Intra- and interindividual variation in the power law exponent for area summation

Subjects were required to indicate the magnitudes of subjective sums of areas of regular 20-sided polygons by three methods—generation of a polygon of matching area, estimation of the ratio of the subjective sum to a fixed area, and magnitude estimation. Power law exponents were estimated for each subject by maximum likelihood estimation. A very large amount of interindividual exponent variation was observed. Exponents resulting from magnitude estimation were consistent on the average with previous results, but exponents from area matching and ratio estimation were much larger and close to unity. When subjects were given feedback on the discrepancy between their responses and physical area summation, they readily shifted their exponents toward unity.     

直感判断力:理解儒学的心之能力

Mou Zongsan incorrectly uses Kant’s practical reason to interpret Confucianism. The saying that “what is it that we have in common in our minds? It is the li 理 (principles) and the yi 义 (righteousness)” reveals how Mencius explains the origin of li and yi through a theory of common sense. In “the li and the yi please our minds, just as the flesh of beef and mutton and pork please our mouths,” “please” is used twice, proving aesthetic judgment is necessary to understanding Mencius. An analysis of Zhu Xi and Wang Yangming’s ideas will show that Confucianism should be interpreted by appealing to aesthetic judgment, and a discussion of Kant’s theory of judgment and Gadamer’s critique of Kant’s theory will support the same point. The conclusion is that Chinese moral philosophy should be interpreted through aesthetic judgment.     

Reliability for the law of comparative judgment

A variance-components analysis is presented for paired comparisons in terms of three components:s, the scale value of the stimuli;d, a deviation from the linear model specified by the law of comparative judgment; andb, a binomial error component. Estimates are given for each of the three variances, _s ² , _d ² , and _b ² . Several coefficients, analogous to reliability coefficients, based on these three variances are indicated. The techniques are illustrated in a replicated comparison of handwriting specimens.This research was jointly supported in part by Princeton University, the Office of Naval Research under contract Nonr-1858(15), and the National Science Foundation under grant NSF G-642, and in part by Educational Testing Service. Reproduction in whole or in part is permitted for any purpose of the United States Government.Thanks are due to Ledyard Tucker and Frederic Lord for valuable suggestions on the development presented here.     

A revised law of comparative judgment

In contrast to Thurstone's Law of Comparative Judgment, a model in which a comparison pair and its complement are assumed to give rise to two different distributions of differences is considered. The consequences of this revised model on scaling problems is developed.The research in this article was supported jointly by the Army, Navy, and Air Force under contract with the Massachusetts Institute of Technology.     

The power law repealed: The case for an exponential law of practice

The power function is treated as the law relating response time to practice trials. However, the evidence for a power law is flawed, because it is based on averaged data. We report a survey that assessed the form of the practice function for individual learners and learning conditions in paradigms that have shaped theories of skill acquisition. We fit power and exponential functions to 40 sets of data representing 7,910 learning series from 475 subjects in 24 experiments. The exponential function fit better than the power function in all the unaveraged data sets. Averaging produced a bias in favor of the power function. A new practice function based on the exponential, the APEX function, fit better than a power function with an extra, preexperimental practice parameter. Clearly, the best candidate for the law of practice is the exponential or APEX function, not the generally accepted power function. The theoretical implications are discussed.     

Stevens' power law and time perception: effect of filled intervals, duration of the standard, and number of presentations of the standard

Subjects estimated the duration of five different time intervals, either filled or unfilled, in relation to one or two standard intervals. Half of the subjects were presented with the standard at the beginning of the experiment only, and half were presented with the standard before every interval. The five estimations were then used to calculate, for each subject, the exponent in Stevens' Power Law which describes the form of the power relationship. The resulting over-all exponent of .91, although nearly linear, was significantly different from 1.0. The data were then analyzed by a 2 X 2 X 2 analysis of variance which showed a significant interaction between presenting the standard either once or before each interval with the duration of the standard.     

The presence of a nonresponding effector increases inhibition of return

Inhibition of return (IOR) refers to the performance disadvantage for targets presented at an exogenously cued location, relative to an uncued location, at relatively long cue-target onset asynchronies. In this experiment, we investigated the influence on IOR of a nonresponding effector (i.e., the index finger of the nonresponding hand) placed on a response key in a simple-RT task. With peripheral cues and targets, IOR and spatial stimulus-response compatibility effects were larger when the nonresponding hand was placed on a response key. IOR—the slowed responding to go signals at the cued location—was accompanied by a lower false alarm rate when no-go signals were presented there. These findings provide direct evidence for a motoric component to IOR wherein some portion of the inhibition is observed as a criterion shift against responding to the cued location.     

