Interactive scaling with individual subjects

A metric multidimensional scaling (MDS) procedure based on computer-subject interaction is developed, and an experiment designed to validate the procedure is presented. The interactive MDS system allows generalization of current MDS systems in two directions: (a) very large numbers of stimuli may be scaled; and (b) the scaling is performed with individual subjects, facilitating the investigation of individual as well as group processes. The experiment provided positive support for the interactive MDS system. Specifically, (a) individual data are amenable to meaningful interpretation, and they provide a tentative basis for quantitative investigation; and (b) grouped data provide meaningful interpretive and quantitative results which are equivalent to results from standard paired-comparisons methods.This report was supported in part by a PHS research grant, No. M-10006, from the National Institute of Mental Health, in part by a Science Development grant, No. GU-2059, from the National Science Foundation, both granted to the Psychometric Laboratory at the University of North Carolina, and in part by a PHS research grant, No. MH-16474, from the National Institute of Mental Health, Public Health Service, granted to the second author. The major portion of this research was performed while the second author was the L. L. Thurstone Distinguished Fellow at the Psychometric Laboratory of the University of North Carolina while on leave from the University of Southern California. The authors are indebted to Amnon Rapoport and Thomas S. Wallsten for their critical evaluations of an earlier version of this report. While this paper was entirely a cooperative effort on the part of both authors, the first author was primarily responsible for the algorithms, and the second for developing the mathematical model.     

Generalizability of scores influenced by multiple sources of variance

Generalizability theory concerns the adequacy with which a universe score can be inferred from a set of observations. In this paper the theory is applied to a universe in which observations are classifiable according to two independent variable aspects of the measuring procedure. Several types of universe scores are developed and the variance components ascertained for each type. The composition of expected observed-score variance and the adequacy of inference to a particular type of universe score is a function of the procedure used in gathering data. A generalizability study provides estimates of variance components which can be used in designing an efficient procedure for a particular decision purpose.This study was conducted under Grant M-1839 from the National Institute of Mental Health while the authors were on the staff of the University of Illinois. Dr. Rajaratnam shared responsibility for the technical report of July, 1961 on which this paper is based. The present revision was made subsequent to her death in 1963. The present addresses of the other authors are: Goldine C. Gleser, Department of Psychiatry, Central Clinic, Cincinatti, 29, Ohio; Lee J. Cronbach, School of Education, Stanford University.     

Gödel’s Natural Deduction

This is a companion to a paper by the authors entitled “Gödel on deduction”, which examined the links between some philosophical views ascribed to Gödel and general proof theory. When writing that other paper, the authors were not acquainted with a system of natural deduction that Gödel presented with the help of Gentzen’s sequents, which amounts to Ja?kowski’s natural deduction system of 1934, and which may be found in Gödel’s unpublished notes for the elementary logic course he gave in 1939 at the University of Notre Dame. Here one finds a presentation of this system of Gödel accompanied by a brief reexamination in the light of the notes of some points concerning his interest in sequents made in the preceding paper. This is preceded by a brief summary of Gödel’s Notre Dame course, and is followed by comments concerning Gödel’s natural deduction system.     

Extensive measurement and ratio functions

Extensive measurement theory is developed in terms of theratio of two elements of an arbitrary (not necessarily Archimedean) extensive structure; thisextensive ratio space is a special case of a more general structure called aratio space. Ratio spaces possess a natural family of numerical scales (r-scales) which are definable in non-representational terms; ther-scales for an extensive ratio space thus constitute a family of numerical scales (extensive r-scales) for extensive structures which are defined in a non-representational manner. This is interpreted as involving arelational theory of quantity which contrasts in certain respects with thequalitative theory of quantity implicit in standard representational extensive measurement theory. The representational properties of extensiver-scales are investigated, and found to coincide withweak extensive measurement in the sense of Holman. This provides support for the thesis (developed in a separate paper) that weak extensive measurement is a more natural model of actual physical extensive scales than is the standard model using strong extensive measurement. Finally, the present apparatus is applied to slightly simplify the existing necessary and sufficient conditions for strong extensive measurement.This work was supported by a Junior Faculty Summer Research Fellowship awarded by the University Research Council of the University of Oklahoma for the summer of 1985.     

