Abstract
The power law (y =ax −b) has been shown to provide a good description of data collected in a wide range of fields in psychology. R. B. Anderson and Tweney (1997) suggested that the model’s data-fitting success may in part be artifactual, caused by a number of factors, one of which is the use of improper data averaging methods. The present paper follows up on their work and explains causes of the power law artifact. A method for studying the geometric relations among responses generated by mathematical models is introduced that shows the artifact is a result of the combined contributions of three factors: arithmetic averaging of data that are generated from a nonlinear model in the presence of individual differences.
Article PDF
Similar content being viewed by others
References
Anderson, J. R., &Schooler, L. J. (1991). Reflections of the environment in memory.Psychological Science,2, 396–408.
Anderson, R. B., &Tweney, R. D. (1997). Artifactual power curves in forgetting.Memory & Cognition,25, 724–730.
Ashby, F. G., Maddox, W. T., &Lee, W. W. (1994). On the dangers of averaging across subjects when using multidimensional scaling or the similarity-choice model.Psychological Science,5, 144–151.
Bates, D. M., &Watts, D. G. (1988).Nonlinear regression analysis and its applications. New York: Wiley.
Bryk, A. S., &Raudenbush, S. W. (1987). Application of hierarchical linear models to assessing change.Psychological Bulletin,101, 147–158.
Bryk, A. S., &Raudenbush, S. W. (1992).Hierarchical linear models: Applications and data analysis methods. Newbury Park, CA: Sage.
Bryk, A. S., Raudenbush, S. W., &Congdon, R. (1992).Hierarchical linear model and nonlinear modeling with HLM/2L and HLM/3L programs. Chicago: Scientific Software.
Cudeck, R. (1996). Mixed effects models in the study of individual differences with repeated measures data.Multivariate Behavioral Research,31, 371–403.
Ebbinghaus, H. (1964).Memory: A contribution to experimental psychology (H. A. Ruger & C. E. Bussenius, Trans). New York: Dover. (Original work published in 1885)
Estes, W. K. (1956). The problem of inference from curves based on group data.Psychological Review,53, 134–140.
Kim, C. (1998).Incorporating individual differences in mathematical psychology. Unpublished doctoral dissertation, Ohio State University. Melton, A. W. (1936). The end-spurt in memorization curves as an artifact of the averaging of individual curves. Psychological Monographs, 47 (2, Whole No. 202).
Newell, A., &Rosenbloom, P. S. (1981). Mechanisms of skill acquisition and the law of practice. In J. R. Anderson (Ed.),Cognition skills and their acquisition (pp. 1–55). Hillsdale, NJ: Erlbaum.
Peterson, L. R., &Peterson, M. J. (1959). Short-term retention of individual verbal items.Journal of Experimental Psychology,58, 193–198.
Rubin, D. C., &Wenzel, A. E. (1996). One hundred years of forgetting: A quantitative description of retention.Psychological Review,103, 734–760.
Siegler, R. S. (1987). The perils of averaging data over strategies: An example from children’s addition.Journal of Experimental Psychology: General,116, 250–264.
Singh, R. (1996). Subtractive versus ratio model of “fair” allocation: Can the group level analyses be misleading?Organizational Behavior & Human Decision Processes,68, 123–144.
Stevens, S. S. (1971). Neural events and the psychophysical law.Science,170, 1043–1050.
Stevenson, M. K. (1993). Decision making with long-term consequences: Temporal discounting for single and multiple outcomes in the future.Journal of Experimental Psychology: General,122, 3–22.
Vonesh, E. F., &Chinchilli, V. M. (1997).Linear and nonlinear models for the analysis of repeated measurements. New York: Marcel Dekker.
Wickens, T. D. (1998). On the form of the retention function: Comment on Rubin and Wenzel (1996): A qualitative description of retention.Psychological Review,105, 379–386.
Wixted, J. T., &Ebbesen, E. B. (1991). On the form of forgetting.Psychological Science,2, 409–415.
Wixted, J. T., &Ebbesen, E. B. (1997). Genuine power curves in forgetting: A quantitative analysis of individual subject forgetting functions.Memory & Cognition,25, 731–739.
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper is based on a presentation at the thirty-first annual meeting of the Society for Mathematical Psychology in August 1998. This research was partly supported by NIMH Grant MH57472 to I.J.M. and M.A.P.
Rights and permissions
About this article
Cite this article
Myung, I.J., Kim, C. & Pitt, M.A. Toward an explanation of the power law artifact: Insights from response surface analysis. Memory & Cognition 28, 832–840 (2000). https://doi.org/10.3758/BF03198418
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.3758/BF03198418