A hierarchical approach for fitting curves to response time measurements
 Jeffrey N. Rouder,
 Francis Tuerlinckx,
 Paul Speckman,
 Jun Lu,
 Pablo Gomez
 … show all 5 hide
Abstract
Understanding how response time (RT) changes with manipulations has been critical in distinguishing among theories in cognition. It is well known that aggregating data distorts functional relationships (e.g., Estes, 1956). Less well appreciated is a second pitfall: Minimizing squared errors (i.e., OLS regression) also distorts estimated functional forms with RT data. We discuss three properties of RT that should be modeled for accurate analysis and, on the basis of these three properties, provide a hierarchical Weibull regression model for regressing RT onto covariates. Hierarchical regression model analysis of lexical decision task data reveals that RT decreases as a power function of word frequency with the scale of RT decreasing 11% for every doubling of word frequency. A detailed discussion of the model and analysis techniques are presented as archived materials and may be downloaded from www.psychonomic.org/archive.
 blah RouderPBR2008.zip (976KB)
 Abramowitz, M., Stegun, I. A. (1965) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, New York
 Adelman, J. S., Brown, G. D. A., Quesada, J. F. (2006) Contextual diversity, not word frequency, determines wordnaming and lexical decision times. Psychological Science 17: pp. 814823 CrossRef
 Anderson, J. R. (1982) Acquisition of cognitive skill. Psychological Review 89: pp. 369406 CrossRef
 Andrews, S., Heathcote, A. (2001) Distinguishing common and taskspecific processes in word identification: A matter of some moment?. Journal of Experimental Psychology: Learning, Memory, & Cognition 27: pp. 514544 CrossRef
 Baayen, R. H., Tweedie, F. J., Schreuder, R. (2002) The subjects as a simple random effect fallacy: Subject variability and morphological family effects in the mental lexicon. Brain & Language 81: pp. 5565 CrossRef
 Busemeyer, J. R., Townsend, J. T. (1993) Decision field theory: A dynamiccognitive approach to decision making in an uncertain environment. Psychological Review 100: pp. 432459 CrossRef
 Bush, R. R., Mosteller, F. (1951) A mathematical model for simple learning. Psychological Review 58: pp. 313323 CrossRef
 Clark, H. H. (1973) The languageasfixedeffect fallacy: A critique of language statistics in psychological research. Journal of Verbal Learning & Verbal Behavior 12: pp. 335359 CrossRef
 Cleveland, W. S. (1981) Lowess: A program for smoothing scatterplots by robust locally weighted regression. American Statistician 35: pp. 5454 CrossRef
 Dosher, B. A., Lu, Z.L. (2007) The functional form of performance improvements in perceptual learning: Learning rates and transfer. Psychological Science 18: pp. 531539 CrossRef
 Estes, W. K. (1956) The problem of inference from curves based on grouped data. Psychological Bulletin 53: pp. 134140 CrossRef
 Gelfand, A. E., Smith, A. F. M. (1990) Sampling based approaches to calculating marginal densities. Journal of the American Statistical Association 85: pp. 398409 CrossRef
 Gomez, P., Perea, M., Ratcliff, R. (2007) Diffusion model of the go/nogo task. Journal of Experimental Psychology: General 136: pp. 389413 CrossRef
 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: pp. 392406 CrossRef
 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: pp. 185207 CrossRef
 Jeffreys, H. (1961) Theory of probability. Oxford University Press, Clarendon Press, Oxford
 Kučera, H., Francis, W. N. (1967) Computational analysis of present Day American English. Brown University Press, Providence, RI
 Logan, G. D. (1988) Toward an instance theory of automatization. Psychological Review 95: pp. 492527 CrossRef
 Logan, G. D. (1992) Shapes of reactiontime distributions and shapes of learning curves: A test of the instance theory of automaticity. Journal of Experimental Psychology: Learning, Memory, & Cognition 18: pp. 883914 CrossRef
