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.     

Examining the validity of numerical ratios in loudness fractionation

Direct psychophysical scaling procedures presuppose that observers are able to directly relate a numerical value to the sensation magnitude experienced. This assumption is based on fundamental conditions (specified by Luce, 2002), which were evaluated experimentally. The participants' task was to adjust the loudness of a 1-kHz tone so that it reached a certain prespecified fraction of the loudness of a reference tone. The results of the first experiment suggest that the listeners were indeed able to make adjustments on a ratio scale level. It was not possible, however, to interpret the nominal fractions used in the task as "true" scientific numbers. Thus, Stevens's (1956, 1975) fundamental assumption that an observer can directly assess the sensation magnitude a stimulus elicits did not hold. In the second experiment, the possibility of establishing a specific, strictly increasing transformation function that related the overt numerals to the latent mathematical numbers was investigated. The results indicate that this was not possible for the majority of the 7 participants.     

Partial masking in electrocutaneous sensation: a model for sensation matching, with applications to loudness recruitment

A model for partial masking and other threshold-elevation effects is presented in the context of a sensation-matching paradigm. The model is applied to an electrocutaneous experiment in which the subjects adjusted stimulus intensity on the right-hand fingertip to match sensation levels of standard stimuli presented to the left fingertip. Concurrent mechanical stimulation on the right fingertip masked sensation magnitude in a way consistent with the model. Similarities between this tactile masking effect and analogous auditory phenomena are explored. When applied to loudness matching, the model describes the general shape of loudness contours and it shows that the steep slopes observed in auditory masking and "recruitment" can be a consequence of a threshold shift alone, without a supranormal growth in loudness. The model also shows that a small response bias can distort plots of sensation matching, leading to the suggestion that some varieties of loudness recruitment may not have a sensory basis.     

Sensory and cognitive factors in judgments of loudness

One thousand Hertz tones were presented at equal or unequal intensities to the two ears. In a binaural-summation experiment, the presentation of components was simultaneous, the auditory system integrated the components automatically, and the subjects judged the loudness of the unitary sensation. In two cognitive-summation experiments, the presentation of components was successive, and the subjects had to integrate the two sensations consciously to judge their "total loudness." Results of all three experiments are consistent with models of linear summation of "loudness," but the loudness scales differ in the two tasks: The scales that underlie binaural summation and cognitive summation are nonlinearly related. This outcome suggests two nested processes: First, the auditory system transduces stimulus energy to loudness sensations by means of a nonlinear function; second, tasks that require subjects to judge combinational relations between sensations may impose additional nonlinear transformations on the sensations before the latter are combined.     

The perceptual basis of loudness ratio judgments

In Experiment 1, subjects were required to estimateloudness ratios for 45 pairs of tones. Ten 1,200-Hz tones, differing only in intensity, were used to generate the 45 distinct tone pairs. In Experiment 2, subjects were required to directly compare two pairs of tones (chosen from among the set of 45) and indicate which pair of tones had the greaterloudness ratio. In both Experiments 1 and 2, the subjects’ judgments were used to rank order the tone pairs with respect to their judged loudness ratios. Nonmetric analyses of these rank orders indicated that both magnitude estimates of loudness ratios and direct comparisons of loudness ratios were based on loudnessintervals ordifferences where loudness was a power function of sound pressure. These experiments, along with those on loudness difference judgments (Parker & Schneider, 1974; Schneider, Parker, & Stein, 1974), support Torgerson’s (1961) conjecture that there is but one comparative perceptual relationship for ioudnesses, and that differences in numerical estimates for loudness ratios as opposed to loudness intervals simply reflect different reporting strategies generated by the two sets of instructions.     

"Ratio" and "difference" judgments for length, area, and volume: are there two classes of sensory continua?

Subjects were required to judge ratios and differences of (a) line length for pairs of lines, (b) area for pairs of squares, and (c) volume for pairs of cubes. Nonmetric analyses of these judgments indicated that all subjects were able to make consistent ratio judgments for all three continua. Many of the subjects, when asked to judge subjective differences, however, performed as if they were judging subjective ratios rather than differences. The data for the few subjects who appeared to be judging subjective differences were not consistent across subjects and conditions. Previous studies of ratio and difference judgments of loudness and heaviness, on the other hand, showed the opposite pattern, in that subjects most often behaved as if they were judging sensory differences when asked to judge sensory ratios. We propose that ratio judgments are more natural to perceptual continua along which stimuli are easily "decomposed" into a number of smaller perceptual units.     

