Abstract
Under certain circumstances, it is theoretically important to decide whether a difference between two conditions in mean reaction time (RT) results from a relatively uniform slowing of all the responses in the slower condition or from a mixture of some slowed trials with some unslowed ones. This article describes a likelihood ratio test that can be used to differentiate between these two possibilities and reports computer simulations examining the power and Type I error rate of the test under conditions similar to those encountered in RT research. A freely available computer program, called MIXTEST, can be used both to carry out the likelihood ratio test and to conduct simulations evaluating the performance of the test within various settings.
Article PDF
Similar content being viewed by others
References
Allen, P. A., Smith, A. F., Vires-Collins, H., &Sperry, S. (1998). The psychological refractory period: Evidence for age differences in attentional time-sharing.Psychology & Aging,13, 218–229.
Borger, R. (1963). The refractory period and serial choice-reactions.Quarterly Journal of Experimental Psychology,15, 1–12.
Chen, H., &Chen, J. (2001). The likelihood ratio test for homogeneity in the finite mixture models.Canadian Journal of Statistics,29, 201–216.
Chernoff, H. (1954). On the distribution of the likelihood ratio.Annals of Mathematical Statistics,25, 573–578.
Chhikara, R. S., &Folks, J. L. (1989).The inverse Gaussian distribution: Theory, methodology, and applications. New York: Dekker.
Cox, D. R., &Miller, H. D. (1965).The theory of stochastic processes. London: Chapman & Hall.
Dawson, M. R. W. (1988). Fitting the ex-Gaussian equation to reaction time distributions.Behavior Research Methods, Instruments, & Computers,20, 54–57.
Diederich, A. (1995). Intersensory facilitation of reaction time: Evaluation of counter and diffusion coactivation models.Journal of Mathematical Psychology,39, 197–215.
Dixon, P. (2003). Thep-value fallacy and how to avoid it.Canadian Journal of Experimental Psychology,57, 189–202.
Donders, F. C. (1969). Over de snelheid van psychische processen [On the speed of mental processes] (W. G. Koster, Trans.). In W. G. Koster (Ed.),Attention and performance II (pp. 412–431). Amsterdam: North-Holland. (Original work published 1868)
Dudewicz, E. J., &Mishra, S. N. (1988).Modern mathematical statistics. New York: Wiley.
Eriksen, B. A., &Eriksen, C. W. (1974). Effects of noise letters upon the identification of a target letter in a nonsearch task.Perception & Psychophysics,16, 143–149.
Everitt, B. S., &Hand, B. J. (1981).Finite mixture distributions. London: Chapman & Hall.
Fazio, R. H. (1990). A practical guide to the use of response latency in social psychological research. In C. Hendrick & M. S. Clark (Eds.),Research methods in personality and social psychology (Review of personality and social psychology, Vol. 11, pp. 74–97). Newbury Park, CA: Sage.
Franz, E. A., &Miller, J. O. (2002). Effects of response readiness on reaction time and force output in people with Parkinson’s disease.Brain,125, 1733–1750.
Gottlob, L. R. (2004). Location cuing and response time distributions in visual attention.Perception & Psychophysics,66, 1293–1302.
Green, D. M., &Swets, J. A. (1966).Signal detection theory and psychophysics. New York: Wiley.
Heathcote, A. (1996). RTSYS: A DOS application for the analysis of reaction time data.Behavior Research Methods, Instruments, & Computers,28, 427–445.
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.
Heathcote, A., Popiel, S. J., &Mewhort, D. J. K. (1991). Analysis of response-time distributions: An example using the Stroop task.Psychological Bulletin,109, 340–347.
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.
Hoel, P. G. (1962).Introduction to mathematical statistics (3rd ed.). New York: Wiley.
Hohle, R. H. (1965). Inferred components of reaction times as functions of foreperiod duration.Journal of Experimental Psychology,69, 382–386.
Hohle, R. H., &Gholson, B. (1968). Choice reaction times with equally and unequally probable alternatives.Journal of Experimental Psychology,78, 95–98.
Juhel, J. (1993). Should we take the shape of reaction time distributions into account when studying the relationship between RT and psychometric intelligence?Personality & Individual Differences,15, 357–360.
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, 167–190.
Luce, R. D. (1986).Response times: Their role in inferring elementary mental organization. Oxford: Oxford University Press.
Mattes, S., Ulrich, R., &Miller, J. O. (1997). Effects of response probability on response force in simple RT.Quarterly Journal of Experimental Psychology,50A, 405–420.
McLachlan, G. J., &Basford, K. E. (1988).Mixture models. New York: Dekker.
Mewhort, D. J. K., Braun, J. G., &Heathcote, A. (1992). Response time distributions and the Stroop task: A test of the Cohen, Dunbar, and McClelland (1990) model.Journal of Experimental Psychology: Human Perception & Performance,18, 872–882.
