Psychonomic Bulletin & Review

, Volume 16, Issue 5, pp 798–817 | Cite as

Psychological interpretation of the ex-Gaussian and shifted Wald parameters: A diffusion model analysis

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Abstract

A growing number of researchers use descriptive distributions such as the ex-Gaussian and the shifted Wald to summarize response time data for speeded two-choice tasks. Some of these researchers also assume that the parameters of these distributions uniquely correspond to specific cognitive processes. We studied the validity of this cognitive interpretation by relating the parameters of the ex-Gaussian and shifted Wald distributions to those of the Ratcliff diffusion model, a successful model whose parameters have well-established cognitive interpretations. In a simulation study, we fitted the ex-Gaussian and shifted Wald distributions to data generated from the diffusion model by systematically varying its parameters across a wide range of plausible values. In an empirical study, the two descriptive distributions were fitted to published data that featured manipulations of task difficulty, response caution, and a priori bias. The results clearly demonstrate that the ex-Gaussian and shifted Wald parameters do not correspond uniquely to parameters of the diffusion model. We conclude that researchers should resist the temptation to interpret changes in the ex-Gaussian and shifted Wald parameters in terms of cognitive processes. Supporting materials may be downloaded from http://pbr.psychonomic-journals .org/content/supplemental.

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References

  1. 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.CrossRefGoogle Scholar
  2. 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.CrossRefGoogle Scholar
  3. Blough, D. S. (1988). Quantitative relations between visual search speed and target—distractor similarity. Perception & Psychophysics, 43, 57–71.CrossRefGoogle Scholar
  4. Blough, D. S. (1989). Contrast as seen in visual search reaction times. Journal of the Experimental Analysis of Behavior, 52, 199–211.PubMedCrossRefGoogle Scholar
  5. Brooks, S. P., & Gelman, A. (1998). General methods for monitoring convergence of iterative simulations. Journal of Computational & Graphical Statistics, 7, 434–455.CrossRefGoogle Scholar
  6. 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
  7. Carpenter, R. H. S., & Williams, M. L. L. (1995). Neural computation of log likelihood in control of saccadic eye movements. Nature, 377, 59–62.PubMedCrossRefGoogle Scholar
  8. Emerson, P. L. (1970). Simple reaction time with Markovian evolution of Gaussian discriminal processes. Psychometrika, 35, 99–110.CrossRefGoogle Scholar
  9. Epstein, J. N.. Conners, C. K.. Hervey, A. S.. Tonev, S. T.. Arnold, L. E.. Abikoff, H. B.. et al. (2006). Assessing medication effects in the MTA study using neuropsychological outcomes. Journal of Child Psychology & Psychiatry, 47, 446–456.CrossRefGoogle Scholar
  10. Farrell, S., & Ludwig, C. J. H. (2008). Bayesian and maximum likelihood estimation of hierarchical response time models. Psychonomic Bulletin & Review, 15, 1209–1217.CrossRefGoogle Scholar
  11. Gelman, A., & Hill, J. (2007). Data analysis using regression and multilevel/ hierarchical models. Cambridge: Cambridge University Press.Google Scholar
  12. Gholson, B., & Hohle, R. H. (1968a). Choice reaction times to hues printed in conflicting hue names and nonsense words. Journal of Experimental Psychology, 76, 413–418.PubMedCrossRefGoogle Scholar