Stevens vs Fechner: A plea for dismissal of the case

Fechner's Law is based on the assumption that jnd's can be concatenated in order to get a measure of sensation. Stevens contended this assumption; he argued that measurement of sensation by measuring jnd's is ‘indirect’. His ‘direct’ methods, however, appear to be much less direct than originally assumed. Especially Stevens' (implicit) assumption that subjects use numbers on an absolute scale appears to be unwarranted. Ratio scaling is therefore nothing more than cross-modality matching between the number scale and another modality. This view has two consequences. Firstly, it is quite natural to find power functions by cross-modality matching, even when Fechner's Law holds for both modalities. Secondly, if Stevens' power law is true, there will be an overestimation of all exponents by a factor 2.5. All typical power functions will be concave downward after correction for this effect. The discussion boils down to a discrimination between two quite similar curve families: log functions and power functions with an exponent below 0.6. The practical significance of this discussion is doubted.     

Time errors and differential sensation weighting

Sixteen pairs of successive tones, with different amplitude combinations, were presented with 16 combinations of tone duration and interstimulus interval. A separate group of 12 subjects was assigned to each presentation condition and made comparative loudness judgments for each of the pairs. Perceived within-pair loudness differences were scaled by a Thurstonian method using the subjective width of the "equal" category as the unit. The scale differences were well described by weighted linear combinations of the sensation magnitudes of the tones in the pairs. The time error can be regarded as an effect of this differential weighting. For the longer interstimulus intervals, the weight of the second tone was the greater, causing the usual inverse relation between time error and stimulus intensity level. For the shorter interstimulus intervals, these effects were reversed. An analysis of the pattern of weights led to the development of two models, one of which is a generalization of Michels and Helson's time error model. The weights could be interpreted as reflecting the differential efficiency of the loudness information from the two compared stimuli.     

Stimulus context and absolute magnitude estimation: a study of individual differences

The effect of stimulus context on absolute-magnitude-estimation (AME) judgments was examined by determining whether the loudness judgment of a tone is influenced by the intensities of other tones presented within the session. A group of 18 subjects was tested in separate sessions in which they judged stimuli within either a low (10-60 dB SL) or a high (40-90 dB SL) range of intensities. Examination of the results of individual subjects revealed that judgments of stimuli common to the two ranges were, in most subjects, unaffected or only slightly affected by the position of the range. The judgments of 2 subjects who failed to follow the instructions, however, showed very large context effects due to changing the stimulus range. The results of a second experiment, in which 22 subjects judged the loudness of tones within either a narrow (35-65 dB SL) or a wide (20-80 dB SL) range, revealed that, in all but 1 subject, the width of the range had no systematic effect on the loudness judgments of stimuli common to both ranges. This was also true 1 month later when 16 of the subjects returned to the laboratory to judge the loudness of tones within an even wider range of 10-90 dB SL. It was concluded that AME judgments are relatively insensitive to the potential biasing influences of stimulus context.     

Judgments of average magnitude: Analyses in terms of the functional measurement and two-stage models

Subjects made magnitude estimates of the average loudness of pairs of 1,000-Hz tones varying in sound pressure. A test of fit of an averaging model employing an analysis of variance suggested that the judgments were internally consistent. However, estimates of the parameters of a two-stage model based on the assumption that power transformations were imposed in both input and output implied a nonlinear output function, inconsistent with the averaging model. Additional analyses employing a nonmetric scaling solution also suggested that output was nonlinear, indicating that this implication was not an artifact of the strong assumptions of the two-stage model. Large differences were found among the output functions of individual subjects, and it was suggested that these may have inflated the error term in the analysis of variance, reducing its power to detect violations of the additive model. Similar analyses were performed on data from judgments of average grayness collected by Weiss (1972).     