General estimators for the reliability of qualitative data

We study a proportional reduction in loss (PRL) measure for the reliability of categorical data and consider the general case in which each ofN judges assigns a subject to one ofK categories. This measure has been shown to be equivalent to a measure proposed by Perreault and Leigh for a special case when there are two equally competent judges, and the correct category has a uniform prior distribution. We consider a general framework where the correct category is assumed to have an arbitrary prior distribution, and where classification probabilities vary by correct category, judge, and category of classification. In this setting, we consider PRL reliability measures based on two estimators of the correct category—the empirical Bayes estimator and an estimator based on the judges' consensus choice. We also discuss four important special cases of the general model and study several types of lower bounds for PRL reliability.Bruce Cooil is Associate Professor of Statistics, and Roland T. Rust is Professor and area head for Marketing, Owen Graduate School of Management, Vanderbilt University. The authors thank three anonymous reviewers and an Associate Editor for their helpful comments and suggestions. This work was supported in part by the Dean's Fund for Faculty Research of the Owen Graduate School of Management, Vanderbilt University.     

Bias and runs in dice throwing and recording: A few million throws

An experimenter threw individually 219 different dice of four different brands and recorded even and odd outcomes for one block of 20,000 trials for each die—4,380,000 throws in all. The resulting data on runs offer a basis for comparing the observed properties of such a physical randomizing process with theory and with simulations based on pseudo-random numbers and RAND Corporation random numbers. Although generally the results are close to those forecast by theory, some notable exceptions raise questions about the surprise value that should be associated with occurrences two standard deviations from the mean. These data suggest that the usual significance level may well actually be running from 7 to 15 percent instead of the theoretical 5 percent.The data base is the largest of its kind. A set generated by one brand of dice contains 2,000,000 bits and is the first handmade empirical data of such size to fail to show a significant departure from ideal theory in either location or scale.The analysis was facilitated by a National Science Foundation grant GS-341 and and its continuation GS-2044X. It forms part of a larger study of data analysis.Mr. Longcor is from Waukegan, Illinois; the other authors are from Harvard University. Dr. Iversen has moved to the University of Michigan.     

观察式学习条件下类别维度与家族相似性类别学习

在观察式学习条件下考察类别维度数量（三维度类别、六维度类别、九维度类别）,类别维度结构（线性分离结构、非线性分离结构）的变化对家族相似性类别学习的影响。结果表明：观察式学习条件下,类别维度数量的变化对样例学习无影响,对特征学习有影响,表现在特征正确再认数量上有差异,但特征再认正确率上无差异;类别维度结构的变化对样例学习有影响,对特征学习无影响,表现在样例正确再认数目上线性结构大于非线性结构。     

Thirty categorization results in search of a model

One category structure dominated in the shift toward exemplar-based theories of categorization. Given the theoretical burden on this category structure, the authors reanalyzed 30 of its uses over 20 years in 8 articles. The authors suggest 4 conclusions. (a) This category structure may encourage exemplar-memorization processes because of its poor structure, the learning difficulties it causes, and its small, memorizable exemplar sets. Its results may only generalize narrowly. (b) Exemplar models have an advantage in fitting these 30 data sets only because they reproduce a performance advantage for training items. Other models fit equally well if granted this capacity. (c) A simpler exemplar process than assumed by exemplar models suffices to explain these data sets. (d) An important qualitative result predicted by exemplar theory is not found overall and possibly should not even be expected. The authors conclude that the data produced by this category structure do not clearly support exemplar theory.     

