Hostname: page-component-6766d58669-zlvph Total loading time: 0 Render date: 2026-05-22T06:57:00.698Z Has data issue: false hasContentIssue false
  • English
  • Français

A Hierarchical Bayesian Statistical Framework for Response Time Distributions

Published online by Cambridge University Press:  01 January 2025

Jeffrey N. Rouder ,
Dongchu Sun ,
Paul L. Speckman ,
Jun Lu  and
Duo Zhou
Jeffrey N. Rouder*
Affiliation:
University of Missouri-Columbia
Dongchu Sun
Affiliation:
University of Missouri-Columbia
Paul L. Speckman
Affiliation:
University of Missouri-Columbia
Jun Lu
Affiliation:
University of Missouri-Columbia
Duo Zhou
Affiliation:
University of Missouri-Columbia
*
Requests for reprints should be sent to Jeffrey N. Rouder, Department of Psychological Sciences, 210 McAlester Hall, University of Missouri, Columbia, MO 65211. E-Mail: jeff@missouri.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper provides a statistical framework for estimating higher-order characteristics of the response time distribution, such as the scale (variability) and shape. Consideration of these higher order characteristics often provides for more rigorous theory development in cognitive and perceptual psychology (e.g., Luce, 1986). RT distribution for a single participant depends on certain participant characteristics, which in turn can be thought of as arising from a distribution of latent variables. The present work focuses on the three-parameter Weibull distribution, with parameters for shape, scale, and shift (initial value). Bayesian estimation in a hierarchical framework is conceptually straightforward. Parameter estimates, both for participant quantities and population parameters, are obtained through Markov Chain Monte Carlo methods. The methods are illustrated with an application to response time data in an absolute identification task. The behavior of the Bayes estimates are compared to maximum likelihood (ML) estimates through Monte Carlo simulations. For small sample size, there is an occasional tendency for the ML estimates to be unreasonably extreme. In contrast, by borrowing strength across participants, Bayes estimation “shrinks” extreme estimates. The results are that the Bayes estimators are more accurate than the corresponding ML estimators.

Keywords

Bayesian analysishierarchical modelsresponse timeMCMCWeibull distribution

Information

Type
Theory And Methods
Information
Psychometrika , Volume 68 , Issue 4 , December 2003 , pp. 589 - 606
Copyright
Copyright © 2003 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

We are grateful to Michael Stadler who allowed us use of his data. This research is supported by (a) National Science Foundation Grant SES-0095919 to J. Rouder, D. Sun, and P. Speckman, (b) University of Missouri Research Board Grant 00-77 to J. Rouder, (c) National Science Foundation grant DMS-9972598 to Sun and Speckman, and (d) a grant from the Missouri Department of Conservation to D. Sun.

