Empirical evaluation of the axioms of multiplicativity, commutativity, and monotonicity in ratio production of area

Although direct scaling methods have been widely used in the behavioral sciences since the 1950s, theoretical approaches which could clarify the implicit assumptions inherent in Stevens' ratio scaling approach were developed only recently. Today, it is generally accepted that the axioms of commutativity and multiplicativity are fundamental to the subjects' ratio scaling behavior. Therefore, both axioms were evaluated in ratio production of area. Participants were required to adjust the area of a variable circle to prescribed ratio production factors. The results are in accordance with previous empirical findings: commutativity was satisfied, whereas multiplicativity failed to hold. Additionally, the validity of the monotonicity property was analyzed, which postulates that the subjects' adjustments in a ratio production experiment preserve the mathematical order of the ratio production factors. Monotonicity was satisfied empirically, which is consistent with all the current theories of ratio scaling.     

Stevens’ Direct Scaling Methods and the Uniqueness Problem

Stevens postulated that we can use the responses of a participant in a ratio scaling experiment directly to construct a psychophysical function representing the participant's sensations. Although Stevens' methods of constructing measurement scales are widely used in the behavioral sciences, the problem of which scale type is appropriate to describe ratio scaling data is still unresolved. To deal with this problem, we develop a theoretical framework to specify the scale type attained by Stevens' direct scaling methods. It is shown, under fairly mild background assumptions, that the behavioral axioms presented in this paper are necessary and sufficient for the psychophysical functions to be ordinal-, interval-, log-interval-, or ratio-scales. Furthermore, suggestions on how to test these behavioral axioms are provided. Requests for reprints should be sent to thomas.     

Extensions of Narens' theory of ratio magnitude estimation

Stevens postulated that the responses of a participant in a ratio scaling experiment can be used directly to construct a psychophysical function. Today, it is generally accepted that the axioms of commutativity and multiplicativity are crucial for the interpretation of the subjects' ratio scaling behaviour. Empirical findings provide evidence that commutativity holds, whereas multiplicativity fails to hold across different sensory modalities. This shows that, in principle, Stevens' direct scaling methods yield measurements on a ratio scale level, but that the numerals occurring in a ratio scaling experiment cannot be taken at face value. Thus, Narens and others introduced a transformation function f, which converts the numerals used in an experiment into the latent mathematical numbers. The aim of the present paper is to specify the (unknown) shape of the transformation function f, by analysing different extensions of the multiplicative property. The results provide evidence that f is either a power function or a logarithmic function.     

On the relation between similarity and ratio estimates

Summary Data obtained in four sets of experiments involving pitch, heaviness, greyness, and circular area were re-analyzed. It was found (1) that similarity estimates S _ijare a power function of stimulus ratios S _ijwith the exponent n _s,(2) that ratio estimates q _ijare also a power function of stimulus ratios S _ijwith the exponent n _q(i.e., Stevens' power law), (3) that the exponent n of similarity estimates as power function of ratio estimates is equal to the ratio n _s/n _q,and (4) that, inversely, the exponent m of ratio estimates as power function of similarities is equal to the ratio n _q/n _s.The investigation was supported by a grant from the Swedish Council for Social Science Research.     

On the psychophysical law and estimation procedures in psychophysical scaling

A general formulation of the power law is presented which has two special features: (1) negative exponents are admissible; and (2) the log law is a special limiting case. Estimation procedures, which provide joint estimates of the exponent and the absolute threshold, are derived for the direct ratio scaling methods. A solution is provided for theaveraging problem for ratio production and bisection scaling, two methods generating observations on the physical scale, and Monte Carlo methods are used to evaluate the resulting estimators.     

Stevens' power law and time perception: effect of filled intervals, duration of the standard, and number of presentations of the standard

Subjects estimated the duration of five different time intervals, either filled or unfilled, in relation to one or two standard intervals. Half of the subjects were presented with the standard at the beginning of the experiment only, and half were presented with the standard before every interval. The five estimations were then used to calculate, for each subject, the exponent in Stevens' Power Law which describes the form of the power relationship. The resulting over-all exponent of .91, although nearly linear, was significantly different from 1.0. The data were then analyzed by a 2 X 2 X 2 analysis of variance which showed a significant interaction between presenting the standard either once or before each interval with the duration of the standard.     

