Sequential effects in successive ratio estimation

Sequential effects are examined in four successive ratio estimation (RE) experiments. The procedure in successive RE is identical to that for magnitude estimation (ME), but the task in successive RE is to estimate the ratio of the current to the previous sensation magnitude, and not the separate magnitudes of the sensations. A positive stimulus context effect was found in successive RE for several continua, in agreement with results previously found for ME. The residual autocorrelation for successive RE was zero in many cases, but in some cases negative autocorrelation was found, which is in contrast to the positive autocorrelation that is typically found for ME and other magnitude scaling procedures. It is shown that, when the role of perceptual error is recognized, negative autocorrelation is predicted by a classical model of ratio estimation. Some aspects of response bias are also discussed.     

Variability and sequential effects in cross-modality matching of area and loudness

Individual subjects' performance was examined for cross-modality matching (CMM) of loudness to visual area, as well as for magnitude estimation (ME) of the component continua. Average exponents of power functions relating response magnitude to stimulus intensity were .73 for area, .20 for loudness, and 2.44 for CMM. Predictions of the CMM exponent based on ME were higher than the empirical values, whereas more accurate predictions were made from magnitude production exponents obtained in a previous study. Sequential dependencies were assessed by comparing the response on trial n to the response on trial n--1. The coefficient of variation of the response ratio Rn/Rn-1 was systematically related to the stimulus ratio Sn/Sn-1 for both area and loudness. The coefficient was lowest for ratios near 1 and increased for larger or smaller values. For CMM, the coefficient of variation appeared to be independent of stimulus ratios. The correlation between log Rn and log Rn-1 was also related to Sn/Sn-1 for both ME and CMM. The correlation was highest when Sn/Sn-1 was 1 and dropped to 0 with increasing stimulus separation, but CMM yielded a shallower function than ME.     

The range effect as a function of stimulus set,presence of a standard,and modulus

The intramodal range effect (an inverse relationship between stimulus range and exponent in Stevens’s power law) has been well documented, but its conditions have not been tested. Both the estimates of stimulus magnitudes and their exponents are affected by context, stimulus location, and different standards and moduli, but how these variables might interact with the variable of stimulus range has not been studied. In the present research, exponents were derived from magnitude estimates of line length for each of three different stimulus: ranges at two different locations on the scale of length, with or without a modulus. Moduli of 50 and 500 permitted an analysis of the effect of response magnitude on the range effect. Because different ranges had stimulus values in common, the effect of range and location on exponents from those common values could be determined. Exponents decreased as stimulus range increased, but only in the free-modulus condition. For that condition, exponents derived from magnitude estimates of only the common stimuli also showed the range effect and response magnitude did not influence the range effect. Exponents were higher for stimulus ranges at the lower location, but location does not appear to contribute to the range effect. Although the range effect is not explained, the conditions under which it holds and some factors that may influence it are considered.     

Empirical evaluation of a model of global psychophysical judgments: III. A form for the psychophysical function and intensity filtering

In part I, a concept of ratio estimation is defined and it is shown that if such estimates depend only upon the physical ratio of the signal to the reference signal, the psychophysical function must be a power function. Assuming the same exponents for each component, an invariance condition, equivalent to a sum of power functions, is studied empirically for binaural loudness. It is fully or partially sustained for 19 of 22 respondents. Since failures may be attributable to different exponents in the two ears, the ratio of the two exponents is estimated but that fails to explain the failures. Other possible explanations are suggested. In part II, an intensity filtering model is presented, accounting for the phenomenon where monaural loudness matches show a bias depending on the matching ear. We show (a) that the existence of such a bias does not alter the prior experimental results; and (b) assuming the power function, that five respondents attenuate the opposite ear and two enhance it.     

