Model Selection versus Model Averaging in Dose Finding Studies

Phase II dose finding studies in clinical drug development are typically conducted to adequately characterize the dose response relationship of a new drug. An important decision is then on the choice of a suitable dose response function to support dose selection for the subsequent Phase III studies. In this paper we compare different approaches for model selection and model averaging using mathematical properties as well as simulations. Accordingly, we review and illustrate asymptotic properties of model selection criteria and investigate their behavior when changing the sample size but keeping the effect size constant. In a large scale simulation study we investigate how the various approaches perform in realistically chosen settings. Finally, the different methods are illustrated with a recently conducted Phase II dosefinding study in patients with chronic obstructive pulmonary disease.Comment: Keywords and Phrases: Model selection; model averaging; clinical trials; simulation stud

The impact of model detail on power grid resilience measures

Peer reviewedPreprin

Optimal designs for active controlled dose finding trials with efficacy-toxicity outcomes

Nonlinear regression models addressing both efficacy and toxicity outcomes are increasingly used in dose-finding trials, such as in pharmaceutical drug development. However, research on related experimental design problems for corresponding active controlled trials is still scarce. In this paper we derive optimal designs to estimate efficacy and toxicity in an active controlled clinical dose finding trial when the bivariate continuous outcomes are modeled either by polynomials up to degree 2, the Michaelis- Menten model, the Emax model, or a combination thereof. We determine upper bounds on the number of different doses levels required for the optimal design and provide conditions under which the boundary points of the design space are included in the optimal design. We also provide an analytical description of the minimally supported $D$-optimal designs and show that they do not depend on the correlation between the bivariate outcomes. We illustrate the proposed methods with numerical examples and demonstrate the advantages of the $D$-optimal design for a trial, which has recently been considered in the literature.Comment: Keywords and Phrases: Active controlled trials, dose finding, optimal design, admissible design, Emax model, Equivalence theorem, Particle swarm optimization, Tchebycheff syste

