Group Sequential and Adaptive Designs for Three-Arm 'Gold Standard' Non-Inferiority Trials

This thesis deals with the application of group sequential and adaptive methodology in three-arm non-inferiority trials for the case of normally distributed outcomes. Whenever feasible, use of the three-arm design including a test treatment, an active control and a placebo, is recommended by the health authorities. Nevertheless, especially from an ethical point of view, it is desirable to keep the placebo group size as small as possible. After giving a short introduction to two-arm non-inferiority trials, we investigate a hierarchical single-stage testing procedure for three-arm trials which starts by assessing the superiority comparison between test and placebo and then proceeds to the test versus control non-inferiority comparison. Based on formulas for the overall power we derive optimal sample size allocations that minimise the overall sample size. Interestingly, the placebo group size turns out to be very low under the optimal allocation. The optimal fixed sample size designs will then serve both as a starting point and a benchmark for the designs determined later. Subsequently, a general group sequential design for three-arm non-inferiority trials is presented that aims at further minimising the required sample sizes. By choosing different rejection boundaries for the two comparisons we obtain designs with quite different properties. The influence of the boundaries on the operating characteristics such as the expected sample sizes is investigated by means of a comprehensive comparison to the optimal fixed design. Moreover, approximately optimal boundaries are derived for different optimisation criteria such as minimising the placebo group size. It turns out that the implementation of group sequential methodology can further improve the optimal fixed designs, where the potential early termination of the placebo arm is a key advantage that can make the trial more acceptable for patients. After this, the group sequential testing procedure is extended to adaptive designs that allow data-dependent design changes at the interim analysis. In this context, we discuss optimal mid-trial decision-making based on the observed interim data, with a special focus on sample size re-calculation. In doing so, we will make use of the conditional power and the Bayesian predictive power. Our investigations show the advantages of the proposed adaptive designs over the optimal fixed designs. In particular, the possibility to adapt the sample sizes at interim can help to deal with uncertainties regarding the treatment effects, that often exist in the planning stage of three-arm non-inferiority trials. We conclude with a discussion of the results and an outlook on possible future work

Tristan da Cunha hotspot : Mantle plume or shallow plate tectonics?

Tristan da Cunha is a small volcanic island in the South Atlantic Ocean close to the Mid- Atlantic Ridge. It is part of an area, which is characterized by widely scattered seamounts and small islands at the western end of the Walvis Ridge - Tristan/Gough hotspot track. Tristan da Cunha represents the end member of a classical hotspot track with an underlying plume: The active volcanic island Tristan da Cunha at the youngest end of the track is linked to the Cretaceous Etendeka flood basalt province in northwestern Namibia at its oldest end. But the genesis of the island itself has so far been puzzling. It is hotly debated if the island sits actually above a deep-seated mantle plume or if it is caused by shallow plate tectonics. To understand the Tristan da Cunha hotspot, a multi-disciplinary geophysical study has been conducted in 2012 and 2013 on board the German research vessel Maria S. Merian to acquire marine magnetotelluric and seismological data. The aim was to reveal the upper mantle structure with electrical density and velocity perturbations. Within this study I focused on the seismological dataset. At first, I performed a P-wave finite-frequency tomography with cross-correlated travel time residuals of teleseismic earthquakes. This allows to resolve the upper mantle structure beneath the island in terms of velocity perturbations and clarifies the existence of a mantle plume. I also investigated the local seismicity in the Tristan region to identify tectono-magmatic processes at the Mid-Atlantic Ridge and Close to the islands and seamounts. Moreover, I compared and combined my tomographic results with electromagnetic results to identify zones of partial melt and to understand plume processes in the upper mantle beneath the Tristan da Cunha hotspot. The tomographic results provide evidence for the existence of the Tristan conduit southwest of the archipelago. Its shape is cylindrical with a radius ca. 100 km down to a depth of 250 km. The structure ramifies in narrow veins below that depth. A recent link from the conduit to a seamount chain shows, that melt is channelled towards seamounts and islands in the study area. High seismicity within an oceanic plate segment north of Tristan da Cunha can be related to the internal stresses of the fragment. Differently directed forces act at the northern boundary of this plate. An earthquake free zone coincides spatially with the location of the Tristan mantle plume. This indicates a ductile regime in the lithosphere above the plume. Furthermore, hints for an incipient ridge jump towards a parallel line to the actual location of the Tristan plume were found. Several earthquakes were localised close to the archipelago of Tristan da Cunha. The locations of these earthquakes are related to young surface eruptions like small volcanic cones or seamounts

An optimal linear solver for the Jacobian system of the extreme type-II Ginzburg--Landau problem

