Nonmetric tests of ratio vs. subtractive theories of stimulus comparison

Theories of stimulus comparison were tested by examining ordinal properties of data obtained with six scaling tasks. Subjects judged simple “ratios” or “differences” of stimulus pairs constructed from a factorial design. In four additional tasks, the same judges also compared relations between pairs of stimulus pairs, judging “ratios of ratios,” “ratios of differences,” “differences of ratios,” and “differences of differences.” The data were consistent with a subtractive theory, which asserts that two stimuli are compared by subtraction, regardless of the task, but that judges can compare two stimulus differences by either a ratio or a difference. All six tasks could be related by the subtractive theory using a single set of scale values. Other simple theories, including the theory that “ratio” judgments can be represented by a ratio model, could not reproduce the six rank orders (of the six sets of data) using a single set of scale values.     

Loudness “ratios” and “differences” involve the same psychophysical operation

Subjects judged both “atios” of loudness and “differences” in loudness between pairs of tones that varied in intensity. The pairs were constructed from factorial designs, permitting separation of stimulus and response scaling for each subject. Ratings of “differences” and estimations of “ratios” were monotonically related, inconsistent with the hypothesis that subjects perform both subtractive and ratio operations on a common scale. Instead, the data suggest that both tasks involve the same psychophysical comparison operation with different response transformations. If the operation can be represented by the subtractive model, then category ratings involve a nearly linear transformation and magnitude estimations involve a nearly exponential transformation.     

Scale convergence as a criterion for rescaling: Information integration with difference,ratio, and averaging tasks

Ss lifted pairs of weights simultaneously, one in each hand, and judged either the difference, ratio, or average heaviness of the two weights. Data for the difference and ratio tasks were in general agreement with subtractive and ratio models, but the averaging data showed discrepancies from the constant-weight averaging model similar to those reported in previous psychophysical research. Rescaling was ruled out for the averaging data, because responses to pairs of equal weight were a linear function of subtractive model scale values derived from the difference task data. Scale values for the ratio and difference task data were related exponentially, as were the responses to the pairs, consistent with Torgerson’s conjecture that Ss do not distinguish “differences” from “ratios.” They appear to use the same composition rule but different output functions, depending on the procedures for responding. The scale convergence criterion can thus prevent inappropriate rescaling when a model fails and can dictate rescaling even when a model fits.     

Measurement and the mental map

In order to test between subtractive and ratio theories of stimulus comparison, judges were asked to estimate “ratios” and “differences” of easterliness and westerliness of U.S. cities. “Difference” judgments fit the subtractive model, and “ratio” judgments fit the ratio model. However, “ratios” and “differences” were monotonically related, contrary to the theory that judges compute both relations on a common scale. Results are consistent with the theory that there is but one operation for both “ratios” and “differences.” To assume that the single operation is a ratio requires the complex interpretation that easterliness and westerliness are nonlinearly related. A simpler interpretation is provided by a subtractive theory, in which all four types of judgments are monotonically related to subjective differences on a single cognitive map.     

Digit identity influences numerical estimation in children and adults

Learning the meanings of Arabic numerals involves mapping the number symbols to mental representations of their corresponding, approximate numerical quantities. It is often assumed that performance on numerical tasks, such as number line estimation (NLE), is primarily driven by translating from a presented numeral to a mental representation of its overall magnitude. Part of this assumption is that the overall numerical magnitude of the presented numeral, not the specific digits that comprise it, is what matters for task performance. Here we ask whether the magnitudes of the presented target numerals drive symbolic number line performance, or whether specific digits influence estimates. If the former is true, estimates of numerals with very similar magnitudes but different hundreds digits (such as 399 and 402) should be placed in similar locations. However, if the latter is true, these placements will differ significantly. In two studies (N = 262), children aged 7–11 and adults completed 0–1000 NLE tasks with target values drawn from a set of paired numerals that fell on either side of “Hundreds” boundaries (e.g., 698 and 701) and “Fifties” boundaries (e.g., 749 and 752). Study 1 used an atypical speeded NLE task, while Study 2 used a standard non‐speeded NLE task. Under both speeded and non‐speeded conditions, specific hundreds digits in the target numerals exerted a strong influence on estimates, with large effect sizes at all ages, showing that the magnitudes of target numerals are not the primary influence shaping children's or adults’ placements. We discuss patterns of developmental change and individual difference revealed by planned and exploratory analyses.     

