Memory & Cognition

, Volume 28, Issue 5, pp 832–840 | Cite as

Toward an explanation of the power law artifact: Insights from response surface analysis

  • In Jae MyungEmail author
  • Cheongtag Kim
  • Mark A. Pitt


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.


Response Surface Response Curve Power Function Retention Interval Exponential Model 
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  1. Anderson, J. R., &Schooler, L. J. (1991). Reflections of the environment in memory.Psychological Science,2, 396–408.CrossRefGoogle Scholar
  2. Anderson, R. B., &Tweney, R. D. (1997). Artifactual power curves in forgetting.Memory & Cognition,25, 724–730.Google Scholar
  3. 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.CrossRefGoogle Scholar
  4. Bates, D. M., &Watts, D. G. (1988).Nonlinear regression analysis and its applications. New York: Wiley.CrossRefGoogle Scholar
  5. Bryk, A. S., &Raudenbush, S. W. (1987). Application of hierarchical linear models to assessing change.Psychological Bulletin,101, 147–158.CrossRefGoogle Scholar
  6. Bryk, A. S., &Raudenbush, S. W. (1992).Hierarchical linear models: Applications and data analysis methods. Newbury Park, CA: Sage.Google Scholar
  7. 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.Google Scholar
  8. Cudeck, R. (1996). Mixed effects models in the study of individual differences with repeated measures data.Multivariate Behavioral Research,31, 371–403.CrossRefGoogle Scholar
  9. Ebbinghaus, H. (1964).Memory: A contribution to experimental psychology (H. A. Ruger & C. E. Bussenius, Trans). New York: Dover. (Original work published in 1885)Google Scholar
  10. Estes, W. K. (1956). The problem of inference from curves based on group data.Psychological Review,53, 134–140.Google Scholar
  11. 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).Google Scholar
  12. 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.Google Scholar
  13. Peterson, L. R., &Peterson, M. J. (1959). Short-term retention of individual verbal items.Journal of Experimental Psychology,58, 193–198.CrossRefPubMedGoogle Scholar
  14. Rubin, D. C., &Wenzel, A. E. (1996). One hundred years of forgetting: A quantitative description of retention.Psychological Review,103, 734–760.CrossRefGoogle Scholar
  15. 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.CrossRefGoogle Scholar
  16. 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.CrossRefGoogle Scholar
  17. Stevens, S. S. (1971). Neural events and the psychophysical law.Science,170, 1043–1050.CrossRefGoogle Scholar
  18. 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.CrossRefGoogle Scholar
  19. Vonesh, E. F., &Chinchilli, V. M. (1997).Linear and nonlinear models for the analysis of repeated measurements. New York: Marcel Dekker.Google Scholar
  20. 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.CrossRefGoogle Scholar
  21. Wixted, J. T., &Ebbesen, E. B. (1991). On the form of forgetting.Psychological Science,2, 409–415.CrossRefGoogle Scholar
  22. 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.Google Scholar

Copyright information

© Psychonomic Society, Inc. 2000

Authors and Affiliations

  1. 1.Department of PsychologyOhio State UniversityColumbus

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