Lester E. Krueger

Reconciling Fechner and Stevens: Toward a unified psychophysical law

How does subjective magnitude, S . increase as physical magnitude or intensity, I , increases? Direct ratings (magnitude scales; partition or category scales) can be fitted by the power function, S = aI b , in which S equals I raised to a power or exponent, b , and multiplied by a measure constant, a . The exponent is typically about twice as large for the magnitude scale (Stevens) as for the corresponding partition or category scale, but the higher exponent may be explained by the overly expansive way people use numbers in making magnitude estimations. The partition or category scale and the adjusted (for the use of number) magnitude scale for a given modality or condition generally agree with the neurelectric scale and the summated just noticeable difference (jnd) scale. A unified psychophysical law is proposed in which each jnd has the same subjective magnitude for a given modality or condition, subjective magnitude increases as approximately a power function of physical magnitude with the exponent ranging from near 0 to 1 (compressive function), and subjective magnitude depends primarily on peripheral sensory processes, that is, no nonlinear central transformations occur. An undue reliance on Webers law blinded Fechner to the fact that the true psychophysical scale is approximately a power function. Rejecting Webers law, which is not valid, means that we no longer have to choose between letting the summated jnd scale be a logarithmic function (Fechners law) and introducing a nonlinear central transformation to make it into a power function (Brentano–Ekman-Teghtsoonians law). Fechner and Stevens erred equally about the true psychophysical power function, whose exponent lies halfway between that of Fechner (an exponent approaching zero) and that of Stevens. To be reconciled, Fechnerians must give up the assumptions that Webers law is valid and that the jnd has the same subjective magnitude across modalities and conditions; Stevensians must give up the assumption that the unadjusted (for the use of number) magnitude scale is a direct measure of subjective magnitude.

The category effect in visual search depends on physical rather than conceptual differences.

Is a target letter (digit) more readily detected in a digit (letter) background because of conceptual-level differences between letters and digits or because of physical-level differences? When letters and digits were matched on physical features, both for the target set and the background set, no letter vs. digit category effect was found. With physical differences eliminated, search was faster and more accurate for letter than for digit targets and distractors, presumably because of the greater familiarity of letters. Presenting the same characters in normal and mirrorreversed orientation, which also minimized featural differences between categories, produced only a small normal vs. reversed category effect. In a normal background, a reversed target lost much more from its unfamiliarity than it gained from mismatching the background. The present results indicate that the category effect vanishes when only conceptual level differences are present.

Perceived numerosity: A comparison of magnitude production, magnitude estimation, and discrimination judgments

In previous studies, modalities with a higher Weber fraction have tended to have a lower power-function exponent. Within a modality, however, the Weber fraction and power-function exponent for individual subjects were unrelated, and the present study largely confirms this finding for the numerosity dimension. More important than discriminability in the judgment of numerosity were cognitive factors. A single feedback trial considerably reduced intersubject variability on the magnitude-estimation exponent, although it failed to eliminate individual differences completely (precue and postcue exponents correlated signigicantly, r=+.50). Intrasubject variability, by contrast, seemingly did not involve the underlying exponent. As in previous studies, numerosity generally was underestimated and the power-function exponent was 1.08 for magnitude production and .80 for precue magnitude estimation. Contrary to previous results, however, males and females did not differ in exponent, perhaps because the present procedure allowed self-selection of individuals more interested in numerosity tasks.

Why 2+2=5 looks so wrong: on the odd-even rule in sum verification

Subjects judged whether the proposed product for two multipliers was true or false. Each equation could be judged as plausible or implausible because a product must be even if any of its multipliers is even; otherwise, it must be odd. A proposed product that violates the required odd-even status of the product—that is, deviates from the correct product, whether odd or even, by an odd value (e.g., splits of ±l, ±3, ±5)—can be rejected as false before normal processing is completed (i.e., before the correct product is retrieved and compared with the proposed product). Subjects were indeed faster and more accurate in rejecting a split of +1 or +3 than a split of +2 or +4, and this effect increased as the number of even multipliers in the pair increased. Subjects did not use the odd-even rule when either multiplier was 0 or 1 (Experiment 2), perhaps because other rules are available to bypass normal processing in those cases. A similar odd-even rule is used in sum verification (Krueger &Hallford), and use of the odd-even rules may help to explain why oddness and evenness are such salient features of numbers as abstract concepts (Shepard, Kilpatric, & Cunningham).

