Stevens’ Direct Scaling Methods and the Uniqueness Problem

Stevens postulated that we can use the responses of a participant in a ratio scaling experiment directly to construct a psychophysical function representing the participant's sensations. Although Stevens' methods of constructing measurement scales are widely used in the behavioral sciences, the problem of which scale type is appropriate to describe ratio scaling data is still unresolved. To deal with this problem, we develop a theoretical framework to specify the scale type attained by Stevens' direct scaling methods. It is shown, under fairly mild background assumptions, that the behavioral axioms presented in this paper are necessary and sufficient for the psychophysical functions to be ordinal-, interval-, log-interval-, or ratio-scales. Furthermore, suggestions on how to test these behavioral axioms are provided. Requests for reprints should be sent to thomas.     

What do sensory scales measure?

It is indeed an honour to have drawn Professor Stevens' fire (1964), in view of the magnitude of his research achievement in the field of sensory scaling. If I shall differ with him on certain of his conclusions, it is in full awareness of the debt we all owe him for the experimentation and thinking which has provided much of the factual basis and theoretical stimulus to further research and development in this area.

Professor Stevens (1960, 1964) advocates a psychophysical power law, while I have questioned it (Treisman, 1961, 1962, 1963a, 1963b, 1964a). I shall here examine the main points of disagreement lying behind these differing positions.     

Bain on Neural Networks

In his bookMind and body(1873), Bain set out an account in which he related the processes of associative memory to the distribution of activity in neural groupings—or neural networks as they are now termed. In the course of this account, Bain anticipated certain aspects of connectionist ideas that are normally attributed to 20th-century authors—most notably Hebb (1949). In this paper we reproduce Bain's arguments relating neural activity to the workings of associative memory which include an early version of the principles enshrined in Hebb's neurophysiological postulate. Nonetheless, despite their prescience, these specific contributions to the connectionist case have been almost entirely ignored. Eventually, Bain came to doubt the practicality of his own arguments and, in so doing, he seems to have ensured that his ideas concerning neural groupings exerted little or no influence on the subsequent course of theorizing in this area.     

A reanalysis of Lord’s statistical treatment of football numbers

Stevens’ theory of admissible statistics [Stevens, S. S. (1946). On the theory of scales of measurement. Science, 103, 677680] states that measurement levels should guide the choice of statistical test, such that the truth value of statements based on a statistical analysis remains invariant under admissible transformations of the data. Lord [Lord, F. M. (1953). On the statistical treatment of football numbers. American Psychologist, 8, 750-751] challenged this theory. In a thought experiment, a parametric test is performed on football numbers (identifying players: a nominal representation) to decide whether a sample from the machine issuing these numbers should be considered non-random. This is an apparently illegal test, since its outcomes are not invariant under admissible transformations for the nominal measurement level. Nevertheless, it results in a sensible conclusion: the number-issuing machine was tampered with. In the ensuing measurement-statistics debate Lord’s contribution has been influential, but has also led to much confusion. The present aim is to show that the thought experiment contains a serious flaw. First it is shown that the implicit assumption that the numbers are nominal is false. This disqualifies Lord’s argument as a valid counterexample to Stevens’ dictum. Second, it is argued that the football numbers do not represent just the nominal property of non-identity of the players; they also represent the amount of bias in the machine. It is a question about this property-not a property that relates to the identity of the football players-that the statistical test is concerned with. Therefore, only this property is relevant to Lord’s argument. We argue that the level of bias in the machine, indicated by the population mean, conforms to a bisymmetric structure, which means that it lies on an interval scale. In this light, Lord’s thought experiment-interpreted by many as a problematic counterexample to Stevens’ theory of admissible statistics-conforms perfectly to Stevens’ dictum.     

A Meaningful Justification for the Representational Theory of Measurement

Measurement, the means by which numbers enter science, is of fundamental importance to modern science. The relationship between its qualitative and quantitative aspects has generated many theories and much controversy. In 19th century geometry similar developments led mathematician Felix Klein to devise a theory for unifying qualitative and quantitative approaches to geometry. Klein's theory, which today is called the Erlanger program, was based on transformation groups. In this article, the Erlanger program is given a new foundation based on mathematical logic and is extended to science. The current dominant theory of measurement in the literature, the representational theory, is then justified in terms of the new foundation for the Erlanger program. Certain inferential techniques used in dimensional analysis and the related technique of possible psychophysical laws are also given justifications in terms of the new foundation.     

