Symmetry,scale types,and generalizations of classical physical measurement

A review is presented of a number of recent results concerning fundamental measurement structures with a particular emphasis on generalizations of physical measurement. Relational structures are classified in terms of richness and redundancy of their automorphism groups (i.e., in terms of their symmetries). By means of this classification, the possible types of measurement scales are described, and the possibilities for structures with concatenation operations (either associative or nonassociative and either positive or intensive) and for conjoint structures whose automorphisms factor into component automorphisms are explicitly spelled out. Certain conjoint structures with concatenation operations on components are viewed as generalizations of classical physical measurement and some of their most important algebraic and representational properties are explored.     

A mathematical analysis of a natural class of partitions of a graph

Any finite graph (nondirected, no loops) can be coupled with its complementary graph producing what we term a completed graph. This paper studies completed graphs in the context of a structural concept called stratification that is motivated by theoretical work in social psychology and sociology. A completed graph is stratified in case its node (vertex) set can be divided into two nonempty subsets with exactly one of the two types of ties holding between every pair of nodes from distinct sets. A completed graph is totally stratified in case every one of its nontrivial completed subgraphs is stratified. The first part of the paper relates the concept of stratification to the familiar graph property of connectedness. In particular, a completed graph is totally stratified if and only if it does not have a four point completed subgraph that is connected in both the original graph relation and its complementary relation. Using the concept of stratification, completed graphs can be decomposed uniquely into taxonomic structures, that is, nested sets of partitions. An algorithm for the decomposition based on the work with the concept of stratification is developed. The relationship between the notion of stratification and the ideas in balance theory is examined, and stratification is viewed as a generalization of Davis' notion of clustering. It turns out that the concepts of clique, status, and structural equivalence used throughout the social networks literature can be defined in an interesting way for completed graphs. Cliques are sets of nodes with similar internal links and statuses are sets of nodes with similar external links. Structurally equivalent sets are both cliques and statuses. The concepts of status and structural equivalence are closely related to the decomposition algorithm. In particular, for any totally stratified completed graph, the set of all statuses and the set of all maximal structurally equivalent sets can be generated by mathematical operations performed on the taxonomic decomposition of the completed graph. Finally some of our resuls are informally related to the blockmodeling approach to analyzing social network data discussed in several previous Journal of Mathematical Psychology articles.     

Male alcoholics,secondary psychopathy and Eysenck's theory of personality

The scores of several samples of male alcoholics and others on the MAC Scale (an MMPI-derived scale for the differentiation of male alcohol abusers from non-substance-abusing male psychiatric outpatients) are summarized, and the responses of the false negatives and false positives from the scale's standardization samples are analyzed. The following 7conclusions were reached: (1) There are two broad classes of male alcoholics—those who fall into the E + N + quadrant of secondary psychopathy and those who fall into the E - N + quadrant of neurotic introversion; (2) Unselected runs of male alcoholics populate these two quadrants in a quite stable ratio of 17 to 3, respectively; (3) Pari passu, Eysenck's placement of alcoholics in the E + N + quadrant will be upheld in approximately 85% of the cases while those who hold that alcoholics are ‘dysthymics who also drink too much’ will find themselves in error in approximately similar degree.     

An approach to geometry of visual space with no a priori mapping functions: Multidimensional mapping according to Riemannian metrics

A method of multidimensional mapping is described which constructs a configuration of points {P_i} in a Euclidean map of Riemannian space of constant curvature (hyperbolic, Euclidean, and elliptic) from the dissimilarity matrix (d_ij). The method was applied to the distance matrix in visual space where stimulus points Q_i were either small light points in the dark or small black points in the illuminated field surrounded by white curtains and d_ij represent scaled values of perceptual distances. Configuration of points {Q_i} were at intersections of parallel or distance alleys and horopters for the subject in the horizontal plane of the eye level. In contrast to the theoretical equations for {Q_i} by Luneburg and Blank, no a priori assumption on mapping functions between {Q_i} and {P_i} is necessary in this procedure to fit theoretical curves to {P_i} in the Euclidean map. The data were accounted for better by equations in the hyperbolic plane than by ones in the Euclidean plane. Discussions are made on robustness of Euclidean representation and on how to approach geometry of visual space as a dynamic entity under more natural conditions than the traditional frameless condition for alley and horopter experiments.     

The response ratio hypothesis for magnitude estimation

Assume that each presentation of a signal produces two independent random variable representations and that the ratio of responses on successive trials of a magnitude estimation experiment are proportional to the ratio of a representation from the present trial, which representation is then lost, to the remaining one from the previous trial. The mean response to a particular signal depends on the mean of the representation used, but in general exhibits drift over trials and sequential effects due to the preceeding trial; the mean response ratio does not exhibit drift, but it has a simple form only when there are no sequential effects; however, a modified mean ratio function has a simple form. A model suggested by D. V. Cross is a special case of this one. Simple timing and counting models for the representations fail to exhibit sequential effects, contrary to considerable data. However, data of the authors have suggested a version of the timing model in which the sample size of the representation varies by an order of magnitude depending on how close the signal is to the preceding one; this hypothesis accounts for the observed sequential effects and other aspects of the data.     

