Are model parameters linked to processing stages? An empirical investigation for the ex-Gaussian, ex-Wald, and EZ diffusion models

  • Tobias RiegerEmail author
  • Jeff Miller
Original Article


In previous research, the parameters of the ex-Gaussian distribution have been subject to a wide variety of interpretations. The present study investigated whether the ex-Gaussian model is capable of distinguishing effects on separate processing stages (i.e., pre-motor vs. motor). In order to do so, we used datasets where the locus of effect was quite clear. Specifically, we analyzed data from experiments comparing hand vs. foot responses—presumably differing in the motor stage—and from experiments in which the lateralized readiness potential was used to localize experimental effects into premotor vs. motor processes. Moreover, we broadened the scope to two other descriptive RT models: the ex-Wald and EZ diffusion models. To the extent possible with each of these models, we reanalyzed the RT data of 19 clearly localized experimental effects from 12 separate experiments reported in seven previously published articles. Unfortunately, we did not find a clear pattern of results for any of the models, with no clear link between effects on one of the model’s parameters and effects on different processing stages. The present results suggest that one should resist the temptation to associate specific processing stages with individual parameters of the ex-Gaussian, ex-Wald, and EZ diffusion models.



This research was conducted while the first author was carrying out a research internship at the University of Otago. Tobias Rieger was supported by the mobility program (PROMOS) of the German Academic Exchange Service (DAAD).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.

Informed consent

Informed consent was obtained from all individual participants included in the study.

Supplementary material

426_2019_1176_MOESM1_ESM.docx (218 kb)
Supplementary material 1 (DOCX 218 kb)