Psychological relations and psychophysical scales: On the status of “direct” psychophysical measurement

Arguments, originally developed some fifteen years ago, concerning the sort of “direct” psychophysical measurement advocated by S. S. Stevens are reproduced here in order to facilitate access by a wider audience. Considerations of measurement theory, biological adaptation, perceptual invariance, and neurophysiological noise, as well as various empirical results, suggest (a) that it is primarily the relations—particularly the ratios—between stimulus magnitudes and not the individual magnitudes themselves that are represented within the organism and, accordingly, (b) that “direct” psychophysical measurement, by itself, uniquely determines neither a psychological magnitude of any one stimulus nor a psychophysical law relating such psychological magnitudes to corresponding physical magnitudes. What such measurement can uniquely determine, evidently, is a psychological parameter characterizing each sensory continuum.     

C. S. Lewis Surprised and Humanized by Joy

Abstract : C. S. Lewis' views on sexuality and the relationship between men and women changed radically through his relationship with Joy, his friend and later wife. Lewis makes a sharp distinction between love and friendship in his writings. This article shows how his concrete experience of love and grief with Joy transformed his understanding of relationships between men and women, and, eventually, the meaning of being human and its implications for Christianity.     

A theory of appropriate statistics

A formal theory of appropriateness for statistical operations is presented which incorporates features of Stevens' theory of appropriate statistics and Suppes' theory of empirical meaningfulness. It is proposed that a statistic be regarded as appropriate relative to statements made about it in case the truths of these statements are invariant under permissible transformations of the measurement scale. It is argued that the use of inappropriate statistics leads to the formulation of statements which are either semantically meaning-less or empirically nonsignificant.This research was supported in part by each of the following grants: National Science Foundation Grant GS-333 to the University of Oregon; National Science Foundation Grant to the Institute of Human Learning, University of California, Berkeley; and National Institute of Mental Health Grant MH-08055-01 (under the direction of Ernest W. Adams), also to the Institute of Human Learning. Work on this project was carried out in part during Robert F. Fagot's tenure as Public Health Service Special Fellow (No. MSP-15800) at the University of California, Berkeley, 1962-63; and during Richard E. Robinson's tenure as National Science Foundation Science Faculty Fellow at Stanford University, 1962–63.     

Must the bearer of a right have the concept of that to which he has a right?

Stevens critiques what he calls the "conceptual possession requirement" as defined by Michael Tooley in his article, "Abortion and infanticide," Philosophy and Public Affairs 1972 Fall; 2(1): 37-65. Tooley argued that a being has a right to something only if it has the concept of that thing. He claimed that fetuses and infants do not have a serious right to life because they have no concept of a continuing self, and that therefore abortion and infanticide are morally permissible. Stevens contends that the conceptual possession requirement is not valid, offers a counterexample to Tooley's argument, and defends his reasoning against what he presumes might be Tooley's counterarguments.     

Symmetry, direct measurement, and Torgerson's conjecture

Two forms of direct measurement are considered in the article: a strong form in which ratio productions named by number words are interpreted veridically as the numerical ratios they name; and a weak form in which the ratio productions named by number words may have interpretations as ratios that are different from numerical ratios they name. Both forms assume that the responses to instructions to produce ratios are represented numerically by ratios, and thus the word “ratio”—and supposedly the participants concept associated with it—is being “directly” represented. The strong form additionally “directly represents” the number mentioned in the instruction by itself. The article provides an axiomatic theory for the numerical representations produced by both forms. This theory eliminates the need for assuming anything is being “directly represented,” allowing for a purely behavioral approach to ratio production data. It isolates two critical axioms for empirical testing. An measurement-theoretic explanation is provided for the puzzling empirical phenomenon that subjects do not distinguish between ratios and differences in a variety of direct measurement tasks.     

