Two operations for "ratios" and "differences" of distances on the mental map

Ss judged "ratios of distances" and "differences of distances" between pairs of U.S. cities. Results fit the theory that Ss used two comparison processes as instructed. A ratio scale of distances between cities was constructed from the 2 rank orders. From this scale, an interval scale of the city locations on an east-west continuum was derived. This scale agrees with the subtractive model fit to "ratios" and "differences" of easterliness and westerliness, and it also agrees with multidimensional scaling of judged distances between cities. These findings are consistent with the theory that Ss use subtraction when instructed to judge either "ratios" or "differences," but that they can use both ratio and difference operations when the stimuli (in this case, distances) constitute a ratio scale on the subjective continuum.     

Measurement and the mental map

In order to test between subtractive and ratio theories of stimulus comparison, judges were asked to estimate “ratios” and “differences” of easterliness and westerliness of U.S. cities. “Difference” judgments fit the subtractive model, and “ratio” judgments fit the ratio model. However, “ratios” and “differences” were monotonically related, contrary to the theory that judges compute both relations on a common scale. Results are consistent with the theory that there is but one operation for both “ratios” and “differences.” To assume that the single operation is a ratio requires the complex interpretation that easterliness and westerliness are nonlinearly related. A simpler interpretation is provided by a subtractive theory, in which all four types of judgments are monotonically related to subjective differences on a single cognitive map.     

Pervasiveness and magnitude of context effects: evidence for the relativity of absolute magnitude estimation

To test the assertion that absolute magnitude estimation serves to minimize context effects, two experiments were conducted in which area stimuli were judged under differing conditions. In Experiment 1, four groups of subjects made magnitude estimations of triangles ranging in area from 1.5 to 3,072 cm2. No standard or modulus was used, and instructions were similar to those used in absolute magnitude estimation experiments. Each group first judged a different subrange of the stimuli (1.5-24; 48-768; 6-96; or 192-3,072 cm2) before making judgments of the remaining stimuli. In Experiment 2, two groups of subjects made magnitude estimations of triangles ranging in area from 1.5 to 12,288 cm2, with each group first judging a different subrange of stimuli (1.5-24 cm2 or 768-12,288 cm2). The design and instructions were virtually identical to those used in absolute magnitude estimation experiments. Our results indicate that the wording of the instructions is not crucial and that judgments are influenced in two ways that are not predicted by proponents of absolute magnitude estimation. First, the power functions fit to the initially presented subranges (e.g., 1.5-24 cm2), which were judged without contextual effects produced by previously presented stimuli, were inconsistent with one another. Second, judgments of the remaining stimuli were influenced by the subrange of stimuli judged initially. The prevalence of context effects in both experiments, in spite of instructional differences, suggests that although one should avoid using a standard and modulus, there is little else to be gained by adopting the absolute magnitude estimation procedure.     

Weight of evidence supports one operation for "ratios" and "differences" of heaviness

This paper investigates an apparent contradiction between recent studies of "ratios" and "differences" of heaviness. Birnbaum and Veit (1974) found a single rank order for judgments in the two tasks, whereas Rule, Curtis, and Mullin (1981), who used a different stimulus set, procedure, and experimental design, reported two orders. To investigate the cause of this discrepancy, the present study manipulated the experimental design using the same stimuli and procedure as Rule et al. (1981). In one experiment (within-subject designs), each subject judged all combinations of the standard and comparison stimulus; in the other experiment (between-subjects designs) each subject received only one standard, and different groups of subjects were given different standards. "Ratios" and "differences" of heaviness were monotonically related for the majority of subjects who judged all combinations of standards and comparisons. Variations in the modulus and response examples did not affect the rank order of "ratios" within subjects. These results suggest that the contradiction in results is due to the difference in experimental design rather than differences in stimuli or procedure. In the between-subjects designs, the rank order of the "ratio" judgments depended on the standards and examples. Both previous and present results are consistent with the theory that subjects use one operation, subtraction, for both tasks and that the judgment function varies with between-subjects manipulations of the standard, examples, and modulus.     

Subjective ratios and differences in perceived heaviness

In previous studies, judgments of ratios and differences in subjective magnitude have yielded similar orders, consistent with a hypothesis that a single perceived relation underlies both judgment tasks. In the present research, 15 subjects estimated heaviness differences between 28 pairs of eight weights and each of 8 groups of 10 subjects evaluated heaviness ratios of eight variable stimuli with respect to a different standard stimulus. Presenting stimuli that were equally spaced on a cube-root scale of weight enhanced expected ordinal discrepancies between ratio and difference estimates, and employing independent groups for each standard stimulus in ratio estimation eliminated a possible bias due to varying standards within the presentation sequence. Differences in orders of ratio and difference estimates together with differences in scales obtained from non-metric analyses in terms of a difference model indicated that the judgments were based on two perceived relations that are ordinally consistent with arithmetic operations of ratios and differences. A ratio scale of heaviness was derived from the combined orders of subjective ratios and differences.     