References

Ahrens, J., Dieter, U. (1974). Computer methods for sampling from gamma, beta, poisson and binomial distributions. Computing, 12, 223–246.CrossRefGoogle Scholar
Ahrens, J., Dieter, U. (1982). Generating gamma variates by a modified rejection technique. Communications of the Association for Computing Machinery (ACM), 25, 47–54.10.1145/358315.358390CrossRefGoogle Scholar
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.Google ScholarPubMed
Ashby, F. G., Tien, J.-Y., Balikrishnan, J.D. (1993). Response time distributions in memory scanning. Journal of Mathematical Psychology, 37, 526–555.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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 and Performance, 10, 340–357.Google ScholarPubMed
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, 32–55.10.1037/0096-3445.128.1.32CrossRefGoogle ScholarPubMed
Berger, J.O., Sun, D. (1993). Bayesian analysis for the poly-Weibull distribution. Journal of the American Statistical Association, 88, 1412–1418.CrossRefGoogle Scholar
Bowman, K.O., Shenton, L.R. (1988). Properties of estimators for the gamma distribution. New York, NY: Marcel Dekker.Google Scholar
Busemeyer, J.R., Townsend, J.T. (1993). Decision field theory: A dynamic-cognitive approach to decision making in an uncertain environment. Psychological Review, 100, 432–459.10.1037/0033-295X.100.3.432CrossRefGoogle Scholar
Cave, C., Squire, L. (1992). Intact and long-lasting repetition priming in amnesia. Journal of Experimental Psychology: Learning, Memory, & Cognition, 18, 509–520.Google ScholarPubMed
Dey, D., Ghosh, S., Mallick, B. (2000). Generalized linear models: A Bayesian perspective. New York, NY: Marcel Dekker.CrossRefGoogle Scholar
Dolan, C., van der Maas, H., & Mollenaar, C. M. (in press). A framework for ML estimation of parameters of (mixtures of) common reaction time distributions given optional truncation or censoring. Behavioral Research Methods, Instruments, and Computers.Google Scholar
Dzhafarov, E.N. (1992). The structure of simple reaction time to step-function signals. Journal of Mathematical Psychology, 36, 235–268.10.1016/0022-2496(92)90038-9CrossRefGoogle Scholar
Fox, J.P., Glas, C.A.W. (2001). Bayesian estimation of a multilevel IRT model using Gibbs sampling. Psychometrika, 66, 271–288.10.1007/BF02294839CrossRefGoogle Scholar
Gelfand, A., Smith, A.F.M. (1990). Sampling based approaches to calculating marginal densities. Journal of the American Statistical Association, 85, 398–409.10.1080/01621459.1990.10476213CrossRefGoogle Scholar
Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B. (1995). Bayesian data analysis. London, U.K.: Chapman and Hall.10.1201/9780429258411CrossRefGoogle Scholar
Gelman, A., Rubin, D.B. (1992). Inference from iterative simulation using multiple sequences (with discussion). Statistical Science, 7, 457–511.10.1214/ss/1177011136CrossRefGoogle Scholar
Gilks, W., Wild, P. (1992). Adaptive rejection sampling for Gibbs sampling. Applied Statistics, 41, 337–348.CrossRefGoogle Scholar
Hasher, L., Zacks, R.T. (1979). Automatic and effortful processes in memory. Journal of Experimental Psychology: General, 108, 356–388.10.1037/0096-3445.108.3.356CrossRefGoogle Scholar
Heathcote, A. (1996). RTSYS: A DOS application for the analysis of reaction time data. Behavioral Research Methods, Instruments, and Computers, 28, 427–445.10.3758/BF03200523CrossRefGoogle Scholar
Heathcote, A., Popiel, S. J., Mewhort, D. J. (1991). Analysis of response time distributions: An example using the Stroop task. Psychological Bulletin, 109, 340–347.10.1037/0033-2909.109.2.340CrossRefGoogle Scholar
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.Google Scholar
Hohle, R.H. (1965). Inferred components of reaction time as a function of foreperiod duration. Journal of Experimental Psychology, 69, 382–386.10.1037/h0021740CrossRefGoogle ScholarPubMed
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. Irvine, CA: University of California.Google Scholar
Jacoby, L.L. (1991). A process dissociation framework: Separating automatic fromm intentional uses of memory. Journal of Memory and Language, 30, 513–541.10.1016/0749-596X(91)90025-FCrossRefGoogle Scholar
Jiang, Y., Rouder, J.N., & Speckman, P.L. (in press). A note on the sampling properties of the Vincentizing (quantile averaging) procedure. Journal of Mathematical Psychology.Google Scholar
Johnson, N.L., Kotz, S., Balakrishnan, N. (1994). Continuous univariate distributions, Volume 2 2nd ed., New York, NY: John Wiley & Wiley.Google Scholar
Kreft, I.G.G., de Leeuew, J. (1998). Introducing multilevel modeling. London, U.K.: Sage.10.4135/9781849209366CrossRefGoogle Scholar
Link, S.W. (1975). The relative judgement theory of two choice response time. Journal of Mathematical Psychology, 12, 114–135.10.1016/0022-2496(75)90053-XCrossRefGoogle Scholar
Logan, G.D. (1988). Towards an instance theory of automization. Psychological Review, 95, 492–527.10.1037/0033-295X.95.4.492CrossRefGoogle Scholar
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.Google ScholarPubMed
Luce, R.D. (1986). Response times. New York, NY: Oxford University Press.Google Scholar