  13. Gholson, B., & Hohle, R. (1968b). Verbal reaction times to hues and hue names and forms and form names. Perception & Psychophysics, 3, 191–196.CrossRefGoogle Scholar
  14. Gomez, P.. Ratcliff, R., & Perea, M. (2007). A model of the go/no-go task. Journal of Experimental Psychology: General, 136, 389–413.CrossRefGoogle Scholar
  15. Gordon, B., & Carson, K. (1990). The basis for choice reaction time slowing in Alzheimer’s disease. Brain & Cognition, 13, 148–166.CrossRefGoogle Scholar
  16. Heathcote, A. (2004). Fitting Wald and ex-Wald distributions to response time data: An example using functions for the S-PLUS package. Behavior Research Methods, Instruments, & Computers, 36, 678–694.CrossRefGoogle Scholar
  17. 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.CrossRefGoogle Scholar
  18. Hockley, W. E. (1982). Retrieval processes in continuous recognition. Journal of Experimental Psychology: Learning, Memory, & Cognition, 8, 497–512.CrossRefGoogle 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.PubMedCrossRefGoogle Scholar
  21. Kieffaber, P. D.. Kappenman, E. S.. Bodkins, M.. Shekhar, A., O’Donnell, B. F., & Hetrick, W. P. (2006). Switch and maintenance of task set in schizophrenia. Schizophrenia Research, 84, 345–358.PubMedCrossRefGoogle Scholar
  22. Lee, M. D. (2008). Three case studies in the Bayesian analysis of cognitive models. Psychonomic Bulletin & Review, 15, 1–15.CrossRefGoogle Scholar
  23. Leth-Steensen, C.. King Elbaz, Z., & Douglas, V. I. (2000). Mean response times, variability and skew in the responding of ADHD children: A response time distributional approach. Acta Psychologica, 104, 167–190.PubMedCrossRefGoogle Scholar
  24. Luce, R. D. (1986). Response times: Their role in inferring elementary mental organization. New York: Oxford University Press.Google Scholar
  25. Lunn, D. J.. Thomas, A.. Best, N., & Spiegelhalter, D. (2000). WinBUGS—A Bayesian modelling framework: Concepts, structure, and extensibility. Statistics & Computing, 10, 325–337.CrossRefGoogle Scholar
  26. 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.PubMedCrossRefGoogle Scholar
  27. McGill, W. [J.] (1963). Stochastic latency mechanisms. In R. D. Luce, R. R. Bush, & E. Galanter (Eds.), Handbook of mathematical psychology (Vol. 1, pp. 309–360). New York: Wiley.Google Scholar
  28. McGill, W. J., & Gibbon, J. (1965). The general-gamma distribution and reaction times. Journal of Mathematical Psychology, 2, 1–18.CrossRefGoogle Scholar
  29. Myung, I. J. (2003). Tutorial on maximum likelihood estimation. Journal of Mathematical Psychology, 47, 90–100.CrossRefGoogle Scholar
  30. Okada, R. (1971). Decision latencies in short-term recognition memory. Journal of Experimental Psychology, 90, 27–32.PubMedCrossRefGoogle Scholar
  31. Penner-Wilger, M.. Leth-Steensen, C., & LeFevre, J.-A. (2002). Decomposing the problem-size effect: A comparison of response time distributions across cultures. Memory & Cognition, 30, 1160–1167.CrossRefGoogle Scholar
  32. 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.Google Scholar
  33. Possamaï, C.-A. (1991). A responding hand effect in a simple-RT precueing experiment: Evidence for a late locus of facilitation. Acta Psychologica, 77, 47–63.PubMedCrossRefGoogle Scholar
  34. Ratcliff, R. (1978). A theory of memory retrieval. Psychological Review, 85, 59–108.CrossRefGoogle Scholar
  35. Ratcliff, R. (1993). Methods for dealing with reaction time outliers. Psychological Bulletin, 114, 510–532.PubMedCrossRefGoogle Scholar
  36. Ratcliff, R. (2002). A diffusion model account of response time and accuracy in a brightness discrimination task: Fitting real data and failing to fit fake but plausible data. Psychonomic Bulletin & Review, 9, 278–291.CrossRefGoogle Scholar