Auditory sensory storage in relation to the growth of sensation and acoustic information extraction

A brief, vivid phase of auditory sensory storage that outlasts the stimulus could be used in perception in two ways: First, all of the neural activity resulting from the stimulus, including that of the sensory store, could contribute to a sensation of growing loudness; second, the sensory store could permit the continued extraction of information about the sound's acoustic properties. This study includes a task for which these two processes lead to different predictions; a third prediction is based on the two processes combined. The task required loudness judgments for two brief tones presented with a variable intertone interval. The results of Experiments 1-3 were as one would expect if both the growth of sensation and information extraction contributed to the pattern of loudness judgments. Experiment 4 strengthened the two-process account by demonstrating the separability of the two processes. Approaches to mathematical modeling of these results are discussed.     

Binaural summation after learning psychophysical functions for loudness

Do response-related processes affect perceptual processes? Sometimes they may: Algom and Marks (1990) produced different loudness exponents by manipulating stimulus range, and thereby also modified the rules of loudness summation determined by magnitude scaling. The present study manipulated exponents by having a dozen subjects learn prescribed power functions with exponents of 0.3, 0.6, or 1.2 (re sound pressure). Subjects gave magnitude estimates of the loudness of binaural signals during training, and of monaural and binaural signals after training. During training, subjects’ responses followed the nominal functions reasonably well. Immediately following training, subjects applied the numeric response scales uniformly to binaural and monaural signals alike; the implicit monaural-binaural loudness matches, and thus the basic rules underlying binaural summation, were unaffected by the exponent learned. Comparison of these results with those of Algom and Marks leads us to conclude that changing stimulus range likely influences underlying perceptual events, whereas “calibrating” a loudness scale through pretraining leaves the perceptual processes unaffected.     

Vibrotactile loudness addition

Os adjusted the intensity of vibration at a single locus on the right hand to a value equal in vibratory loudness to various patterns of vibration on the left hand. The patterns were created by 1 to 5 equated vibration generators, varied with respect to sensation level and distances among the vibrators. The results were: (a) increasing from 1 to 5 vibrators produced a doubling in vibratory loudness, (b) neither loudness level of the components nor distance among vibrators had any effect on the slope of the overall loudness growth function. as also adjusted the intensity of a white noise to equal in magnitude the patterns of vibration presented (a) to the left hand as before and (b) to loci distributed over the surface of the body. The results were the same as those obtained using a single vibrator as standard. The specific loci stimulated did not appear to have any effect on vibrotactile loudness addition.     

Does stimulus context affect loudness or only loudness judgments?

Marks (1988) reported that when equal-loudness matches were inferred from magnitude estimates of loudness for tones of two different frequencies, the matches were affected by changes in the stimulus intensity range at both frequencies. Marks interpreted these results as reflecting the operation of response biases in the subjects' estimates; that is, the effect of range was to alter subjects' judgments but not necessarily the perception of loudness itself. We investigated this effect by having subjects choose which of two tone pairs defined the larger loudness interval. By using tones of two frequencies, and varying their respective intensity ranges, we reproduced Marks' result in a procedure devoid of numerical responses. When the tones at one frequency are all soft, but the tones at the other frequency are not all soft, cross-frequency loudness matches are different from those obtained with other intensity range combinations. This suggests that stimulus range affects the perception of loudness in addition to whatever effects it may have on numerical judgments of loudness.     

Vibrotactile loudness addition

Os adjusted the intensity of vibration at a single locus on the right hand to a value equal in vibratory loudness to various patterns of vibration on the left hand. The patterns were created by 1 to 5 equated vibration generators, varied with respect to sensation level and distances among the vibrators. The results were: (a) increasing from 1 to 5 vibrators produced a doubling in vibratory loudness, (b) neither loudness level of the components nor distance among vibrators had any effect on the slope of the overall loudness growth function. Os also adjusted the intensity of a white noise to equal in magnitude the patterns of vibration presented (a) to the left hand as before and(b) to loci distributed over the surface of the body. The results were the same as those obtained using a single vibrator as standard. The specific loci stimulated did not appear to have any effect on vibrotactile loudness addition.     

