, Volume 81, Issue 1, pp 201–216 | Cite as

Human Psychophysical Functions, an Update: Methods for Identifying their form; Estimating their Parameters; and Evaluating the Effects of Important Predictors

  • Diana E. Kornbrot


Stevens’ power law for the judgments of sensation has a long history in psychology and is used in many psychophysical investigations of the effects of predictors such as group or condition. Stevens’ formulation \(\varPsi = {aP}^{n}\), where \(\varPsi \) is the psychological judgment, P is the physical intensity, and \(n\) is the power law exponent, is usually tested by plotting log \((\varPsi )\) against log (P). In some, but by no means all, studies, effects on the scale parameter, \(a\), are also investigated. This two-parameter model is simple but known to be flawed, for at least some modalities. Specifically, three-parameter functions that include a threshold parameter produce a better fit for many data sets. In addition, direct non-linear computation of power laws often fit better than regressions of log-transformed variables. However, such potentially flawed methods continue to be used because of assumptions that the approximations are “close enough” as to not to make any difference to the conclusions drawn (or possibly through ignorance the errors in these assumptions). We investigate two modalities in detail: duration and roughness. We show that a three-parameter power law is the best fitting of several plausible models. Comparison between this model and the prevalent two parameter version of Stevens’ power law shows significant differences for the parameter estimates with at least medium effect sizes for duration.


magnitude estimation magnitude production  Stevens’ power law duration roughness individual differences 



I would like to thank Joseph Glicksohn, Robert Teghtsoonian, and a third anonymous reviewer together with Editor Alberto Maydeu-Olivare for extremely perceptive and constructive comments on earlier versions of this Ms. I would also like to thank members of the International Society for Psychophysics for helpful discussions and email comments.

Supplementary material

11336_2014_9418_MOESM1_ESM.xlsx (181 kb)
Supplementary material 1 (xlsx 180 KB)


