Bias effects on magnitude and ratio estimation power function exponents

A bias model of relative judgment was used to derive a ratio estimation (RE) power function, and its effectiveness in providing estimates of exponents free of the effects of standards was evaluated. The RE bias model was compared with the simple RE power function that ignores bias. Results showed that when bias was not taken into account, estimates of exponents exhibited the usual effects of standards observed in previous research. However, the introduction of bias parameters into the RE power function virtually eliminated these effects. Exponents calculated from "equal-range segments" (e.g., low stimulus range vs. high stimulus range) judged by magnitude estimation (ME) were examined: the effects of equal-range segments on exponents were much stronger for ME than standards were for RE, using the bias model.     

Neutral vs ego-orienting instructions: Effects on judgments of magnitude estimation

Three experiments were performed to examine the relative constancy of the exponent in the psychophysical power law under varying motivating conditions. The method of magnitude estimation was used to obtain judgments of apparent tactual roughness or of apparent area size of squares. Patterns of the qualitative observations of the three Es and of the various exponents for the six groups of Ss indicated that neutral instructions and “ego-orienting” instructions, which were perceived as unbelievable coming from an equal fellow student, both yielded exponents identical to those reported in the literature. Believable ego-orienting instructions given by an E of clearly perceived higher social status produced a statistically significantly lower exponent than neutral. Intermediate conditions, wherein Ss apparently disbelieved both types of instructions, but assumed that the superior-status E was “analyzing” them, yielded exponents of intermediate size. Results and supplementary trend analyses are discussed as possible, highly sensitive indicators of motivational impacts on sensory judgments.     

Cross-modality matching functions generated by magnitude estimation

It is possible to generate cross-modality matching functions by having subjects make magnitude estimates of sets of stimuli appropriate to different modalities. The sets are interspersed among each other in the same test session and judged on a common absolute scale of sensory magnitude. An appropriate statistical device locates stimulus levels that appear, on the average, to match. The method is fast, efficient, circumvents the need for continuous stimulus adjustment, and holds promise for the study of the individual as well as the average psychophysical function. To illustrate its potential uses, advantages, and limitations, we used the method to generate cross-modality matching functions relating loudness and brightness. Compared to the scales of loudness and brightness generated by the magnitude estimations of the same stimuli, the matching functions (1) conform better to power functions, (2) may show less variation in slope (exponent), and (3) show far less variation in absolute magnitude (position).     

Unmasking the magnitude estimation response

The fuzzy judgement model of Ward (1979) predicts an inverse relation between the amount of stimulus information available to subjects and the magnitude of sequential dependencies on previous stimuli and responses in psychophysical scaling tasks. Ward confirmed this prediction for magnitude estimations of interdot distance for previous responses but not for previous stimuli, although the inverse relation has been repeatedly reported for both the previous stimuli and responses in absolute identification (e.g., Mori, 1989). This paper further explores this seemingly puzzling contradiction. A magnitude estimation of loudness experiment was conducted in which the amount of stimulus information available to subjects was manipulated by a modified version of informational masking (Watson, 1987). An absolute-identification-with-feedback experiment was also conducted to check the effectiveness of the informational masking in reducing the amount of stimulus information. The results of the magnitude estimation experiment show a striking similarity with those of Ward and generalize the failure of sequential dependencies on previous stimuli to vary inversely with stimulus information. An additional assumption that judgement strategies are altered under low-information conditions is necessary to explain this result.     

Matching as a multivariate power law: Frequency of behavior versus frequency and magnitude of reinforcement

Two groups of food-deprived rats responded for access to liquid reinforcers on a wide range of concurrent variable-interval schedules. During the first part of the experiment, one group chose between two different sucrose solutions; for a second group, the choice involved a sucrose solution and a solution of condensed milk. In the latter part of the experiment, the two groups chose between identical sucrose solutions. A variety of multivariate equations were applied to data obtained from both conditions. The best-fitting equation proved to be a multivariate power function, which is an extension of Herrnstein's basic matching law.     

Scaling of apparent accentedness by magnitude estimation and sensory modality matching

In each of three experiments, 24 students judged the accentedness present in the speech of eight Spanish-English bilinguals.Ss gave magnitude estimations and also squeezed a hand dynamometer to indicate the amount of accentedness in the reading of an English passage by each of the speakers. There was significant agreement amongSs regarding the speech samples with each scaling method, and interscale agreement was good. Power functions fitted to the data had exponents falling in the range expected from earlier psychophysical studies. Scale values correlated significantly with the frequency of accented pronunciations by the speakers as judged by two independent judges. The use of these scaling methods for future research on linguistic features of accent and on the relation between accent and language attitudes is discussed.     