Bias in proportion judgments: the cyclical power model

When participants make part-whole proportion judgments, systematic bias is commonly observed. In some studies, small proportions are overestimated and large proportions underestimated; in other studies, the reverse pattern occurs. Sometimes the bias pattern repeats cyclically with a higher frequency (e.g., overestimation of proportions less than .25 and between .5 and .75; underestimation otherwise). To account for the various bias patterns, a cyclical power model was derived from Stevens' power law. The model proposes that the amplitude of the bias pattern is determined by the Stevens exponent, beta (i.e., the stimulus continuum being judged), and that the frequency of the pattern is determined by a choice of intermediate reference points in the stimulus. When beta < 1, an over-then-under pattern is predicted; when beta > 1, the under-then-over pattern is predicted. Two experiments confirming the model's assumptions are described. A mixed-cycle version of the model is also proposed that predicts observed asymmetries in bias patterns when the set of reference points varies across trials.     

Evaluation of three sensitivity measures in magnitude estimation tasks

Taking into account the studies about the measure of sensitivity in magnitude estimation tasks, we analyze the three most common measures used in this topic: Pearson's product-moment correlation between the logarithm of the stimulus and the logarithm of the response (R), the exponent of Stevens' power function (K), and the measure "M" proposed by Garriga-Trillo. Using a sample of participants greater than usual in psychophysical studies (180 participants), we designed an experiment with two sets of stimuli with different stimulus ranges. In each of these sets, we used two kinds of stimuli (line segments and squares). Our conclusions were: (1) we rejected the use of K as a sensitivity measure because the results provided by this index were the opposite of those expected when we compared the two stimulus ranges. (2) We also rejected the use of M because this measure is a linear transformation of Kendall's coefficient of concordance. (3) Lastly, we suggest the mathematical transformation proposed by Fisher to achieve a normal distribution, and recommend this transformation as the best sensitivity measure.     

BRIGHTNESS SCALES FOR MONOCHROMATIC LIGHT

By means of amethod of ratio estimation, scale values were obtained for the subjective brightness of various physical intensities of monochromatic light of various wave lengths. In a second experiment the scale was constructed by a method of magnitude estimation. The brightness functions were studied by plotting the scale values against stimulus intensity for each wave length. The two experiments gave essentially the same results. It was shown: (1) Brightness of monochromatic light is a power function of stimulus intensity. The exponent of the function is approximately one-third for all wave lengths. (2) Properties of the brightness functions can explain certain empirical relations between brightness, hue and saturation.     

Range and regression effects in magnitude scaling

The relation between power law exponents obtained by magnitude estimation and magnitude production was studied for both loudness and perceived distance. While the results confirm the usual finding of higher values for production for relatively large stimulus ranges, just the opposite occurs when the stimulus range is short, necessitating a revision of the Stevens-Greenbaum regression principle. The relation between range and exponent was explored, both for the case in which several intensities are presented for judgment and for the simpler case of only two intensities. In both cases, a power relation was described relating stimulus ratios to judgmental ratios, with exponents containing both range-dependent and range-independent components.     

A note on cross-modality matching

”Stevens' law”, that the psychophysical law is a power function, is often taken to be confirmed by results of cross-modality matching experiments, for it predicts both the fact that cross-modality matching experiments yield power functions, and the exponents of these power functions. Both these predictions, however, follow from a more general form of the psychophysical law, of which Fechner's law is a special case. In view of this, an interpretation of cross-modality matching based on stimulus discriminability rather than sensation magnitude is proposed.     

Empirical evaluation of axioms fundamental to Stevens's ratio-scaling approach: I. Loudness production

Stevens's direct scaling methods rest on the assumption that subjects are capable of reporting or producing ratios of sensation magnitudes. Only recently, however, did an axiomatization proposed by Narens (1996) specify necessary conditions for this assumption that may be put to an empirical test. In the present investigation, Narens's central axioms of commutativity and multiplicativity were evaluated by having subjects produce loudness ratios. It turned out that the adjustments were consistent with the commutativity condition; multiplicativity (the fact that consecutive doubling and tripling of loudness should be equivalent to making the starting intensity six times as loud), however, was violated in a significant number of cases. According to Narens's (1996) axiomatization, this outcome implies that although in principle a ratio scale of loudness exists, the numbers used by subjects to describe sensation ratios may not be taken at face value.     

SCALES FOR SUBJECTIVE DISTANCE

By means of the method of ratio estimation, scale values were obtained for subjective distance. In three experiments different stimulus ranges of the objective distances were used. It was found: ( 1 ) Subjective distance is a power function of the objective distance. ( 2 ) The exponent of the function varies with the stimulus range. With increasing stimulus range the exponent has a tendency to decrease. It is conceivable that the change of the exponent may be explained by an adaptation of the subjective range to the stimulus range.     

Tests of the psychological meaning of the power law.