Nonmetric tests of ratio vs. subtractive theories of stimulus comparison

Theories of stimulus comparison were tested by examining ordinal properties of data obtained with six scaling tasks. Subjects judged simple “ratios” or “differences” of stimulus pairs constructed from a factorial design. In four additional tasks, the same judges also compared relations between pairs of stimulus pairs, judging “ratios of ratios,” “ratios of differences,” “differences of ratios,” and “differences of differences.” The data were consistent with a subtractive theory, which asserts that two stimuli are compared by subtraction, regardless of the task, but that judges can compare two stimulus differences by either a ratio or a difference. All six tasks could be related by the subtractive theory using a single set of scale values. Other simple theories, including the theory that “ratio” judgments can be represented by a ratio model, could not reproduce the six rank orders (of the six sets of data) using a single set of scale values.     

Power functions of loudness magnitude estimations and auditory brainstem evoked responses

The correspondence between subjective and neural response to change in acoustic intensity was considered by deriving power functions from subjective loudness estimations and from the amplitude and latency of auditory brainstem evoked response components (BER). Thirty-six subjects provided loudness magnitude estimations of 2-sec trains of positive polarity click stimuli, 20/sec, at intensity levels ranging from 55 to 90 dB in 5-dB steps. The loudness power function yielded an exponent of .48. With longer trains of the same click stimuli, the exponents of BER latency measures ranged from -.14 for wave I to -.03 for later waves. The exponents of BER amplitude-intensity functions ranged from .40 to .19. Although these exponents tended to be larger than exponents previously reported, they were all lower than the exponent derived from the subjective loudness estimates, and a clear correspondence between the exponents of the loudness and BER component intensity functions was not found.     

Loudness “ratios” and “differences” involve the same psychophysical operation

Subjects judged both “atios” of loudness and “differences” in loudness between pairs of tones that varied in intensity. The pairs were constructed from factorial designs, permitting separation of stimulus and response scaling for each subject. Ratings of “differences” and estimations of “ratios” were monotonically related, inconsistent with the hypothesis that subjects perform both subtractive and ratio operations on a common scale. Instead, the data suggest that both tasks involve the same psychophysical comparison operation with different response transformations. If the operation can be represented by the subtractive model, then category ratings involve a nearly linear transformation and magnitude estimations involve a nearly exponential transformation.     

Stimulus discriminability in the magnitude estimation and category rating of loudness

Thirty-three Ss made category ratings and magnitude estimations of 10 auditory stimuli differing in loudness. The results from each task were examined in terms of the response uncertainty conditional upon each stimulus. The results did not support the suggestion that category judgments are influenced by the relative discriminability of stimuli in a way which is not characteristic of magnitude estimates, but were found to be consistent with the subjective Standard hypothesis. It is argued that the observed quasi-logarithmic relationship between category scale and ratio scale values reflects the constraints placed upon responses in category rating tasks.     

"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.     

“Ratios” and “differences” in perceived sweetness intensity

For a number of perceptual continua, it has been shown in previous studies that subjects use only one quantitative comparison between two sensory impressions of a pair of stimuli, irrespective of whether they are instructed to judge “ratios” or “differences”. This comparison can be described by algebraic subtraction. The present study was designed to investigate whether this one-operation theory for psychophysical judgment also applied to the sensory continuum of sweetness. Subjects were presented with pairs of fructose solutions, and judged “ratios” of, or “differences" in, perceived sweetness intensities. The pairs were constructed on the basis of a factorial judgment design. The results showed that the reported “differences” could be adequately described by a difference response model, and that the reported “ratios” could be adequately described by a ratio response model. However, the reported “ratios” and reported “differences” were monotonically related, and the marginal means of the log-transformed response matrix of “ratios” were a linear function of the marginal means of the response matrix of “differences”. These results are incompatible with the notion that subjects judged differences when instructed to judge “differences”, but ratios when instructed to judge “ratios”. The consistency of the ratio response model with “ratio” judgments is probably caused by a comparative operation based on “differences” in combination with an exponential response output function. It may be concluded that subjects judge only “differences”, and not “ratios”, between perceived sweetness intensities.     

PSYCHOPHYSICAL SCALES OF THE ODOR INTENSITY OF AMYL ACETATE

The paper presents psychophysical scales of amyl acetate in benzyl benzoate sniffed from cotton. Four scales obtained by direct scaling procedures, ratio estimation and magnitude estimation, yield functions of the form R = kSⁿ , with n ranging from 0.39 to 0.57. These data support earlier findings that the intensity of smell is a negatively accelerated function of stimulus intensity. In addition, comparison of magnitude and category scales indicate that the subjective intensity of smell is a prothetic continuum. Finally, further analysis of subjective ratios as a function of stimulus ratios again shows that ( a ) the magnitude scale is a ratio scale and ( b ) the function obtained conforms to the power law.     

Magnitude estimation of apparent sums and differences

Os first scaled two continua by magnitude estimation: apparent area of circles and loudness of 1,000-Hz tones. They then gave magnitude estimations of apparent sums and apparent differences for IS pairs of stimuli on each of the two continua. The scales for sums and differences were in some cases nearly linearly related to the power function obtained when the same as scaled the underlying continuum. However, systematic departure from linearity was the usual result. The power law exponents obtained were generally smaller than those usually reported for the two sensory continua.     