Understanding text after severe closed-head injury: assessing inferences and memory operations with a think-aloud procedure

A think-aloud method was used to examine the content of information available to working memory during narrative comprehension in a CHI population. Twenty severe CHI participants (>1 year post-injury) and 20 controls talked aloud after they read each sentence of story narratives. Trabasso and Magliano's (1996a) verbal protocol analysis was then used to code for the production of inferential and non-inferential clauses and the memory operations that supported inferential clause production. We found that CHI and control groups produced a comparable number of clauses, that inferences dominated narrative comprehension, and that both groups produced more explanatory inferences than predictive or associative inferences. Despite these qualitative similarities, the CHI group demonstrated poorer comprehension, generated proportionately fewer inferences, relied less on retrieval as a memory source for explanatory inferences, and produced more non-inferential clauses and associative inferences.     

The Harvard Pigeon Lab under Herrnstein

The history of the Harvard Pigeon Lab is a history of two periods of remarkable productivity, the first under Skinner's leadership and the second under Herrnstein's. In each period, graduate students flocked to the leader and then began stimulating one another. Chance favored Herrnstein's leadership, too, because an unusually large number of graduate students were admitted in the fall of 1962. In each period, productivity declined as the leader lost interest in the laboratory and withdrew. Directly and indirectly, the laboratory finally died as a result of the cognitive "revolution." Skinner and his students saw the possibility of a natural science of behavior and set about establishing that science based on concepts such as response rate, stimulus control, and schedules of reinforcement. Herrnstein and his students saw that the science could be quantitative and set about making it so, with relative response rate, the matching law, and the psychophysics of choice (analogous to S. S. Stevens' psychophysics). The history might provide a golden research opportunity for someone interested in the impact of such self-organizing research groups on the progress of science.     

Theorizing the Temporal Aspect of Ikigai or Life Worth Living among Japanese University Students: a Mixed-Methods Approach

Although cross-cultural and non-Western studies have advanced our knowledge on well-being, many studies have adopted English words including ‘happiness’ as their guiding concepts, which may have limited and biased their insight. The current study is part of a larger mixed-methods project that theorizes how Japanese university students pursue ikigai or a life worth living. The first qualitative study, based on 27 photo-elicitation interviews, generated a grounded theory of houkousei, or life directionality. Our qualitative findings suggested that when students formed explicit associations among the past, present, and future, they gained strong ikigai feelings. These associations were developed either cognitively by mentally associating existing present experiences with the past or future, or behaviourally by strategically choosing current experiences more pertinent to the past or future than alternatives. These actions resulted in two subjective states: life legacy and life momentum. Life legacy was the perception that one’s past had meaningfully contributed to his or her present experiences, life, and self. Life momentum meant the belief that one’s present experiences were helping him or her achieve the desired future. Lastly, having defining past experiences and setting clear goals both facilitated the associative actions. To further validate this theory, we collected online survey data from a national sample of 672 Japanese students. Our quantitative results, based on partial least squares structural equation modelling (PLS-SEM), largely supported our theoretical model. Our findings are discussed in light of the ikigai and eudaimonic well-being literatures.

     

A neural model of hippocampal–striatal interactions in associative learning and transfer generalization in various neurological and psychiatric patients

Building on our previous neurocomputational models of basal ganglia and hippocampal region function (and their modulation by dopamine and acetylcholine, respectively), we show here how an integration of these models can inform our understanding of the interaction between the basal ganglia and hippocampal region in associative learning and transfer generalization across various patient populations. As a common test bed for exploring interactions between these brain regions and neuromodulators, we focus on the acquired equivalence task, an associative learning paradigm in which stimuli that have been associated with the same outcome acquire a functional similarity such that subsequent generalization between these stimuli increases. This task has been used to test cognitive dysfunction in various patient populations with damages to the hippocampal region and basal ganglia, including studies of patients with Parkinson’s disease (PD), schizophrenia, basal forebrain amnesia, and hippocampal atrophy. Simulation results show that damage to the hippocampal region—as in patients with hippocampal atrophy (HA), hypoxia, mild Alzheimer’s (AD), or schizophrenia—leads to intact associative learning but impaired transfer generalization performance. Moreover, the model demonstrates how PD and anterior communicating artery (ACoA) aneurysm—two very different brain disorders that affect different neural mechanisms—can have similar effects on acquired equivalence performance. In particular, the model shows that simulating a loss of dopamine function in the basal ganglia module (as in PD) leads to slow acquisition learning but intact transfer generalization. Similarly, the model shows that simulating the loss of acetylcholine in the hippocampal region (as in ACoA aneurysm) also results in slower acquisition learning. We argue from this that changes in associative learning of stimulus–action pathways (in the basal ganglia) or changes in the learning of stimulus representations (in the hippocampal region) can have similar functional effects.     

