Abstract
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.
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Notes
Schwarz (2001) interprets the diffusion process as an accumulation of noisy partial information over time. Thus, the ex-Wald model has one additional noise parameter which is just an arbitrary scaling parameter usually set to one for the parameter estimations. See Schwarz (2001) for further methodological and mathematical details.
For the mathematical derivation of the EZ diffusion model and more details about prerequisites, assumptions, and parameter recovery, see Wagenmakers et al. (2007).
Comparisons of EZ diffusion model parameters were not possible for the experiments from Miller and Ulrich (1998), Miller and Low (2001), or Miller (2012), because those experiments used simple, go/no-go, 4-choice, and 6-choice tasks, whereas the EZ diffusion model is only applicable to 2-choice tasks.
We also ran a parallel set of analyses including participants who were excluded from the published analyses due to issues with EEG recording, and these analyses produced very similar results.
We also separately checked the results for the drift rate v and the boundary separation a and found that the effects on TD were mostly driven by v. These detailed results can be found in the online supplement.
We also checked whether parameters of the shifted Wald (Heathcote, 2004)—where the non-Wald parameter is a constant shift rather than an exponential—can be linked to processing stages. Though the results across the hand vs. foot experiments were rather consistent, there was no clear pattern of results for the LRP experiments. For details see the online supplement.
We also fitted the EZ2 diffusion model (Grasman, Wagenmakers, & van der Maas, 2009)—an extension to the EZ diffusion model that allows for variations in starting point z—to the same datasets as the EZ diffusion model. However, the obtained pattern of results was no more systematic regarding effects on different stages than the result pattern produced by the EZ diffusion model, so we did not include it.
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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).
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Appendices
Appendix 1
Parameter search process
The parameter search process was carried out using the same basic methods for both the ex-Gaussian and ex-Wald models. First, for each observed RT distribution (i.e., combination of participant and condition), individual RTs were excluded using the same criteria as in the original analyses (e.g., training blocks, minimum RT, maximum RT), and error trials were always excluded. Then, for each observed RT distribution and each model, we conducted 22 different searches for the best-fitting (i.e., maximum likelihood) parameter values, using different starting values in order to increase the probability that the search process would find the global maxima rather than ending at local maxima. We then selected the parameter values that provided the best fit across all searches for each participant x condition x model combination and used them in the further analyses.
The 22 sets of starting values for each run of the parameter search routine were chosen depending on the mean and variance of the particular set of RTs being fit. Specifically, the starting value of the exponential parameter was set to one of 22 percentages of the observed RT variance (i.e., 0.1%, 1%, from 10% to 95% in 5% steps, 99%, and 99.9%). Given that value for the exponential component, the starting values of the non-exponential components were then set so that the starting ex-Gaussian or ex-Wald convolution would have mean and variance equal to the observed values for the RTs being fit. For all parameter searches, we placed a lower limit of three on the minimum value of σ for the ex-Gaussian and ex-Wald distributions, based on initial findings that the estimation process occasionally resulted in unreasonably small σ values (i.e., predicting essentially exponential RT distributions). Tarantino et al. (2013) reported a similar problem in their parameter estimations and chose to replace the unreasonable estimates with the average values for the corresponding condition.
Appendix 2
Parameter estimates per condition
See Table 4.
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Rieger, T., Miller, J. Are model parameters linked to processing stages? An empirical investigation for the ex-Gaussian, ex-Wald, and EZ diffusion models. Psychological Research 84, 1683–1699 (2020). https://doi.org/10.1007/s00426-019-01176-4
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DOI: https://doi.org/10.1007/s00426-019-01176-4