Use of numerical symbols by the chimpanzee (Pan troglodytes): Cardinals, ordinals, and the introduction of zero

An adult female chimpanzee with previous training in the use of Arabic numerals 1–9 was introduced to the meaning of "zero" in the context of three different numerical tasks. The first two were cardinal tasks where the subject was required either to select numerals corresponding to the number of items presented on a computer screen (productive use of numerals) or to match sets of the appropriate size to numerals presented as samples (receptive use). The third task addressed the ordinal meaning of the same symbols where the subject was required to respond to numerals sequentially, arranging them into an ascending series. The subject mastered the recognition of the meaning of zero in all three tasks. However, details of her usage of the symbol revealed that transfer of the meaning between different kinds of tasks was incomplete, suggesting that the level of abstraction characteristic of human numerical ability was not attained in the chimpanzee. Over the course of acquisition leading to the high levels of accuracy eventually observed, the newly introduced zero appeared to shift along the length of a continuous numerical scale toward the lower end, while confusions with 1 remained the most frequently encountered mistakes. Such patterns of error thus suggest that Ai's understanding of the meaning of zero in relation to the rest of the number symbols was not consistent with an "absence of items versus presence of items" scheme. Electronic Publication     

Subtractive versus Ratio Model of “Fair” Allocation: Can the Group Level Analyses Be Misleading?

Two models of equity judgments are ratio and subtraction. Proponents of the former assume a linear relationship between the subjective feelings of equity and their overt expressions; those of the latter assume a monotonic relationship. Consequently, the ratio and subtractive rules are tested with the raw and monotonically rescaled data, respectively. I evaluated these two approaches with managers and students from India. Experiment 1 varied merit and pay of two persons and obtained judgments of difference between unfairness to them. Experiments 2 and 3 manipulated two inputs of two persons and studied “fair” reward for them. I analyzed both the raw and rescaled data at the group and individual levels. The group analyses supported the ratio model; the individual analyses showed that majority was consistent with the subtractive model. Discrepant results from these analyses were due to individual differences in the models employed and use of the response scale. Implications of the findings are discussed for cross-cultural and developmental research in “fair” allocation.     

Selective attention in the speeded classification and comparison of multidimensional stimuli

Selective attention was studied when subjects were required to make either speeded classifications of single stimuli or comparisons of pairs of multidimensional stimuli. Experiment 1 established that subjects were able to attend selectively to form when the irrelevant dimension of size or shading varied in a speeded-classification (card-sorting) task. Experiment 2 confirmed this finding in a discrete-trials task. However, subjects werenot able to filter out irrelevant dimensional disparity in a comparably designed simultaneous-comparison (i.e., “same”-“different”) task. Mean “same” reaction time increased monotonically with increases in disparity between the two stimuli on the irrelevant dimension. Experiment 3 also revealed a monotonie increase in “same” RT as a function of irrelevant disparity in a successive-comparison task. These results were discussed in terms of a normalization model proposed by Dixon and Just (1978) in which it is assumed that a subject equates the two stimuli on the irrelevant dimension before deciding that they are the same along the relevant dimension. It was concluded that: (1) although subjects can efficiently filter out irrelevant disparity in a speeded-classification task, interference due to irrelevant disparity is obtained in the comparison tasks, (2) a common process such as normalization does not necessarily underlie performance in the speeded-classification and comparison tasks, (3) the ability to attend selectively to a stimulus dimension may be task determined as well as stimulus determined, and (4) contrary to the Dixon and Just proposal, normalization of irrelevant disparity occurs in a comparison task, even when the relevant dimension is represented as a separate encoding feature.     

Comma N' cents in pricing: The effects of auditory representation encoding on price magnitude perceptions

Numbers and prices can be processed and encoded in three different forms: 1) visual [based on their written form in Arabic numerals (e.g., 72)], 2) verbal [based on spoken word-sounds (e.g., “seventy” and “two”), and 3) analog (based on judgments of relative “size” or amount (e.g., more than 70 but less than 80)]. In this paper, we demonstrate that including commas (e.g., $1599 vs. $1599) and cents (e.g., $1599.85 vs. $1599) in a price's Arabic written form (i.e., how it is perceived visually) can change how the price is encoded and represented verbally in a consumer's memory. In turn, the verbal encoding of a written price can influence assessments of the numerical magnitude of the price. These effects occur because consumers non-consciously perceive that there is a positive relationship between syllabic length and numerical magnitude. Three experiments are presented demonstrating this important effect.     

