Stevens' Power Law - Criticisms

Criticisms

Stevens generally collected magnitude estimation data from multiple observers, averaged the data across subjects, and then fitted a power function to the data. Because the fit was generally reasonable, he concluded the power law was correct. This approach ignores any individual differences that may obtain and indeed it has been reported that the power relationship does not always hold as well when data are considered separately for individual respondents (Green & Luce 1974).

Another issue is that the approach provides neither a direct test of the power law itself nor the underlying assumptions of the magnitude estimation/production method.

Stevens' main assertion was that using magnitude estimations/productions respondents were able to make judgements on a ratio scale (i.e., if x and y are values on a given ratio scale, then there exists a constant k such that x = ky). In the context of axiomatic psychophysics, (Narens 1996) formulated a testable property capturing the implicit underlying assumption this assertion entailed. Specifically, for two proportions p and q, and three stimuli, x, y, z, if y is judged p times x, z is judged q times y, then t = pq times x should be equal to z. This amounts to assuming that respondents interpret numbers in a veridical way. This property was unambiguously rejected (Ellermeier & Faulhammer 2000)(Zimmer 2005). Without assuming veridical interpretation of numbers, (Narens 1996) formulated another property that, if sustained, meant that respondents could make ratio scaled judgments, namely, if y is judged p times x, z is judged q times y, and if y' is judged q times x, z' is judged p times y', then z should equal z'. This property has been sustained in a variety of situations (Ellermeier & Faulhammer 2000)(Zimmer 2005).

Because Stevens fit power functions to data, his method did not provide a direct test of the power law itself. (Luce 2002), under the condition that respondents' numerical distortion function and the psychophysical functions could be separated, formulated a behavioral condition equivalent to the psychophysical function being a power function. This condition was confirmed for just over half the respondents and the power form was found to be a reasonable approximation for the rest (Steingrimsson & Luce 2006).

It has also been questioned, particularly in terms of signal detection theory, whether any given stimulus is actually associated with a particular and absolute perceived intensity; i.e. one that is independent of contextual factors and conditions. Consistent with this, Luce (1990, p. 73) observed that "by introducing contexts such as background noise in loudness judgements, the shape of the magnitude estimation functions certainly deviates sharply from a power function".