This paper considers the extreme type-II Ginzburg--Landau equations, a nonlinear PDE model for describing the states of a wide range of superconductors. Based on properties of the Jacobian operator and an AMG strategy, a preconditioned Newton--Krylov method is constructed. After a finite-volume-type discretization, numerical experiments are done for representative two- and three-dimensional domains. Strong numerical evidence is provided that the number of Krylov iterations is independent of the dimension $n$ of the solution space, yielding an overall solver complexity of O(n)

Fluctuation based interpretable analysis scheme for quantum many-body snapshots

Microscopically understanding and classifying phases of matter is at the heart of strongly-correlated quantum physics. With quantum simulations, genuine projective measurements (snapshots) of the many-body state can be taken, which include the full information of correlations in the system. The rise of deep neural networks has made it possible to routinely solve abstract processing and classification tasks of large datasets, which can act as a guiding hand for quantum data analysis. However, though proven to be successful in differentiating between different phases of matter, conventional neural networks mostly lack interpretability on a physical footing. Here, we combine confusion learning with correlation convolutional neural networks, which yields fully interpretable phase detection in terms of correlation functions. In particular, we study thermodynamic properties of the 2D Heisenberg model, whereby the trained network is shown to pick up qualitative changes in the snapshots above and below a characteristic temperature where magnetic correlations become significantly long-range. We identify the full counting statistics of nearest neighbor spin correlations as the most important quantity for the decision process of the neural network, which go beyond averages of local observables. With access to the fluctuations of second-order correlations -- which indirectly include contributions from higher order, long-range correlations -- the network is able to detect changes of the specific heat and spin susceptibility, the latter being in analogy to magnetic properties of the pseudogap phase in high-temperature superconductors. By combining the confusion learning scheme with transformer neural networks, our work opens new directions in interpretable quantum image processing being sensible to long-range order.Comment: 15+3 page

Preconditioned Recycling Krylov subspace methods for self-adjoint problems

The authors propose a recycling Krylov subspace method for the solution of a sequence of self-adjoint linear systems. Such problems appear, for example, in the Newton process for solving nonlinear equations. Ritz vectors are automatically extracted from one MINRES run and then used for self-adjoint deflation in the next. The method is designed to work with arbitrary inner products and arbitrary self-adjoint positive-definite preconditioners whose inverse can be computed with high accuracy. Numerical experiments with nonlinear Schr\"odinger equations indicate a substantial decrease in computation time when recycling is used

Plasmons in Two-Dimensional Topological Insulators

We analyze collective excitations in models of two-dimensional topological insulators using the random phase approximation. In a two-dimensional extension of the Su-Schrieffer-Heeger model, edge plasmonic excitations with induced charge-density distributions localized at the boundaries of the system are found in the topologically non-trivial phase, dispersing similarly as one-dimensional bulk plasmons in the conventional Su-Schrieffer-Heeger chain. For two-dimensional bulk collective modes, we reveal regimes of enhanced inter-band wave function correlations, leading to characteristic hardening and softening of inter- and intra-band bulk plasmonic branches, respectively. In the two-dimensional Haldane Chern insulator model, chiral, uni-directional edge plasmons in nano-ribbon architectures are observed, which can be characterized by an effective Coulomb interaction cross section. Bulk collective excitations in the two-dimensional Haldane model are shown to be originated by single-particle band structure details in different topological phases

Familienexterne Nachfolge: Das Zusammenfinden von Übergebern und Übernehmern

Nach Schätzungen des IfM Bonn steht in 71.000 Familienunternehmen pro Jahr die Regelung der Nachfolge an. In welche Hände die von der Nachfolge betroffenen Unternehmen gehen werden, hängt von verschiedenen Faktoren (wie der Existenz von zur Nachfolge geeigneten Kindern oder der Attraktivität des Unternehmens) ab. Familienexterne Nachfolgelösungen gewinnen dabei in den letzten Jahren zunehmend an Bedeutung. Die Realisierung einer familienexternen Nachfolge gestaltet sich allerdings deutlich schwieriger als die Realisierung einer familieninternen Lösung - vor allem in der so genannten Matchingphase. Diese ist durch das Finden geeigneter Übernahmeinteressierter bzw. Unternehmen und die Verhandlungen der beiden Parteien über Details der Nachfolge (niedergelegt in einem Übernahmevertrag) gekennzeichnet. Im Zentrum der Untersuchung stand die Analyse der Matchingphase sowohl aus Sicht der Eigentümer als auch der Übernahmeinteressierten. Zusätzlich wurden Experten herangezogen, die aufgrund ihrer eigenen Rolle im Matchingprozess über ein entsprechendes Erfahrungswissen verfügen. Die Befragungen belegen wie erwartet, dass die Suche nach einem familienexternen Nachfolger bzw. einem Unternehmen problembehaftet ist. Allerdings erweist sich die an die Suche anschließende Verhandlungsphase als kritischer. --