Comparative judgments with numerical reference points

A model of subjective magnitude comparisons is explored, which assumes that subjects compare symbolic stimulus magnitudes with respect to a reference point. The reference point may be established implicitly by the question (e.g., “Which is larger?” vs “Which is smaller?”) or be presented explicitly (e.g., “Choose the stimulus closer to X.”). The model was tested in five experiments in which subjects judged which of two comparison digits was closer to (or further from) a reference digit. Regression analyses in three experiments revealed that reaction time depended on the ratio of the distances from the comparison items to the reference point. The other two experiments provided evidence that subjects can strategically vary the processes by which they compare stimuli to a reference point. The results indicated that subjects can perform various types of “analog arithmetic” using either the linear number scale or a nonlinear scale of subjective digit magnitude.     

Reversible operations in the balance scale task

The development of two types of reversible operations was examined in three tasks involving a two-armed balance scale. The first type of reversible operation, compensation between dimensions, was shown to undergo considerable development between eighth grade and college. Even eighth graders were inconsistent in applying compensation to those versions of the balance scale task that required it. Some subjects, primarily preschoolers, used the given information in a way that was the opposite of that required for correct answers, a result that has been reported for other tasks requiring compensation. A second type of reversible operation, termed “correspondence of patterns”, was also tested and it was tentatively concluded that the responses of the modal college student are not fully reversible in this sense. This finding is also similar to results obtained in judgments of time, speed, and distance and in social judgment studies with college students.     

“Ratios” and “differences” in perceived sweetness intensity

For a number of perceptual continua, it has been shown in previous studies that subjects use only one quantitative comparison between two sensory impressions of a pair of stimuli, irrespective of whether they are instructed to judge “ratios” or “differences”. This comparison can be described by algebraic subtraction. The present study was designed to investigate whether this one-operation theory for psychophysical judgment also applied to the sensory continuum of sweetness. Subjects were presented with pairs of fructose solutions, and judged “ratios” of, or “differences" in, perceived sweetness intensities. The pairs were constructed on the basis of a factorial judgment design. The results showed that the reported “differences” could be adequately described by a difference response model, and that the reported “ratios” could be adequately described by a ratio response model. However, the reported “ratios” and reported “differences” were monotonically related, and the marginal means of the log-transformed response matrix of “ratios” were a linear function of the marginal means of the response matrix of “differences”. These results are incompatible with the notion that subjects judged differences when instructed to judge “differences”, but ratios when instructed to judge “ratios”. The consistency of the ratio response model with “ratio” judgments is probably caused by a comparative operation based on “differences” in combination with an exponential response output function. It may be concluded that subjects judge only “differences”, and not “ratios”, between perceived sweetness intensities.     

2~5岁学前儿童数概念的发展水平及过程——理解者水平理论的视角

本研究从儿童数概念发展的理解者水平模型的理论视角,对100名2～5岁学前儿童的数概念发展水平进行划分,并比较不同水平儿童对后继函数的理解和掌握,探讨儿童数概念的发展过程。结果表明：儿童在4岁时基本达到了数概念发展的最高水平即基数原则水平,已经理解了后继函数,能够把它的方向性和单位性的变化,对应到数数序列的数词上。而2～3岁的儿童还处于子集水平,该水平的儿童和基数原则水平的儿童相比,对后继函数的理解存在差异。但后继函数的发展不是全或者无的,儿童积累的数词越多,后继函数发展得越好。     

Empirical evaluation of a model of global psychophysical judgments: IV. Forms for the weighting function

Understanding the psychological interpretation of numerals is of both practical and theoretical interest. In classical magnitude estimation, respondents match numerals to sensations and in magnitude production they select sensations that stand in a prescribed numerical ratio to a given standard. The present work focusses on evaluating several possible, and related, forms for the function W formulating the distortion of numerals. The main form, of which a power function is a special case, is the Prelec exponential/power representation. Behavioral equivalents to power and to Prelec functions are formulated, tested, and rejected. It is argued that either the mathematical form or the assumption W(1)=1 is wrong. Whereas, the axiomatic literature has focussed exclusively on the former inference, we explore the alternate that W(1)≠1. Behavioral axioms are formulated in each case and experimentally tested. We conclude that most respondents satisfy a general power function and that those who do not, satisfy the general Prelec function.     