  1. Allan, L. G. (1983). Magnitude estimation of temporal intervals. Perception and Psychophysics, 33(1), 29–42. doi: 10.3758/BF03205863.CrossRefPubMedGoogle Scholar
  2. Anderson, N. H. (1970). Functional measurement and psychophysical judgment. Psychological Review, 77(3), 153.CrossRefPubMedGoogle Scholar
  3. Beck, A., Ward, C., Mendelson, M., Mock, J., & Erbaugh, J. (1961). An inventory for measuring depression. Archives of General Psychiatry, 4(6), 561–571. doi: 10.1016/S0010-440X(61)80020-5.CrossRefPubMedGoogle Scholar
  4. Borg, G. (1962). Physical performance and perceived exertion (Vol. XI). Lund, Sweden: Gleerup.Google Scholar
  5. Borg, G. (1998). Borg’s perceived exertion and pain scales. Champaign, IL: Human Kinetics.Google Scholar
  6. Borg, G., Hassmen, P., Lagerstrum, M. (1987). Perceived exertion related to heart rate and blood lactate during arm and leg exercise. European Journal of Applied Physiology and Occupational Physiology, 56(6), 679–685. doi: 10.1007/BF00424810.
  7. Borg, G., Lindblad, I., & Holmgren, A. (1981). Quantitative evaluation of chest pain. Acta Medica Scandinavica, 209(S644), 43–45. doi: 10.1111/j.0954-6820.1981.tb03117.x.CrossRefGoogle Scholar
  8. Borg, G., Van Den Burg, M., Hassmen, P., Kaijser, L., & Tanaka, S. (1987). Relationships between perceived exertion hr and hla in cycling running and walking. Scandinavian Journal of Sports Sciences, 9(3), 69–78.Google Scholar
  9. Borg, G. V., & Marks, L. (1983). Twelve meanings of the measure constant in psychophysical power functions. Bulletin of the Psychonomic Society, 21(1), 73–75. doi: 10.3758/BF03329958.CrossRefGoogle Scholar
  10. Edwards, A. L. (1983). Techniques of attitude scale construction. New York, NY: Irvington.Google Scholar
  11. Edwards, W. (1954). The theory of decision making. Psychological Bulletin, 51(4), 380–417.CrossRefPubMedGoogle Scholar
  12. Eisler, H. (1976). Experiments on subjective duration 1868–1975: A collection of power function exponents. Psychological Bulletin, 83(6), 1154–1171. doi: 10.1037/0033-2909.83.6.1154.CrossRefPubMedGoogle Scholar
  13. Eisler, H., & Eisler, A. D. (1992). Time perception: Effects of sex and sound intensity on scales of subjective duration. Scandinavian Journal of Psychology, 33(4), 339–358. doi: 10.1111/j.1467-9450.1992.tb00923.x.CrossRefPubMedGoogle Scholar
  14. Ekman, G. (1959). Weber’s law and related functions. The Journal of Psychology, 47(2), 343–352.CrossRefGoogle Scholar
  15. Florentine, M., & Epstein, M. (2006). To honor stevens and repeal his law (for the auditory system. Paper presented at the Fechner Day 2006. Proceedings of the 22nd Annual Meeting of the International Society for Psychophysics, St. Albans, UK.
  16. Fucci, D., Petrosino, L., Hallowell, B., Andra, L., & Wilcox, C. (1997). Magnitude estimation scaling of annoyance in response to rock music: Effects of sex and listeners’ preference. Perceptual and Motor Skills, 84(2), 663–670.CrossRefPubMedGoogle Scholar
  17. Galanter, E. (1962). The direct measurement of utility and subjective probability. The American Journal of Psychology, 75(2), 208–220.
  18. Galanter, E. (1990). Utility Functions for Nonmonetary Events. The American Journal of Psychology, 103(4), 449–470.
  19. Kahneman, D., & Tversky, A. (1979). Prospect theory. Economerica, 47, 263. doi: 10.2307/1914185.CrossRefGoogle Scholar
  20. Kornbrot, D. E., Donnelly, M., & Galanter, E. (1981). Estimates of utility function parameters from signal-detection experiments. Journal of Experimental Psychology-Human Perception and Performance, 7(2), 441–458. doi: 10.1037/0096-1523.7.2.441.CrossRefGoogle Scholar
  21. Kornbrot, D. E., Msetfi, R. M., & Grimwood, M. J. (2013). Time perception and depressive realism: judgment type, psychophysical functions and bias. PLoS ONE, 8(8), e71585. doi: 10.1371/journal.pone.0071585.
  22. Kornbrot, D. E., Penn, P., Petrie, H., Furner, S., & Hardwick, A. (2007). Roughness perception in haptic virtual reality for sighted and blind people. [Article]. Perception & Psychophysics, 69, pp. 502–512. doi: 10.3758/BF03193907,
  23. Likert, R., Roslow, S., & Murphy, G. (1993). A simple and reliable method of scoring the Thurstone attitude scales. Personnel Psychology, 46(3), 689–690. doi: 10.1111/j.1744-6570.1993.tb00893.x.CrossRefGoogle Scholar
  24. Lindsay, D. R. J., & Norman, D. A. (1977). Human information processing (3rd ed.). London: Academic.Google Scholar
  25. Marks, L. E., & Cain, W. S. (1972). Perception of intervals and magnitudes for three prothetic continua. Journal of Experimental Psychology, 94(1), 6–17. doi: 10.1037/h0032746.CrossRefPubMedGoogle Scholar
  26. Marks, L. E., & Stevens, J. C. (1966). Individual brightness functions. Perception and Psychophysics, 1(1), 17–24. doi: 10.3758/BF03207815.CrossRefGoogle Scholar