  37. Ratcliff, R.. Gomez, P., & McKoon, G. (2004). A diffusion model account of the lexical decision task. Psychological Review, 111, 159–182.PubMedCrossRefGoogle Scholar
  38. Ratcliff, R., & McKoon, G. (2008). The diffusion decision model: Theory and data for two-choice decision tasks. Neural Computation, 20, 873–922.PubMedCrossRefGoogle Scholar
  39. Ratcliff, R., & Murdock, B. B. (1976). Retrieval processes in recognition memory. Psychological Review, 83, 190–214.CrossRefGoogle Scholar
  40. Ratcliff, R., & Rouder, J. N. (2000). A diffusion model account of masking in two-choice letter identification. Journal of Experimental Psychology: Human Perception & Performance, 26, 127–140.CrossRefGoogle Scholar
  41. Ratcliff, R.. Thapar, A.. Gomez, P., & McKoon, G. (2004). A diffusion model analysis of the effects of aging in the lexical-decision task. Psychology & Aging, 19, 278–289.CrossRefGoogle Scholar
  42. Ratcliff, R.. Thapar, A., & McKoon, G. (2001). The effects of aging on reaction time in a signal detection task. Psychology & Aging, 16, 323–341.CrossRefGoogle Scholar
  43. Ratcliff, R.. Thapar, A., & McKoon, G. (2003). A diffusion model analysis of the effects of aging on brightness discrimination. Perception & Psychophysics, 65, 523–535.CrossRefGoogle Scholar
  44. Ratcliff, R.. Thapar, A., & McKoon, G. (2004). A diffusion model analysis of the effects of aging on recognition memory. Journal of Memory & Language, 50, 408–424.CrossRefGoogle Scholar
  45. Rohrer, D. (1996). On the relative and absolute strength of a memory trace. Memory & Cognition, 24, 188–201.CrossRefGoogle Scholar
  46. Rohrer, D. (2002). The breadth of memory search. Memory, 10, 291–301.PubMedCrossRefGoogle Scholar
  47. Rohrer, D., & Wixted, J. T. (1994). An analysis of latency and interresponse time in free recall. Memory & Cognition, 22, 511–524.CrossRefGoogle Scholar
  48. Rotello, C. M., & Zeng, M. (2008). Analysis of RT distributions in the remember—know paradigm. Psychonomic Bulletin & Review, 15, 825–832.CrossRefGoogle Scholar
  49. Rouder, J. N.. Lu, J.. Speckman, P.. Sun, D., & Jiang, Y. (2005). A hierarchical model for estimating response time distributions. Psychonomic Bulletin & Review, 12, 195–223.CrossRefGoogle Scholar
  50. 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.CrossRefGoogle Scholar
  51. Schmiedek, F.. Oberauer, K.. Wilhelm, O.. Süß, H.-M., & Wittmann, W. W. (2007). Individual differences in components of reaction time distributions and their relations to working memory and intelligence. Journal of Experimental Psychology: General, 136, 414–429.CrossRefGoogle Scholar
  52. Schouten, J. F., & Bekker, J. A. M. (1967). Reaction time and accuracy. Acta Psychologica, 27, 143–153.PubMedCrossRefGoogle Scholar
  53. Schwarz, W. (2001). The ex-Wald distribution as a descriptive model of response times. Behavior Research Methods, Instruments, & Computers, 33, 457–469.CrossRefGoogle Scholar
  54. Schwarz, W. (2002). On the convolution of inverse Gaussian and exponential random variables. Communications in Statistics: Theory & Methods, 31, 2113–2121.CrossRefGoogle Scholar
  55. Shiffrin, R. M.. Lee, M. D.. Kim, W., & Wagenmakers, E.-J. (2008). A survey of model evaluation approaches with a tutorial on hierarchical Bayesian methods. Cognitive Science, 32, 1248–1284.CrossRefGoogle Scholar
  56. Smith, P. L. (1995). Psychophysically principled models of visual simple reaction time. Psychological Review, 102, 567–593.CrossRefGoogle Scholar