Single judgments of numerosity

In previous studies, subjects generally underestimated the number of elements present in a display. To eliminate the range and intertrial effects that arise when several displays are judged in succession and that might have produced the underestimation, subjects in the present study judged only a single display. The single judgments fitted a power function having an exponent of .83, which is consistent with previous data. Single judgments of loudness, area, and duration, by contrast, have produced abnormally low exponents apparently because the built-in scale unit, or modulus, available on numerosity is lacking for other modalities. The tendency to underestimate numerosity was much stronger for female than for male subjects.

Intertrial effects of same-different judgements

Specific intertrial effects (repetition effects) and general intertrial effects (refractoriness or persisting attention to the preceding trial) were studied with the same-different judgment task, which dissociates the effects of response repetition and stimulus repetition. Response repetition alone did not facilitate performance. Stimulus repetition did aid performance, but mainly when accompanied by response repetition. Subjects tended to avoid the normal comparison process by using the (invalid!) “bypass rule” (Fletcher and Rabbitt, 1978): repeat the response if the stimulus or some aspect thereof (letter contents, size, position) is repeated from the preceding trial, otherwise change the response. As to general effects, partial refractoriness was evident at response execution, but not at earlier processing stages. Mean RT increased, but errors decreased, as the response-stimulus interval (RSI) between trials decreased. Presenting a new letter pair immediately after the preceding response produced a delay, but subjects used the waiting time, while the response system recovered or was redirected to the present trial, to improve the accuracy of their decision.

Further evidence for priming in perceptual matching: Temporal, not spatial, separation enhances the fast-same effect

Sequential (vs. simultaneous) presentation of two letters in a physical-identity matching task enhanced the fast-same effect, but did not reduce the preponderance of false-“different” errors or the effect of visual similarity. Thus, the sequential enhancement of the fast-same effect involves an increased efficiency in encoding (d’) owing to letter repetition, as Proctor claimed, rather than a criterion shift (β), and it involves the visual code rather than the name code. The increased efficiency in detecting sameness with sequential presentation might result from spatial separation (e.g., reduced lateral interference and self-termination), though, rather than temporal separation (e.g., priming), However, such spatial factors as letter size and interletter spacing had no consistent effect on the speed advantage forsame pairs, and it was concluded that temporal, not spatial, separation enhances the fast-same effect on sequential trials, Consistent with the response-competition model (Eriksen, O’Hara, & Eriksen), responses were slower and more errorful to similar than to dissimilardifferent pairs, and were equally fast to dissimilar andsame pairs on simultaneous trials.

Same-different judgments under high speed stress: missing-feature principle predominates in early processing.

Three well-practiced subjects madesame-different judgments of letter pairs under varying degrees of speed stress. In contrast to the preponderance of false-different errors found using the same method without speed stress (Chignell & Krueger), there were more false-same responses, particularly on similardifferent pairs. The early, or developing, percept looks very similar from one letter to another, apparently because real differences between letters frequently are missed, whereas spurious differences due to missing or unresolved features at encoding or comparison are simply disregarded (missing-feature principle) rather than interpreted as featural mismatches (internal-noise principle). Distinguishing features are added quite late (300 + msec) in processing, just as Eriksen, O’Hara, and Eriksen proposed.

The word-superiority effect: Is its locus visual-spatial or verbal?

Hemispheric specialization was used to study letter detection. Better detection of letters in words than nonwords was found only in the right hemifield, both for horizontal (Experiment I) and vertical arrays (Experiment II). These results indicate that the present variant of the word-superiority effect has a verbal locus (left hemisphere) rather than a visual-spatial locus (right hemisphere).

Repeating the target neither speeds nor slows its detection: Evidence for independent channels in letter processing.

A target letter at a predesignated location typically is identified less readily when extraneous letters are added to the display. This disruption has been attributed to lateral interference via interactive or inhibitory channels or to attempts to encode the string as a unit. In the present study, subjects saw a single letter (e.g., B), a repeated-letter string (e.g., BBBB), or an extraneous-letters string (e.g., BCLD) and had to decide whether the leftmost letter in the string matched a target letter. Since trials were blocked by string type, letter position did not have to be discriminated on repeated-letter trials, nor was response competition present on those trials. With normal letter spacing, RT was virtually the same on repeated-letter trials as on single-letter trials. (Increasing the letter spacing in Experiment 3 did produce a slight, but nonsignificant, 22-msec increment on the repeated-letter trials.) The results indicate that individual letters are perceived as such just as well when presented in a group as when presented individually and thus provide support for the parallel, independent-channels model.