An application of a dynamic model of judgment to magnitude production

A dynamic model of judgment, together with a model of stimulus context effects, is applied to magnitude production (MP) and magnitude estimation (ME) experiments. Participants' responses in MP were correlated across trials, as is typically found for ME. The magnitudeof the autocorrelation, however, was small, which suggests that participants in MP tend to rely more heavily on a long-term frame of reference. Second, a stimulus context effect found for ME did not appear for MP, most likely because of the different nature of the task (i.e., intermediate values of the stimulus were heard while the participant produced a response). A fit of an earlier regression model, on the other hand, suggests that the number presented on the previous trial in MP has a large contrastive effect on the current response.The present model offers a different view of this result, in that it shows that a negative coefficient for the earlier model is consistent with a positive judgmental effect. The regression effect noted by Stevens and Greenbaum (1966), which is a value of the estimated ME exponent that is smaller than the inverse of the estimated MP exponent, was also found; it i s shown that the effect did not arise from bias in estimation.     

On bias in magnitude scaling and some conjectures of Stevens

Bias in magnitude scaling can be viewed as involving deviations of judgments from proportionality. A model of bias is shown to provide a theoretical basis for Stevens's conjecture about geometrically averaging magnitude estimation and magnitude production exponents in order to obtain an estimate of the psychophysical exponent. An overlooked result is that one can also obtain an estimate of the magnitude of the bias. Examples from several well-known studies are presented. The bias is also shown to vary in response to experimental manipulation of the stimulus range. Aspects of predicting exponents across experiments are clarified, and a new prediction is examined. The model of bias fills some theoretical gaps in magnitude scaling and clarifies underlying assumptions and predictions.     

Stimulus range,number of categories,and the “virtual” exponent

Three different stimulus modalities (line length, number, and sound pressure) were judged by magnitude scaling techniques and by 7-, 15-, 31-, and 75-point category scales. All of the 40 subjects were given the same number stimuli, but two different sound-pressure ranges were presented (each to 20 subjects) and four different line-length ranges were presented (each to 10 subjects). Analyses of lack of fit for various simple functions were performed to determine bestfitting functions. The simple power function was often found to be an adequate fit to the data for all the response modalities used, although all of the response modalities were sensitive to changes in stimulus range. For simple power functions, the category-scale exponent was a function of both the range of stimuli and the number of categories provided. Category scales did not always produce exponents smaller than those obtained with magnitude estimation, which calls into question the concept of a virtual exponent for category scales.     

Power laws for the perception of rotation and the oculogyral illusion

Magnitude estimation was used to measure subjective motion for two indicators of vestibular function. Twelve as made estimates of 5-sec pulses of angular acceleration across the range of angular acceleration × time (at) =10-150 deg/sec. Results were: (1) the power law describes subjective motion for all individual as, (2) the power function exponent (1.41) for the perception of rotation is slightly greater than the exponent (1.25) for the oculogyral illusion, (3) a significant number of as gave higher exponents for the perception of rotation, and (4) the magnitude estimates of the oculogyral illusion and perception of rotation were highly correlated within and across as.     

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.     

The role of visual-auditory “compellingness” in the ventriloquism effect: Implications for transitivity among the spatial senses

A magnitude estimation response procedure was used to evaluate the strength of visualauditory intersensory bias effects under conditions of spatial discrepancy. Maj or variables were the cognitive compellingness of the stimulus situation and instructions as to the unity or duality of the perceptual event. With a highly compelling stimulus situation and single-event instructions, subjects showed a very high visual bias of audition, a significant auditory bias of vision, and a sum of bias effects that indicated that their perception was fully consonant with the assumption of a single perceptual event. This finding reopens the possibility that the spatial modalities function as a transitive system, an outcome that Pick, Warren, and Hay (1969) had expected but did not obtain. Furthermore, the results support the model for intersensory interaction proposed by Welch and Warren (1980) with respect to the susceptibility of intersensory bias effects to several independent variables. Finally, a new means of assessing intersensory bias effects by the use of spatial separation threshold was demonstrated.     

