A family of chance-corrected association coefficients for metric scales

A chance-corrected version of the family of association coefficients for metric scales proposed by Zegers and ten Berge is presented. It is shown that a matrix with chance-corrected coefficients between a number of variables is Gramian. The members of the chance-corrected family are shown to be partially ordered. The author is obliged to Jos ten Berge for helpful comments.     

Comparative evaluation of two superior stopping rules for hierarchical cluster analysis

A split-sample replication stopping rule for hierarchical cluster analysis is compared with the internal criterion previously found superior by Milligan and Cooper (1985) in their comparison of 30 different procedures. The number and extent of overlap of the latent population distributions was systematically varied in the present evaluation of stopping-rule validity. Equal and unequal population base rates were also considered. Both stopping rules correctly identified the actual number of populations when there was essentially no overlap and clusters occupied visually distinct regions of the measurement space. The replication criterion, which is evaluated by clustering of cluster means from preliminary analyses that are accomplished on random partitions of an original data set, was superior as the degree of overlap in population distributions increased. Neither method performed adequately when overlap obliterated visually discernible density nodes.This research was supported in part by NIMH grant 5R01 MH 32457 14.     

Proof-Theoretic Modal PA-Completeness III: The Syntactic Proof

This paper is the final part of the syntactic demonstration of the Arithmetical Completeness of the modal system G; in the preceding parts [9] and [10] the tools for the proof were defined, in particular the notion of syntactic countermodel. Our strategy is: PA-completeness of G as a search for interpretations which force the distance between G and a GL-LIN-theorem to zero. If the GL-LIN-theorem S is not a G-theorem, we construct a formula H expressing the non G-provability of S, so that ⊢_GL-LIN ∼ H and so that a canonical proof T of ∼ H in GL-LIN is a syntactic countermodel for S with respect to G, which has the height θ(T) equal to the distance d(S, G) of S from G. Then we define the interpretation ξ of S which represents the proof-tree T in PA. By induction on θ(T), we prove that ⊢_PA S^ξ and d(S, G) > 0 imply the inconsistency of PA. This revised version was published online in June 2006 with corrections to the Cover Date.     

Cognitive Task Analysis for Implicit Knowledge About Visual Representations With Similarity Learning Methods

Visual representations are prevalent in STEM instruction. To benefit from visuals, students need representational competencies that enable them to see meaningful information. Most research has focused on explicit conceptual representational competencies, but implicit perceptual competencies might also allow students to efficiently see meaningful information in visuals. Most common methods to assess students’ representational competencies rely on verbal explanations or assume explicit attention. However, because perceptual competencies are implicit and not necessarily verbally accessible, these methods are ill‐equipped to assess them. We address these shortcomings with a method that draws on similarity learning, a machine learning technique that detects visual features that account for participants’ responses to triplet comparisons of visuals. In Experiment 1, 614 chemistry students judged the similarity of Lewis structures and in Experiment 2, 489 students judged the similarity of ball‐and‐stick models. Our results showed that our method can detect visual features that drive students’ perception and suggested that students’ conceptual knowledge about molecules informed perceptual competencies through top‐down processes. Furthermore, Experiment 2 tested whether we can improve the efficiency of the method with active sampling. Results showed that random sampling yielded higher accuracy than active sampling for small sample sizes. Together, the experiments provide the first method to assess students’ perceptual competencies implicitly, without requiring verbalization or assuming explicit visual attention. These findings have implications for the design of instructional interventions that help students acquire perceptual representational competencies.     

Metric and nonmetric plotting models

A problem with standard printer scatterplot routines is that points are not adequately labeled. Four methods of generating printer plots are presented. All four methods allow multi-character point labels, and three of the methods can guarantee that no portion of the labels are hidden. This is accomplished by using varying amounts of metric and nonmetric distance information to locate the points.The author wishes to thank Warren Sarle and Forrest Young who provided many valuable suggestions during the development of PROC IDPLOT.The methods and procedure described in this paper were developed by the author at the University of North Carolina as a part of a software development project funded by SAS Institute Inc.     