The purpose of this study was to establish a theoretical framework for Stevens' empirically derived power law. Three models were proposed to explain the power law. They respectively outline how sensory, stimulus, and response variables determine the judgmental behavior in a psychophysical task. A correlational study on individual differences in exponents was carried out to test the predictions derived from each model. The use of four different sensory continua and four scaling procedures provided the experimental means of manipulating the sensory, stimulus, and response variables in the scaling situation. The results showed that response variables are important determinants of judgmental behavior in psychophysical scaling. These findings suggest that subjects' responses to stimulus intensities in a scaling task are largely cognitive.     

A model of the influence of processing effort and number of changes on subjective time perception

The paper is concerned with a model for subjective time estimation as determined by processing effort and number of changes. In a previous paper, a proposal was made that should solve a problem of confounded variables in the area of time estimation. An experimental design was developed to allow for the independent manipulation of processing effort and number of changes. It is shown that this proposal is valid only under the assumption that Stevens' law holds with exponent one. Therefore, another way of conceptualizing this problem is advanced in which one asks for the relative impact of processing effort and change on the parameters of Stevens' law. It can be demonstrated that this law yields considerable fit despite interindividual differences.     

METHODOLOGICAL NOTE ON SCALES OF GUSTATORY INTENSITY

E kman , G. Methodological note on scales of gustatory intensity. Scand. J. Psychol ., 1961, 2, 185–100.—The subjective salt intensity of seven concentrations of sodium chloride in water was measured by the method of ratio estimation and by three variants of the method of magnitude estimation. The four scales thus constructed were not in agreement. The magnitude scales varied systematically with the stimulus used as standard. The average magnitude scale was in good agreement with the scale constructed by the method of ratio estimation. A combined scale was constructed from all the data. This scale is a power function of stimulus concentration. The exponent of the function is 1.59. The function includes an additive constant, which may indicate either the lack of an absolute zero point of the scale, or the presence of a basic sensation without external stimulation.     

Extensive measurement with incomparability

Standard theories of extensive measurement assume that the objects to be measured form a complete order with respect to the relevant property. In this paper, representation and uniqueness theorems are presented for a theory that departs radically from this completeness assumption. It is first shown that any quasi-order on a countable set can be represented by vectors of real numbers. If such an order is supplemented by a concatenation operator, yielding a relational structure that satisfies a set of axioms similar to the standard axioms for an extensive structure, we obtain a scale possessing the crucial properties of a ratio scale. Incomparability is thus compatible with extensive measurement. The paper ends with a brief discussion on some possible applications and developments of this result.     

The influence of object familiarity on magnitude estimates of apparent size.

Three magnitude-estimation experiments were used to determine the exponents of the power function relating size judgments and physical size for two-dimensional familiar and unfamiliar stimuli. The exponent of the power function was used to index the effect of familiar size on perceived size under a variety of conditions, from full-cue to reduced-cue viewing conditions. Although the value of the exponents varied across the three experiments, within each experiment the exponent of the familiar stimulus was not significantly different from that of the unfamiliar stimulus, indicating that familiar size does not influence the rate of growth of perceived size. The results of a fourth experiment excluded a possible explanation of the findings of experiments 1-3 in terms of subjects responding to relative angular size as a consequence of the successive presentation of the different-sized representations of the familiar stimulus. Taken together, the present findings are consistent with the hypothesis that the influence of familiar size on estimates of size mainly reflects the intrusion of nonperceptual processes in spatial responses.     

Identification variability as a measure of loudness: an application to gender differences.

It is well known that discrimination response variability increases with stimulus intensity, closely related to Weber's Law. It is also an axiom that sensation magnitude increases with stimulus intensity. Following earlier researchers such as Thurstone, Garner, and Durlach and Braida, we explored a new method of exploiting these relationships to estimate the power function exponent relating sound pressure level to loudness, using the accuracy with which listeners could identify the intensity of pure tones. The log standard deviation of the normally distributed identification errors increases linearly with stimulus range in decibels, and the slope, a, of the regression is proportional to the loudness exponent, n. Interestingly, in a demonstration experiment, the loudness exponent estimated in this way is greater for females than for males.     

Empirical evaluation of axioms fundamental to Stevens’s ratio-scaling approach: I. Loudness production

Stevens’s direct scaling methods rest on the assumption that subjects are capable of reporting or producing ratios of sensation magnitudes. Only recently, however, did an axiomatization proposed by Narens (1996) specify necessary conditions for this assumption that may be put to an empirical test. In the present investigation, Narens’s central axioms ofcommutativity andmultiplicativity were evaluated by having subjects produce loudness ratios. It turned out that the adjustments were consistent with the commutativity condition; multiplicativity (the fact that consecutive doubling and tripling of loudness should be equivalent to making the starting intensity six times as loud), however, was violated in a significant number of cases. According to Narens’s (1996) axiomatization, this outcome implies that although in principle a ratio scale of loudness exists, the numbers used by subjects to describe sensation ratios may not be taken at face value.