An empirical test of two psychophysical models

Two theoretical relationships between sensitivity measures (Weber fractions, Ekman fractions, and their logarithms) and the exponents of the psychophysical power function were tested empirically with the brightness attribute. One model was based on Weber and Ekman fractions, the other on the logarithms of these measures. The stimulus parameters were time interval between standard and comparison targets and position of the standard in the luminance series. Weber fractions were based on data obtained by the method of constant stimuli, whereas Ekman fractions and exponents were based on data obtained by magnitude estimation. The results were in closer agreement with the theoretical predictions generated by the logarithmic model when group data were analyzed. With individual subjects, a detailed correspondence between fact and theory was not found with either model.     

Shift in stimulus range and the exponent of the power function for loudness

The exponent of the power function for loudness was tracked over the course of 60 trials with one stimulus range and compared to the exponent over the course of 60 subsequent trials with a different stimulus range. Three stimulus sets were used: (1) weak, a short range of relatively soft tones (45-55 dBA); (2) strong, a short range of relatively loud tones (64-74 dBA); and (3) complete, a longer range of soft to loud tones (40-90 dBA). All pairs of stimulus sets were tested, together with three control conditions in which no shift in range occurred. Ten subjects were run in each of the nine groups. For preshift trials, the mean exponent was lowest for the strong stimulus series, highest for the weak series, and at an intermediate value for the complete series. These differences were all significant. Following a shift in stimulus range, the weak series still yielded the highest exponent, but the exponents were not reliably different for the complete and strong series. Postshift exponents also depended significantly on the preshift range experienced by the subjects. These effects were not confined to the period immediately following the shift in range, but persisted for up to 60 trials.     

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.     

Children use categories to maximize accuracy in estimation

The present study tests a model of category effects upon stimulus estimation in children. Prior work with adults suggests that people inductively generalize distributional information about a category of stimuli and use this information to adjust their estimates of individual stimuli in a way that maximizes average accuracy in estimation (see Huttenlocher, Hedges & Vevea, 2000). However, little is known about the developmental origin of this cognitive process. In the present study, 5- and 7-year-old children viewed stimuli that varied in size and reproduced each from memory. Consistent with the predictions of a Bayesian model of category effects on estimation, responses were adjusted toward the central value of the stimulus distribution. Additionally, the dispersion of the stimulus distribution affected the pattern of bias and variability of responses in a way that is predicted by the model. The results suggest that, like adults, children use categories for increasing average accuracy in estimating inexact stimuli.     

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.     

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.     

Apparent duration of long meaningful events and meaningless intervals

Time estimates of 1 1/2-, 5¼-, and 14 1/2-min intervals were obtained from 12 American graduate students and 12 Indian graduate students by the methods of verbal estimation and cross-modality matching. Material presented during stimulus intervals varied in degree of meaningfulness. Each subject was tested on 4 successive days with basically the same material in order to determine the effects of repetition. The relationship between perceived and physical time was found to follow Stevens’ power law, and confidence limits of exponents obtained in this study include the exponents previously reported for short durations. Neither actual judgments nor exponents were affected by cultural background or by cognitive factors such as memory for material presented in the interval, familiarity, complexity, degree of meaningfulness, and repetition. It had previously been reported that time judgments were dependent on these cognitive factors. In light of the present research, it is necessary to review and replicate those studies which support a cognitive view of time perception.     

Memory mechanisms and the psychophysical scaling of duration

Two experiments were performed to examine the suggestion that underlying memory mechanisms may be revealed in the form of the psychophysical function for duration. In experiment 1 a broad range of durations, with fine spacing at the lower end, was employed to bring out any transition in function that might reflect a change from 'ionic' memory to short-term memory. Estimation in conventional time-units (Verbal Estimation) was also compared with unit-free estimation (Magnitude Estimation). In experiment 2 Verbal Estimation was compared with the Production method, for a different range of stimulus values, and with varying interval content. Contrary to earlier claims, memory mechanisms were not found to be reflected in the values of power exponents for subjective duration. The value of the search for such functions is questioned, as simple linear plots fit the data at least as well.