Meyer, D. E., Osman, A. M., Irwin, D. E., &Yantis, S. (1988). Modern mental chronometry.Biological Psychology,26, 3–67.
Meyer, D. E., Yantis, S., Osman, A. M., &Smith, J. E. K. (1985). Temporal properties of human information processing: Tests of discrete versus continuous models.Cognitive Psychology,17, 445–518.
Miller, J. O. (1982). Divided attention: Evidence for coactivation with redundant signals.Cognitive Psychology,14, 247–279.
Miller, J. [O.] (1998). Cupid: A program for computations with probability distributions.Behavior Research Methods, Instruments, & Computers,30, 544–545.
Neisser, U. (1963). Decision time without reaction time: Experiments in visual scanning.American Journal of Psychology,76, 376–385.
Pashler, H. E. (1991). Shifting visual attention and selecting motor responses: Distinct attentional mechanisms.Journal of Experimental Psychology: Human Perception & Performance,17, 1023–1040.
Pashler, H. E. (1994). Dual-task interference in simple tasks: Data and theory.Psychological Bulletin,116, 220–244.
Pashler, H. E., &Johnston, J. C. (1989). Chronometric evidence for central postponement in temporally overlapping tasks.Quarterly Journal of Experimental Psychology,41A, 19–45.
Rao, C. R. (1965).Linear statistical inference and its applications. New York: Wiley.
Ratcliff, R. (1978). A theory of memory retrieval.Psychological Review,85, 59–108.
Ratcliff, R., Gomez, P., &McKoon, G. (2004). A diffusion model account of the lexical decision task.Psychological Review,111, 159–182.
Ratcliff, R., &Rouder, J. N. (1998). Modeling response times for two-choice decisions.Psychological Science,9, 347–356.
Rosenbrock, H. H. (1960). An automatic method for finding the greatest or least value of a function.Computer Journal,3, 175–184.
Rouder, J. N., &Speckman, P. L. (2004). An evaluation of the Vincentizing method of forming group-level response time distributions.Psychonomic Bulletin & Review,11, 419–427.
Schwarz, W. (1993). A diffusion model of early visual search: Theoretical analysis and experimental results.Psychological Research,55, 200–207.
Schwarz, W. (1994). Diffusion, superposition, and the redundanttargets effect.Journal of Mathematical Psychology,38, 504–520.
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.
Smith, M. C. (1969). The effect of varying information on the psychological refractory period.Acta Psychologica,30, 220–231.
Smith, P. L. (2000). Stochastic dynamic models of response time and accuracy: A foundational primer.Journal of Mathematical Psychology,44, 408–463.
Sternberg, S. (1966). High-speed scanning in human memory.Science,153, 652–654.
Titterington, D. M., Smith, A. F. M., &Makov, U. E. (1985).Statistical analysis of finite mixture distributions. New York: Wiley.
Tombu, M., &Jolicoeur, P. (2002). All-or-none bottleneck versus capacity sharing accounts of the psychological refractory period phenomenon.Psychological Research,66, 274–286.
Townsend, J. T., &Ashby, F. G. (1983).The stochastic modeling of elementary psychological processes. Cambridge: Cambridge University Press.
Usher, M., Olami, Z., &McClelland, J. L. (2002). Hick’s law in a stochastic race model with speed—accuracy tradeoff.Journal of Mathematical Psychology,46, 704–715.
Van Zandt, T. (2000). How to fit a response time distribution.Psychonomic Bulletin & Review,7, 424–465.
Wang, Y. H. (2000). Fiducial intervals: What are they?American Statistician,54, 105–111.
Welford, A. T. (1959). Evidence of a single-channel decision mechanism limiting performance in a serial reaction task.Quarterly Journal of Experimental Psychology,11, 193–210.
Wenger, M. J., &Townsend, J. T. (2000). Basic response time tools for studying general processing capacity in attention, perception, and cognition.Journal of General Psychology,127, 67–99.
Wilks, S. S. (1938). The large-sample distribution of the likelihood ratio for testing composite hypotheses.Annals of Mathematical Statistics,9, 60–62.
Yantis, S., Meyer, D. E., &Smith, J. E. K. (1991). Analyses of multinomial mixture distributions: New tests for stochastic models of cognition and action.Psychological Bulletin,110, 350–374.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by a grant from the Marsden Fund, administered by the Royal Society of New Zealand. I thank Ann Reynolds and Wolfgang Schwarz for constructive comments on earlier versions of the article, the software, and the software documentation.
Rights and permissions
About this article
Cite this article
Miller, J. A likelihood ratio test for mixture effects. Behavior Research Methods 38, 92–106 (2006). https://doi.org/10.3758/BF03192754
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.3758/BF03192754