Herramientas digitales para la modelización matemática colaborativa en línea

[EN] To enable collaborative modeling activities online digital tools are essential. In this paper we present a holistic and adaptable concept for the development and implementation of modeling activities – which could especially be fruitful in times of homeschooling and distance learning. The concept is based on two digital tools: Jupyter Notebooks and a communication platform with video conferences.We carried out this concept in the context of two types of modeling activities: guided modeling days, where the students work on previously prepared and didactically developed digital learning material, and modeling weeks, in which the students work on open problems from research and industry very freely. In this paper the usage of Jupyter Notebook in modeling activities is presented and illustrated with the example of the optimization of a solar power plant. On top, we share our experiences in online modeling activities with high-school students in Germany.[ES] Para facilitar las actividades de modelización colaborativa en línea, las herramientas digitales son esenciales. En este trabajo presentamos un concepto holístico y adaptable para el desarrollo y la implementación de actividades de modelización – que podría ser especialmente provechoso en tiempos de educación a distancia. El concepto se basa en dos herramientas digitales: Jupyter Notebooks y una plataforma de comunicación con videoconferencia. Realizamos este concepto en el contexto de dos tipos de actividades de modelización matemática: días de modelización guiada, en los que los alumnos trabajan con material de aprendizaje digital previamente preparado y desarrollado didácticamente, y semanas de modelización, en las que los alumnos trabajan en problemas abiertos de la investigación o de la industria de forma libre. Se presenta el uso de Jupyter Notebook en las actividades de modelización y se ilustra con el ejemplo de la optimización de una planta solar. Además, compartimos nuestras experiencias en actividades de modelización en línea con estudiantes de secundaria en Alemania.Schönbrodt, S.; Wohak, K.; Frank, M. (2022). Digital Tools to Enable Collaborative Mathematical Modeling Online. Modelling in Science Education and Learning. 15(1):151-174. https://doi.org/10.4995/msel.2022.16269OJS151174151Blum, W. (2015). Quality Teaching of Mathematical Modelling: What Do We Know, What Can We Do? In S. J. Cho (Ed.), The Proceedings of the 12th International Congress on Mathematical Education (pp. 73-96). Cham: Springer International Publishing. https://doi.org/10.1007/978-3-319-12688-3_9Blum, W., & Borromeo Ferri, R. (2009). Mathematical Modelling: Can it Be Taught and Learnt? Journal of Mathematical Modelling and Application, 1 (1), 45-58.Blum, W., Galbraith, P., Henn, H.-W., & Niss, M. (2007). Modelling and Applications in Mathematics Education. New York: Springer. https://doi.org/10.1007/978-0-387-29822-1Blum, W., & Lei, D. (2007). How do students and teachers deal with modelling problems? In C. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical Modelling (ICTMA 12): Education, Engineering and Economics (pp. 222-231). Chichester: Horwood Publishing. https://doi.org/10.1533/9780857099419.5.221Borromeo Ferri, R. (2006, 04). Theoretical and empirical differentiations of phases in the modeling process. ZDM, 38(2), 86-95. doi: 10.1007/BF02655883 https://doi.org/10.1007/BF02655883Bruffee, K. (1995). Sharing Our Toys: Cooperative Learning versus Collaborative Learning. Change, 27 (1), 12-18. https://doi.org/10.1080/00091383.1995.9937722Computer-Based Maths. (n.d.). The Computational Thinking Process Poster. www.computationalthinking.org/helix. (accessed: 2021-01-23)Frank, M., Richter, P., Roeckerath, C., & Schönbrodt, S. (2018). Wie funktioniert eigentlich GPS? - Ein Computergestützter Modellierungsworkshop [How does GPS actually work? - A Computer-Supported Modeling Workshop]. In Greefrath, G. and Siller, S. (Ed.), Digitale Werkzeuge, Simulationen und mathematisches Modellieren [Digital tools, simulations and mathematical modeling] (pp. 137-163). Wiesbaden: Springer-Verlag. https://doi.org/10.1007/978-3-658-21940-6_7Frey, K. (2012). Die Projektmethode: Der Weg zum bildenden Tun [The project method: the path to educational action] (12th ed.; U. Schäfer, Ed.). Weinheim: Beltz.Gerhard, M., Hattebuhr, M., Schönbrodt, S., & Wohak, K. (2021). Aufbau und Einsatzmöglichkeiten des Lehr- und Lernmaterials [Structure and possible applications of the teaching and learning material]. In M. Frank & C. Roeckerath (Eds.), Neue Materialien für einen realitätsbezogenen Mathematikunterricht 9 [New materials for reality-based mathematics teaching 9]. Springer Spektrum.Greefrath, G., & Siller, H.-S. (2018). Digitale Werkzeuge, Simulationen und mathematisches Modellieren [Digital tools, simulations and mathematical modeling]. In Greefrath, G. and Siller, S. (Ed.), Digitale Werkzeuge, Simulationen und mathematisches Modellieren [Digital tools, simulations and mathematical modeling] (pp. 3-22). Wiesbaden: Springer-Verlag. https://doi.org/10.1007/978-3-658-21940-6_1Golub, J. (1988). Focus on Collaborative Learning. Urbana, Illinois: National Council of Teachers of English.Johnson, D., & Johnson, R. (1989). Cooperation and Competition: Theory and Research. Interaction Book Company.Johnson, D., & Johnson, R. (2014). Using technology to revolutionize cooperative learning: An opinion. Frontiers in Psychology, 5 , 1-3. https://doi.org/10.3389/fpsyg.2014.01156Panitz, T. (1999a). Collaborative versus cooperative learning: A comparison of the two concepts which will help us understand the underlying nature of interactive learning. ERIC Document Reproduction Service No. ED448443.Panitz, T. (1999b). The Motivational Benefits of Cooperative Learning. New directions for teaching and learning, 78. https://doi.org/10.1002/tl.7806Roberts, T. (2004). Preface. In T. Robert (Ed.), Online Collaborative Learning. Hershey, London: Information Science Publishing.Nason R. and Woodruff E. (2004). Online Collaborative Learning in Mathematics: Some Necessary Innovations. Online Collaborative Learning. Robert T.S (Ed.) pp 103-131 Information Science Publishing, Hershey (London) https://doi.org/10.4018/978-1-59140-174-2.ch005Siller, H.