  1. American Psychiatric Association. (2013). Diagnostic and statistical manual of mental disorders: DSM-5 (5th ed.). Washington, DC: American Psychiatric Pub.Google Scholar
  2. Arnold, N. R., Bröder, A., & Bayen, U. J. (2014). Empirical validation of the diffusion model for recognition memory and a comparison of parameter-estimation methods. Psychological Research, 79(5), 882–898. Scholar
  3. 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(1), 32–55. Scholar
  4. Balota, D. A., Tse, C.-S., Hutchison, K. A., Spieler, D. H., Duchek, J. M., & Morris, J. C. (2010). Predicting conversion to dementia of the Alzheimer’s type in a healthy control sample: The power of errors in Stroop color naming. Psychology and Aging, 25(1), 208–218. Scholar
  5. Balota, D. A., & Yap, M. J. (2011). Moving beyond the mean in studies of mental chronometry: The power of response time distributional analyses. Current Directions in Psychological Science, 20(3), 160–166. Scholar
  6. Brown, S. D., & Heathcote, A. (2008). The simplest complete model of choice response time: Linear ballistic accumulation. Cognitive Psychology, 57(3), 153–178. Scholar
  7. Burbeck, S. L., & Luce, R. D. (1982). Evidence from auditory simple reaction times for both change and level detectors. Perception & Psychophysics, 32(2), 117–133. Scholar
  8. Buzy, W. M., Medoff, D. R., & Schweitzer, J. B. (2009). Intra-individual variability among children with ADHD on a working memory task: An ex-Gaussian approach. Child Neuropsychology, 15(5), 441–459. Scholar
  9. Castellanos, F. X., Sonuga-Barke, E. J., Milham, M. P., & Tannock, R. (2006). Characterizing cognition in ADHD: Beyond executive dysfunction. Trends in Cognitive Sciences, 10(3), 117–123. Scholar
  10. Christie, L. S., & Luce, R. D. (1956). Decision structure and time relations in simple choice behavior. Bulletin of Mathematical Biophysics, 18(2), 89–112. Scholar
  11. Deecke, L., Grözinger, B., & Kornhuber, H. (1976). Voluntary finger movement in man: Cerebral potentials and theory. Biological Cybernetics, 23(2), 99–119. Scholar
  12. Dutilh, G., Annis, J., Brown, S. D., Cassey, P., Evans, N. J., Grasman, R. P. P. P., et al. (2018). The quality of response time data inference: A blinded, collaborative assessment of the validity of cognitive models. Psychonomic Bulletin & Review. Scholar
  13. Epstein, J. N., Langberg, J. M., Rosen, P. J., Graham, A., Narad, M. E., Antonini, T. N., et al. (2011). Evidence for higher reaction time variability for children with ADHD on a range of cognitive tasks including reward and event rate manipulations. Neuropsychology, 25(4), 427–441. Scholar
  14. Gholson, B., & Hohle, R. H. (1968a). Choice reaction times to hues printed in conflicting hue names and nonsense words. Journal of Experimental Psychology, 76(3, Pt.1), 413–418. Scholar
  15. Gholson, B., & Hohle, R. H. (1968b). Verbal reaction times to hues vs hue names and forms vs form names. Perception & Psychophysics, 3(3), 191–196. Scholar
  16. Gmehlin, D., Fuermaier, A. B. M., Walther, S., Debelak, R., Rentrop, M., Westermann, C., et al. (2014). Intraindividual variability in inhibitory function in adults with ADHD—An ex-Gaussian approach. PLoS ONE, 9(12), 1–19. Scholar
  17. Gomez, P., Ratcliff, R., & Childers, R. (2015). Pointing, looking at, and pressing keys: A diffusion model account of response modality. Journal of Experimental Psychology: Human Perception and Performance, 41(6), 1515–1523. Scholar
  18. Gordon, B., & Carson, K. (1990). The basis for choice reaction time slowing in Alzheimer’s disease. Brain and Cognition, 13(2), 148–166. Scholar
  19. Grasman, R. P. P. P., Wagenmakers, E.-J., & van der Maas, H. L. J. (2009). On the mean and variance of response times under the diffusion model with an application to parameter estimation. Journal of Mathematical Psychology, 53(2), 55–68. Scholar
  20. Hackley, S. A., & Valle-Inclan, F. (1998). Automatic alerting does not speed late motoric processes in a reaction-time task. Nature, 391(6669), 786–788. Scholar
  21. 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(4), 678–694. Scholar
  22. Heathcote, A., Popiel, S. J., & Mewhort, D. (1991). Analysis of response time distributions: An example using the Stroop task. Psychological Bulletin, 109(2), 340–347. Scholar
  23. Hervey, A. S., Epstein, J. N., Curry, J. F., Tonev, S., Arnold, L. E., Conners, C. K., et al. (2006). Reaction time distribution analysis of neuropsychological performance in an ADHD sample. Child Neuropsychology, 12(2), 125–140. Scholar
  24. Hohle, R. H. (1965). Inferred components of reaction times as functions of foreperiod duration. Journal of Experimental Psychology, 69(4), 382–386. Scholar
  25. Izawa, J., Pekny, S. E., Marko, M. K., Haswell, C. C., Shadmehr, R., & Mostofsky, S. H. (2012). Motor learning relies on integrated sensory inputs in ADHD, but over-selectively on proprioception in Autism spectrum conditions. Autism Research, 5(2), 124–136. Scholar
