Perceptual and Motor Skills, 1992, 7 5 , 1205-1206.
O Perceptual and Motor Skills 1992
FECHNER'S PSYCHOPHYSICAL LAW AS A SPECIAL CASE STEVENS' THREE-PARAMETER POWER LAW '
OF
TARALD 0. KVALSETH University of Minnesotu
Summary.-This paper shows that Fechner's law of psychophysics is a special case of Stevens' power law with three parameters. Consequently, at least as good over-all fit to experimental data is always obtained, at the cost of one additional parameter, by using Stevens' power law involving an extra parameter added to the response variable.
The relationship between the stimulus intensity (3) of some physical attribute and the corresponding (mean) magnitude of the subjective esrimate or response (R) is frequently described in terms of two classical psychophysical laws: Stevens' power law and Fechner's logarithmic law (Stevens, 1975). Accordingly, R is (1) a power (geometric) function of S or (2) a Linear function of the logarithm of S. I t has been pointed out that, under certain special conditions, one is a special case of the other (Ekrnan, 1964; Kvalseth, 1981). The purpose of the present communication is to show that a three-parameter form of Stevens' power law always includes Fechner's law as a special case. O n e frequently used form of the power law is given by -
-
where a, 6 , and c are unknown parameters to be estimated from experimental data. For moderate to large values of S, the relationship in Equation 1 without the constant c generally provides good fits to experimental data; however, when the data include s m d S values, substantially better fits may be achieved by including c (alternatively, c is sometimes subtracted from S in Equation 1 and considered to be some threshold value of S ; see, e.g., Falmagne, 1986). To prove that Fechner's law is a particular case of Equation 1, we simply need to use the following alternative expression for Equation 1:
where the new parameters a and are defined in terms of the original a, 6 , and c as (Y = a + c and /3 = ab. By letting b go to zero in Equation 2, and using the well-known L'Hbspital's rule to deal with the intermediate 010 term, we see that Equation 2 reduces to
'Address enquiries to T. 0. KvBlseth, Department of Mechanical Engineering, University of Mhnesota, Minneapolis, MN 55455.
where "log" stands for the natural logarithm. Equation 3 is precisely Fechner's law. Consequently, we have shown that Fechner's law is indeed a special case of Stevens' power law in Equation 1. As a consequence of the fact that Equation 3 is a special case of Equation 1, Fechner's law can never achieve fits to experimental data that are superior to those of the three-parameter form of Stevens' power law in Equation 1 for any given curve-fitting (regression) procedure. Of course, in general, the mere fact that one model contains one more parameter than another model is not sufficient to account for such differences in fits between two models. Even if Equation 3 gives a highly respectable fit to some given set of experimental data, that is, when imposing the restriction that b = 0 (i.e., b approaches zero) in Equation 2 , the fit of Equation 1 to those same data will be at least as good since no restriction is imposed on the b parameter. REFERENCES EKMAN, G. IS the power law a special case of Fechner's law? Pmceptual and Motor Skills, 1964, 19, 730. FALMAGNE, J. C. Psychophysical measurement and theory. I n K. R. Boff, L. Kaufman, & J. P. Thomas (Eds.), Handbook o/perception and human performance. Vol. 1. Sensory processes and perception. New York: Wiley. Pp. 1.1-1.66. K V A L S E ~ - T. I , 0. Is Fechner's logarithmic law a special case of Stevens' power law? Perceptual and Moior Skillr, 1981, 5 2 , 617-618. STEVENS, S. S. Psychopbysics: introduction to its perceptual, neural and social prospects. New York: W i e y , 1975. Accepted October 8, 1992.
O Perceptual and Motor Skills 1992
FECHNER'S PSYCHOPHYSICAL LAW AS A SPECIAL CASE STEVENS' THREE-PARAMETER POWER LAW '
OF
TARALD 0. KVALSETH University of Minnesotu
Summary.-This paper shows that Fechner's law of psychophysics is a special case of Stevens' power law with three parameters. Consequently, at least as good over-all fit to experimental data is always obtained, at the cost of one additional parameter, by using Stevens' power law involving an extra parameter added to the response variable.
The relationship between the stimulus intensity (3) of some physical attribute and the corresponding (mean) magnitude of the subjective esrimate or response (R) is frequently described in terms of two classical psychophysical laws: Stevens' power law and Fechner's logarithmic law (Stevens, 1975). Accordingly, R is (1) a power (geometric) function of S or (2) a Linear function of the logarithm of S. I t has been pointed out that, under certain special conditions, one is a special case of the other (Ekrnan, 1964; Kvalseth, 1981). The purpose of the present communication is to show that a three-parameter form of Stevens' power law always includes Fechner's law as a special case. O n e frequently used form of the power law is given by -
-
where a, 6 , and c are unknown parameters to be estimated from experimental data. For moderate to large values of S, the relationship in Equation 1 without the constant c generally provides good fits to experimental data; however, when the data include s m d S values, substantially better fits may be achieved by including c (alternatively, c is sometimes subtracted from S in Equation 1 and considered to be some threshold value of S ; see, e.g., Falmagne, 1986). To prove that Fechner's law is a particular case of Equation 1, we simply need to use the following alternative expression for Equation 1:
where the new parameters a and are defined in terms of the original a, 6 , and c as (Y = a + c and /3 = ab. By letting b go to zero in Equation 2, and using the well-known L'Hbspital's rule to deal with the intermediate 010 term, we see that Equation 2 reduces to
'Address enquiries to T. 0. KvBlseth, Department of Mechanical Engineering, University of Mhnesota, Minneapolis, MN 55455.
where "log" stands for the natural logarithm. Equation 3 is precisely Fechner's law. Consequently, we have shown that Fechner's law is indeed a special case of Stevens' power law in Equation 1. As a consequence of the fact that Equation 3 is a special case of Equation 1, Fechner's law can never achieve fits to experimental data that are superior to those of the three-parameter form of Stevens' power law in Equation 1 for any given curve-fitting (regression) procedure. Of course, in general, the mere fact that one model contains one more parameter than another model is not sufficient to account for such differences in fits between two models. Even if Equation 3 gives a highly respectable fit to some given set of experimental data, that is, when imposing the restriction that b = 0 (i.e., b approaches zero) in Equation 2 , the fit of Equation 1 to those same data will be at least as good since no restriction is imposed on the b parameter. REFERENCES EKMAN, G. IS the power law a special case of Fechner's law? Pmceptual and Motor Skills, 1964, 19, 730. FALMAGNE, J. C. Psychophysical measurement and theory. I n K. R. Boff, L. Kaufman, & J. P. Thomas (Eds.), Handbook o/perception and human performance. Vol. 1. Sensory processes and perception. New York: Wiley. Pp. 1.1-1.66. K V A L S E ~ - T. I , 0. Is Fechner's logarithmic law a special case of Stevens' power law? Perceptual and Moior Skillr, 1981, 5 2 , 617-618. STEVENS, S. S. Psychopbysics: introduction to its perceptual, neural and social prospects. New York: W i e y , 1975. Accepted October 8, 1992.