  57. Spieler, D. H. (2001). Modelling age-related changes in information processing. European Journal of Cognitive Psychology, 13, 217–234.Google Scholar
  58. 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.CrossRefGoogle Scholar
  59. 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 & Performance, 26, 506–526.CrossRefGoogle Scholar
  60. Sternberg, S. (1966). High-speed scanning in human memory. Science, 153, 652–654.PubMedCrossRefGoogle Scholar
  61. Thapar, A.. Ratcliff, R., & McKoon, G. (2003). A diffusion model analysis of the effects of aging on letter discrimination. Psychology & Aging, 18, 415–429.CrossRefGoogle Scholar
  62. Townsend, J. T., & Ashby, F. G. (1983). The stochastic modeling of elementary psychological processes. Cambridge: Cambridge University Press.Google Scholar
  63. Vandekerckhove, J., & Tuerlinckx, F. (2007). Fitting the Ratcliff diffusion model to experimental data. Psychonomic Bulletin & Review, 14, 1011–1026.CrossRefGoogle Scholar
  64. Vandekerckhove, J., & Tuerlinckx, F. (2008). Diffusion model analysis with MATLAB: A DMAT primer. Behavior Research Methods, 40, 61–72.PubMedCrossRefGoogle Scholar
  65. Voss, A.. Rothermund, K., & Voss, J. (2004). Interpreting the parameters of the diffusion model: An empirical validation. Memory & Cognition, 32, 1206–1220.CrossRefGoogle Scholar
  66. Wagenmakers, E.-J. (2009). Methodological and empirical developments for the Ratcliff diffusion model of response times and accuracy. European Journal of Cognitive Psychology, 21, 641–671.CrossRefGoogle Scholar
  67. Wagenmakers, E.-J.. Ratcliff, R.. Gomez, P., & McKoon, G. (2008). A diffusion model account of criterion shifts in the lexical decision task. Journal of Memory & Language, 58, 140–159.CrossRefGoogle Scholar
  68. Wagenmakers, E.-J.. van der Maas, H. L. J.. Dolan, C. V., & Grasman, R. P. P. P. (2008). EZ does it! Extensions of the EZ-diffusion model. Psychonomic Bulletin & Review, 15, 1229–1235.CrossRefGoogle Scholar
  69. Wagenmakers, E.-J.. van der Maas, H. L. J., & Grasman, R. P. P. P. (2007). An EZ-diffusion model for response time and accuracy. Psychonomic Bulletin & Review, 14, 3–22.CrossRefGoogle Scholar
  70. Wald, A. (1947). Sequential analysis. New York: Wiley.Google Scholar
  71. Wickelgren, W. A. (1977). Speed—accuracy tradeoff and information processing dynamics. Acta Psychologica, 41, 67–85.CrossRefGoogle Scholar
  72. Wixted, J. T.. Ghadisha, H., & Vera, R. (1997). Recall latency following pure- and mixed-strength lists: A direct test of the relative strength model of free recall. Journal of Experimental Psychology: Learning, Memory, & Cognition, 23, 523–538.CrossRefGoogle Scholar
  73. Wixted, J. T., & Rohrer, D. (1993). Proactive interference and the dynamics of free recall. Journal of Experimental Psychology: Learning, Memory, & Cognition, 19, 1024–1039.CrossRefGoogle Scholar
  74. Yap, M. J., & Balota, D. A. (2007). Additive and interactive effects on response time distributions in visual word recognition. Journal of Experimental Psychology: Learning, Memory, & Cognition, 33, 274–296.CrossRefGoogle Scholar
  75. Yap, M. J.. Balota, D. A.. Cortese, M. J., & Watson, J. M. (2006). Single- versus dual-process models of lexical decision performance: Insights from response time distributional analysis. Journal of Experimental Psychology: Human Perception & Performance, 32, 1324–1344.CrossRefGoogle Scholar

Copyright information

© Psychonomic Society, Inc. 2009

Authors and Affiliations

  1. 1.Department of PsychologyUniversity of AmsterdamAmsterdamThe Netherlands

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