 McCullagh, P., Nelder, J. A. (1989) Generalized linear models. Chapman & Hall, London
 McCusker, L. M. (1977). Some determinants of word recognition: Frequency. Paper presented at the 24th Annual Convention of the Southwestern Psychological Association, Fort Worth, TX.
 Morton, J. (1969) The interaction of information in word recognition. Psychological Review 76: pp. 165178 CrossRef
 Murray, W. S., Forster, K. I. (2004) Serial mechanisms in lexical access: The rank hypothesis. Psychological Review 111: pp. 721756 CrossRef
 Press, W. H., Teukolsky, S. A., Vetterling, W. T., Flannery, B. P. (1992) Numerical recipes in C: The art of scientific computing. Cambridge University Press, Cambridge
 Ratcliff, R. (1978) A theory of memory retrieval. Psychological Review 85: pp. 59108 CrossRef
 Ratcliff, R., Rouder, J. N. (1998) Modeling response times for decisions between two choices. Psychological Science 9: pp. 347356 CrossRef
 Raudenbush, S. W., Bryk, A. S. (2002) Hierarchical linear models: Applications and data analysis methods. Sage, Thousand Oaks, CA
 Rickard, T. C. (2004) Strategy execution in cognitive skill learning: An itemlevel test of candidate models. Journal of Experimental Psychology: Learning, Memory, & Cognition 30: pp. 6582 CrossRef
 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: pp. 359378 CrossRef
 Rouder, J. N. (2005) Are unshifted distributional models appropriate for response time?. Psychometrika 70: pp. 377381 CrossRef
 Rouder, J. N., Lu, J. (2005) An introduction to Bayesian hierarchical models with an application in the theory of signal detection. Psychonomic Bulletin & Review 12: pp. 573604 CrossRef
 Rouder, J. N., Lu, J., Speckman, P. L., Sun, D., Jiang, Y. (2005) A hierarchical model for estimating response time distributions. Psychonomic Bulletin & Review 12: pp. 195223 CrossRef
 Rouder, J. N., Sun, D., Speckman, P. L., Lu, J., Zhou, D. (2003) A hierarchical Bayesian statistical framework for response time distributions. Psychometrika 68: pp. 587604 CrossRef
 Shepard, R. N., Metzler, J. (1971) Mental rotation of three dimensional objects. Science 171: pp. 701703 CrossRef
 Skrondal, A., RabeHesketh, S. (2004) Generalized latent variable modeling: Multilevel, longitudinal, and structural equation models. CRC Press, Boca Raton, FL CrossRef
 Snijders, T. A. B., Bosker, R. J. (1999) Multilevel analysis: An introduction to basic and advanced multilevel modeling. Sage, London
 Spiegelhalter, D. J., Best, N. G., Carlin, B. P., Linde, A. (2002) Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society: Series B 64: pp. 583639 CrossRef
 Breukelen, G. J. P. (2005) Psychometric modeling of response speed and accuracy with mixed and conditional regression. Psychometrika 70: pp. 359391 CrossRef
 Noortgate, W., Boeck, P., Meulders, M. (2003) Crossclassification multilevel logistic models in psychometrics. Journal of Educational & Behavioral Statistics 28: pp. 369386 CrossRef
 Wagenmakers, E.J., Brown, S. (2007) On the linear relation between the mean and the standard deviation of a response time distribution. Psychological Review 114: pp. 830841 CrossRef
 Title
 A hierarchical approach for fitting curves to response time measurements
 Journal

Psychonomic Bulletin & Review
Volume 15, Issue 6 , pp 12011208
 Cover Date
 20081201
 DOI
 10.3758/PBR.15.6.1201
 Print ISSN
 10699384
 Online ISSN
 15315320
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 Authors

 Jeffrey N. Rouder ^{(1)}
 Francis Tuerlinckx ^{(2)}
 Paul Speckman ^{(1)}
 Jun Lu ^{(3)}
 Pablo Gomez ^{(4)}
 Author Affiliations

 1. Department of Psychological Sciences, University of Missouri, 210 McAlester Hall, 65211, Columbia, MO
 2. University of Leuven, Leuven, Belgium
 3. American University, Washington, DC
 4. DePaul University, Chicago, Illinois