Evaluation of the long-term performance of water-analyzing laboratories

After 7 years of successful operation, the database of the proficiency testing (PT) scheme for water analysis organized by the Institute for Agrobiotechnology (IFA), Tulln contains nearly 4,000 data sets from over 300 interlaboratory comparison samples. About 70 analytical parameters (major ions, metals, trace ions, herbicides, volatile halogenated hydrocarbons, organochlorine pesticides and polycyclic aromatic hydrocarbons) were covered by the PT scheme. The data were evaluated using robust statistics in order to determine a set of coefficients of variation (CV) for each analytical parameter. Concentration and time dependence of the CV were checked. The CV were combined to obtain standard deviations for proficiency assessment (Z-score criteria). Furthermore, a viewer program was designed to facilitate monitoring of the analytical performance of participating laboratories.     

Enhancing students’ fraction magnitude knowledge: A study with students in early elementary education

The idea of magnitude is central to understanding fractional numbers. To investigate this relationship, we implemented a design research project in an urban school in the northeast of the US to examine the potential of a measuring perspective and the mathematical notion of fraction-of-quantity to enhance second-grade students’ conceptual understanding of fraction magnitude. We used ideas from the history of mathematics and mathematics education within a cultural-historical framework to define fractions and construct tasks. The research team consisted of a university professor, two doctoral students, one of whom was an administrator of the municipal board of education, eight elementary school teachers, and a parent. The research sessions involved 35 students divided into two classes, meeting one hour per session twice a week for 12 weeks or 24 hours. The students manipulated non-symbolic or non-numeric manipulatives (Cuisenaire rods) and learned to talk about specific relations they perceived among them. Through physical manipulations and discourse, students developed the idea that a fraction reports a multiplicative comparison between two commensurable quantities of the same kind. Our results indicate that second-grade students appropriated the concept of the magnitude of fractions-of-quantity and, based on mental manipulations of evoked non-numeric images, constructed symbolic expressions involving fractional comparisons.     

Heteroscedasticity and nonnormality

In this paper we consider the problem of comparing several means under heteroscedasticity and nonnormality. By combining Huber's M-estimators with the Brown-Forsythe test , several robust procedures were developed; these procedures were compared through computer simulation studies with-the Tan-Tabatabai procedure which was developed by combining Tiku's MML estimators with the Brown-Forsythe test . The numerical results indicate clearly that the Tan-Tabatabai procedure is considerably more powerful than tests based on Huber's M-estimators over a wide range of nonnormal distributions     

Flippable Pairs and Subset Comparisons in Comparative Probability Orderings

We show that every additively representable comparative probability order on n atoms is determined by at least n–1 binary subset comparisons. We show that there are many orders of this kind, not just the lexicographic order. These results provide answers to two questions of Fishburn et al. (Math. Oper. Res. 27:227–243, 2002). We also study the flip relation on the class of all comparative probability orders introduced by Maclagan. We generalise an important theorem of Fishburn, Pekeč and Reeds, by showing that in any minimal set of comparisons that determine a comparative probability order, all comparisons are flippable. By calculating the characteristics of the flip relation for n=6 we discover that the polytopes associated with the regions in the corresponding hyperplane arrangement can have no more than 13 facets and that there are 20 regions whose associated polytopes have 13 facets. All the neighbours of the 20 comparative probability orders which correspond to those regions are representable. Research partially supported by the N.Z. Centres of Research Excellence Fund (grant UOA 201).     

Comparing the Likelihood Functions of Phylogenetic Trees

DNA sequence data provide a good source of information on the evolutionary history of organisms. Among the proposed methods, the maximum likelihood methods require an explicit probabilistic model of nucleotide substitution that makes the assumption clear. However, procedures for testing hypotheses on topologies have not been well developed. We propose a revised version of the maximum likelihood estimator of a tree and derive some of its properties. Then we present tests to compare given trees and to derive the most likely candidates for the true topology, applying to maximum likelihoods the notion of contrast, as defined in the framework of the analysis of variance, and the procedures used in multiple comparison. Finally, an example is presented.     

Expected cost bounds for the selection and ordering procedures based on binary-type questions

The problems considered in this paper are similar to those studied by Arrow, Pesotchinsky and Sobel (1981). We search for the optimal methods of ordering of a sample or of identifying sample order statistics by asking binary type questions and employing the criterion which defines the cost of a question as the number of responses. The asymptotic results show that our procedures with simple algorithms are comparable (in terms of cost) with the best known methods for ordering by using pairwise comparisons.     

Clusterwise aggregation of relations

In this paper we handle the general problem of finding q(> 1) central relations on a set of objects which best fit the information contained in a finite number of given relations on that set. The proposed CAR (clusterwise aggregation of relations) algorithm allows one to consider the well-known situation of determining a single central relation as a special case (q = 1) and takes into account the fact that the representation of appropriately selected subsets of relations by different central relations can provide additional insights into whether different clusters or segments of relations exist in the given set of relations. Two examples demonstrate the usefulness of the suggested approach.     

Judgement by pairwise comparisons

This article gives an alternative solution to the problem of W. Gaul and M. Schader (Vol. 4, no. 4 pp. 273–282).     

Vergleichsmessung der spezifischen Aktivität einer 90Sr/90Y-Lösung zwischen dem ASMW Berlin und dem WNIIM Leningrad

In der Arbeit wird über eine internationale Vergleichsmessung berichtet, die die beteiligten Institutionen zur Überprüfung der Genauigkeit ihrer Normalmeβeinrichtungen zur Darstellung der Aktivitätseinheit β-strahlender Nuklide durchführten. Verschiedene Korrektionsfaktoren sowie die Verfahren ihrer Ermittlung werden diskutiert. Die Abweichung der Ergebnisse beträgt 0,7% und liegt damit innerhalb der angegebenen Fehlergrenzen.