood estimation.Journal of Mathematical Psychology,47, 90–100.
Myung, I. J., Kim, C., &Pitt, M. A. (2002). Toward an explanation of the power law artifact: Insights from response surface analysis.Memory & Cognition,28, 832–840.
Myung, I. J., &Pitt, M. A. (1997). Applying Occam’s razor in modeling cognition: A Bayesian approach.Psychonomic Bulletin & Review,4, 79–95.
Nelder, J. A., &Mead, R. (1965). A simplex method for function minimization.Computer Journal,7, 308–313.
Peruggia, M., Van Zandt, T., &Chen, M. (2002). Was it a car or a cat I saw? An analysis of response times for word recognition. In C. Gatsonis, R. E. Kass, A. Carriquiry, A. Gelman, D. Higdon, D. K. Pauler, & I. Verdinelli (Eds.),Case studies in Bayesian statistics (Vol. 6, pp. 319–334). New York: Springer-Verlag.
Phillips, L. D., &Edwards, W. (1966). Conservatism in a simple probability inference task.Journal of Experimental Psychology,72, 346–354.
Pitt, M. A., Myung, I. J., &Zhang, S. (2002). Toward a method of selecting among computational models of cognition.Psychological Review,109, 472–491.
Poltrock, S. E. (1989). A random walk model of digit comparison.Journal of Mathematical Psychology,33, 131–162.
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., &Rouder, J. N. (1998). Modeling response times for decisions between two choices.Psychological Science,9, 347–356.
Ratcliff, R., &Rouder, J. N. (2000). A diffusion model analysis of letter masking.Journal of Experimental Psychology: Human Perception & Performance,26, 127–140.
Rickard, T. C. (1997). Bending the power law: A CMPL theory of strategy shifts and the automatization of cognitive skills.Journal of Experimental Psychology: General,126, 288–311.
Riefer, D. M., &Batchelder, W. H. (1988). Multinomial modeling and the measure of cognitive processes.Psychological Review,95, 318–339.
Rouder, J. N. (2000). Assessing the roles of change discrimination and luminance integration: Evidence for a hybrid race model of perceptual decision making in luminance discrimination.Journal of Experimental Psychology: Human Perception & Performance,26, 359–378.
Rouder, J. N. (2001). Testing evidence accrual models by manipulating stimulus onset.Journal of Mathematical Psychology,45, 334–354.
Rouder, J. N. (2004). Modeling the effects of choice-set size on the processing of letters and words.Psychological Review,111, 80–93.
Rouder, J. N., &Batchelder, W. H. (1998). Multinomial models for measuring storage and retrieval processes in paired-associate learning. In C. Dowling, F. Roberts, & P. Theuns (Eds.),Recent progress in mathematical psychology (pp. 195–225). Hillsdale, NJ: Erlbaum.
Rouder, J. N., Lu, J., Sun, D., Speckman, P. L., & Morey, R. D. (2003, November).A statistical model of skill acquisition. Poster presented at the 20th Annual Meeting of the Psychonomic Society, Vancouver.
Rouder, J. N., Ratcliff, R., &McKoon, G. M. (2000). A neural network model of priming in object recognition.Psychological Science,11, 13–19.
Rouder, J. N., &Speckman, P. L. (2004). An evaluation of the Vincentizing method of forming group-level response time distributions.Psychonomic Bulletin & Review,11, 419–427.
Rouder, J. N., Sun, D., Speckman, P. L., Lu, J., &Zhou, D. (2003). A hierarchical Bayesian statistical framework for response time distributions.Psychometrika,68, 589–606.
Schneider, W., &Shiffrin, R. M. (1977). Controlled and automatic human information processing: I. Detection, search, and attention.Psychological Review,84, 1–66.
Schwarz, W. (2001). The ex-Wald distribution as a descriptive model of response times.Behavior Research Methods, Instruments, & Computers,33, 457–469.
Sheu, C.-F., &O’Curry, S. L. (1998). Simulation-based Bayesian inference using BUGS.Behavior Research Methods, Instruments, & Computers,30, 232–237.
Smith, D. G., &Mewhort, D. J. K. (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.
Speckman, P. L., &Rouder, J. N. (2004). A comment on Heathcote, Brown, and Mewhort’s QMLE method for response time distributions.Psychonomic Bulletin & Review,11, 574–576.
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.
Sternberg, S. (1966). High-speed scanning in human memory. Science,153, 652–654.
Tanner, M. A. (1993).Tools for statistical inference: Methods for the exploration of posterior distributions and likelihood functions. Berlin: Springer-Verlag.
Theeuwes, J. (1992). Perceptual selectivity for color and form.Perception & Psychophysics,51, 599–606.
Theeuwes, J. (1994). Stimulus-driven capture and attentional set: Selective search for color and visual abrupt onsets.Journal of Experimental Psychology: Human Perception & Performance,20, 799–806.
Thomas, E. A. C., &Ross, B. (1980). On appropriate procedures for combining probability distributions within the same family.Journal of Mathematical Psychology,21, 136–152.
Townsend, J. T., &Ashby, F. G. (1983).The stochastic modeling of elementary psychological processes. Cambridge: Cambridge University Press.
Townsend, J. T., &Nozawa, G. (1995). On the spatio-temporal properties of elementary perception: An investigation on parallel, serial, and coactive theories.Journal of Mathematical Psychology,39, 321–359.
Treisman, A. M., &Gelade, G. (1980). A feature-integration theory of attention.Cognitive Psychology,12, 97–136.
Tversky, A., &Kahneman, D. (1990). Judgements under uncertainty: Heuristics and biases. In P. K. Moser (Ed.),Rationality in action: Contemporary approaches (pp. 171–188). New York: Cambridge University Press.
Ulrich, R., &Miller, J. (1994). Effects of truncation of reaction time analysis.Journal of Experimental Psychology: General,123, 34–80.
Van Zandt, T. (2000). How to fit a response time distribution.Psychonomic Bulletin & Review,7, 424–465.
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.
Vickers, D. (1980). Discrimination. In A. Welford (Ed.),Reaction times (pp. 25–72). London: Academic Press.
Vincent, S. B. (1912). The function of vibrissae in the behavior of the white rat.Behavioral Monographs,1(Whole No. 5).
Wang, X. H., Bradlow, E. T., &Wainer, H. (2002). A general Bayesian model for testlets: Theory and applications.Applied Psychological Measurement,26, 109–128.
Wickens, T. D., &Keppel, G. (1983). On the choice of design and test statistic in the analysis of experiments with sampled materials.Journal of Verbal Learning & Verbal Behavior,22, 296–309.
Yellott, J. I. (1969). Probability learning with noncontingent success.Journal of Mathematical Psychology,6, 541–575.
Yellott, J. I. (1971). Correction for fast guessing and the speed-accuracy tradeoff in choice reaction time.Journal of Mathematical Psychology,8, 159–199.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by National Science Foundation Grant SES-0095919 to J.N.R., D.S., and P.S., by University of Missouri Research Board Grant 00-77 to J.N.R., and by National Science Foundation Grant DMS-9972598 to D.S. and P.S.
Rights and permissions
About this article
Cite this article
Rouder, J.N., Lu, J., Speckman, P. et al. A hierarchical model for estimating response time distributions. Psychonomic Bulletin & Review 12, 195–223 (2005). https://doi.org/10.3758/BF03257252
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3758/BF03257252