  26. Jackson, J. D., Balota, D. A., Duchek, J. M., & Head, D. (2012). White matter integrity and reaction time intraindividual variability in healthy aging and early-stage Alzheimer disease. Neuropsychologia, 50(3), 357–366. Scholar
  27. Kinoshita, S., & Hunt, L. (2008). RT distribution analysis of category congruence effects with masked primes. Memory & Cognition, 36(7), 1324–1334. Scholar
  28. Kóbor, A., Takács, Ádám, Bryce, D., Szűcs, D., Honbolygó, F., Nagy, P., & Csépe, V. (2015). Children with ADHD show impairments in multiple stages of information processing in a Stroop task: An ERP study. Developmental Neuropsychology, 40(6), 329–347. Scholar
  29. Lee, R. W. Y., Jacobson, L. A., Pritchard, A. E., Ryan, M. S., Yu, Q., Denckla, M. B., et al. (2015). Jitter reduces response-time variability in ADHD: An ex-Gaussian analysis. Journal of Attention Disorders, 19(9), 794–804. Scholar
  30. Lerche, V., & Voss, A. (2017). Experimental validation of the diffusion model based on a slow response time paradigm. Psychological Research. Scholar
  31. Leth-Steensen, C., Elbaz, Z. K., & Douglas, V. I. (2000). Mean response times, variability, and skew in the responding of ADHD children: a response time distributional approach. Acta Psychologica, 104(2), 167–190. Scholar
  32. Low, K. A., Miller, J., & Vierck, E. (2002). Response slowing in Parkinson’s disease: a psychophysiological analysis of premotor and motor processes. Brain, 125(9), 1980–1994. Scholar
  33. Luce, R. (1986). Response times: Their role in inferring elementary mental organization. Oxford: Oxford University Press.Google Scholar
  34. Matzke, D., & Wagenmakers, E.-J. (2009). Psychological interpretation of the ex-Gaussian and shifted Wald parameters: A diffusion model analysis. Psychonomic Bulletin & Review, 16(5), 798–817. Scholar
  35. McGill, W. J. (1963). Stochastic latency mechanisms. In R. D. Luce, R. R. Bush, & E. Galanter (Eds.), Handbook of mathematical psychology (pp. 309–360). New York: Wiley.Google Scholar
  36. McGill, W. J., & Gibbon, J. (1965). The general-gamma distribution and reaction times. Journal of Mathematical Psychology, 2(1), 1–18. Scholar
  37. Miller, J. (2012). Selection and preparation of hand and foot movements: Cz activity as a marker of limb system preparation. Psychophysiology, 49(5), 590–603. Scholar
  38. Miller, J. (2017). Psychophysiological measurement of backward response activation in the prioritized processing paradigm. Journal of Experimental Psychology: Human Perception and Performance, 43(5), 941–953. Scholar
  39. Miller, J., Brookie, K., Wales, S., Wallace, S., & Kaup, B. (2018). Embodied cognition: Is activation of the motor cortex essential for understanding action verbs? Journal of Experimental Psychology. Learning, Memory, and Cognition, 44(3), 335–370. Scholar
  40. Miller, J., & Low, K. (2001). Motor processes in simple, go/no-go, and choice reaction time tasks: a psychophysiological analysis. Journal of Experimental Psychology: Human Perception and Performance, 27(2), 266–289. Scholar
  41. Miller, J., & Ulrich, R. (1998). Locus of the effect of the number of alternative responses: Evidence from the lateralized readiness potential. Journal of Experimental Psychology: Human Perception and Performance, 24(4), 1215–1231. Scholar
  42. Miller, J., Ulrich, R., & Rinkenauer, G. (1999). Effects of stimulus intensity on the lateralized readiness potential. Journal of Experimental Psychology: Human Perception and Performance, 25(5), 1454–1471. Scholar
  43. Moutsopoulou, K., & Waszak, F. (2012). Across-task priming revisited: Response and task conflicts disentangled using ex-Gaussian distribution analysis. Journal of Experimental Psychology: Human Perception and Performance, 38(2), 367–374. Scholar
  44. Osman, A., & Moore, C. M. (1993). The locus of dual-task interference: psychological refractory effects on movement-related brain potentials. Journal of Experimental Psychology: Human Perception and Performance, 19(6), 1292–1312. Scholar
  45. Osman, A., Moore, C. M., & Ulrich, R. (1995). Bisecting RT with lateralized readiness potentials: Precue effects after LRP onset. Acta Psychologica, 90(1–3), 111–127. Scholar
  46. Possamaï, C.-A. (1991). A responding hand effect in a simple-RT precueing experiment: Evidence for a late locus of facilitation. Acta Psychologica, 77(1), 47–63. Scholar
  47. Praamstra, P., & Seiss, E. (2005). The neurophysiology of response competition: Motor cortex activation and inhibition following subliminal response priming. Journal of Cognitive Neuroscience, 17(3), 483–493. Scholar
  48. Ratcliff, R. (1978). A theory of memory retrieval. Psychological Review, 85(2), 59–108. Scholar
  49. Ratcliff, R. (1979). Group reaction time distributions and an analysis of distribution statistics. Psychological Bulletin, 86(3), 446–461. Scholar