Measurement foundations for multiattribute psychophysical theories based on first order polynomials

The class of first order polynomial measurement representations is defined, and a method for proving the existence of such representations is described. The method is used to prove the existence of first order polynomial generalizations of expected utility theory, difference measurement, and additive conjoint measurement. It is then argued that first order polynomial representations provide a deep and far reaching characterization of psychological invariance for subjective magnitudes of multiattributed stimuli. To substantiate this point, two applications of first order polynomial representation theory to the foundations of psychophysics are described. First, Relation theory, a theory of subjective magnitude proposed by Shepard (Journal of Mathematical Psychology, 1981, 24, 21–57) and Krantz (Journal of Mathematical Psychology, 1972, 9, 168–199), is generalized to a theory of magnitude for multiattributed stimuli. The generalization is based on a postulate of context invariance for the constituent uniattribute magnitudes of a multiattribute magnitude. Second, the power law for subjective magnitude is generalized to a multiattribute version of the power law. Finally, it is argued that a common logical pattern underlies multiattribute generalizations of psychological theories to first order polynomial representations. This abstract pattern suggests a strategy for theory construction in multiattribute psychophysics.     

Judgments of differences and ratios of numerals

Each subject performed two tasks, dividing a line segment so that either (a) theratio of subjective lengths corresponded to the ratio of the magnitudes of two numerals or (b) thedifference in length was proportional to the numerical difference. Had subjects actually performed two operations on the same scale, the responses would have been nonmonotonically related. Instead, data for the two tasks were nearly identical and ordinally compatible with either a ratio or a subtractive model. The ratio model implied scale values for numerals that were a positively accelerated function of numerical value, inconsistent with previous results. With a nonlinear response function for graphic length, the subtractive model fit well, yielding scale values that were a negatively accelerated function of numerical value and a linear function of previously obtained scales. These results, together with other recent findings, suggest that subjects may perform the same operation in spite of instructions to judge “ratios” or “differences” and that this operation can be best represented by a subtractive model.     

Direct quantitative judgments of sums and a two-stage model for psychophysical judgments

Judged magnitudes of differences between stimuli have previously been shown to support a two-stage interpretation of magnitude estimation, in which input transformations and output transformations are each describable as power functions. In an effort to provide support for the model independent of the difference estimation procedure. the present investigation employed two additional judgment tasks. We obtained magnitude judgments and category judgments of the combined magnitudes (sums) of paired weights from two groups of Ss. Values of the inferred input exponent k calculated from the two sets of data were very similar and were also remarkably similar to the exponent previously calculated from magnitude estimations of differences between weights. The output exponent calculated from magnitude judgments of sums described a concave upward function; however. the similar function describing category judgments was essentially linear. These results show that the inferred input exponent is not the result of the difference estimation task, and in addition provides support for the contention that the interval scale may be a less biased sensory measure than the magnitude scale. The introduction of an additive constant to the model improved its fit to the data but the rule by which it was introduced made very little difference.     

Torgerson’s conjecture and Luce’s magnitude production representation imply an empirically false property

The prediction presented is based upon the empirically well sustained magnitude production representation that arose in both of Luce’s global psychophysical theories for subjective intensity of binary and unary continua coupled with Torgerson’s (1961) conjecture that respondents fail to distinguish subjective differences from subjective ratios. When applied to eqisections and fractionation the conjecture implies that the cognitive distortion function of the magnitude production representation is the identity function, which is firmly rejected by existing data.     

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.     

On the form and location of the Psychometric Bisection Function for time

The Psychometric Bisection Function for time relates the discriminability of intermediate duration stimuli to a short and long training duration. Bisection Functions for animals (R. M. Church & M. Z. Deluty, Journal of Experimental Psychology: Animal Behavior Processes, 1977, 3, 216–228) confirm Weber's Law and also show indifference between short and long reports at the geometric mean of the training durations. Two discrimination processes are studied which, in combination with different constructions of the subjective time scale, result in Bisection Functions which differ in form and location. The two discrimination processes use a likelihood ratio rule or a similarity rule to compare intermediate durations to the training durations. These rules in combination with two different constructions of the subjective time scale result in four models which conform to Weber's Law. For one of the scales subjective time is a power function of real time with the scalar property on variance (Scalar Timing). For the other, subjective time is a logarithmic function of real time with constant variance (Log Timing). Both Log and Scalar Timing assume normality on the subjective scale. Only three of these models also entail the geometric mean at the indifference point. The exception is Scalar Timing with the likelihood ratio discrimination rule. This model entails indifference at approximately the harmonic mean of the training stimuli. Variants of the remaining three models differ theoretically but alternatives are difficult to discriminate empirically. A contrast is provided by a Poisson Timing subjective scale in which variance increases directly with the mean. This scale results in indifference at the geometric mean for both discrimination rules but violates Weber's Law in both cases.