Nested models of relative judgment: Applications to a similarity averaging model

A theoretical analysis of the Eisler and Ekman (1959) model of similarity judgments for unidimensional continua is presented, based on a general model of relative judgment. This general model assumes that judgments are mediated by perceived relations of pairs of stimuli, that there exists a transformation of the judgmental response that is a function of the sensory ratio of the two stimuli, and that response bias operates in a multiplicative manner. Three structural conditions are presented, each imposing constraints on the structure of observed judgments. The structural conditions define threenested models of relative judgment, with the second a weakened version of the first, and the third a weakened version of the second. The special virtue of the general model is that it is applicable to a variety of judgmental tasks (e.g., ratio estimation, similarity, pair comparison), the key being derivation of theresponse transformation conforming to the structural conditions. The structural conditions thus constitute necessary conditions for several different judgmental models. The theory was first applied with success to ratio estimation judgments (Fagot, 1978), and this paper applies the general model to the Eisler and Ekman similarity “averaging” model. Empirical tests were carried out on published data for pitch, darkness, visual area, and heaviness judgments. Although the strong form of the model presented by Eisler and Ekman was rejected, weakened versions were generally supported by the data. These results were similar to those obtained for ratio estimation (Fagot, 1978), and are interpreted to be very promising for the general model of relative judgment.     

"Fair" Allocations of Pay and Workload: Tests of a Subtractive Model with Nonlinear Judgment Function

An allocation is commonly regarded as "fair" if the ratio of outcome of two persons equals the ratio of their respective inputs. In allocation of pay, however, a subtractive model with sigmoidal-shaped judgment function has recently been reported. The present research further tested the plausibility of the subtractive model in allocations of pay and workload. Subjects decided either pay (N = 17) or workload (N = 17) for Person A using three pieces of information - the pay of B and the input of A and B or the workload of B and the pay of A and B. As hypothesized, factorial plots of the input (or pay) of Persons A and B across the low, moderate, and high levels of pay (or workload) of Person B yielded the diverging, slanted barrel, and converging patterns, respectively. Monotonic transformations of the data rescaled nonparallelism to perfect parallelism. The judgment function had a sigmoidal shape characterized by "floor" and "ceiling" effects. These results support the subtractive model but question the ratio model of "fair" allocations.     

Extensions of Narens' theory of ratio magnitude estimation

Stevens postulated that the responses of a participant in a ratio scaling experiment can be used directly to construct a psychophysical function. Today, it is generally accepted that the axioms of commutativity and multiplicativity are crucial for the interpretation of the subjects' ratio scaling behaviour. Empirical findings provide evidence that commutativity holds, whereas multiplicativity fails to hold across different sensory modalities. This shows that, in principle, Stevens' direct scaling methods yield measurements on a ratio scale level, but that the numerals occurring in a ratio scaling experiment cannot be taken at face value. Thus, Narens and others introduced a transformation function f, which converts the numerals used in an experiment into the latent mathematical numbers. The aim of the present paper is to specify the (unknown) shape of the transformation function f, by analysing different extensions of the multiplicative property. The results provide evidence that f is either a power function or a logarithmic function.     

Decision bias and personnel selection strategies

The purpose of this study was to examine the influence of two decisional biases—framing and cost salience—on personnel selection decisions. One hundred twenty-eight graduate and undergraduate students participated in a personnel selection simulation. Framing was manipulated by inducing participants to use either a “rejecting” strategy (identify those applicants whom you would not interview) or an “accepting” strategy (list those applicants whom you would interview). Cost salience was manipulated by making selection-related costs either implicit or explicit. Results showed that “accepting” strategy subjects selected less applicants to be interviewed than “rejecting” strategy subjects, but only when selection-related costs were made salient. More time was required for subjects to make their selection decisions when selection-related costs were made salient. Framing and cost salience also influenced the success probability thresholds used by subjects to select applicants. Limitations of this research and directions for future study were discussed.     

On the theory of biased bisection operations and their inverses

A theory of fundamental measurement involving multiple operations is presented. The operations are bisection, and two inverses of the bisection operation called right and left displacement, analogous to “displacement” of an interval “up” or “down” a scale. Following Pfanzagl, “biased” operations are permitted, and linear, reflexive representations are presented for the biased operations. Results are obtained on the constructability of unbiased scaling operations, and several methods of constructing standard sequences of equally spaced stimuli using biased operations, are described. A striking property of biased linear representation models is derived, showing that if bias is present, indefinite iterations of one or the other of the two displacement operations will converge to asymptote. Preliminary data support this surprising convergence property.     

The Variation of Hand Steadiness with Physical Stress

The performance of 12 subjects in a hand steadiness task was compared at different work loads. Hand steadiness expressed in the form of values of hand shakiness was found not to increase linearly with work intensity. On the contrary, the relation between hand shakiness and activation level, induced by physical work, was found to be positively accelerated by an exponent of 1.6. Subjective effort measured by a ratio estimation method grew, likewise, according to a positively accelerating function with an exponent of about 1.6, while subjective effort according to the “RPE” category scale, like heart rate, grew linearly with work load.