A contribution to the study of the metric and Euclidean structures of dissimilarities

This paper is concerned with the geometric properties of dissimilarity coefficients defined on finite sets and especially with their Euclidean nature. We present several particular transformations which preserve Euclideanarity and we complete, through the study of a one-parameter family, the current knowledge of the metric and Euclidean structure of coefficients based on binary data. These results are directly deduced from two theorems which prove the positive semi-definite status of some quadratic forms which play a large role in some definitions of dissimilarity commonly used.The authors wish to thank B. Fichet for his helpful suggestions, the associate Editor and an anonymous reviewer for comments and highly constructive criticisms on earlier drafts of the paper.     

Proof-Theoretic Modal PA-Completeness I: A System-Sequent Metric

This paper is the first of a series of three articles that present the syntactic proof of the PA-completeness of the modal system G, by introducing suitable proof-theoretic objects, which also have an independent interest. We start from the syntactic PA-completeness of modal system GL-LIN, previously obtained in [7], [8], and so we assume to be working on modal sequents S which are GL-LIN-theorems. If S is not a G-theorem we define here a notion of syntactic metric d(S, G): we calculate a canonical characteristic fomula H of S (char(S)) so that _G H (S) and _GL-LIN H, and the complexity of H gives the distance d(S, G) of S from G. Then, in order to produce the whole completeness proof as an induction on this d(S, G), we introduce the tree-interpretation of a modal sequent Q into PA, that sends the letters of Q into PA-formulas describing the properties of a GL-LIN-proof P of Q: It is also a d(*, G)-metric linked interpretation, since it will be applied to a proof-tree T of H with H = char(S) and ( H) = d(S, G).     

Automatic grading and hinting in open-ended text questions

Open-ended text questions provide better assessment of learner’s knowledge, but analysing answers for this kind of questions, checking their correctness, and generating of detailed formative feedback about errors for the learner are more difficult and complex tasks than for closed-ended questions like multiple-choice.The analysis of answers for open-ended questions can be performed on different levels. Analysis on character level allows to find errors in characters’ placement inside a word or a token; it is typically used to detect and correct typos, allowing to differ typos from actual errors in the learner’s answer. The word-level or token-level analysis allows finding misplaced, extraneous, or missing words in the sentence. The semantic-level analysis is used to capture formally the meaning of the learner’s answer and compare it with the meaning of the correct answer that can be provided in a natural or formal language. Some systems and approaches use analysis on several levels.The variability of the answers for open-ended questions significantly increases the complexity of the error search and formative feedback generation tasks. Different types of patterns including regular expressions and their use in questions with patterned answers are discussed. The types of formative feedback and modern approaches and their capabilities to generate feedback on different levels are discussed too.Statistical approaches or loosely defined template rules are inclined to false-positive grading. They are generally lowering the workload of creating questions, but provide low feedback. Approaches based on strictly-defined sets of correct answers perform better in providing hinting and answer-until-correct feedback. They are characterised by a higher workload of creating questions because of the need to account for every possible correct answer by the teacher and fewer types of detected errors.The optimal choice for creating automatised e-learning courses are template-based open-ended question systems like OntoPeFeGe, Preg, METEOR, and CorrectWriting which allows answer-until-correct feedback and are able to find and report various types of errors. This approach requires more time to create questions, but less time to manage the learning process in the courses once they are run.     

Point-Free Geometry and Verisimilitude of Theories

A metric approach to Popper’s verisimilitude question is proposed which is related to point-free geometry. Indeed, we define the theory of approximate metric spaces whose primitive notions are regions, inclusion relation, minimum distance, and maximum distance between regions. Then, we show that the class of possible scientific theories has the structure of an approximate metric space. So, we can define the verisimilitude of a theory as a function of its (approximate) distance from the truth. This avoids some of the difficulties arising from the known definitions of verisimilitude.