David V. Cross

The relation between size and apparent heaviness

A formula for the size-weight illusion was derived from the Stevens and Rubin (1970) finding that heaviness functions form a family of power functions that converge at a common point in the vicinity of the heaviest weight that can be lifted. Magnitude estimations of the apparent heaviness of 42 plastic cylinders varying in size and weight were obtained from 20 subjects, who were allowed to use both hands to lift the weights. It was predicted that this would increase the maximum weight that could be lifted, which would, in turn, enhance the magnitude of the illusion (the dependence of heaviness on size). The results supported this and other predictions of the model concerning the dependence of the illusion on weight as well as volume.

The role of regression and range effects in determination of the power function for electric shock

A scale for the apparent intensity of electric shock applied to the forearm was derived from cross-modality matching functions relating noise level, number, and force of handgrip to both line length and shock. For each response mode, the effects of psychophysical regression were estimated from the line judgments and used to make adjustments in the corresponding shock judgments. For shocks ranging from (1.0 to 5.5 mA, combined estimates of subjective magnitude were found to grow as the 2.26 power of the stimulating current.

Sequential dependencies and regression in psychophysical judgments

AbstractA tendency for judgments of stimulus magnitude to be biased in the direction of the value of the immediately preceding stimulus is found in magnitude estimations of loudness. This produces a bias in the empirical psychophysical function that results in underestimation of the exponent of the unbiased function presumed to relate number and stimulus intensity, N = aSn. The biased judgment can be represented as a power product of focal and preceding stimulus intensity, Nij= aS m Sjb. A bias-free estimate of the correct exponent, n, can be obtained from the relation n = m + b.

Discounting in judgments of delay and probability

Subjects drew lines proportional in length to their subjective valuation of various amounts of money, available immediately and certainly, relative to a standard amount ranging across groups from

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

1,000,000 to

Prior experience and patterning in a prisoner's dilemma game

10. They also drew lines proportional to their subjective valuation of standard amounts with delays ranging from 1 day to 50 years and with probabilities ranging from 1/10 to 1/10,000,000. Amounts of certain‐immediate moneyequivalent (in terms of drawn line length) to delayed or probabilistic money were determined. The delay and probability discount functions thereby obtained were hyperbolic in form, rather than exponential, consistent with previous findings. Large money amounts were valued higherwhentheyweredelayedbyadaythanwhenavailableimmediately.Steepness of delay discounting was not systematically related to standard money amount but probabilistic discounting was steeper for higher standard amounts than for lower amounts. Some of these results diAer from those obtained with choice procedures. Possible reasons for the diAerences are discussed. Copyright #2000 John Wiley & Sons, Ltd.

The Psychophysical Scaling and Validation of a Political Support Scale

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.

Some Technical Notes on Psychophysical Scaling

State University of New York at Stony Brook, USAABSTRACTFemale college students first played a pseudo-prisoner’s dilemma (PPD) gamewith the experimenter, who followed a fixed strategy. In the first experiment theexperimenter’s strategies for di•erent groups of subjects were: (a) play tit-for-tat;(b) play randomly; (c) always cooperate; (d) always defect (‘cooperation’ and‘defection’, defined as in an actual prisoner’s dilemma game). Only the tit-for-tatgroup increased cooperation over trials; other groups decreased cooperation.After playing the PPD with the experimenter, subjects played an actual prisoner’sdilemma (PD) game with each other. In the PD game, subjects began cooperatingmoderately but cooperation deteriorated regardless of what the experimenter’sstrategy had been in the earlier (PPD) game. In a second experiment, subjectsagain played a PPD game with the experimenter and then played a PD game witheach other. Half played one trial at a time as in the first experiment while halfplayed in patterns of four trials at a time. In the PD game, patterning of trialsretarded the development of mutual defection regardless of previous experience.The cooperation-preserving e•ect of patterning of trials in this social task iscompared with similar e•ects on individual tasks involving self-control and risk-aversion. #1998 John Wiley & Sons, Ltd.

Lotteries and the Time Horizon

The aims of this paper are twofold: first, to illustrate the utility of psychophysical scaling methods in building reliable and valid scales of social opinion, and second, to report research findings on the development and validation of a scale for measuring support for political phenomena. The scaling experiments reported here provide compelling evidence of the successful extension of psychophysics from the measurement of sensory phenomena to the measurement of political opinion. In contrast to alternative scaling techniques, these psychophysical scaling methods provide a means to develop interval or ratio scales and assign meaningful values to the scale items. A thirty-item adjectival scale is tested and found to be a reliable, regression-free, and psychophysically valid instrument for measuring the degree of support for political institutions, policies, and roles. The range of the General Support Scale is approximately 100:1. Intervally spaced items may be selected from this scale for use in a survey instrument.

The calibration and cross-modal validation of ratio scales of political opinion in survey research☆

A simple theory of psychophysical scaling is presented. It is based on the power law and a multiplicative error model that treats magnitude judgments as lognormal variates. Conventional techniques for estimating power-law exponents are reviewed along with a solution to the problem of estimating parameters of psychophysical relationships when both variables in the relationship are measured with error. This problem arises when validating ratio scales of opinion by matching sensory magnitudes to social stimuli and thereby producing psychophysical cross-modality matching functions. The regression problem is reconsidered in this context and a method for dealing with it is proposed.