-S., & Greefrath, G. (2010). Mathematical Modelling in Class regarding to Technology. In Proceedings of the 6th CERME conference (pp. 2136-2145). (CERME-Proceedings)Greefrath G.and Siller H-St (2018). Digitale Werkzeuge, Simulationen und mathematisches Modellieren (Digital tools, simulations and mathematical modeling). Digitale Werkzeuge, Simulationen und mathematisches Modellieren (Digital tools, simulations and mathematical modeling). Greefrath G. and Siller S. (Eds.) pp. 3-22. Springer-Verlag (Wiesbaden) https://doi.org/10.1007/978-3-658-21940-6_1Hänze, M., Schmidt-Weigand, F., & Staudel, L. (2010). Gestufte Lernhilfen [Staggered learning aids]. In S. Boller & R. Lau (Eds.), Innere Differenzierung in der Sekundarstufe II. Ein Praxishandbuch für Lehrer/innen [Inner differentiation in upper secondary education. A practical handbook for teachers] (pp. 63-73). Weinheim: Beltz.Kaiser, G., & Schwarz, B. (2010). Authentic Modelling Problems in Mathematics Education - Examples and Experiences. Journal fur Mathematik-Didaktik, 31 , 51-76. https://doi.org/10.1007/s13138-010-0001-3Krajcik J.S. and Blumenfeld Ph.C. (2005). Project-Based Learning. The Cambridge Handbook of the Learning Sciences. Sawyer, R. Keith (Ed.) pp 317-334. Cambridge Handbooks in Psychology. Cambridge University Press (Cambridge) doi:10.1017/CBO9780511816833.020 https://doi.org/10.1017/CBO9780511816833.020Ludwig, M. (1997). Projekte im Mathematikunterricht des Gymnasiums [Projects in mathematics lessons of the high school] (phdthesis). Julius-Maximilians-Universitöt Würzburg. https://doi.org/10.1007/BF03338857Maaß, K. (2010). Classifiation Scheme for Modelling Tasks. Journal fur Mathematik-Didaktik, 31 (2), 285-311. doi: 10.1007/s13138-010-0010-2 https://doi.org/10.1007/s13138-010-0010-2Bock, W., & Bracke, M. (2015). Applied School Mathematics - Made in Kaiserslautern. In H. Neuntzer & D. Prätzel-Wolters (Eds.), Currents in industrial mathematics: From concepts to research to education (pp. 403-437). Berlin, Heidelberg: Springer. https://doi.org/10.1007/978-3-662-48258-2Kronberg, R., York-Barr, J., Arnold, K., Gombos, S., Truex, S., Vallejo, B., & Stevenson, J. (1997). Differentiated Teaching & Learning in Heterogeneous Classrooms: Strategies for Meeting the Needs of All Students. Washington D.C.: ERIC Clearinghouse. Retrieved from https://eric.ed.gov/?id=ED418538Stahl, G., Koschmann, T., & Suthers, D. (2006). Computer-supported collaborative learning: An historical perspective. In R. Sawyer (Ed.), Cambridge handbook of the learning sciences (pp. 409-426). Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511816833.025Niss, M. (1992). Applications and modelling in school mathematics - directions for future development. Roskilde: IMFUFA Roskilde Universitetscenter.Schmidt, L. (2019). Machine Learning: automatische Bilderkennung mit Mathematik?! - Ein Lehr-Lern- Modul im Rahmen eines mathematischen Modellierungstages für Schülerinnen und Schüler der Sekundarstufe II [Machine Learning: automatic image recognition with mathematics?! - A teaching-learning module in the context of a mathematical modeling day for high school students]. www.cammp.online Masterthesis4druck.pdf. (Master's thesis, RWTH Aachen, accessed: 2021-02-23)Schönbrodt, S., & Frank, M. (2020). Schüler/innen forschen zu erneuerbaren Energien - Optimierung eines Solarkraftwerks [Students research on renewable energies - Optimization of a solar power plant]. In H.-S. Siller, W. Weigel, & J. F. Worler (Eds.), Beiträge zum Mathematikunterricht [Contributions to mathematics education] (pp. 1534-1534). Münster: WTM-Verlag.Schönbrodt, S. (2019). Maschinelle Lernmethoden für Klassifizierungsprobleme - Perspektiven für die mathematische Modellierung mit Schülerinnen und Schülern [Machine learning methods for classification problems - perspectives for mathematical modeling with students]. Wiesbaden: Springer Spektrum. https://doi.org/10.1007/978-3-658-25137-6Vos, P. (2011). What is 'authentic' in the teaching and learning of mathematical modelling? In G. Kaiser, W. Blum, R. Borromeo Ferri, & G. Stillman (Eds.), Trends in Teaching and Learning of Mathematical Modelling, ICTMA 14 (pp. 713-722). Dordrecht: Springer. https://doi.org/10.1007/978-94-007-0910-2_68Winter, H. (1995). Mathematikunterricht und Allgemeinbildung [Mathematics education and general education]. Mitteilungen der Gesellschaft für Didaktik der Mathematik, 61 , 37-46. Retrieved 23 January, 2021, from https://ojs.didaktik-der-mathematik.de/index.php/mgdm/article/view/69/80Wohak, K., & Frank, M. (2019). Complex Modeling: Insights into our body through computer tomography - perspectives of a project day on inverse problems. In U. T. Jankvist, M. van den Heuvel-Panhuizen, & M. Veldhuis (Eds.), Eleventh Congress of the European Society for Research in Mathematics Education (pp. 4815-4822). Utrecht: Freudenthal Group.Wohak, K., Sube, M., Schönbrodt, S., Frank, M., & Roeckerath, C. (2021). Authentische und relevante Modellierung mit Schülerinnen und Schülern an nur einem Tag?! [Authentic and relevant modeling with students in just one day?!]. In M. Bracke, M. Ludwig, & K. Vorhölter (Eds.), Modellierungsprojekte mit Schülerinnen und Schülern. Realitätsbezüge im Mathematikunterricht [Modeling projects with students. Reality references in mathematics lessons] (pp. 37-50). Wiesbaden: Springer Spektrum. https://doi.org/10.1007/978-3-658-33012-5_4Vorholter K. and Freiwald J. (2022). Concept and structure of the Hamburg Modeling Days Modelling in Science Education and Learning. (In this issue).Hattebuhr M. and Frank M. (2022). Compartment models to study human impact on climate change Modelling in Science Education and Learning. (In this issue)