  50. Ratcliff, R., & Smith, P. L. (2004). A comparison of sequential sampling models for two-choice reaction time. Psychological Review, 111(2), 333–367. Scholar
  51. Ridderinkhof, K. R., Scheres, A., Oosterlaan, J., & Sergeant, J. A. (2005). Delta plots in the study of individual differences: new tools reveal response inhibition deficits in AD/HD that are eliminated by methylphenidate treatment. Journal of Abnormal Psychology, 114(2), 197–215. Scholar
  52. Rohrer, D. (1996). On the relative and absolute strength of a memory trace. Memory & Cognition, 24(2), 188–201. Scholar
  53. Rohrer, D. (2002). The breadth of memory search. Memory, 10(4), 291–301. Scholar
  54. Rohrer, D., & Wixted, J. T. (1994). An analysis of latency and interresponse time in free recall. Memory & Cognition, 22(5), 511–524. Scholar
  55. Rosenbrock, H. (1960). An automatic method for finding the greatest or least value of a function. The Computer Journal, 3(3), 175–184. Scholar
  56. 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(3), 414–429. Scholar
  57. Schwarz, W. (2001). The ex-Wald distribution as a descriptive model of response times. Behavior Research Methods, Instruments, & Computers, 33(4), 457–469. Scholar
  58. Singh, T., Laub, R., Burgard, J. P., & Frings, C. (2018). Disentangling inhibition-based and retrieval-based aftereffects of distractors: Cognitive versus motor processes. Journal of Experimental Psychology: Human Perception and Performance, 44(5), 797–805. Scholar
  59. Smulders, F. T., Kok, A., Kenemans, J. L., & Bashore, T. R. (1995). The temporal selectivity of additive factor effects on the reaction process revealed in ERP component latencies. Acta Psychologica, 90(1–3), 97–109. Scholar
  60. Smulders, F. T., & Miller, J. O. (2012). The lateralized readiness potential. The Oxford Handbook of Event-Related Potential Components. Scholar
  61. 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(2), 461–479. Scholar
  62. 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(2), 506–526. Scholar
  63. Steinhauser, M., & Hübner, R. (2009). Distinguishing response conflict and task conflict in the Stroop task: Evidence from ex-Gaussian distribution analysis. Journal of Experimental Psychology: Human Perception and Performance, 35(5), 1398–1412. Scholar
  64. Stroop, J. R. (1935). Studies of interference in serial verbal reactions. Journal of Experimental Psychology, 18(6), 643–662.Google Scholar
  65. Tamm, L., Narad, M. E., Antonini, T. N., O’Brien, K. M., Hawk, L. W., & Epstein, J. N. (2012). Reaction time variability in ADHD: A review. Neurotherapeutics, 9(3), 500–508. Scholar
  66. Tarantino, V., Cutini, S., Mogentale, C., & Bisiacchi, P. S. (2013). Time-on-task in children with ADHD: An ex-Gaussian analysis. Journal of the International Neuropsychological Society, 19(7), 820–828. Scholar
  67. Tse, C.-S., Balota, D. A., Yap, M. J., Duchek, J. M., & McCabe, D. P. (2010). Effects of healthy aging and early stage dementia of the Alzheimer’s type on components of response time distributions in three attention tasks. Neuropsychology, 24(3), 300–315. Scholar
  68. Van Zandt, T. (2000). How to fit a response time distribution. Psychonomic Bulletin & Review, 7(3), 424–465. Scholar
  69. Van Zandt, T. (2002). Analysis of response time distributions. Stevens’ Handbook of Experimental Psychology, 4, 461–516. Scholar
  70. Vaurio, R. G., Simmonds, D. J., & Mostofsky, S. H. (2009). Increased intra-individual reaction time variability in attention-deficit/hyperactivity disorder across response inhibition tasks with different cognitive demands. Neuropsychologia, 47(12), 2389–2396. Scholar
  71. Verleger, R., Kuniecki, M., Möller, F., Fritzmannova, M., & Siebner, H. R. (2009). On how the motor cortices resolve an inter-hemispheric response conflict: An event-related EEG potential-guided TMS study of the flankers task. European Journal of Neuroscience, 30(2), 318–326. Scholar
  72. Voss, A., Rothermund, K., & Voss, J. (2004). Interpreting the parameters of the diffusion model: An empirical validation. Memory & Cognition, 32(7), 1206–1220. Scholar
  73. Wagenmakers, E.-J., Van Der Maas, H. L., & Grasman, R. P. (2007). An EZ-diffusion model for response time and accuracy. Psychonomic Bulletin & Review, 14(1), 3–22. Scholar
  74. Whelan, R. (2008). Effective analysis of reaction time data. The Psychological Record, 58(3), 475–482. Scholar
  75. 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, and Cognition, 23(3), 523–538. Scholar
  76. Wixted, J. T., & Rohrer, D. (1993). Proactive interference and the dynamics of free recall. Journal of Experimental Psychology. Learning, Memory, and Cognition, 19(5), 1024–1039. Scholar

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Authors and Affiliations

  1. 1.Department of Psychology and ErgonomicsTechnische Universität BerlinBerlinGermany
  2. 2.Department of PsychologyUniversity of OtagoDunedinNew Zealand

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