Abstract
In simple, go/no-go, and choice reaction time (RT) tasks, responses are faster to two redundant targets than to a single target. This redundancy gain has been explained in terms of a race model assuming that whichever target is processed faster determines RT (Raab, 1962). Miller (1982) presented a race model inequality to test the race model by comparing the RT distributions of single and redundant target conditions. Here, we present simulations indicating that the standard tests of this inequality (for a description of the testing algorithm, see Ulrich, Miller, & Schröter, 2007) are afflicted with systematic biases and Type I error accumulation. Systematic biases tend to produce violations of the race model inequality, but they decrease as the numbers of observations increase. Reasonably unbiased tests of the race model inequality are obtained for sample sizes of at least 20 for each target condition. In addition, Type I error accumulates because of testing the inequality at multiple percentiles. To reduce Type I error, the race model inequality should be tested in a restricted range of percentiles, preferably in the percentile range 10% to 25%.
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Billingsley, P. (1979).Probability and measure. New York: Wiley.
Colonius, H. (1990). Possibly dependent probability summation of reaction time.Journal of Mathematical Psychology, 34, 253–275.
Devroye, L. (1986).Non-uniform random variate generation. New York: Springer.
Egeth, H. E., &Mordkoff, J. T. (1991). Redundancy gain revisited: Evidence for parallel processing of separable dimensions. In J. R. Pomerantz & G. R. Lockhead (Eds.),The perception of structure (pp. 131–140). Washington, DC: American Psychological Association.
Fréchet, M. (1951). Sur les tableaux de correlation dont les marges sont données.Annales de l’Université de Lyon: Sec. A. Series 3, 14, 53–57.
Gilchrist, W. G. (2000).Statistical modeling with quantile functions. Boca Raton, FL: Chapman & Hall/CRC.
Gondan, M., Lange, K., Rösler, F., &Röder, B. (2004). The redundant target effect is affected by modality switch costs.Psychonomic Bulletin & Review, 11, 307–313.
Hazen, A. (1914). Storage to be provided in impounding reservoirs for municipal water supply.Transactions of the American Society of Civil Engineers, 77, 1539–1669.
Hershenson, M. (1962). Reaction time as measure of intersensory facilitation.Journal of Experimental Psychology, 63, 289–293.
Hyndman, R. J., &Fan, Y. (1996). Sample quantiles in statistical packages.American Statistician, 50, 361–365.
Krummenacher, J., Müller, H. J., &Heller, D. (2001). Visual search for dimensionally redundant pop-out targets: Evidence for parallelcoactive processing of dimensions.Perception & Psychophysics, 63, 901–917.
Luce, R. D. (1986).Response times: Their role in inferring elementary mental organization. Oxford: Oxford University Press.
Maris, G., &Maris, E. (2003). Testing the race model inequality: A nonparametric approach.Journal of Mathematical Psychology, 47, 507–514.
Miller, J. O. (1982). Divided attention: Evidence for coactivation with redundant signals.Cognitive Psychology, 14, 247–279.
Miller, J. O. (1986). Timecourse of coactivation in bimodal divided attention.Perception & Psychophysics, 40, 331–343.
Miller, J. O. (1991). Channel interaction and the redundant-targets effect in bimodal divided attention.Journal of Experimental Psychology: Human Perception & Performance, 17, 60–169.
Miller, J. O. (2006). A likelihood ratio test for mixture effects.Behavior Research Methods, 38, 92–106.
Mordkoff, J. T., &Miller, J. O. (1993). Redundancy gains and coactivation with two different targets: The problem of target preferences and the effects of display frequency.Perception & Psychophysics, 53, 527–535.
Mordkoff, J. T., &Yantis, S. (1991). An interactive race model of divided attention.Journal of Experimental Psychology: Human Perception & Performance, 17, 520–538.
Parzen, E. (1960).Modern probability theory and its application. New York: Wiley.
Raab, D. H. (1962). Statistical facilitation of simple reaction times.Transactions of the New York Academy of Sciences, 24, 574–590.
Schröger, E., & Widmann, A. (1998). Speeded responses to audiovisual signal changes result from bimodal integration.Psychophysiological Research, 35, 755–759.
Schwarz, W. (2001). The ex-Wald distribution as a descriptive model of response times.Behavior Research Methods, Instruments, & Computers, 33, 457–469.
Schwarz, W. (2002). On the convolution of inverse Gaussian and exponential random variables.Communications in Statistics: Theory & Methods, 31, 2113–2121.
Ulrich, R., &Giray, M. (1986). Separate-activation models with variable base times: Testability and checking of cross-channel dependency.Perception & Psychophysics, 39, 248–254.
Ulrich, R., Miller, J., &Schröter, H. (2007). Testing the race model inequality: An algorithm and computer programs.Behavior Research Methods, 39, 291–302.
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This research was supported by a grant from the G. A. Lienert Foundation to A.K. and by a grant from The Marsden Fund administered by the Royal Society of New Zealand. We thank Wolfgang Schwarz and two anonymous reviewers for helpful comments on earlier versions of the manuscript.
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Kiesel, A., Miller, J. & Ulrich, R. Systematic biases and Type I error accumulation in tests of the race model inequality. Behavior Research Methods 39, 539–551 (2007). https://doi.org/10.3758/BF03193024
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DOI: https://doi.org/10.3758/BF03193024