The rasch model,the law of comparative judgment and additive conjoint measurement

Relationships between the Rasch model and both the law of comparative judgment and additive conjoint measurement are discussed. The distance between the ability of Persona and the difficult of Itemi is, in the Rasch model, the baseline value corresponding to the probability thata will respond correctly toi, where this probability is interpreted as the area under a logistic curve (which is substantially equivalent to the normal curve) and is thus an application of the law of comparative judgment. Under certain assumptions, the Rasch model is also a special case of additive conjoint measurement and, properly reinterpreted, may be usefully applied in contexts other than individual differences.     

The law of comparative judgment in the successive intervals and graphic rating scale methods

The law of comparative judgment is applied to the successive intervals and graphic rating scale methods. A procedure for estimating the modal discriminal process and discriminal dispersion of the stimuli, as well as the value of the boundaries of the intervals on the continuum, is given. From the estimated values it is possible to determine the theoretical proportions and to compare them with the actual experimental proportions. The agreement between these values is an indication of the adequacy of the assumptions made.     

The law of categorical judgment (corrected) extended: a note on Rosner and Kochanski (2009)

Rosner and Kochanski (2009) noticed an inconsistency in the mathematical statement of the Law of Categorical Judgment and derived "the valid equation, the Law of Categorical Judgment (Corrected)" (p. 125). The purpose of this comment is to point out that the law can be corrected in many different ways, leading to substantially different equations. The different versions have different consequences for the predicted distributions of the responses and, hence, for fitting real data. Some of these consequences are unexpected and sometimes undesirable. Researchers should be aware of the different possibilities as they may lead to pronouncedly different accounts of given data.     

A solution for case iii of the law of comparative judgment

A solution for Case III of the Law of Comparative Judgment, modeled after Thurstone's solution for Case IV but eliminating the restrictive assumption of relatively equal discriminal dispersions, is developed.     

Concerning the psychophysical power law

It is fortunate that, despite our theorizings, most of our facts manage to stay stubbornly as they were. I am encouraged by that thought as it bears on a recent theoretical attempt by Treisman (1964) to do over the psychophysical law. I should like, more in explication than in rebuttal, to comment on three major points relating to the law in question.     

Stimulus sequence and the exponent of the power function for loudness

In two experiments, 15 and 13 subjects estimated the loudness of 12 sound-pressure levels (38-104 dB; 6-dB intervals) of a 1000-Hz tone by the method of magnitude estimation with a modulus assigned to the first stimulus presented. The tone duration was 1 sec. and the interstimulus interval was 6 sec. The presentation order was systematically ascending-descending in one experiment and balanced-irregular in the other. The results indicate that (1) loudness is a power function of sound pressure with an exponent of 0.60 for the systematic order and 0.29 for the irregular order. (2) For both the irregular and systematic orders, a large step-size (12 or 18 dB) between the stimulus on Trial n and on Trial n-1 (or n-3) results in a slight assimilation effect. This also occurs for the small step-size (6 dB) in the irregular order. (3) The size of momentary exponents (based on two points, Trials n and n-1 or n-3) depends on the sound pressures of successive stimuli, whether the steps are positive or negative, and whether the stimuli have been presented in systematic or irregular order. For positive steps, the momentary exponent is lower for a soft tone (Trial n) than for a loud tone, whereas for negative steps the momentary exponent is lower for a loud tone than for a soft tone. These effects ar more pronounced when these stimuli are presented in an irregular order. A relative judgment model is offered for magnitude estimation. It assumes that subjects judge the loudness of a stimulus in terms of three reference markers: the minimum and maximum sound pressures as well as the sound pressure of the previous stimulus.     

Estimation in the power law

Most psychophysicists neglect to consider how error should be characterized in applications of the power law. Failures of the power law to agree with certain theoretical predictions may be due to bias of parameter estimates. A power law with lognormal product structure is proposed and approximately unbiased parameter estimates are given for several common estimation situations.