Stability of individual loudness functions obtained by magnitude estimation and production

A correlational analysis of individual magnitude estimation and production exponents at the same frequency was perfor.med, as well as an analysis of individual exponents produced in different sessions by the same procedure across frequency(250, 1, 000, and 3, 000 Hz). Taken together, results show, first, that individual exponent differences do not decrease by counterbalancing magnitude estimation with magnitude production, and, second, that individual exponent differences remain stable over time despite changes in stimulus frequency. Further results disclose that although individual magnitude estimation and production exponents do not necessarily obey the .6 power law, it is possible to predict the slope (exponent) of an equal-sensation function averaged for a group of listeners from individual magnitude estimation and production data. Assuming that individual listeners with sensorineural hearing loss also produce stable and reliable magnitude functions, it is also shown that the slope of the loudness-recruitment function measured by magnitude estimation and production can be predicted for individuals with bilateral losses of long duration. Thus, results obtained in normal and in pathological ears suggest that individual listeners can produce loudness judgments that reveal, albeit indirectly, the input-output characteristic of the auditory system.     

Color sensation of normal and anomalous trichromats measured by magnitude estimation

Summary Colors perceived by normal, protanomalous, and deuteranomalous trichromats were scaled by magnitude estimation. As a result, opponent-colors functions were obtained which show the hue sensations produced by spectral lights. For color-normal observers, the loci of maximal hue sensations were different in some aspects from those expected from current opponent-colors theories.The results obtained from the anomalous observers confirm the view that the causes of prot- and of deuteranomalous vision consist in reverse shifts of the long- and middle-wavelength cone sensitivities respectively. It was found that anomalous trichromats perceive less red in the violet spectrum than do normal subjects. This finding may be explained by the hypothesis that short-wave-length red is produced by a secondary maximum of the long-wavelength cone excitation. In the experiment, the main defect of anomalous trichromats, compared with normal subjects, consisted in the high instability of hue perception and hue recognition. This instability may be explained by an unfavorable signal/noise relation created by the similarity of the two long-wavelength receptor sensibilities in the anomalous red/green systems.     

Individual differences in magnitude estimation of loudness

Twenty-two male and female subjects, aged 15 to 31 years, participated in two sessions, 11 weeks apart, of magnitude estimations of loudness. Stable individual differences in the exponent of the psychophysical power law, ψ=k?ⁿ, were shown. The correlation between subjects’ exponents of the first and second sessions was +.59. The generality of these findings and the origin of the individual differences were discussed.     

Effect of reference number on magnitude estimation

The psychophysical function obtained by the method of magnitude estimation was influenced by the reference number (modulus) assigned to a “standard” line and the position of this standard in the range of comparison stimuli. Data from two experiments with judgments of apparent length of lines show how both variables systematically affect the slope of the power function. AllowingO to choose his own reference numbers, even though these numbers varied among as, tended to produce less variability in slope than ifE imposed fixed reference numbers forO to use.     

The response ratio hypothesis for magnitude estimation

Assume that each presentation of a signal produces two independent random variable representations and that the ratio of responses on successive trials of a magnitude estimation experiment are proportional to the ratio of a representation from the present trial, which representation is then lost, to the remaining one from the previous trial. The mean response to a particular signal depends on the mean of the representation used, but in general exhibits drift over trials and sequential effects due to the preceeding trial; the mean response ratio does not exhibit drift, but it has a simple form only when there are no sequential effects; however, a modified mean ratio function has a simple form. A model suggested by D. V. Cross is a special case of this one. Simple timing and counting models for the representations fail to exhibit sequential effects, contrary to considerable data. However, data of the authors have suggested a version of the timing model in which the sample size of the representation varies by an order of magnitude depending on how close the signal is to the preceding one; this hypothesis accounts for the observed sequential effects and other aspects of the data.     

Two classes of models for magnitude estimation

One class of models assumes that presentation of a signal results in an internal representation as a random variable. Depending on whether the signal is close to or far from the preceding signal, the variance of the representation is smaller or larger. Responses are determined largely by this random variable; however, when the signal is close to the preceding one, the response is generated by modifying the representation multiplicatively by some function of the ratio of the previous response to its representation. Power and linear functions are explored. The form of the random variable is assumed to be that arising from either the timing or the counting model operating on a Poisson process. Detailed analyses are carried out successfully only for the timing model with neural sample sizes independent of intensity; however, the data require the sample to increase with intensity. The linear response function coupled with the constant sample size counting model appears somewhat viable, but detailed calculations are very difficult to carry out. The second class of models postulates a power function relation between magnitude estimates and signals intensity for which the exponent is a Gaussian distributed random variable and the unit is the product of two log normal random variables. Again we assume an attention band such that succesive stimuli that are widely separated in intensity lead to independent samples of the random variables while a variety of assumptions is explored for successive stimuli that are near each other in intensity. Although they each give rise to the qualitative features of the data, estimates of parameters are sufficiently inconsistent that we are led to reject all of the submodels studied.