On the relation theory of "direct" psychophysics and its publication history

The relation theory attempts to account for the ratio judgments of the "direct" psychophysics founded by S. S. Stevens. Its publication history is reviewed and features of the main contributions are compared.     

Weakly Associative Relation Algebras with Polyadic Composition Operations

In this paper we introduced various classes of weakly associative relation algebras with polyadic composition operations. Among them is the class RWA of representable weakly associative relation algebras with polyadic composition operations. Algebras of this class are relativized representable relation algebras augmented with an infinite set of operations of increasing arity which are generalizations of the binary relative composition. We show that RWA is a canonical variety whose equational theory is decidable.     

Why Freud and Jung Can’t Speak: A Neurological Proposal

This article provides a modern neurological explanation for the theoretical differences in psychoanalytic concepts and techniques between Freud and Jung. Specifically, the article contrasts their analytic skills as arising in the left and right brain, respectively. Modern neurological techniques reveal unique brain functions that explain many of the visionary and so-called mystic phenomena discussed by Jung. Many of his psychoanalytic concepts can be traced to right brain function. Modern research and philosophic analyses also provide light on Freud’s research method and its limitations.Barbara Stevens Barnum, RN, PhD, FAAN, former editor of Nursing Leadership Forum and Nursing & Health Care, is presently a writer/consultant. Before retiring, her positions at Columbia University included holding the directorship, Division of Health Services, Sciences and Education at Teachers College, where she also held the Stewart Chair in the Department of Nursing Education. Dr. Barnum has written in areas of nursing management, theory, education, and aspects of spirtuality/holistic nursing. Her books include: Nursing theory: Analysis, application, evaluation, Spirituality in nursing: From traditional to New Age, and The New Healers: Minds and Hands in Complementary Medicine.     

正结合演算

从结构推理的观点来看,结合演算是一种很弱的逻辑,因为它仅容纳一种结构规则,即“结合规则”。正结合演算作为一种“正命题逻辑”,是结合演算的基础。本文构建了正结合演算结构推理系统BL和对应的公理系统B,阐述了结合演算拒斥“交换规则”、“收缩规则”和“弱化规则”的理论意义和应用价值,证明了系统BL和系统B的等价性。     

Extensive measurement with incomparability

Standard theories of extensive measurement assume that the objects to be measured form a complete order with respect to the relevant property. In this paper, representation and uniqueness theorems are presented for a theory that departs radically from this completeness assumption. It is first shown that any quasi-order on a countable set can be represented by vectors of real numbers. If such an order is supplemented by a concatenation operator, yielding a relational structure that satisfies a set of axioms similar to the standard axioms for an extensive structure, we obtain a scale possessing the crucial properties of a ratio scale. Incomparability is thus compatible with extensive measurement. The paper ends with a brief discussion on some possible applications and developments of this result.     

Bias in proportion judgments: the cyclical power model

When participants make part-whole proportion judgments, systematic bias is commonly observed. In some studies, small proportions are overestimated and large proportions underestimated; in other studies, the reverse pattern occurs. Sometimes the bias pattern repeats cyclically with a higher frequency (e.g., overestimation of proportions less than .25 and between .5 and .75; underestimation otherwise). To account for the various bias patterns, a cyclical power model was derived from Stevens' power law. The model proposes that the amplitude of the bias pattern is determined by the Stevens exponent, beta (i.e., the stimulus continuum being judged), and that the frequency of the pattern is determined by a choice of intermediate reference points in the stimulus. When beta < 1, an over-then-under pattern is predicted; when beta > 1, the under-then-over pattern is predicted. Two experiments confirming the model's assumptions are described. A mixed-cycle version of the model is also proposed that predicts observed asymmetries in bias patterns when the set of reference points varies across trials.