Signal/noise ratios for domain-referenced tests

Using the assumption of randomly parallel tests and concepts from generalizability theory, three signal/noise ratios for domain-referenced tests are developed, discussed, and compared. The three ratios have the same noise but different signals depending upon the kind of decision to be made as a result of measurement. It is also shown that these ratios incorporate a definition of noise or error which is different from the classical definition of noise typically used to characterize norm-referenced tests. Part of the work leading to this paper was accomplished while the authors were assistant professors at the State University of New York at Stony Brook.     

The advantages of mathematical modeling over traditional methods in the analysis of category clustering

A mathematical model for the analysis of category clustering is developed and testd. The model, which can be applied to categories of any size, is an extension of a two-item statistical model developed by Batchelder and Riefer (Psychological Review, 1980, 87, 375–397), and is equivalent to their model when categories consist of two items. The model is based on a current theory of clustering which postulates that the learning of a list of category items occurs on different hierarchical levels. Two category list-learning experiments are presented, and the data from these experiments are analyzed using the general statistical model. The first experiment reveals that the probabilities of storing and retrieving a cluster increase with category size, while the learning of items as singletons decreases. The effects of within-category spacing indicate that the storage of clusters decreases while cluster retrievability increases with an increase in input spacing. In the second experiment, the storage and retrieval of clusters are shown to be unaffected by whether the presentation of items is uncued or cued with the name of the category. However, the association of items decreases and the learning of items as singletons increases with uncued presentation. In the final sections, the general statistical model is compared to other methods for the measurement of category clustering. The model is shown to be superior to numerical indices of clustering, since these measures are not based on any theory of clustering, and because unitary measures cannot capture the multiprocess nature of categorized recall. The model is also argued to have certain advantages over other mathematical models that have been applied to category clustering, since these models cannot account for situations in which a portion of the items are clustered while others are learned singularly.     

Sequences in countable nonstandard models of the natural numbers

Two different equivalence relations on countable nonstandard models of the natural numbers are considered. Properties of a standard sequence A are correlated with topological properties of the equivalence classes of the transfer of A. This provides a method for translating results from analysis into theorems about sequences of natural numbers.Most of the results in this paper appeared in the author's Ph.D. dissertation at the University of Colorado, 1985.     

Identity, place, and bystander intervention: social categories and helping after natural disasters

The authors developed a Self-Categorization Theory (SCT) approach to bystander behavior. Participants were 100 undergraduates at an English university. The authors made either a European or a British identity salient. Participants then rated their likelihood of offering both financial and political help after natural disasters in Europe and South America. When European (but not British) identity was salient, participants were less likely to offer help for disasters in South America than Europe. They were also more likely to offer financial help after disasters in Europe when European non-British identity was salient. There were no differences in levels of emotional response to disasters by identity salience. Results indicate that social category relations rather than geographical proximity or emotional reaction are most important in increasing helping behavior after natural disasters.     

A Categorical Approach to Probability Theory

First, we discuss basic probability notions from the viewpoint of category theory. Our approach is based on the following four “sine quibus non” conditions: 1. (elementary) category theory is efficient (and suffices); 2. random variables, observables, probability measures, and states are morphisms; 3. classical probability theory and fuzzy probability theory in the sense of S. Gudder and S. Bugajski are special cases of a more general model; 4. a good model allows natural modifications. Second, we show that the category ID of D-posets of fuzzy sets and sequentially continuous D-homomorphisms allows to characterize the passage from classical to fuzzy events as the minimal generalization having nontrivial quantum character: a degenerated state can be transported to a nondegenerated one. Third, we describe a general model of probability theory based on the category ID so that the classical and fuzzy probability theories become special cases and the model allows natural modifications. Finally, we present a modification in which the closed unit interval [0,1] as the domain of traditional states is replaced by a suitable simplex.     

