A General Approach to Two-Stage Tests (Dissertation Premium) - Softcover

Inhaltsangabe

In classical, "fixed-sample" statistics, a fixed set of data is at hand. No analysis is performed before all data is available. In contrast, flexible designs allow data gathering and data analysis to overlap. As a field of research in applied mathematical statistics, flexible designs have a history of some seventy-five years. Group sequential and adaptive designs are the object of particularly vivid research activity. A unifying framework at least for parts of the theory appears desirable. Valuable work has already been done in this direction; we hope that we can contribute to this development.

Our aim is to enhance the understanding of adaptive two-stage tests. We begin with a concise historical overview of some important related concepts, including group sequential designs, alpha-spending functions, conditional error functions and p-value combinations. We highlight the relationships between these concepts. In particular, published examples have suggested a correspondence between conditional error functions and p-value combinations. This is the starting point for our main contributions. We present a formal link between the two approaches. Based on this, we develop a general framework for two-stage tests. Common examples are covered as special cases. The idea is to view a two-stage test as a function family in the unit square, satisfying minimal technical requirements. The proposed framework is theoretically sound and geometrically intuitive. We also present a sensible overall p-value notion that covers different previously proposed definitions.

Our framework provides great flexibility, and to explore this flexibility is our second major topic. We present software tools that allow a user-friendly implementation of different two-stage tests. Comfortable visualization routines for conditional error functions are particularly useful. By means of these tools, we examine different tests systematically for their structural properties. One of these tests is newly invented. Finally, we investigate the power of the selected tests by simulation.

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