  27. Marks, L. E., & Stevens, J. C. (1968). The form of the psychophysical function near threshold. Perception and Psychophysics, 4(5), 315–318. doi: 10.3758/BF03210523.CrossRefGoogle Scholar
  28. McGraw, A. P., Larsen, J. T., Kahneman, D., & Schkade, D. (2010). Comparing gains and losses. Psychological Science, 21(10), 1438–1445. doi: 10.1177/0956797610381504.CrossRefPubMedGoogle Scholar
  29. Mountcastle, V. B., Poggio, G. F., & Werner, G. (1963). The relation of thalamic cell response to peripheral stimuli varied over an intensive continuum. Journal of Neurophysiology, 26, 807–834.PubMedGoogle Scholar
  30. Norman, G. (2010). Likert scales, levels of measurement and the “laws” of statistics. Advances in health sciences education, 15(5), 625–632. doi: 10.1007/s10459-010-9222-y.CrossRefPubMedGoogle Scholar
  31. Sellin, T., & Wolfgang, M. E. (1964). The measurement of delinquency. New York, NY: Wiley.Google Scholar
  32. Sellin, T., & Wolfgang, M. E. (1978). The measurement of delinquency (2nd ed.). New York, NY: Wiley.Google Scholar
  33. Steingrimsson, R. (2011). Evaluating a model of global psychophysical judgments for brightness: II. Behavioral properties linking summations and productions. Attention. Perception and Psychophysics, 73(3), 872–885. doi: 10.3758/s13414-010-0067-5.CrossRefGoogle Scholar
  34. Steingrimsson, R., & Luce, R. D. (2005a). Evaluating a model of global psychophysical judgments. I: Behavioral properties of summations and productions. Journal of Mathematical Psychology, 49(4), 290–307. doi: 10.3758/APP.71.8.1916.CrossRefGoogle Scholar
  35. Steingrimsson, R., & Luce, R. D. (2005). Evaluating a model of global psychophysical judgments. II: Behavioral properties linking summations and productions. Journal of Mathematical Psychology, 49(4), 308–319. doi: 10.1016/ Scholar
  36. Steingrimsson, R., & Luce, R. D. (2006). Empirical evaluation of a model of global psychophysical judgments: III. A form for the psychophysical function and intensity filtering. Journal of Mathematical Psychology, 50(1), 15–29. doi: 10.1016/ Scholar
  37. Steingrimsson, R., & Luce, R. D. (2007). Empirical evaluation of a model of global psychophysical judgments: IV. Forms for the weighting function. Journal of Mathematical Psychology, 51(1), 29–44. doi: 10.1016/ Scholar
  38. Steingrimsson, R., & Luce, R. D. (2012). Predictions from a model of global psychophysics about differences between perceptual and physical matches. Attention Perception and Psychophysics, 74(8), 1668–1680. doi: 10.3758/s13414-012-0334-8.CrossRefGoogle Scholar
  39. Stevens, J. C. (1974). Families of converging power functions in psychophysics. In H. R. Moskowitz, B. Scharf, & J. C. Stevens (Eds.), Sensation and measurement: Papers in honor of S. S. Stevens. Oxford, England: D. Reidel.Google Scholar
  40. Stevens, J. C. (1990). Perceived roughness as a function of body locus. Attention Perception and Psychophysics, 47(3), 298–304. doi: 10.3758/BF03205004.CrossRefGoogle Scholar
  41. Stevens, J. C., & Marks, L. E. (1980). Cross-modality matching functions generated by magnitude estimation. Perception and Psychophysics, 27(5), 379–389. doi: 10.3758/BF03204456.CrossRefPubMedGoogle Scholar
  42. Stevens, J. C., & Marks, L. E. (1999). Stevens power law in vision: Exponents, intercepts, and thresholds. Fechner Day, 99, 82–87.Google Scholar
  43. Stevens, S. S. (1946). On the theory of scales of measurement. Science, 103, 677–680.
  44. Stevens, S. S. (1961). To honour Fechner and repeal his law. Science, 133, 80–86. doi: 10.1126/science.133.3446.80.CrossRefPubMedGoogle Scholar
  45. Stevens, S. S. (1975). Psychophysics. New York, NY: Wiley.Google Scholar
  46. Stevens, S. S., & Galanter, E. H. (1957). Ratio scales and category scales for a dozen perceptual continua. Journal of Experimental Psychology, 54, 377–411. doi: 10.1037/h0043680.CrossRefPubMedGoogle Scholar
  47. Stevens’ Power Law. Retrieved August 18, 2013 from
  48. Teghtsoonian, R. (1971). On the exponents in Stevens’ law and the constant in Ekman’s law. Psychological Review, 78, 71–80.CrossRefPubMedGoogle Scholar
  49. Teghtsoonian, R. (2012). The standard model for perceived magnitude: A framework for (almost) everything known about it. American Journal of Psychology, 125(2), 165–174. doi: 10.5406/amerjpsyc.125.2.0165.CrossRefPubMedGoogle Scholar
  50. Tversky, A. (1967). Additivity, utility and subjective probability. Journal of Mathematical Psychology, 4, 175–202.CrossRefGoogle Scholar
  51. West, R., Ward, L., & Khosla, R. (2000). Constrained scaling: The effect of learned psychophysical scales on idiosyncratic response bias. Attention Perception and Psychophysics, 62(1), 137–151. doi: 10.3758/BF03212067.CrossRefGoogle Scholar

Copyright information

© The Psychometric Society 2014

Authors and Affiliations

  1. 1.Department of PsychologyUniversity of HertfordshireHertfordshireUK

Personalised recommendations