Stevens' power law

Stevens' power law is a proposed relationship between the magnitude of a physical stimulus and its perceived intensity or strength. It is widely considered to supersede the Weber-Fechner law on the basis that it describes a wider range of sensations, although critics argue the validity of the law is contingent on the virtue of approaches to the measurement of perceived intensity that are employed in relevant experiments.

The theory is named after psychophysicist Stanley Smith Stevens (1906–1973). Although the idea of a power law had been suggested by 19th century researchers, Stevens is credited with reviving the law and publishing a body of psychophysical data to support it in 1957.

The general form of the law is:
 * $$\psi(I) = k I ^a $$

where $$\psi$$ is the psychophysical function caputuring sensation, k is a constant, I is the magnitude of the physical stimulus, and a is an exponent. The value of a is dependent on the type of stimulation.

The table to the right lists the exponents reported by Stevens.

Methods
The principal methods used by Stevens to measure the perceived intensity of a stimulus were magnitude estimation and magnitude production. In magnitude estimation with a standard, the experimenter presents a stimulus called a standard and assigns it a number called the modulus. For subsequent stimuli, subjects report numerically their perceived intensity relative to the standard so as to preserve the ratio between the sensations and the numerical estimates (e.g., a sound perceived twice as loud as the standard should be given a number twice the modulus). In magnitude estimation without a standard (usually just magnitude estimation), subjects are free to choose their own standard, assigning any number to the first stimulus and all subsequent ones with the only requirement being that the ratio between sensations and numbers is preserved. In magnitude production a number and a reference stimulus is given and subjects produce a stimulus that is perceived as that number times the reference. Also used is cross-modality matching, which generally involves subjects altering the magnitude of one physical quantity, such as the brightness of a light, so that its perceived intensity is equal to the perceived intensity of another type of quantity, such as warmth or pressure.

Criticisms
Stevens generally collected magnitude estimation data from multiple observers, averaged the data across subjects, and then fitted the data to a power function. Because the fit was generally reasonable, he concluded the power law. 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 does not provide a direct test of the power law itself. In the context of axiomatic psychophysics, Naren's (1986) formulated what he thought were the underlying assumptions of Stevens' magnitude estimation/production method. He showed that Stevens had implicitly assumed that respondents used numbers in a veridical way, i.e., without any cognitive distortion. Subsequent empirical evaluations showed Stevens' assumption to be wrong (Ellermeier & Faulhammer, 2000; Zimmer, 2005). Taken these results into account, Luce (2002) formulated a testable behavioral axiom equivalent to the psychphysical function being a power function and Steingrimsson and Luce (2006) found it to hold well for a bit over half the respondents and to be a good approximation for the rest.

The methods of magnitude estimation and product have been criticized for not meeting requirements of measurment. Stevens' approach entails an implicit assumption that reported or produced magnitudes exhibit certain consistencies normally required of measurements. Specifically, consider three stimuli x, y, and z. If it is reported, for example, that the ratio of perceived intensity of z:y is 2:1, and of y:x is also 2:1, then it should be reported that the ratio of perceived intensity of z:x is 4:1. By presenting a series of pairs of stimuli to subjects, it is therefore possible to empirically test the internal consistency of the relations between magnitudes obtained from such experimental procedures. Narens (1996) formally stated and tested these assumptions, and reported negative results.

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.