Guano: The White Gold of the Seabirds

The term “Guano” applies to natural mineral deposits consisting of excrements, eggshells and carcasses of dead seabirds found in almost rainless, hot-dry climatic regions and corresponding fertilizers. Guanos are classified according to age, genesis, geographical origin and chemical composition. Main types are nitrogen- and phosphate Guanos. Phosphate Guanos require a calcareous subsoil for the development, while nitrogen Guanos are formed only under the special climatic conditions of the subtropical-edge tropical high pressure belt with coastal deserts. The most significant nitrogen Guano is the Peru-Guano, which has been used over 2000 years as agricultural fertilizer in Peru. In Europe the application of Guano as fertilizer emerged in the 1840 as “Guano boom” and lasted until the early twentieth century when Guano was replaced by industrial manufactured fertilizers. Only a small quantity is still exported to Europe as additive to organic/mineral fertilizers, more for image boosting than for effect

Finite temperature Casimir effect of massive fermionic fields in the presence of compact dimensions

We consider the finite temperature Casimir effect of a massive fermionic field confined between two parallel plates, with MIT bag boundary conditions on the plates. The background spacetime is $M^{p+1}\times T^q$ which has $q$ dimensions compactified to a torus. On the compact dimensions, the field is assumed to satisfy periodicity boundary conditions with arbitrary phases. Both the high temperature and the low temperature expansions of the Casimir free energy and the force are derived explicitly. It is found that the Casimir force acting on the plates is always attractive at any temperature regardless of the boundary conditions assumed on the compact torus. The asymptotic limits of the Casimir force in the small plate separation limit are also obtained.Comment: 10 pages, accepted by Phys. Lett.