Category use and category learning

Categorization models based on laboratory research focus on a narrower range of explanatory constructs than appears necessary for explaining the structure of natural categories. This mismatch is caused by the reliance on classification as the basis of laboratory studies. Category representations are formed in the process of interacting with category members. Thus, laboratory studies must explore a range of category uses. The authors review the effects of a variety of category uses on category learning. First, there is an extensive discussion contrasting classification with a predictive inference task that is formally equivalent to classification but leads to a very different pattern of learning. Then, research on the effects of problem solving, communication, and combining inference and classification is reviewed.     

Domain differences in the structure of artifactual and natural categories

In three experiments, different methodologies, measures, and items were employed to address the question of whether, and to what extent, membership in a semantic category is all or none (i.e., absolute) or a matter of degree (i.e., graded). Resemblance theory claims that categorization is based on similarity, and because similarity is graded, category membership may also be graded. Psychological essentialism asserts that categorization is based on the presumption of the category essence. Because artifactual (e.g., FURNITURE) and natural (e.g., FRUIT) categories have different sorts of essences, artifacts and natural kinds may be categorized in qualitatively different manners. The results converged onthe finding of a robust domain difference in category structure: Artifactual categories were more graded than natural categories. Furthermore, typicality reliably predicted absolute category membership, but failed to predict graded category membership. These results suggest that resemblance theory and psychological essentialism may provide a concerted account of representation and categorization across domains.     

类别学习多重系统理论的研究与展望

介绍了国外学者为验证类别学习的多重系统理论所做的行为实验研究及主要结果,包括反馈训练、延迟反馈、反应位置、间断的类别刺激分布等因素对信息整合的类别学习的影响,和数字Stroop任务、序列记忆扫描任务、类别数量等因素对基于规则的类别学习的影响;同时,作者指出了实验中没有很好地解决两种类别结构之间的难度差异等问题,并提出了今后需要进一步研究的课题     

A note on natural numbers objects in monoidal categories

The internal language of a monoidal category yields simple proofs of results about a natural numbers object therein.     

Supernatural beliefs,natural kinds,and conceptual structure

This article presents cross-cultural evidence in support of the notion that adults' natural kind concepts are theory based but may be informed by knowledge/belief systems other than the biological. Three groups of subjects from western Nigeria--rural, urban, and elite--participated in the study. Subjects heard stories describing alterations of appearance; that is, one natural kind was made to resemble another in both ritual and nonritual contexts. Subjects then were required to judge the identity of the altered item and to give an explanation for the category judgment. It was predicted that subjects would make more nonpreservation-of-identity category judgments supported by supernatural explanations in the ritual contexts and that subjects' use of supernatural explanations would reflect the extent of their engagement with the supernatural. The first prediction was borne out; the second prediction was only partially supported. Discussion of the results emphasizes the importance of exploring the role of sociocultural factors in conceptual structure.     

Numerical methods for polynomial models in nonlinear factor analysis

The basic concepts of nonlinear factor analysis are introduced and some extensions of the general theory are developed. An elementary account of the class of multiple-factor polynomial models is presented, using more elementary algebraic methods than have been employed in earlier accounts of this theory. Working formulas are developed for the multiple-factor polynomial model without product terms.Some empirical results are presented.This work was carried out while the author was a Visiting Research Fellow at the Educational Testing Service, Princeton, N. J. It made use of computer facilities at Princeton University that are supported in part by National Science Foundation Grant NSF-Gp579.     

Why Numbers Are Sets

I follow standard mathematical practice and theory to argue that the natural numbers are the finite von Neumann ordinals. I present the reasons standardly given for identifying the natural numbers with the finite von Neumann's (e.g., recursiveness; well-ordering principles; continuity at transfinite limits; minimality; and identification of n with the set of all numbers less than n). I give a detailed mathematical demonstration that 0 is { } and for every natural number n, n is the set of all natural numbers less than n. Natural numbers are sets. They are the finite von Neumann ordinals.