Madden, D.J., Gottlob, L.R., Denny, L.L., Turkington, T.G., Provenzale, J.M., Hawk, T.C., Coleman, R.E. (1999). Aging and recognition memory: Changes in regional cerebral blood flow associated with components of reaction time distributions. Journal of Cognitive Neuroscience, 11, 511–520.10.1162/089892999563571CrossRefGoogle ScholarPubMed
Mitchell, D., Brown, A. (1988). Persistent repetition priming in picture naming and its dissociation from recognition memory. Journal of Experimental Psychology: Learning, Memory, & Cognition, 14, 213–222.Google ScholarPubMed
Nelder, J.A., Mead, R. (1965). A simplex method for function minimization. Computer Journal, 7, 308–313.10.1093/comjnl/7.4.308CrossRefGoogle Scholar
Petrusic, W.M., Baranski, J.V., Kennedy, R. (1999). Similarity comparisons with remembered and perceived magnitudes: Memory psychophysics and fundamental measurement. Memory & Cognition, 26, 1041–1055.10.3758/BF03201182CrossRefGoogle Scholar
Plourde, C.E., Besner, D. (1997). On the locus of the word frequency effect in visual word recognition. Canadian Journal of Experimental Psychology, 51, 181–194.10.1037/1196-1961.51.3.181CrossRefGoogle Scholar
Posner, M.I. (1978). Chronometric explorations of the mind. Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, F.P. (1992). Numerical recipes in C: The art of scientific computing 2nd ed., Cambridge, England: Cambridge University Press.Google Scholar
Ratcliff, R. (1978). A theory of memory retrieval. Psychological Review, 85, 59–108.10.1037/0033-295X.85.2.59CrossRefGoogle Scholar
Ratcliff, R. (1979). Group reaction time distributions and an analysis of distribution statistics. Psychological Bulletin, 86, 446–461.10.1037/0033-2909.86.3.446CrossRefGoogle Scholar
Ratcliff, R., Rouder, J.N. (1998). Modeling response times for decisions between two choices. Psychological Science, 9, 347–356.10.1111/1467-9280.00067CrossRefGoogle Scholar
Ratcliff, R., Rouder, J. (2000). A diffusion model analysis of letter masking. Journal of Experimental Psychology: Human Perception and Performance, 26, 127–140.Google Scholar
Rouder, J.N. (1996). Premature sampling in random walks. Journal of Mathematical Psychology, 40, 287–296.CrossRefGoogle Scholar
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 and Performance, 26, 359–378.Google ScholarPubMed
Rouder, J.N. (2001). Testing evidence accrual models by manipulating stimulus onset. Journal of Mathematical Psychology, 45, 334–354.10.1006/jmps.2000.1319CrossRefGoogle ScholarPubMed
Schnieder, W., Shiffrin, R.M. (1977). Controlled and automatic human information processing: I. Detection, search, and attention. Psychological Review, 84, 1–66.10.1037/0033-295X.84.1.1CrossRefGoogle Scholar
Smith, P.L. (1995). Multiple detector models of visual simple reaction time. Psychological Review, 102, 567–593.CrossRefGoogle Scholar
Smith, R.L., Naylor, J.C. (1987). A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution. Applied Statistician, 36, 358–369.10.2307/2347795CrossRefGoogle Scholar
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 and Performance, 22, 461–479.Google ScholarPubMed
Spieler, D.H., Balota, D.A., Faust, M.E. (2000). Levels of selective attention revealed through analyses of response time distributions. Journal of Experimental Psychology: Human Perception and Performance, 26, 506–526.Google ScholarPubMed
Sternberg, S. (1966). High-speed scanning in human memory. Scuebce, 153, 652–654.Google ScholarPubMed
Tanner, M.A. (1993). Tools for statistical inference: Methods for the exploration of posterior distributions and likelihood functions. Berlin, Germany: Springer.10.1007/978-1-4684-0192-9CrossRefGoogle Scholar
Thomas, E.A.C., Ross, B. (1980). On appropriate procedures for combining probability distributions within the same family. Journal of Mathematical Psychology, 21, 136–152.10.1016/0022-2496(80)90003-6CrossRefGoogle Scholar
Townsend, J.T., Ashby, F.G. (1983). Stochastic modeling of elementary psychological processes. Cambridge, U.K.: Cambridge University Press.Google Scholar
Treisman, A.M., Gelade, G. (1980). A feature-integration theory of attention. Cognitive Psychology, 12, 97–136.10.1016/0010-0285(80)90005-5CrossRefGoogle ScholarPubMed
Ulrich, R., Miller, J.O. (1993). Information processing models generating lognormally distributed reaction times. Journal of Mathematical Psychology, 37, 513–525.10.1006/jmps.1993.1032CrossRefGoogle Scholar
Van Zandt, T. (2000). How to fit a response time distribution. Psychonomic Bulletin and Review, 7, 424–465.CrossRefGoogle Scholar
Van Zandt, T., Colonius, H., Proctor, R. W. (2000). A comparison of two response time models applied to perceptual matching. Psychonomic Bulletin and Review, 7, 208–256.10.3758/BF03212980CrossRefGoogle ScholarPubMed
Vincent, S.B. (1912). The function of vibrissae in the behavior of the white rat. Behavioral Monographs, 1, No. 5.Google Scholar
Wang, X.H., Bradlow, E.T., Wainer, H. (2002). A general Bayesian model for testlets: Theory and applications. Applied Psychological Measurement, 26, 109–128.10.1177/0146621602026001007CrossRefGoogle Scholar
Wild, P., Gilks, W.R. (1993). Adaptive rejection sampling from log-concave density functions. Applied Statistics, 42, 701–708.CrossRefGoogle Scholar
Wixted, J.T., Rohrer, D. (1993). Proactive interference and the dynamics of free recall. Journal of Experimental Psychology: Learning, Memory, & Cognition, 19, 1024–1039.Google Scholar