Comprehension of gerundive constructions by Broca's aphasics

Eleven diagnosed Broca's aphasics were tested for comprehension of sentences containing gerundive constructions. Results showed that, as a group, these patients were sensitive to grammatical distinctions signaled by “function” words, contrary to the predictions of some current theories about the specific deficits exhibited in this syndrome. Further analysis showed that the patients fell into two subgroups, one which exhibited sensitivity to the distinctions in normal grammar, and another which exhibited no sensitivity to such distinctions. This suggests that the syndrome may have different manifestations in different populations—patient age is suggested as a possible means for distinguishing the two groups. Suggestions for further research into the nature of the comprehension deficits exhibited by Broca's aphasics are made.     

On the scales of measurement

Let $\text{X}$ = 〈X, ≧, R₁, R₂…〉 be a relational structure, 〈$\text{X, ≧}$〉 be a Dedekind complete, totally ordered set, and n be a nonnegative integer. $\text{X}$ is said to satisfy n-point homogeneity if and only if for each x₁,…, x_n, y₁,…, y_n such that x₁ ? x₂ ? … ? x_n and y₁ ? y₂ … ? y_n, there exists an automorphism α of $\text{X}$ such that α(x₁) = y_i. $\text{X}$ is said to satisfy n-point uniqueness if and only if for all automorphisms β and γ of $\text{X}$, if β and γ agree at n distinct points of $\text{X}$, then β and γ are identical. It is shown that if $\text{X}$ satisfies n-point homogeneity and n-point uniqueness, then n ≦ 2, and for the case n = 1, $\text{X}$ is ratio scalable, and for the case n = 2, interval scalable. This result is very general and may in part provide an explanation of why so few scale types have arisen in science. The cases of 0-point homogeneity and infinite point homogeneity are also discussed.     

The acquisition of prenominal modifier sequences

This paper presents a study of children's interpretations of prenominal modifier sequences like the one in ‘the second green ball’. In such phrases the modifier ‘second’ has scope over the rest of the phrase. It is shown that children misinterpret these phrases in such a way that it appears that they give ‘second’ scope only over the head noun of the phrase. A series of six experiments, all designed to test various hypotheses about the motivating factors behind this misinterpretation is described. Experiments 1 and 2 constitute a replication and extension of the work in this area originally done by Roeper (1972). Experiment 3 tests a set of hypotheses about the influence of certain syntactic variables in conditioning the children's misinterpretations. Experiment 4 is an attempt to check an hypothesis about the semantic complexity of phrases like ‘the second green ball’. Experiments 5 and 6 test the children's sensitivity to ordering constraints in different kinds of prenominal modifier sequences. The results are presented in Section 3 and discussed in Section 4. It is argued that the children's misinterpretations of the phrases come about because of their tendency to be conservative about the amount of hierarchically organized structure they will postulate in the absence of evidence that such structure is needed.     

The effect of repetition on search through active memory

Short lists of digits, some of which contained repeated items, were searched in memory. Repeated items were recognized faster than non-repeated items. Search times for non-repeated items in lists with a repeat were slightly faster than those for lists of the same length in which no items were repeated. A serial position effect was found such that items at either the beginning or end of the memory list were responded to faster than items in the middle of the list. Neither the repeated digit effect nor the serial position effect support serial exhaustive search. An empirical equation was found by which response time is predicted by the log proportion of items in a memory list that match the test stimulus. These results are compatible with the notions of neural pathway activation and trace strength.     

Learning to search: From weak methods to domain-specific heuristics

Learning from experience involves three distinct components—generating behavior, assigning credit, and modifying behavior. We discuss these components in the context of learning search heuristics, along with the types of learning that can occur. We then focus on SAGE, a system that improves its search strategies with practice. The program is implemented as a production system, and learns by creating and strengthening rules for proposing moves. SAGE incorporates five different heuristics for assigning credit and blame, and employs a discrimination process to direct its search through the space of rules. The system has shown its generality by learning heuristics for directing search in six different task domains. In addition to improving its search behavior on practice problems, SAGE is able to transfer its expertise to scaled-up versions of a task, and in one case, transfers its acquired search strategy to problems with different initial and goal states.     

From 3 to 20: The early training project: By S. W. Gray,B. K. Ramsey,and R. A. Klaus. Baltimore: Univ. Park Press, 1982. xi + 344 pp., $27.95

A perspective on the controversy over the effects of early childhood intervention is provided. Gray, Ramsey, and Klaus' book is reviewed. The results of the study are commented upon, and implications for the continuing controversy about the effects of early childhood intervention are suggested.