Squares in Fork Arrow Logic

In this paper we show that the class of fork squares has a complete orthodox axiomatization in fork arrow logic (FAL). This result may be seen as an orthodox counterpart of Venema's non-orthodox axiomatization for the class of squares in arrow logic. FAL is the modal logic of fork algebras (FAs) just as arrow logic is the modal logic of relation algebras (RAs). FAs extend RAs by a binary fork operator and are axiomatized by adding three equations to RAs equational axiomatization. A proper FA is an algebra of relations where the fork is induced by an injective operation coding pair formation. In contrast to RAs, FAs are representable by proper ones and their equational theory has the expressive power of full first-order logic. A square semantics (the set of arrows is U×U for some set U) for arrow logic was defined by Y. Venema. Due to the negative results about the finite axiomatizability of representable RAs, Venema provided a non-orthodox finite axiomatization for arrow logic by adding a new rule governing the applications of a difference operator. We address here the question of extending the type of relational structures to define orthodox axiomatizations for the class of squares. Given the connections between this problem and the finitization problem addressed by I. Németi, we suspect that this cannot be done by using only logical operations. The modal version of the FA equations provides an orthodox axiomatization for FAL which is complete in view of the representability of FAs. Here we review this result and carry it further to prove that this orthodox axiomatization for FAL also axiomatizes the class of fork squares.     

Monaural loudness adaptation for middle-intensity middle-frequency signals: the importance of measurement technique

Using the Simple Adaptation technique (SA) and the Ipsilateral Comparison Paradigm (ICP), the authors studied monaural loudness adaptation to a middle-intensity [60 dB(A)] tone at signal frequencies of 250, 1000, and 4000 Hz in the left and right ears. Adaptation effects were absent when the SA procedure was used. However, they were observed uniformly across all frequency values with the ICP, a result that challenges the assertion in the literature, on the basis of SA measures, that loudness adaptation for middle-intensity signals occurs only at frequencies above 4000 Hz. The ICP features periodic intensity modulations (+/-10 dB relative to the base signal) to accommodate listeners' needs for referents by which they can gauge subtle changes in the loudness of the adapting tone, a key component that is missing in the SA method. Adaptation effects in this investigation were similar in both ears, supporting the equal susceptibility assumption common in loudness adaptation studies.     

Expectancy-value models of health behaviour: An analysis by conjoint measurement

Various studies in the health area consistently rejected the multiplicative combination between severity and probability of threat which is predicted by expectancy-value (EV) theories. It is hypothesized here, that this negative evidence may be due to an overly demanding assumption underlying the multiplicative combination, namely, the assumption that people are able to performs trade-offs between expectancies and valences. This hypothesis is tested in two studies in which subjects judged hypothetical health threats. Results from a nonparametric analysis (conjoint measurement) of individual data (Study 1) and an experimental study of trade-off judgments (Study 2) are mostly consistent with the prediction. Unexpectedly, however, an ANOVA of the aggregate data of Study 1 yielded a small, but significant effect consistent with the multiplicative assumption. Whereas this latter result can be interpreted as evidencing an attempt to perform trade-offs, the overall results show as predicted that trade-off judgments are associated with a systematic error component due to the inherent difficulty of this type of judgment.     

Monaural Loudness Adaptation for Middle-Intensity Middle-Frequency Signals: The Importance of Measurement Technique

Using the Simple Adaptation technique (SA) and the Ipsilateral Comparison Paradigm (ICP), the authors studied monaural loudness adaptation to a middle-intensity [60 dB(A)] tone at signal frequencies of 250, 1000, and 4000 Hz in the left and right ears. Adaptation effects were absent when the SA procedure was used. However, they were observed uniformly across all frequency values with the ICP, a result that challenges the assertion in the literature, on the basis of SA measures, that loudness adaptation for middle-intensity signals occurs only at frequencies above 4000 Hz. The ICP features periodic intensity modulations (±10 dB relative to the base signal) to accommodate listeners' needs for referents by which they can gauge subtle changes in the loudness of the adapting tone, a key component that is missing in the SA method. Adaptation effects in this investigation were similar in both ears, supporting the equal susceptibility assumption common in loudness adaptation studies.